diff --git a/AGENTS.md b/AGENTS.md
index 7e8ae57d0..bcf34bab9 100644
--- a/AGENTS.md
+++ b/AGENTS.md
@@ -5,16 +5,108 @@ Issue and its latest comments. `docs/STATUS.md` distinguishes current scope
 from the verbatim historical ledger. Old acquisition instructions, opening
 Issue bodies and publication-portfolio snapshots are not a current run queue.
 
-Owner-delegated focus dated 2026-09-13, reconciled with the latest #650:
-- One active acceptance package: consolidate and independently audit #735's
-  arbitrary-period root theorem, with #613/#736's sharp axial full-law boundary,
-  as one probability manuscript. Return the package to #735.
-- Uniform oblique winding corridors are a possible strengthening within this
-  programme, not a new mandatory parallel project or duplicate census.
-- #636 is a visible representation reserve, paused for new extension this round.
-  Reuse #708/#710/#733/#737 rather than re-commissioning their finite results.
-- #275: original-U candidate maps before more sampling; theory-blocked P1,
-  with its original source/normalizer/moving-root contract retained.
+Owner-delegated focus dated 2026-09-29, replacing the Sep13 one-paper allocation:
+- Primary: joint near-critical rank process. #773/#823 supply finite birth/arm
+  inputs; direct jump-two suppression is not a proof against aggregation of two
+  distinct births. This branch now supplies a two-time density bound and an
+  author triangular noncoalescence corollary. A follow-up supplies square
+  NN/matching input transfer and population-IQR/pivotal-clock comparability.
+  Advance continuum torus-observer identification and the joint law, not
+  another request to derive these bounds. No unique copula or universality yet.
+  The new H(s,t) kernel criterion separates rank-only Markov models from
+  entry-history dependence. Exact L3 square/triangular positive/negative
+  controls are available in insertion AND label time. The L512 memory
+  contrast is modest, retrospective and insertion-clock only; do not promote
+  it to continuum certification or redispatch the old 9! enumerations.
+  L3 completion geometry is now computed: triangular direction does not
+  remove memory, whereas a binary completion phase gives an exact four-state
+  count-clock realization. Square unmarked label rank is Markov but marked
+  axial sectors are not. Use the completion-pivotal hazard identity for the
+  next scale-dependent mechanism question; do not redispatch direction-only
+  closure at L3 or confuse count-clock closure with hidden-count label time.
+  A new independent 140k square-L512 completion block is now delivered,
+  fixed at the old count cutoffs. The pooled survival contrast is smaller;
+  exact-D and exact-(D,nu2) weighted finite-lag contrasts are unresolved.
+  The directional kernel's next-step necessity has now been checked:
+  within-D early histories had +1.629 +/- .391 more completion sites in
+  one post-hoc readout. The later independent safe-insertion block gives
+  -.040 +/- .570 and does NOT reproduce it. Do not retain the earlier
+  selected contrast as an accepted empirical exclusion of direction closure.
+  Advance completion-count successor dynamics or a targeted independent
+  prediction, not an automatic next descriptor or reacquisition of this
+  block. Flow-matched Markov completions do not identify actual history.
+  The next-count question is now answered negatively in finite L4: even
+  (rank,D,nu2) fails weak Markovness from the uniform empty start, on both
+  square and triangular lattices. Safe-pair synergy gives the exact
+  two-step transfer and safe-insertion nu drift. Do not redispatch generic
+  finite closure as missing or automatically enumerate the next L.
+  Its large-size importance remains open; use a specified completion-
+  creation prediction, not equality of the tautological immediate hazard.
+  The first large-size safe-insertion block is now delivered: 140k new
+  L512 prefixes, four probes per eligible prefix, primary same-(D,nu)
+  increment contrast +.00668 +/- .01209, unresolved. Only .560% of probes
+  have nonzero increments. Exact projected-component evaluation now resolves
+  the SAME target on all140k prefixes: -.003449 +/- .000772, SE15.65x smaller.
+  This supports finite L512 failure of enlarged (rank,D,nu) closure; it is
+  post-readout estimator refinement, not independent confirmation or a
+  continuum theorem. Direction-only replication remains unresolved.
+  Owner allocation on Sep30: one support worker owns precision/replication;
+  the main investigator advances mechanism, not further precision rounds.
+  The new clock-preserving geometric source leaves all count arrivals and
+  current exit hazard unchanged but has two-step response
+  -s*Var_safe(degree)/(m*(m-1)). Three specified L4 full-lag responses are
+  already calculated; do not redispatch them. Advance finite-window source
+  propagation and the exact safe-path versus survival-weight decomposition
+  in notes/birth-selection-intervention-20260930.md. Replication is not a
+  prerequisite for these derivations or an excuse for another descriptor.
+  The support worker's other independent140k archive readout is now complete:
+  -.00319937 +/- .00062009, same negative sign, no block pooling. Other archive
+  outcomes were already read; this is first e readout, not new prospective
+  production. Do not redispatch that replication or run another precision
+  block. Main's new fixed735-step source response has its own contract at
+  analysis/geometric-source-window-20260930/contract.json; it reuses the first
+  block's permutations to test finite-window transmission, not replication.
+  That735-step job is now completed: susceptibility=-.01104203 +/-.00015767,
+  same55028risk/140000prefixes;14workers,519.145s onTgFr7R, released toReady.
+  Do not rerun it as a missing tool, new precision block or size ladder.
+  The safe-path/selection coupling is now completed too: same1198cells give
+  safe-reference=-.00253706+/-.00063649, natural=-.00220507+/-.00078709,
+  selection=+.00033199+/-.00028738 (unresolved). All140000births match;
+  97666new conditional continuations, not independent preparations. Main next:
+  distinguish realizable entrance and safe-growth sources with nonredundant
+  same-contract readouts. Do not redispatch this coupling or precision.
+  Y and two-step survival at fixed(k,D,c) are affine. Two first-birth-
+  preserving sources also have proportional static M/Etop and moving-root
+  columns at one p; thermal p plus ONE source rank2 does not identify TWO
+  geometric sources. See notes/safe-transport-selection-readout-20260930.md.
+  A separate one-sided-birth source derivation now maps deltaM=deltaEtop
+  through the moving root; symmetricL4 gives a genuine positive normal
+  response and rank2Jacobian. It is a constructed dynamic-source control,
+  not the missing original norm4/U six-coordinate forward columns.
+  Label hazard derivative explicitly contains
+  mean(c)+2mean(e)-Var(c); fixed-count data are not fixed-label observations.
+- Parallel: #809/#833 source-dependent predictive state and noisy information.
+  Reuse #829--#837 and this branch's all-k deterministic four-state encoding
+  theorem, including actual two-row product queries. Distributed deterministic
+  block codes restore rank four. The stochastic parity follow-up retains
+  tensor rank three but weakens single-readout distinction; its product family
+  also has positive initial memory three. Full physical language, transcripts
+  and feedback are separate contracts. Do not infer reliable memory from rank.
+- Bounded writing: #739 already supplies arbitrary-oblique full-law necessity;
+  #771 adds fixed-p consequences. Do not dispatch that proof as missing again.
+- #275: derive normal projected candidate columns under the ORIGINAL source,
+  normalizer and moving root. E_top production existed; missing model maps do
+  not mean that experiment was never run. Aggregate rank is not original U.
+
+These are attention priorities, never task locks or permission classes. The
+owner authorizes useful local/cloud computation; benchmark the specific job
+and use task-local paths. Owner routing update on 2026-09-30: medium and large
+computations go to Huawei first; keep local work for small derivations, scoring
+and editing. This changes compute placement, not scientific permission. Do not
+reset containers or stop unknown jobs. Current
+machine availability must be checked, not inferred from a dated skill snapshot.
+Coordinate through concise repository updates rather than repeated cross-chat
+messages. Keep completed results available even when their lane is not primary.
 
 Choose work by the mathematical or physical distinction it can settle, not by
 how many PRs, scripts, new widths or compatibility scores it generates. A useful
diff --git a/README.md b/README.md
index 241562607..47b2a043a 100644
--- a/README.md
+++ b/README.md
@@ -1,95 +1,103 @@
 # Matching One
 
-**Exact topology, geometric threshold laws, and the information needed to identify a physical response.**
+**拓扑阈值的几何、两次出生的随机过程，以及观察者能保留多少信息。**
 
-Matching One studies square-lattice site percolation. Its strongest current
-questions are no longer "which exponent fits?" or "which familiar field name
-looks plausible?" They are:
+当前研究入口更新于 **2026-09-30**，阅读截至 #837 的开放研究及最新讨论。
+主分支基线为 `d31fa5fe`；下列较新成果多数仍在开放 PR 中，**进入总览不等于已合并或已被独立证明**。
 
-1. When does a topological balance root locate the infinite-volume threshold,
-   even when the whole finite-volume threshold law does not concentrate?
-2. Which finite states preserve a declared observable under future updates and
-   physical source interventions?
-3. Which additional prediction or measurement can actually distinguish the
-   remaining mechanisms of the original normalized response U?
+## 现在值得集中做什么
 
-**Start with [the research frontier](docs/RESEARCH-FRONTIER.md), then
-[the execution roadmap](docs/ROADMAP.md).** This entry point was reset under the
-owner's research delegation dated 2026-09-13. Historical data, failed tests of
-physical hypotheses, and unmerged branches are retained. Main integration,
-mathematical correctness, independent evidence and publication novelty are
-separate questions.
-
-## The exact observable
-
-For an occupied configuration on a torus,
-
-    r = rank im[H1(occupied complex) -> H1(torus)] in {0,1,2},
-    X = r-1,
-    M = E[X] = P2-P0,       E_top = E[X^2] = P2+P0,
-    F(p) = E_p[r]/2 = (1+M(p))/2.
-
-The balance root is Q(1/2), where Q=F^{-1}. In contrast,
-H=P2/(P0+P2) conditions on the rare non-rank-one sectors; it is NOT F.
-The identity X^3=X defines a two-dimensional nonconstant observable algebra,
-not a two-state dynamical model or a two-field continuum theory.
-These observables sit in the established matching/wrapping/homological
-percolation literature; their names alone are not a novelty claim.
-
-## Three research lanes
-
-| Lane | Current asset | Next result worth obtaining |
+| 注意力 | 已经到达哪里 | 下一项真正增加信息的工作 |
 |---|---|---|
-| **Geometric separation of root and full-law consistency** | [#735](https://github.com/LightChainr/Matching-One/pull/735): arbitrary-period balance roots; [#736](https://github.com/LightChainr/Matching-One/pull/736): sharp axial full-law criterion | Check the supplied proof and novelty once; prove or refute the uniform oblique winding-corridor lemma |
-| **Observable- and source-dependent minimal state** | [#708](https://github.com/LightChainr/Matching-One/pull/708), [#710](https://github.com/LightChainr/Matching-One/pull/710), [#733](https://github.com/LightChainr/Matching-One/pull/733): exact rank closure, visible spectrum, physical site sources | An all-width structural closure theorem or one source-faithful separating response, not another width count |
-| **Original-U physical identifiability** | [#275](https://github.com/LightChainr/Matching-One/issues/275): current assets do not identify the named candidates | Two same-source, same-normalizer forward maps; nuisance-profiled separation before further acquisition |
-
-The linked research PRs are **open/unmerged at this snapshot**; these links do
-not silently install their code on main. The [frontier](docs/RESEARCH-FRONTIER.md)
-pins their commits and distinguishes results from remaining hypotheses.
-
-For axial w-by-m tori with m>=w and w->infinity, the new organizing contrast is
-
-    balance-root consistency:       no aspect-ratio restriction (#718/#735),
-    convergence of every fixed Q(u): log(m)/w -> 0 iff (#736 + #613 inputs).
-
-The iff proof uses critical square-site RSW, finite-product continuity, a
-seam-closed crossing ring and independent transverse bands. It is not derived
-by fitting sizes. Its arbitrary-oblique necessity extension remains open,
-and no near-critical rate or new numerical p_c is claimed.
-
-## What survives from the numerical program
-
-The global orientation-sensitive matching-odd signal and several frozen H4
-transfer comparisons remain substantive finite-size evidence. A successful
-harmonic comparison is not a standalone nonzero detection and not a field
-identification. Single-multiplier full-curve closure and several scalar
-correction models have failed; their raw blocks are not discarded.
-
-The primitive-C3 Gaussian observer has a later exact unit-rotation explanation;
-its old H8 near-alias must not be exported to square-site U. N580's recovered
-covariance and bounded-H8 reanalysis do not establish a bare-aspect physical law.
-See [current adjudications](docs/STATUS.md), not the historical opening text of
-an issue, for these distinctions. N denotes site count; on square-period
-geometries the linear scale is sqrt(N), not N.
-
-## Other valuable assets, without automatic expansion
-
-The cut-network/branching program [#549](https://github.com/LightChainr/Matching-One/pull/549)
-separates complete unbranched survival from branching prediction. Its exact
-class-count lower bound is not a noise-robust effective-dimension theorem.
-Bounded algebraic exclusion and projective inference remain publication units,
-not reasons to grow a polynomial census or claim a new statistical principle.
-The [research atlas](docs/RESEARCH-ATLAS.md) preserves less prominent work and
-closed-unmerged assets; its old execution priorities are historical.
-
-## Reproduction and working rules
-
-[REPRODUCIBILITY.md](REPRODUCIBILITY.md) describes the existing engines and data.
-[GOVERNANCE.md](GOVERNANCE.md) keeps exploration lightweight; raw data and
-chronology are preserved. [AGENTS.md](AGENTS.md) routes new work to the current
-scientific question instead of stale acquisition instructions.
-
-The old README, roadmap, map and status ledger are preserved byte-for-byte in
-[docs/history](docs/history/README.md). This is a navigation and allocation
-reset, not a claim that older experiments were never run.
+| **主线：联合近临界秩过程** | 735步传输已分辨；去掉自然生存重加权后，L512负向历史差仍在 | 区分入口几何与安全演化的真实源响应；先解决读出冗余，不追加精度批次 |
+| **并行：源和噪声约束下的预测状态** | 确定性分布式编码恢复秩 4；随机奇偶编码的乘积查询秩=正初始记忆=3，但单次读出变弱 | 在具体可实现的查询族下比较误差与记忆成本；明确单次、整段轨迹与自适应反馈的差别 |
+| **有界收尾：几何阈值定理** | #739 已提供任意整数周期的 full-law iff；#771 推进固定亚临界自由能 | 整合已给证明、明确概率输入和固定-p 边界；不重新指派“尚未做”的斜向必要性 |
+| **机制研究：原始 U 与 root H4** | 强 root H4 与弱 normalized-U 可以由热切向投影相容；#275 原合同仍缺两个候选的完整映射 | 同源、同正规化下的 normal forward columns / 实际秩投影 Ward 或混合响应；更多精度不能替代缺失的列 |
+
+优先级只分配注意力，**不锁任务、不设置研究许可、不自动否定低优先级方向**。
+理论、已有数据和新计算可以并行。不要把支持性工具的完成数量当成机制空间缩小。
+
+**最新机制判别：** [安全传输／筛选分解已完成](notes/safe-transport-selection-readout-20260930.md)。
+同1198端点格、同权重下，安全参考差 **−0.002537±0.000636**，自然存活差 −0.002205±0.000787；
+筛选贡献 +0.000332±0.000287，尚未分辨。纯筛选不再是这个比较的首选解释，入口/安全演化需要进入模型。
+原14万出生前缀的新条件续接，不是新独立块；不宣称已分离出生与演化，也不把点估计当作解释百分比。
+
+[保时钟几何干预](notes/birth-selection-intervention-20260930.md)
+只重新分配安全站点的到达概率，保持整个占据数过程和当前退出概率不变；两步生存响应
+精确为 `−(m−c) Var_safe(degree)/[m(m−1)]`。三个实际方格构型的全滞后响应已算出，
+不是把局部曲率外推。出生历史差同时被分解为“向后同调保留 × 向前完成”的协方差；
+安全路径权重给出几何演化与生存筛选的分离式。它们是有限机制推导，不是原始 U 或连续极限的识别。
+
+[L512长窗口已实际计算](notes/geometric-source-window-readout-20260930.md)：
+735步生存源导数 **−0.011042±0.000158**，证明该几何作用传到延迟终点；不是有限theta效应或新独立块。
+[第二出生源的移动根映射](notes/one-sided-birth-source-normal-response-20260930.md)还给出
+`delta M=delta E_top` 及不被移动根消去的正控制；对称 L4 例子的 normal E_top 导数约 `1.6853e−4`。
+这提供真实源响应列，但不是原 norm-4/U 源的替代品。
+另有[精确识别限制](notes/safe-transport-selection-readout-20260930.md)：保持首次出生边际的两个几何源，
+在单个p的M/E_top及移动根读出上都共线。热参数＋一个几何源的秩二，不等于两个几何机制已可区分；
+同(k,D,c)下Y与两步生存也只是同一量，下一项必须使用非冗余的同合同读出。
+
+## 三个不能混写的当前判断
+
+1. **两次出生不会在极限中聚合：已有作者证明。**
+   [两时间密度界](notes/two-time-pivotal-density-bound-20260929.md)与
+   [方格转接及总体 IQR 时钟](notes/square-birth-transfer-and-iqr-clock-20260929.md)
+   给出具名概率输入下的子序列无聚合；尚未识别唯一 copula 或普适振幅。
+   本地12.8万＋独立云端56万条过滤是有限数值补充，不是这项证明。
+2. **完成数不能在所有有限格点闭合过程：已有精确反例。**
+   [L4 原均匀源](notes/completion-pair-synergy-L4-20260929.md)中，同方向、同完成数的
+   早晚历史，一步生存相同、两步不同。[协同完成图](notes/completion-pair-synergy-20260929.md)
+   把新增完成数直接接到未来 rank；这并不自动证明 L512 或连续极限仍有可测记忆。
+3. **大尺寸方向模型是否充分：仍开放。**
+   [独立安全插入块](notes/safe-insertion-independent-block-20260929.md)的方向内完成数差
+   `−0.040±0.570` 未复现旧事后 `+1.629±.391`。旧数字保留，但不是已稳定排除模型的证据。
+   同 (D,nu) 安全增量原四探针差 `+0.00668±0.01209` 也未分辨。
+
+当前数值方法：[投影连通分量判据](notes/projected-component-pair-criterion-20260929.md)
+可精确求协同完成对，直接计算同一目标的条件均值 `2e/(m-nu)`，消除安全探针的随机噪声。
+[14万前缀的完整读出](notes/exact-completion-pairs-readout-20260930.md)已完成：
+同 (D,nu) 增量差为 **−0.003449±0.000772**，约4.47个批次SE，标准误缩小15.65倍。
+早出生组的完成创建较少，给出具体两步 rank 传递，而不是新增描述指标。
+这是原数据的事后估计量精化，不增加独立证据；支持 L512 扩展观察者的闭合失败，
+不证明连续极限，也不改变上面的方向-only 未复现结论。
+[标签曲率公式](notes/completion-pair-label-curvature-20260930.md)进一步分开创建和生存选择。
+
+[独立旧档案的固定目标复核](notes/completion-creation-independent-replication-20260930.md)
+现已由支持工作者完成：第二块为 **−0.003199±0.000620**，与前块相近，未合并。
+它是另一随机块上首次 e 读出，其它结果此前已读过，不称为全新前瞻验证。
+735步传输亦已完成；下一项是[安全路径与自然存活的直接耦合](notes/safe-path-selection-coupling-20260930.md)，不是第三轮精度扩张。
+
+需要追溯时：[历史核与 L3 正负控制](notes/birth-history-memory-readout-20260929.md) ·
+[L3 完成相态/方向标记](notes/birth-completion-geometry-20260929.md) ·
+[首个独立几何块](notes/completion-hazard-independent-block-20260929.md) ·
+[确定性编码](notes/deterministic-block-encoding-obstruction-20260929.md) ·
+[随机编码与可靠性](notes/stochastic-parity-block-encoding-20260929.md)。
+
+完整入口：[研究前沿](docs/RESEARCH-FRONTIER.md) ·
+[下一步](docs/ROADMAP.md) · [科学结论](docs/STATUS.md) ·
+[观察者地图](docs/RESEARCH-MAP.md) · [机器可读当前索引](analysis/research_frontier.json)。
+
+## 固定术语：不要把不同问题压成同一个故事
+
+对 honest torus 上的占据子复形，令
+
+    r = rank im[H1(occupied complex) -> H1(torus)] ∈ {0,1,2},
+    X = r-1,       M = E[X] = P2-P0,
+    E_top = E[X²] = P2+P0,       F = E[r]/2 = (1+M)/2.
+
+平衡根为 `Q(1/2)`，`Q=F^{-1}`；`H=P2/(P0+P2)` 是另一个条件观察量。
+`J1,J2` 是**同一均匀随机排列**首次达到秩 1、2 的占据数，`T1,T2` 是同一 iid
+均匀标签过滤的出生时间。`N` 是站点数；方形周期 `L×L` 有 `N=L²`。
+
+有限表、过程极限、CFT 算符、论文新颖性是不同结论。全局 H4、带电扇区、临时闭合前缀、
+固定环面过滤以及 original-U 都有自己的源、几何和读出合同；不得互换。
+
+## 保留哪些历史资产
+
+原始样本、冻结预测、负结果和未合并研究全部保留。
+[历史 atlas](docs/RESEARCH-ATLAS.md) 和
+[历史 ledger](analysis/research_ledger.yaml) 用于查找旧证据，不再作为当前派工单。
+同一随机块的根、斜率、分位数和新投影仍然只是一块数据。
+
+[复现说明](REPRODUCIBILITY.md) · [轻量探索规则](GOVERNANCE.md) ·
+[协作入口](AGENTS.md)。本轮不合并别人的 PR，不关闭或锁定研究 Issue，不创建自动尺寸阶梯。
diff --git a/analysis/birth-completion-geometry-20260929/analyze.py b/analysis/birth-completion-geometry-20260929/analyze.py
new file mode 100644
index 000000000..355e85877
--- /dev/null
+++ b/analysis/birth-completion-geometry-20260929/analyze.py
@@ -0,0 +1,319 @@
+#!/usr/bin/env python3
+"""Exact L3 occupied-set geometry and first-birth history transport.
+
+512 subsets per lattice, dynamic integer prefix counts; no 9! permutation
+rerun and no Monte Carlo. New scientific readout: completion-pivotal count
+and primitive rank-one homology direction.
+"""
+from collections import Counter, defaultdict
+from fractions import Fraction
+import gzip
+import hashlib
+from itertools import combinations
+import json
+import math
+from pathlib import Path
+import platform
+import time
+
+
+HERE = Path(__file__).resolve().parent
+ARCHIVE = HERE.parent / "birth-gap-20260929/data"
+L, N = 3, 9
+FACT = [math.factorial(k) for k in range(N+1)]
+
+
+def primitive(x, y):
+    g = math.gcd(x, y)
+    x, y = x//g, y//g
+    if x < 0 or (x == 0 and y < 0):
+        x, y = -x, -y
+    return x, y
+
+
+def topology(mask, neighbors):
+    """Lifted graph traversal; ambient image span, not max component rank."""
+    potentials = {}
+    basis = None
+    rank = 0
+    for root in range(N):
+        if not mask >> root & 1 or root in potentials:
+            continue
+        potentials[root] = (0, 0)
+        stack = [root]
+        while stack:
+            u = stack.pop()
+            ux, uy = potentials[u]
+            for v, dx, dy in neighbors[u]:
+                if not mask >> v & 1:
+                    continue
+                expected = ux+dx, uy+dy
+                if v not in potentials:
+                    potentials[v] = expected
+                    stack.append(v)
+                else:
+                    vx, vy = potentials[v]
+                    x, y = expected[0]-vx, expected[1]-vy
+                    assert x % L == 0 and y % L == 0
+                    x, y = x//L, y//L
+                    if x or y:
+                        if basis is None:
+                            basis = primitive(x, y)
+                            rank = 1
+                        elif basis[0]*y != basis[1]*x:
+                            rank = 2
+    return rank, basis if rank == 1 else None
+
+
+def bits(mask):
+    return [i for i in range(N) if mask >> i & 1]
+
+
+def analyze(lattice):
+    steps = [(1, 0), (-1, 0), (0, 1), (0, -1)]
+    if lattice == "triangular":
+        steps += [(1, 1), (-1, -1)]
+    neighbors = [[(((v % L + dx) % L)+L*((v//L+dy) % L), dx, dy)
+                  for dx, dy in steps] for v in range(N)]
+    topo = [topology(mask, neighbors) for mask in range(1 << N)]
+    ways = [Counter() for _ in topo]
+    rows, hist = [], Counter()
+    rank_counts = Counter()
+    directions = set()
+    # Proper subsets have smaller integer masks: this is a topological DP order.
+    for mask, (rank, direction) in enumerate(topo):
+        k = mask.bit_count()
+        rank_counts[k, rank] += 1
+        if rank == 1:
+            directions.add(direction)
+            for v in bits(mask):
+                old = mask ^ (1 << v)
+                previous_rank, previous_direction = topo[old]
+                if previous_rank == 0:
+                    ways[mask][k] += FACT[k-1]
+                else:
+                    assert previous_rank == 1 and previous_direction == direction
+                    ways[mask].update(ways[old])
+            assert sum(ways[mask].values()) == FACT[k]
+            pivotal = [v for v in range(N) if not mask >> v & 1
+                       and topo[mask | (1 << v)][0] == 2]
+            nu = len(pivotal)
+            for j1, count in ways[mask].items():
+                hist[j1, k+1, direction] += count*nu*FACT[N-k-1]
+            rows.append({"mask": mask, "occupied": bits(mask), "k": k,
+                         "direction": list(direction), "completion_sites": pivotal,
+                         "nu2": nu, "prefix_birth_counts": dict(sorted(ways[mask].items()))})
+        elif rank == 0 and k < N:
+            direct = sum(topo[mask | (1 << v)][0] == 2
+                         for v in range(N) if not mask >> v & 1)
+            hist[k+1, k+1, None] += FACT[k]*direct*FACT[N-k-1]
+    hist = Counter({key: value for key, value in hist.items() if value})
+    assert sum(hist.values()) == FACT[N]
+    # A single narrow source-convention control, not a repeat of acquisition.
+    filename = f"exact-control-{lattice}-L3-b00.json.gz"
+    raw = (ARCHIVE / filename).read_bytes()
+    archived = json.loads(gzip.decompress(raw))
+    collapsed = Counter()
+    for (j1, j2, direction), count in hist.items():
+        collapsed[j1, j2] += count
+    assert collapsed == Counter({(j1, j2): count for j1, j2, count in archived["histogram"]})
+    assert archived["samples"] == FACT[N]
+
+    profiles = []
+    for k in range(N):
+        at_k = [r for r in rows if r["k"] == k]
+        if not at_k:
+            continue
+        counts = Counter(r["nu2"] for r in at_k)
+        profiles.append({"k": k, "rank_one_configurations": len(at_k),
+                         "nu2_state_counts": dict(sorted(counts.items()))})
+
+    direction_kernels = []
+    for direction in sorted(directions):
+        subhist = {(j1, j2): count for (j1, j2, d), count in hist.items() if d == direction}
+        h = lambda s, t: sum(count for (j1, j2), count in subhist.items() if j1 <= s and j2 > t)
+        violations = []
+        for a, b, c in combinations(range(N+1), 3):
+            A, B, C, D = h(a, b), h(a, c), h(b, c), h(b, b)
+            det = B*D-A*C
+            if det:
+                violations.append({"times": [a, b, c],
+                                   "conditional_covariance": str(Fraction(det, D*D)),
+                                   "survival_difference": str(Fraction(B, A)-Fraction(C-B, D-A))})
+        direction_kernels.append({
+            "direction": list(direction), "non_direct_permutation_count": sum(subhist.values()),
+            "paired_histogram": [[j1, j2, count] for (j1, j2), count in sorted(subhist.items())],
+            "count_time_violations": violations,
+        })
+
+    # Exact decomposition of the previously used one-step witness at (4,5,6).
+    risk_rows = [r for r in rows if r["k"] == 5]
+    by_direction_and_nu = defaultdict(lambda: [0, 0])
+    for r in risk_rows:
+        early = sum(count for j1, count in r["prefix_birth_counts"].items() if j1 <= 4)
+        late = r["prefix_birth_counts"].get(5, 0)
+        by_direction_and_nu[tuple(r["direction"]), r["nu2"]][0] += early
+        by_direction_and_nu[tuple(r["direction"]), r["nu2"]][1] += late
+    early_total = sum(x[0] for x in by_direction_and_nu.values())
+    late_total = sum(x[1] for x in by_direction_and_nu.values())
+    early_mean = Fraction(sum(nu*c[0] for (d, nu), c in by_direction_and_nu.items()), early_total)
+    late_mean = Fraction(sum(nu*c[1] for (d, nu), c in by_direction_and_nu.items()), late_total)
+    contrast = -(early_mean-late_mean)/(N-5)
+
+    # Strong-lumping diagnostic on rank-one rows only. It concerns arbitrary
+    # preparations, not automatically the weak Markov property from empty.
+    coarse = defaultdict(list)
+    row_lookup = {r["mask"]: r for r in rows}
+    for r in rows:
+        signature = Counter()
+        mask = r["mask"]
+        for v in range(N):
+            if mask >> v & 1:
+                continue
+            new = mask | (1 << v)
+            if topo[new][0] == 2:
+                signature["absorbed_rank2"] += 1
+            else:
+                next_row = row_lookup[new]
+                signature[f"rank1_nu{next_row['nu2']}"] += 1
+        coarse[r["k"], tuple(r["direction"]), r["nu2"]].append((mask, tuple(sorted(signature.items()))))
+    failures = []
+    for key, members in sorted(coarse.items()):
+        distinct = {}
+        for mask, signature in members:
+            distinct.setdefault(signature, mask)
+        if len(distinct) > 1:
+            failures.append({"k": key[0], "direction": list(key[1]), "nu2": key[2],
+                             "witnesses": [{"mask": mask, "successor_counts": dict(sig)}
+                                           for sig, mask in list(distinct.items())[:2]]})
+    # Drop the direction and retain only one completion phase at each k.
+    # These rows establish strong closure within rank one, not within rank zero.
+    by_k_nu = defaultdict(list)
+    for (k, direction, nu), members in coarse.items():
+        by_k_nu[k, nu].extend(members)
+    transition_classes = []
+    for (k, nu), members in sorted(by_k_nu.items()):
+        signatures = {sig for mask, sig in members}
+        assert len(signatures) == 1
+        signature = dict(next(iter(signatures)))
+        transition_classes.append({"k": k, "nu2": nu, "states": len(members),
+                                   "successor_counts": signature,
+                                   "denominator": N-k})
+
+    # An exact count-clock weak Markov realization from the empty start.
+    # States are zero, rank-one slow, rank-one fast, two; k is external time.
+    min_nu = {k: min(nu for kk, nu in by_k_nu if kk == k) for k, nu in by_k_nu}
+    def phase(mask):
+        rank = topo[mask][0]
+        if rank == 0:
+            return 0
+        if rank == 2:
+            return 3
+        r = row_lookup[mask]
+        return 1+int(r["nu2"] != min_nu[r["k"]])
+    matrices = []
+    for k in range(N):
+        classes = defaultdict(list)
+        for mask in range(1 << N):
+            if mask.bit_count() == k:
+                classes[phase(mask)].append(mask)
+        matrix = [[Fraction(int(i == j)) for j in range(4)] for i in range(4)]
+        reachable = []
+        for state, masks in classes.items():
+            reachable.append(state)
+            if state == 3:
+                continue
+            counts = Counter()
+            signatures = set()
+            for mask in masks:
+                local = Counter(phase(mask | (1 << v)) for v in range(N) if not mask >> v & 1)
+                counts.update(local)
+                signatures.add(tuple(sorted(local.items())))
+            if state in (1, 2):
+                assert len(signatures) == 1
+            matrix[state] = [Fraction(counts[j], len(masks)*(N-k)) for j in range(4)]
+        assert all(sum(row) == 1 for row in matrix)
+        matrices.append({"k": k, "reachable_states": sorted(reachable),
+                         "matrix": [[str(x) for x in row] for row in matrix]})
+
+    # Reproduce the full paired-birth law using this small chain, without paths.
+    law = {(0, 0): Fraction(1)}
+    chain_hist = Counter()
+    for entry in matrices:
+        k = entry["k"]
+        matrix = [[Fraction(x) for x in row] for row in entry["matrix"]]
+        new_law = defaultdict(Fraction)
+        for (state, j1), mass in law.items():
+            for next_state, probability in enumerate(matrix[state]):
+                if not probability:
+                    continue
+                next_j1 = k+1 if state == 0 and next_state != 0 else j1
+                if next_state == 3:
+                    chain_hist[next_j1, k+1] += mass*probability
+                else:
+                    new_law[next_state, next_j1] += mass*probability
+        law = dict(new_law)
+    assert not law
+    assert chain_hist == Counter({key: Fraction(value, FACT[N]) for key, value in collapsed.items()})
+    return {
+        "lattice": lattice, "L": L, "N": N, "subsets": 1 << N,
+        "archived_histogram_matches": True,
+        "archive_file": "analysis/birth-gap-20260929/data/"+filename,
+        "archive_sha256": hashlib.sha256(raw).hexdigest(),
+        "primitive_unoriented_directions": [list(d) for d in sorted(directions)],
+        "rank_counts": [[k, rank, count] for (k, rank), count in sorted(rank_counts.items())],
+        "completion_profiles": profiles,
+        "direction_kernels": direction_kernels,
+        "risk_k5_entry_cut4": {
+            "grain": "ordered prefixes ending at rank-one occupied set; multiply by 4! for full-permutation counts",
+            "early_prefixes": early_total, "late_prefixes": late_total,
+            "mean_nu2_early": str(early_mean), "mean_nu2_late": str(late_mean),
+            "one_step_survival_difference": str(contrast),
+            "composition": [{"direction": list(d), "nu2": nu, "early_prefixes": counts[0], "late_prefixes": counts[1]}
+                            for (d, nu), counts in sorted(by_direction_and_nu.items())],
+        },
+        "rank_one_k_direction_nu2_strong_lumping_failures": failures,
+        "rank_one_k_nu2_transition_classes": transition_classes,
+        "count_clock_phase_realization": {
+            "states": ["rank0", "rank1_slow", "rank1_fast", "rank2"],
+            "phase": "at fixed k, slow is minimum nu2 among rank-one states; fast is the other value if present",
+            "external_time": "occupied count k, not hidden iid-label count",
+            "claim": "weak Markov realization from the empty uniform-permutation start; rank-one rows strongly close across all configurations",
+            "off_risk_rows": "identity placeholders; not a statement about artificial unreachable preparations",
+            "paired_birth_histogram_reproduced_exactly": True,
+            "transition_matrices": matrices,
+        },
+        "rank_one_configuration_rows": rows,
+    }
+
+
+def main():
+    started = time.perf_counter()
+    cases = [analyze(lattice) for lattice in ("square", "triangular")]
+    report = {
+        "schema": "matching-one.birth-completion-geometry.v1",
+        "as_of": "2026-09-29",
+        "question": "Does rank-one direction remove birth-history memory, and what immediate completion geometry carries it?",
+        "method": "lifted traversal of 512 subsets per lattice; exact prefix dynamic programming and outgoing-site counts",
+        "direction": "primitive unoriented rational homology line; fixed until rank two by image inclusion",
+        "nu2": "number of vacant vertices whose addition changes ambient rank from one to two",
+        "one_step_exit": "nu2/(N-k) conditional on the full occupied configuration at count k",
+        "cases": cases,
+        "no_monte_carlo_or_permutation_enumeration": True,
+        "limitations": ["L3 only; not large-size or continuum sufficiency.",
+                        "One-step hazard sufficiency is not recursive state sufficiency.",
+                        "Strong-lumping failure is not by itself failure of the weak Markov law from the empty state.",
+                        "A homology direction is not the full connectivity or geometric state."],
+        "python": platform.python_version(), "platform": platform.platform(),
+        "wall_seconds": time.perf_counter()-started,
+    }
+    (HERE / "result.json").write_text(json.dumps(report, indent=2)+"\n")
+    for c in cases:
+        print(json.dumps({k: c[k] for k in ("lattice", "primitive_unoriented_directions", "completion_profiles",
+                                           "risk_k5_entry_cut4", "rank_one_k_direction_nu2_strong_lumping_failures")}, indent=2))
+        print("direction violation counts",[len(d["count_time_violations"]) for d in c["direction_kernels"]])
+    print("seconds", report["wall_seconds"])
+
+
+if __name__ == "__main__":
+    main()
diff --git a/analysis/birth-completion-geometry-20260929/exact_marked_clock.py b/analysis/birth-completion-geometry-20260929/exact_marked_clock.py
new file mode 100644
index 000000000..87ea40fb6
--- /dev/null
+++ b/analysis/birth-completion-geometry-20260929/exact_marked_clock.py
@@ -0,0 +1,117 @@
+#!/usr/bin/env python3
+"""Exact iid-label kernels of the persistent rank-one direction sectors.
+
+Transform the new 512-subset/prefix-DP output, without sampling labels or
+rerunning permutation enumeration. SymPy performs rational polynomial work.
+"""
+import hashlib
+import json
+import math
+from pathlib import Path
+import platform
+import time
+
+import sympy as S
+
+
+HERE = Path(__file__).resolve().parent
+
+
+def main():
+    started = time.perf_counter()
+    raw = (HERE / "result.json").read_bytes()
+    source = json.loads(raw)
+    p, q = S.symbols("p q")
+    a, b, c = S.symbols("a b c")
+    A = lambda x: 2*x**3-6*x**2+3*x
+    B = lambda x: -2*x**3+3*x+1
+    cases = []
+    for case in source["cases"]:
+        n = case["N"]
+        assert n == 9
+        total = math.factorial(n)
+        pooled = S.Integer(0)
+        sectors = []
+        for sector in case["direction_kernels"]:
+            hist = sector["paired_histogram"]
+            direction = sector["direction"]
+            def h(k, ell):
+                return sum(count for j1, j2, count in hist if j1 <= k and j2 > ell)
+            kernel = S.expand(sum(
+                S.Rational(h(k, ell), total)*math.comb(n, k)*math.comb(n-k, ell-k)
+                * p**k*(q-p)**(ell-k)*(1-q)**(n-ell)
+                for k in range(n+1) for ell in range(k, n+1)))
+            pooled += kernel
+            if case["lattice"] == "square":
+                if 0 in direction:
+                    orbit = "axial"
+                    closed = 3*p**3*(1-q)**3*((1+q)**3-p**3)
+                    det_closed = -9*a**3*b**3*(1-b)**3*(1-c)**3*(b**3-a**3)*((1+c)**3-(1+b)**3)
+                    mean_nu = 3-3*(1-q)*(1+q)**2/((1+q)**3-p**3)
+                else:
+                    orbit = "diagonal"
+                    closed = 3*p**6*(1-q)**3
+                    det_closed = S.Integer(0)
+                    mean_nu = S.Integer(3)
+            else:
+                assert case["lattice"] == "triangular"
+                orbit = "three_direction_orbit"
+                closed = 3*p**3*(1-q)**3*(A(p)+B(q))
+                det_closed = 9*a**3*b**3*(1-b)**3*(1-c)**3*(A(b)-A(a))*(B(c)-B(b))
+                mean_nu = 3-(1-q)*(3-6*q**2)/(A(p)+B(q))
+            assert S.expand(kernel-closed) == 0
+            H = lambda x, y: kernel.subs({p: x, q: y}, simultaneous=True)
+            determinant = H(a, c)*H(b, b)-H(a, b)*H(b, c)
+            assert S.expand(determinant-det_closed) == 0
+            # This uses the general finite completion-hazard identity, not
+            # a second independent enumeration of the marked geometry.
+            assert S.cancel(-(1-q)*S.diff(kernel, q)/kernel-mean_nu) == 0
+            times = [S.Rational(1, 3), S.Rational(1, 2), S.Rational(2, 3)]
+            x, y, z = times
+            Hab, Hac, Hbb, Hbc = H(x, y), H(x, z), H(y, y), H(y, z)
+            assert Hab > 0 and Hbb > Hab
+            difference = S.cancel(Hac/Hab-(Hbc-Hac)/(Hbb-Hab))
+            coefficients = S.Matrix(n+1, n+1, lambda i, j: kernel.coeff(p, i).coeff(q, j))
+            sectors.append({
+                "direction": direction, "symmetry_orbit": orbit,
+                "kernel_closed_form": str(closed),
+                "coefficient_matrix_rank": int(coefficients.rank()),
+                "markov_determinant": str(det_closed),
+                "markov_identity_for_all_label_triples": det_closed == 0,
+                "cohort_mean_completion_count": str(mean_nu),
+                "specified_label_triple": [str(t) for t in times],
+                "early_minus_late_survival_at_triple": str(difference),
+                "all_closed_form_identities_exact": True,
+            })
+        pooled_closed = (6*p**3*(1-q**2)**3 if case["lattice"] == "square"
+                         else 9*p**3*(1-q)**3*(A(p)+B(q)))
+        assert S.expand(pooled-pooled_closed) == 0
+        cases.append({"lattice": case["lattice"], "L": 3, "N": n,
+                      "unmarked_kernel": str(pooled_closed),
+                      "marked_kernels_sum_to_unmarked": True, "sectors": sectors})
+    output = {
+        "schema": "matching-one.birth-marked-clock.v1",
+        "as_of": "2026-09-29",
+        "source": "analysis/birth-completion-geometry-20260929/result.json",
+        "source_sha256": hashlib.sha256(raw).hexdigest(),
+        "definition": "H_d(p,q)=P(T1<=p,T2>q,D=d), unconditional sector mass; 0<=p<=q<=1",
+        "normalization": "full 9! permutation count, not conditional on visiting the sector",
+        "clock_transform": "multinomial thinning of the same uniform permutation, iid Uniform site labels",
+        "determinant": "H_d(a,c)H_d(b,b)-H_d(a,b)H_d(b,c)",
+        "cohort_mean_completion_count": "-(1-q) partial_q log H_d(p,q), fixed entry cutoff p",
+        "cases": cases,
+        "boundaries": [
+            "Exact L3 statement about a specified observer and clock, not a continuum result.",
+            "Revealing a persistent direction can destroy a weak Markov projection; no contradiction with unmarked rank Markovness.",
+            "Polynomial coefficient rank is not a positive hidden-state dimension.",
+            "Four-state count-clock realization does not establish four-state label-clock closure with hidden count.",
+        ],
+        "python": platform.python_version(), "sympy": S.__version__,
+        "wall_seconds": time.perf_counter()-started,
+    }
+    (HERE / "marked-clock.json").write_text(json.dumps(output, indent=2)+"\n")
+    print(json.dumps(output, indent=2))
+
+
+if __name__ == "__main__":
+    main()
diff --git a/analysis/birth-completion-geometry-20260929/marked-clock.json b/analysis/birth-completion-geometry-20260929/marked-clock.json
new file mode 100644
index 000000000..ba658995b
--- /dev/null
+++ b/analysis/birth-completion-geometry-20260929/marked-clock.json
@@ -0,0 +1,173 @@
+{
+  "schema": "matching-one.birth-marked-clock.v1",
+  "as_of": "2026-09-29",
+  "source": "analysis/birth-completion-geometry-20260929/result.json",
+  "source_sha256": "34ae32ed0961cda535f8dbf1f104341ac7161fa5d15b8c39f838bc8776515b1d",
+  "definition": "H_d(p,q)=P(T1<=p,T2>q,D=d), unconditional sector mass; 0<=p<=q<=1",
+  "normalization": "full 9! permutation count, not conditional on visiting the sector",
+  "clock_transform": "multinomial thinning of the same uniform permutation, iid Uniform site labels",
+  "determinant": "H_d(a,c)H_d(b,b)-H_d(a,b)H_d(b,c)",
+  "cohort_mean_completion_count": "-(1-q) partial_q log H_d(p,q), fixed entry cutoff p",
+  "cases": [
+    {
+      "lattice": "square",
+      "L": 3,
+      "N": 9,
+      "unmarked_kernel": "6*p**3*(1 - q**2)**3",
+      "marked_kernels_sum_to_unmarked": true,
+      "sectors": [
+        {
+          "direction": [
+            0,
+            1
+          ],
+          "symmetry_orbit": "axial",
+          "kernel_closed_form": "3*p**3*(1 - q)**3*(-p**3 + (q + 1)**3)",
+          "coefficient_matrix_rank": 2,
+          "markov_determinant": "-9*a**3*b**3*(1 - b)**3*(1 - c)**3*(-a**3 + b**3)*(-(b + 1)**3 + (c + 1)**3)",
+          "markov_identity_for_all_label_triples": false,
+          "cohort_mean_completion_count": "-(3 - 3*q)*(q + 1)**2/(-p**3 + (q + 1)**3) + 3",
+          "specified_label_triple": [
+            "1/3",
+            "1/2",
+            "2/3"
+          ],
+          "early_minus_late_survival_at_triple": "-1084/250187",
+          "all_closed_form_identities_exact": true
+        },
+        {
+          "direction": [
+            1,
+            -1
+          ],
+          "symmetry_orbit": "diagonal",
+          "kernel_closed_form": "3*p**6*(1 - q)**3",
+          "coefficient_matrix_rank": 1,
+          "markov_determinant": "0",
+          "markov_identity_for_all_label_triples": true,
+          "cohort_mean_completion_count": "3",
+          "specified_label_triple": [
+            "1/3",
+            "1/2",
+            "2/3"
+          ],
+          "early_minus_late_survival_at_triple": "0",
+          "all_closed_form_identities_exact": true
+        },
+        {
+          "direction": [
+            1,
+            0
+          ],
+          "symmetry_orbit": "axial",
+          "kernel_closed_form": "3*p**3*(1 - q)**3*(-p**3 + (q + 1)**3)",
+          "coefficient_matrix_rank": 2,
+          "markov_determinant": "-9*a**3*b**3*(1 - b)**3*(1 - c)**3*(-a**3 + b**3)*(-(b + 1)**3 + (c + 1)**3)",
+          "markov_identity_for_all_label_triples": false,
+          "cohort_mean_completion_count": "-(3 - 3*q)*(q + 1)**2/(-p**3 + (q + 1)**3) + 3",
+          "specified_label_triple": [
+            "1/3",
+            "1/2",
+            "2/3"
+          ],
+          "early_minus_late_survival_at_triple": "-1084/250187",
+          "all_closed_form_identities_exact": true
+        },
+        {
+          "direction": [
+            1,
+            1
+          ],
+          "symmetry_orbit": "diagonal",
+          "kernel_closed_form": "3*p**6*(1 - q)**3",
+          "coefficient_matrix_rank": 1,
+          "markov_determinant": "0",
+          "markov_identity_for_all_label_triples": true,
+          "cohort_mean_completion_count": "3",
+          "specified_label_triple": [
+            "1/3",
+            "1/2",
+            "2/3"
+          ],
+          "early_minus_late_survival_at_triple": "0",
+          "all_closed_form_identities_exact": true
+        }
+      ]
+    },
+    {
+      "lattice": "triangular",
+      "L": 3,
+      "N": 9,
+      "unmarked_kernel": "9*p**3*(1 - q)**3*(2*p**3 - 6*p**2 + 3*p - 2*q**3 + 3*q + 1)",
+      "marked_kernels_sum_to_unmarked": true,
+      "sectors": [
+        {
+          "direction": [
+            0,
+            1
+          ],
+          "symmetry_orbit": "three_direction_orbit",
+          "kernel_closed_form": "3*p**3*(1 - q)**3*(2*p**3 - 6*p**2 + 3*p - 2*q**3 + 3*q + 1)",
+          "coefficient_matrix_rank": 2,
+          "markov_determinant": "9*a**3*b**3*(1 - b)**3*(1 - c)**3*(2*b**3 - 3*b - 2*c**3 + 3*c)*(-2*a**3 + 6*a**2 - 3*a + 2*b**3 - 6*b**2 + 3*b)",
+          "markov_identity_for_all_label_triples": false,
+          "cohort_mean_completion_count": "-(1 - q)*(3 - 6*q**2)/(2*p**3 - 6*p**2 + 3*p - 2*q**3 + 3*q + 1) + 3",
+          "specified_label_triple": [
+            "1/3",
+            "1/2",
+            "2/3"
+          ],
+          "early_minus_late_survival_at_triple": "-1156/716639",
+          "all_closed_form_identities_exact": true
+        },
+        {
+          "direction": [
+            1,
+            0
+          ],
+          "symmetry_orbit": "three_direction_orbit",
+          "kernel_closed_form": "3*p**3*(1 - q)**3*(2*p**3 - 6*p**2 + 3*p - 2*q**3 + 3*q + 1)",
+          "coefficient_matrix_rank": 2,
+          "markov_determinant": "9*a**3*b**3*(1 - b)**3*(1 - c)**3*(2*b**3 - 3*b - 2*c**3 + 3*c)*(-2*a**3 + 6*a**2 - 3*a + 2*b**3 - 6*b**2 + 3*b)",
+          "markov_identity_for_all_label_triples": false,
+          "cohort_mean_completion_count": "-(1 - q)*(3 - 6*q**2)/(2*p**3 - 6*p**2 + 3*p - 2*q**3 + 3*q + 1) + 3",
+          "specified_label_triple": [
+            "1/3",
+            "1/2",
+            "2/3"
+          ],
+          "early_minus_late_survival_at_triple": "-1156/716639",
+          "all_closed_form_identities_exact": true
+        },
+        {
+          "direction": [
+            1,
+            1
+          ],
+          "symmetry_orbit": "three_direction_orbit",
+          "kernel_closed_form": "3*p**3*(1 - q)**3*(2*p**3 - 6*p**2 + 3*p - 2*q**3 + 3*q + 1)",
+          "coefficient_matrix_rank": 2,
+          "markov_determinant": "9*a**3*b**3*(1 - b)**3*(1 - c)**3*(2*b**3 - 3*b - 2*c**3 + 3*c)*(-2*a**3 + 6*a**2 - 3*a + 2*b**3 - 6*b**2 + 3*b)",
+          "markov_identity_for_all_label_triples": false,
+          "cohort_mean_completion_count": "-(1 - q)*(3 - 6*q**2)/(2*p**3 - 6*p**2 + 3*p - 2*q**3 + 3*q + 1) + 3",
+          "specified_label_triple": [
+            "1/3",
+            "1/2",
+            "2/3"
+          ],
+          "early_minus_late_survival_at_triple": "-1156/716639",
+          "all_closed_form_identities_exact": true
+        }
+      ]
+    }
+  ],
+  "boundaries": [
+    "Exact L3 statement about a specified observer and clock, not a continuum result.",
+    "Revealing a persistent direction can destroy a weak Markov projection; no contradiction with unmarked rank Markovness.",
+    "Polynomial coefficient rank is not a positive hidden-state dimension.",
+    "Four-state count-clock realization does not establish four-state label-clock closure with hidden count."
+  ],
+  "python": "3.11.15",
+  "sympy": "1.14.0",
+  "wall_seconds": 0.25175825000042096
+}
diff --git a/analysis/birth-completion-geometry-20260929/result.json b/analysis/birth-completion-geometry-20260929/result.json
new file mode 100644
index 000000000..e139601eb
--- /dev/null
+++ b/analysis/birth-completion-geometry-20260929/result.json
@@ -0,0 +1,9869 @@
+{
+  "schema": "matching-one.birth-completion-geometry.v1",
+  "as_of": "2026-09-29",
+  "question": "Does rank-one direction remove birth-history memory, and what immediate completion geometry carries it?",
+  "method": "lifted traversal of 512 subsets per lattice; exact prefix dynamic programming and outgoing-site counts",
+  "direction": "primitive unoriented rational homology line; fixed until rank two by image inclusion",
+  "nu2": "number of vacant vertices whose addition changes ambient rank from one to two",
+  "one_step_exit": "nu2/(N-k) conditional on the full occupied configuration at count k",
+  "cases": [
+    {
+      "lattice": "square",
+      "L": 3,
+      "N": 9,
+      "subsets": 512,
+      "archived_histogram_matches": true,
+      "archive_file": "analysis/birth-gap-20260929/data/exact-control-square-L3-b00.json.gz",
+      "archive_sha256": "dccb0e132d7b2100bd39c6414f385366354c9099facbecbbaa1e3a5fd8f37a78",
+      "primitive_unoriented_directions": [
+        [
+          0,
+          1
+        ],
+        [
+          1,
+          -1
+        ],
+        [
+          1,
+          0
+        ],
+        [
+          1,
+          1
+        ]
+      ],
+      "rank_counts": [
+        [
+          0,
+          0,
+          1
+        ],
+        [
+          1,
+          0,
+          9
+        ],
+        [
+          2,
+          0,
+          36
+        ],
+        [
+          3,
+          0,
+          78
+        ],
+        [
+          3,
+          1,
+          6
+        ],
+        [
+          4,
+          0,
+          90
+        ],
+        [
+          4,
+          1,
+          36
+        ],
+        [
+          5,
+          0,
+          45
+        ],
+        [
+          5,
+          1,
+          72
+        ],
+        [
+          5,
+          2,
+          9
+        ],
+        [
+          6,
+          1,
+          48
+        ],
+        [
+          6,
+          2,
+          36
+        ],
+        [
+          7,
+          2,
+          36
+        ],
+        [
+          8,
+          2,
+          9
+        ],
+        [
+          9,
+          2,
+          1
+        ]
+      ],
+      "completion_profiles": [
+        {
+          "k": 3,
+          "rank_one_configurations": 6,
+          "nu2_state_counts": {
+            "0": 6
+          }
+        },
+        {
+          "k": 4,
+          "rank_one_configurations": 36,
+          "nu2_state_counts": {
+            "1": 36
+          }
+        },
+        {
+          "k": 5,
+          "rank_one_configurations": 72,
+          "nu2_state_counts": {
+            "2": 72
+          }
+        },
+        {
+          "k": 6,
+          "rank_one_configurations": 48,
+          "nu2_state_counts": {
+            "3": 48
+          }
+        }
+      ],
+      "direction_kernels": [
+        {
+          "direction": [
+            0,
+            1
+          ],
+          "non_direct_permutation_count": 152928,
+          "paired_histogram": [
+            [
+              3,
+              5,
+              2592
+            ],
+            [
+              3,
+              6,
+              5184
+            ],
+            [
+              3,
+              7,
+              5184
+            ],
+            [
+              4,
+              5,
+              7776
+            ],
+            [
+              4,
+              6,
+              15552
+            ],
+            [
+              4,
+              7,
+              15552
+            ],
+            [
+              5,
+              6,
+              31104
+            ],
+            [
+              5,
+              7,
+              31104
+            ],
+            [
+              6,
+              7,
+              38880
+            ]
+          ],
+          "count_time_violations": []
+        },
+        {
+          "direction": [
+            1,
+            -1
+          ],
+          "non_direct_permutation_count": 12960,
+          "paired_histogram": [
+            [
+              6,
+              7,
+              12960
+            ]
+          ],
+          "count_time_violations": []
+        },
+        {
+          "direction": [
+            1,
+            0
+          ],
+          "non_direct_permutation_count": 152928,
+          "paired_histogram": [
+            [
+              3,
+              5,
+              2592
+            ],
+            [
+              3,
+              6,
+              5184
+            ],
+            [
+              3,
+              7,
+              5184
+            ],
+            [
+              4,
+              5,
+              7776
+            ],
+            [
+              4,
+              6,
+              15552
+            ],
+            [
+              4,
+              7,
+              15552
+            ],
+            [
+              5,
+              6,
+              31104
+            ],
+            [
+              5,
+              7,
+              31104
+            ],
+            [
+              6,
+              7,
+              38880
+            ]
+          ],
+          "count_time_violations": []
+        },
+        {
+          "direction": [
+            1,
+            1
+          ],
+          "non_direct_permutation_count": 12960,
+          "paired_histogram": [
+            [
+              6,
+              7,
+              12960
+            ]
+          ],
+          "count_time_violations": []
+        }
+      ],
+      "risk_k5_entry_cut4": {
+        "grain": "ordered prefixes ending at rank-one occupied set; multiply by 4! for full-permutation counts",
+        "early_prefixes": 3456,
+        "late_prefixes": 5184,
+        "mean_nu2_early": "2",
+        "mean_nu2_late": "2",
+        "one_step_survival_difference": "0",
+        "composition": [
+          {
+            "direction": [
+              0,
+              1
+            ],
+            "nu2": 2,
+            "early_prefixes": 1728,
+            "late_prefixes": 2592
+          },
+          {
+            "direction": [
+              1,
+              0
+            ],
+            "nu2": 2,
+            "early_prefixes": 1728,
+            "late_prefixes": 2592
+          }
+        ]
+      },
+      "rank_one_k_direction_nu2_strong_lumping_failures": [],
+      "rank_one_k_nu2_transition_classes": [
+        {
+          "k": 3,
+          "nu2": 0,
+          "states": 6,
+          "successor_counts": {
+            "rank1_nu1": 6
+          },
+          "denominator": 6
+        },
+        {
+          "k": 4,
+          "nu2": 1,
+          "states": 36,
+          "successor_counts": {
+            "absorbed_rank2": 1,
+            "rank1_nu2": 4
+          },
+          "denominator": 5
+        },
+        {
+          "k": 5,
+          "nu2": 2,
+          "states": 72,
+          "successor_counts": {
+            "absorbed_rank2": 2,
+            "rank1_nu3": 2
+          },
+          "denominator": 4
+        },
+        {
+          "k": 6,
+          "nu2": 3,
+          "states": 48,
+          "successor_counts": {
+            "absorbed_rank2": 3
+          },
+          "denominator": 3
+        }
+      ],
+      "count_clock_phase_realization": {
+        "states": [
+          "rank0",
+          "rank1_slow",
+          "rank1_fast",
+          "rank2"
+        ],
+        "phase": "at fixed k, slow is minimum nu2 among rank-one states; fast is the other value if present",
+        "external_time": "occupied count k, not hidden iid-label count",
+        "claim": "weak Markov realization from the empty uniform-permutation start; rank-one rows strongly close across all configurations",
+        "off_risk_rows": "identity placeholders; not a statement about artificial unreachable preparations",
+        "paired_birth_histogram_reproduced_exactly": true,
+        "transition_matrices": [
+          {
+            "k": 0,
+            "reachable_states": [
+              0
+            ],
+            "matrix": [
+              [
+                "1",
+                "0",
+                "0",
+                "0"
+              ],
+              [
+                "0",
+                "1",
+                "0",
+                "0"
+              ],
+              [
+                "0",
+                "0",
+                "1",
+                "0"
+              ],
+              [
+                "0",
+                "0",
+                "0",
+                "1"
+              ]
+            ]
+          },
+          {
+            "k": 1,
+            "reachable_states": [
+              0
+            ],
+            "matrix": [
+              [
+                "1",
+                "0",
+                "0",
+                "0"
+              ],
+              [
+                "0",
+                "1",
+                "0",
+                "0"
+              ],
+              [
+                "0",
+                "0",
+                "1",
+                "0"
+              ],
+              [
+                "0",
+                "0",
+                "0",
+                "1"
+              ]
+            ]
+          },
+          {
+            "k": 2,
+            "reachable_states": [
+              0
+            ],
+            "matrix": [
+              [
+                "13/14",
+                "1/14",
+                "0",
+                "0"
+              ],
+              [
+                "0",
+                "1",
+                "0",
+                "0"
+              ],
+              [
+                "0",
+                "0",
+                "1",
+                "0"
+              ],
+              [
+                "0",
+                "0",
+                "0",
+                "1"
+              ]
+            ]
+          },
+          {
+            "k": 3,
+            "reachable_states": [
+              0,
+              1
+            ],
+            "matrix": [
+              [
+                "10/13",
+                "3/13",
+                "0",
+                "0"
+              ],
+              [
+                "0",
+                "1",
+                "0",
+                "0"
+              ],
+              [
+                "0",
+                "0",
+                "1",
+                "0"
+              ],
+              [
+                "0",
+                "0",
+                "0",
+                "1"
+              ]
+            ]
+          },
+          {
+            "k": 4,
+            "reachable_states": [
+              0,
+              1
+            ],
+            "matrix": [
+              [
+                "1/2",
+                "12/25",
+                "0",
+                "1/50"
+              ],
+              [
+                "0",
+                "4/5",
+                "0",
+                "1/5"
+              ],
+              [
+                "0",
+                "0",
+                "1",
+                "0"
+              ],
+              [
+                "0",
+                "0",
+                "0",
+                "1"
+              ]
+            ]
+          },
+          {
+            "k": 5,
+            "reachable_states": [
+              0,
+              1,
+              3
+            ],
+            "matrix": [
+              [
+                "0",
+                "4/5",
+                "0",
+                "1/5"
+              ],
+              [
+                "0",
+                "1/2",
+                "0",
+                "1/2"
+              ],
+              [
+                "0",
+                "0",
+                "1",
+                "0"
+              ],
+              [
+                "0",
+                "0",
+                "0",
+                "1"
+              ]
+            ]
+          },
+          {
+            "k": 6,
+            "reachable_states": [
+              1,
+              3
+            ],
+            "matrix": [
+              [
+                "1",
+                "0",
+                "0",
+                "0"
+              ],
+              [
+                "0",
+                "0",
+                "0",
+                "1"
+              ],
+              [
+                "0",
+                "0",
+                "1",
+                "0"
+              ],
+              [
+                "0",
+                "0",
+                "0",
+                "1"
+              ]
+            ]
+          },
+          {
+            "k": 7,
+            "reachable_states": [
+              3
+            ],
+            "matrix": [
+              [
+                "1",
+                "0",
+                "0",
+                "0"
+              ],
+              [
+                "0",
+                "1",
+                "0",
+                "0"
+              ],
+              [
+                "0",
+                "0",
+                "1",
+                "0"
+              ],
+              [
+                "0",
+                "0",
+                "0",
+                "1"
+              ]
+            ]
+          },
+          {
+            "k": 8,
+            "reachable_states": [
+              3
+            ],
+            "matrix": [
+              [
+                "1",
+                "0",
+                "0",
+                "0"
+              ],
+              [
+                "0",
+                "1",
+                "0",
+                "0"
+              ],
+              [
+                "0",
+                "0",
+                "1",
+                "0"
+              ],
+              [
+                "0",
+                "0",
+                "0",
+                "1"
+              ]
+            ]
+          }
+        ]
+      },
+      "rank_one_configuration_rows": [
+        {
+          "mask": 7,
+          "occupied": [
+            0,
+            1,
+            2
+          ],
+          "k": 3,
+          "direction": [
+            1,
+            0
+          ],
+          "completion_sites": [],
+          "nu2": 0,
+          "prefix_birth_counts": {
+            "3": 6
+          }
+        },
+        {
+          "mask": 15,
+          "occupied": [
+            0,
+            1,
+            2,
+            3
+          ],
+          "k": 4,
+          "direction": [
+            1,
+            0
+          ],
+          "completion_sites": [
+            6
+          ],
+          "nu2": 1,
+          "prefix_birth_counts": {
+            "3": 6,
+            "4": 18
+          }
+        },
+        {
+          "mask": 23,
+          "occupied": [
+            0,
+            1,
+            2,
+            4
+          ],
+          "k": 4,
+          "direction": [
+            1,
+            0
+          ],
+          "completion_sites": [
+            7
+          ],
+          "nu2": 1,
+          "prefix_birth_counts": {
+            "3": 6,
+            "4": 18
+          }
+        },
+        {
+          "mask": 31,
+          "occupied": [
+            0,
+            1,
+            2,
+            3,
+            4
+          ],
+          "k": 5,
+          "direction": [
+            1,
+            0
+          ],
+          "completion_sites": [
+            6,
+            7
+          ],
+          "nu2": 2,
+          "prefix_birth_counts": {
+            "3": 12,
+            "4": 36,
+            "5": 72
+          }
+        },
+        {
+          "mask": 39,
+          "occupied": [
+            0,
+            1,
+            2,
+            5
+          ],
+          "k": 4,
+          "direction": [
+            1,
+            0
+          ],
+          "completion_sites": [
+            8
+          ],
+          "nu2": 1,
+          "prefix_birth_counts": {
+            "3": 6,
+            "4": 18
+          }
+        },
+        {
+          "mask": 47,
+          "occupied": [
+            0,
+            1,
+            2,
+            3,
+            5
+          ],
+          "k": 5,
+          "direction": [
+            1,
+            0
+          ],
+          "completion_sites": [
+            6,
+            8
+          ],
+          "nu2": 2,
+          "prefix_birth_counts": {
+            "3": 12,
+            "4": 36,
+            "5": 72
+          }
+        },
+        {
+          "mask": 55,
+          "occupied": [
+            0,
+            1,
+            2,
+            4,
+            5
+          ],
+          "k": 5,
+          "direction": [
+            1,
+            0
+          ],
+          "completion_sites": [
+            7,
+            8
+          ],
+          "nu2": 2,
+          "prefix_birth_counts": {
+            "3": 12,
+            "4": 36,
+            "5": 72
+          }
+        },
+        {
+          "mask": 56,
+          "occupied": [
+            3,
+            4,
+            5
+          ],
+          "k": 3,
+          "direction": [
+            1,
+            0
+          ],
+          "completion_sites": [],
+          "nu2": 0,
+          "prefix_birth_counts": {
+            "3": 6
+          }
+        },
+        {
+          "mask": 57,
+          "occupied": [
+            0,
+            3,
+            4,
+            5
+          ],
+          "k": 4,
+          "direction": [
+            1,
+            0
+          ],
+          "completion_sites": [
+            6
+          ],
+          "nu2": 1,
+          "prefix_birth_counts": {
+            "3": 6,
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+          }
+        },
+        {
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+          }
+        },
+        {
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+          }
+        },
+        {
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+          }
+        },
+        {
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+          ],
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+            "5": 48
+          }
+        },
+        {
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+          ],
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+          "prefix_birth_counts": {
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+            "5": 48
+          }
+        },
+        {
+          "mask": 438,
+          "occupied": [
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+          ],
+          "nu2": 3,
+          "prefix_birth_counts": {
+            "3": 72,
+            "4": 360,
+            "5": 288
+          }
+        },
+        {
+          "mask": 440,
+          "occupied": [
+            3,
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+            8
+          ],
+          "k": 5,
+          "direction": [
+            1,
+            0
+          ],
+          "completion_sites": [
+            0,
+            1,
+            2
+          ],
+          "nu2": 3,
+          "prefix_birth_counts": {
+            "3": 12,
+            "4": 60,
+            "5": 48
+          }
+        },
+        {
+          "mask": 448,
+          "occupied": [
+            6,
+            7,
+            8
+          ],
+          "k": 3,
+          "direction": [
+            1,
+            0
+          ],
+          "completion_sites": [],
+          "nu2": 0,
+          "prefix_birth_counts": {
+            "3": 6
+          }
+        },
+        {
+          "mask": 449,
+          "occupied": [
+            0,
+            6,
+            7,
+            8
+          ],
+          "k": 4,
+          "direction": [
+            1,
+            0
+          ],
+          "completion_sites": [
+            3,
+            4
+          ],
+          "nu2": 2,
+          "prefix_birth_counts": {
+            "3": 6,
+            "4": 18
+          }
+        },
+        {
+          "mask": 450,
+          "occupied": [
+            1,
+            6,
+            7,
+            8
+          ],
+          "k": 4,
+          "direction": [
+            1,
+            0
+          ],
+          "completion_sites": [
+            4,
+            5
+          ],
+          "nu2": 2,
+          "prefix_birth_counts": {
+            "3": 6,
+            "4": 18
+          }
+        },
+        {
+          "mask": 451,
+          "occupied": [
+            0,
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+          ],
+          "k": 5,
+          "direction": [
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+          ],
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+            3,
+            4,
+            5
+          ],
+          "nu2": 3,
+          "prefix_birth_counts": {
+            "3": 12,
+            "4": 60,
+            "5": 48
+          }
+        },
+        {
+          "mask": 452,
+          "occupied": [
+            2,
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+            7,
+            8
+          ],
+          "k": 4,
+          "direction": [
+            1,
+            0
+          ],
+          "completion_sites": [
+            3,
+            5
+          ],
+          "nu2": 2,
+          "prefix_birth_counts": {
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+            "4": 18
+          }
+        },
+        {
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+          "occupied": [
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+            3,
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+          ],
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+          }
+        },
+        {
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+          "occupied": [
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+            "5": 48
+          }
+        },
+        {
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+          "occupied": [
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+          ],
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+            3,
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+            5
+          ],
+          "nu2": 3,
+          "prefix_birth_counts": {
+            "3": 72,
+            "4": 360,
+            "5": 288
+          }
+        },
+        {
+          "mask": 456,
+          "occupied": [
+            3,
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+          ],
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+          "direction": [
+            1,
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+          ],
+          "completion_sites": [
+            0,
+            2
+          ],
+          "nu2": 2,
+          "prefix_birth_counts": {
+            "3": 6,
+            "4": 18
+          }
+        },
+        {
+          "mask": 458,
+          "occupied": [
+            1,
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+            "4": 36,
+            "5": 72
+          }
+        },
+        {
+          "mask": 464,
+          "occupied": [
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+          ],
+          "k": 4,
+          "direction": [
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+            0,
+            1
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+            "4": 18
+          }
+        },
+        {
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+          "occupied": [
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+          ],
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+            "5": 72
+          }
+        },
+        {
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+          "occupied": [
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+          ],
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+            0,
+            1,
+            2
+          ],
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+          "prefix_birth_counts": {
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+            "4": 60,
+            "5": 48
+          }
+        },
+        {
+          "mask": 480,
+          "occupied": [
+            5,
+            6,
+            7,
+            8
+          ],
+          "k": 4,
+          "direction": [
+            1,
+            0
+          ],
+          "completion_sites": [
+            1,
+            2
+          ],
+          "nu2": 2,
+          "prefix_birth_counts": {
+            "3": 6,
+            "4": 18
+          }
+        },
+        {
+          "mask": 481,
+          "occupied": [
+            0,
+            5,
+            6,
+            7,
+            8
+          ],
+          "k": 5,
+          "direction": [
+            1,
+            0
+          ],
+          "completion_sites": [
+            1,
+            2,
+            3,
+            4
+          ],
+          "nu2": 4,
+          "prefix_birth_counts": {
+            "3": 12,
+            "4": 36,
+            "5": 72
+          }
+        },
+        {
+          "mask": 488,
+          "occupied": [
+            3,
+            5,
+            6,
+            7,
+            8
+          ],
+          "k": 5,
+          "direction": [
+            1,
+            0
+          ],
+          "completion_sites": [
+            0,
+            1,
+            2
+          ],
+          "nu2": 3,
+          "prefix_birth_counts": {
+            "3": 12,
+            "4": 60,
+            "5": 48
+          }
+        },
+        {
+          "mask": 496,
+          "occupied": [
+            4,
+            5,
+            6,
+            7,
+            8
+          ],
+          "k": 5,
+          "direction": [
+            1,
+            0
+          ],
+          "completion_sites": [
+            0,
+            1,
+            2
+          ],
+          "nu2": 3,
+          "prefix_birth_counts": {
+            "3": 12,
+            "4": 60,
+            "5": 48
+          }
+        },
+        {
+          "mask": 504,
+          "occupied": [
+            3,
+            4,
+            5,
+            6,
+            7,
+            8
+          ],
+          "k": 6,
+          "direction": [
+            1,
+            0
+          ],
+          "completion_sites": [
+            0,
+            1,
+            2
+          ],
+          "nu2": 3,
+          "prefix_birth_counts": {
+            "3": 72,
+            "4": 360,
+            "5": 288
+          }
+        }
+      ]
+    }
+  ],
+  "no_monte_carlo_or_permutation_enumeration": true,
+  "limitations": [
+    "L3 only; not large-size or continuum sufficiency.",
+    "One-step hazard sufficiency is not recursive state sufficiency.",
+    "Strong-lumping failure is not by itself failure of the weak Markov law from the empty state.",
+    "A homology direction is not the full connectivity or geometric state."
+  ],
+  "python": "3.11.15",
+  "platform": "macOS-27.0-arm64-arm-64bit",
+  "wall_seconds": 0.0206800420419313
+}
diff --git a/analysis/birth-gap-20260929/CLOUD-HANDOFF.md b/analysis/birth-gap-20260929/CLOUD-HANDOFF.md
new file mode 100644
index 000000000..5b7a40c3a
--- /dev/null
+++ b/analysis/birth-gap-20260929/CLOUD-HANDOFF.md
@@ -0,0 +1,65 @@
+# Independent cloud block: ready-to-run handoff
+
+Run from the copied task repository root, using only the allocated TV2N0X
+task directory. This subtask did not access or run on the server.
+
+The local engine has already passed all 9! L=3 permutation controls once per
+lattice. Its unchanged SHA256 is
+`72997c726bc734da7695d104fc8fb51ed5672506c967c2a88312bdc092925506`.
+The completed local manifest SHA256 is
+`d223ad486d153239a40c02fb534e03efe33fd9b5dde741f206b32d147c608f4e`.
+
+Acquisition (14 measured-available workers, 140,000 paths per cell):
+
+```bash
+python3 analysis/birth-gap-20260929/run_pilot.py \
+  --compiler g++ \
+  --sizes 64 128 256 512 --batches 14 --samples 10000 --workers 14 \
+  --role cloud-tv2n0x-20260929-v1 \
+  --skip-exact-control \
+  --exact-control-reference 'local L3 square+triangular all 9!; data/run.json sha256=d223ad486d153239a40c02fb534e03efe33fd9b5dde741f206b32d147c608f4e' \
+  --git-base-commit d31fa5fe9584b77595e4a78b9560aa2ff77cf0b3 \
+  --budget-seconds 300 \
+  --output analysis/birth-gap-20260929/cloud-data
+```
+
+This first compiles with the specified compiler, skips only the already-supplied
+exact control, benchmarks 200 permutations per lattice at the largest requested
+L, and stops **before production** if the size-scaled benchmark predicts more
+than 300 seconds at the requested worker count. The estimate is not an enforced
+hard timeout; CPU contention can make actual time longer. Per-batch data and the
+partial manifest are saved as jobs finish.
+
+If the benchmark stops, choose L=64,128,256 and a **different output directory**
+(the existing manifest is never overwritten). Use a new role if a production
+block already began; otherwise the original production seeds have not been used.
+The production namespace is hashed with lattice/L/batch. Different namespace
+means separate pseudorandom streams; changing output path alone does not.
+
+Analyze the cloud block without changing local results:
+
+```bash
+python3 analysis/birth-gap-20260929/analyze.py \
+  --data analysis/birth-gap-20260929/cloud-data \
+  --epsilons .00625 .0125 .025 .05 .1 .2 \
+  --output-json analysis/birth-gap-20260929/cloud-summary.json \
+  --output-md analysis/birth-gap-20260929/CLOUD-SUMMARY.md
+```
+
+This epsilon grid is declared before cloud results. It applies to both hard
+near-diagonal CDF and soft Z(delta), in prescribed-power and empirical-width
+clocks. Source compilation and analysis use only C++17/Python stdlib; no scipy,
+NumPy, OpenMP or installations beyond the parent's compiler provisioning.
+
+The analyzer saves full paired delete-one covariance, re-estimates empirical
+birth widths within every deletion, and explicitly marks reused prior exact
+controls rather than claiming they ran again on the cloud. The optimized beta
+recurrence terminates only its floating-point-underflowed far right tail; at
+L<=512 and epsilon<=.2 the starting binomial mass is representable. It raises
+an error instead of silently substituting a normal approximation outside that
+regime.
+
+Interpretation remains finite-size: a decreasing microscopic atom and a stable
+mean gap do not exclude a partial diagonal atom. The cloud block adds an
+independent larger-size test of the soft near-diagonal behaviour, not a proof
+of square-site exponents or process-limit noncoalescence.
diff --git a/analysis/birth-gap-20260929/SUMMARY.md b/analysis/birth-gap-20260929/SUMMARY.md
new file mode 100644
index 000000000..6987e0bd9
--- /dev/null
+++ b/analysis/birth-gap-20260929/SUMMARY.md
@@ -0,0 +1,121 @@
+# Executed joint-birth pilot — 2026-09-29
+
+128,000 independent-filtration samples; 8 batches × 2,000 per lattice/size.
+Local ARM64 acquisition including compilation, exact controls and benchmarks: 6.89 s, at most 4 concurrent jobs.
+
+## Scale and direct jump
+
+Errors after ± are complete-batch jackknife SE, not confidence limits.
+
+| Lattice | L | Direct 0→2 / 16,000 | E[D]/L^(5/4) | E[D]/mixture IQR |
+|---|---:|---:|---:|---:|
+| square | 16 | 97 | 0.4230 ± 0.0032 | 0.8460 ± 0.0635 |
+| square | 32 | 24 | 0.4206 ± 0.0029 | 0.7808 ± 0.0054 |
+| square | 64 | 8 | 0.4273 ± 0.0026 | 0.7437 ± 0.0068 |
+| square | 128 | 1 | 0.4278 ± 0.0030 | 0.7195 ± 0.0079 |
+| triangular | 16 | 38 | 0.4032 ± 0.0018 | 0.8064 ± 0.0036 |
+| triangular | 32 | 13 | 0.4031 ± 0.0027 | 0.7482 ± 0.0050 |
+| triangular | 64 | 4 | 0.4075 ± 0.0030 | 0.7233 ± 0.0091 |
+| triangular | 128 | 1 | 0.4056 ± 0.0031 | 0.7013 ± 0.0076 |
+
+## Continuous-label near diagonal: P[L^(3/4)(T2−T1) ≤ ε]
+
+W is the pooled J1/J2 interquartile width; zero atom included. Subtract the preceding direct-atom frequency for strictly positive mass.
+
+| Lattice | L | ε=.025 | ε=.05 | ε=.1 | ε=.2 |
+|---|---:|---:|---:|---:|---:|
+| square | 16 | 0.05352 ± 0.00094 | 0.09853 ± 0.00139 | 0.18323 ± 0.00218 | 0.33486 ± 0.00290 |
+| square | 32 | 0.04713 ± 0.00141 | 0.09132 ± 0.00193 | 0.17605 ± 0.00300 | 0.33208 ± 0.00319 |
+| square | 64 | 0.04568 ± 0.00177 | 0.08830 ± 0.00320 | 0.17245 ± 0.00407 | 0.32576 ± 0.00314 |
+| square | 128 | 0.04557 ± 0.00147 | 0.09160 ± 0.00148 | 0.17613 ± 0.00277 | 0.33549 ± 0.00193 |
+| triangular | 16 | 0.05033 ± 0.00114 | 0.09686 ± 0.00185 | 0.18530 ± 0.00228 | 0.34564 ± 0.00129 |
+| triangular | 32 | 0.04878 ± 0.00092 | 0.09536 ± 0.00172 | 0.18421 ± 0.00289 | 0.34440 ± 0.00411 |
+| triangular | 64 | 0.04785 ± 0.00173 | 0.09253 ± 0.00203 | 0.17919 ± 0.00144 | 0.34160 ± 0.00341 |
+| triangular | 128 | 0.04813 ± 0.00185 | 0.09363 ± 0.00218 | 0.17973 ± 0.00291 | 0.33807 ± 0.00390 |
+
+## Exponent-free near diagonal: P[(N+1)(T2−T1)/W ≤ ε]
+
+W is the pooled J1/J2 interquartile width; zero atom included. Subtract the preceding direct-atom frequency for strictly positive mass.
+
+| Lattice | L | ε=.025 | ε=.05 | ε=.1 | ε=.2 |
+|---|---:|---:|---:|---:|---:|
+| square | 16 | 0.03009 ± 0.00220 | 0.05334 ± 0.00409 | 0.09818 ± 0.00764 | 0.18259 ± 0.01393 |
+| square | 32 | 0.02623 ± 0.00097 | 0.05056 ± 0.00147 | 0.09796 ± 0.00201 | 0.18862 ± 0.00308 |
+| square | 64 | 0.02707 ± 0.00111 | 0.05206 ± 0.00235 | 0.10100 ± 0.00396 | 0.19642 ± 0.00449 |
+| square | 128 | 0.02728 ± 0.00104 | 0.05426 ± 0.00133 | 0.10843 ± 0.00175 | 0.20801 ± 0.00329 |
+| triangular | 16 | 0.02639 ± 0.00066 | 0.05015 ± 0.00114 | 0.09650 ± 0.00184 | 0.18463 ± 0.00228 |
+| triangular | 32 | 0.02669 ± 0.00053 | 0.05240 ± 0.00099 | 0.10232 ± 0.00181 | 0.19737 ± 0.00304 |
+| triangular | 64 | 0.02721 ± 0.00140 | 0.05365 ± 0.00206 | 0.10369 ± 0.00269 | 0.20054 ± 0.00319 |
+| triangular | 128 | 0.02832 ± 0.00154 | 0.05534 ± 0.00189 | 0.10771 ± 0.00262 | 0.20563 ± 0.00336 |
+
+## Discrete near diagonal: P[D/L^(5/4) ≤ ε]
+
+W is the pooled J1/J2 interquartile width; zero atom included. Subtract the preceding direct-atom frequency for strictly positive mass.
+
+| Lattice | L | ε=.025 | ε=.05 | ε=.1 | ε=.2 |
+|---|---:|---:|---:|---:|---:|
+| square | 16 | 0.00606 ± 0.00058 | 0.06763 ± 0.00163 | 0.17763 ± 0.00303 | 0.32125 ± 0.00383 |
+| square | 32 | 0.02569 ± 0.00118 | 0.07325 ± 0.00222 | 0.16375 ± 0.00368 | 0.33031 ± 0.00306 |
+| square | 64 | 0.04106 ± 0.00159 | 0.08744 ± 0.00349 | 0.17250 ± 0.00468 | 0.32512 ± 0.00328 |
+| square | 128 | 0.04200 ± 0.00160 | 0.08981 ± 0.00144 | 0.17588 ± 0.00338 | 0.33600 ± 0.00188 |
+| triangular | 16 | 0.00237 ± 0.00039 | 0.06275 ± 0.00185 | 0.17694 ± 0.00337 | 0.33356 ± 0.00191 |
+| triangular | 32 | 0.02569 ± 0.00081 | 0.07575 ± 0.00174 | 0.16975 ± 0.00276 | 0.34431 ± 0.00464 |
+| triangular | 64 | 0.04306 ± 0.00193 | 0.09131 ± 0.00236 | 0.17969 ± 0.00169 | 0.34125 ± 0.00360 |
+| triangular | 128 | 0.04519 ± 0.00187 | 0.09119 ± 0.00225 | 0.17913 ± 0.00305 | 0.33881 ± 0.00403 |
+
+## Discrete exponent-free near diagonal: P[D/W ≤ ε]
+
+W is the pooled J1/J2 interquartile width; zero atom included. Subtract the preceding direct-atom frequency for strictly positive mass.
+
+| Lattice | L | ε=.025 | ε=.05 | ε=.1 | ε=.2 |
+|---|---:|---:|---:|---:|---:|
+| square | 16 | 0.00606 ± 0.00058 | 0.00606 ± 0.00058 | 0.06763 ± 0.00163 | 0.17763 ± 0.00303 |
+| square | 32 | 0.02569 ± 0.00118 | 0.05025 ± 0.00203 | 0.09575 ± 0.00169 | 0.18550 ± 0.00393 |
+| square | 64 | 0.02100 ± 0.00069 | 0.05100 ± 0.00207 | 0.09725 ± 0.00364 | 0.19112 ± 0.00975 |
+| square | 128 | 0.02538 ± 0.00143 | 0.05025 ± 0.00173 | 0.10606 ± 0.00152 | 0.20738 ± 0.00282 |
+| triangular | 16 | 0.00237 ± 0.00039 | 0.00237 ± 0.00039 | 0.06275 ± 0.00185 | 0.17694 ± 0.00337 |
+| triangular | 32 | 0.02569 ± 0.00081 | 0.05250 ± 0.00130 | 0.10150 ± 0.00247 | 0.19369 ± 0.00377 |
+| triangular | 64 | 0.02138 ± 0.00133 | 0.05306 ± 0.00202 | 0.10138 ± 0.00249 | 0.19631 ± 0.00157 |
+| triangular | 128 | 0.02706 ± 0.00165 | 0.05381 ± 0.00193 | 0.10394 ± 0.00217 | 0.20287 ± 0.00308 |
+
+## Soft near-diagonal Z(δ), G=L^(3/4)(T2−T1)
+
+Z(δ)=E[(1−G/δ)+]; same samples and covariance, not additional independent evidence.
+
+| Lattice | L | δ=.025 | δ=.05 | δ=.1 | δ=.2 |
+|---|---:|---:|---:|---:|---:|
+| square | 16 | 0.03005 ± 0.00070 | 0.05312 ± 0.00091 | 0.09728 ± 0.00132 | 0.17906 ± 0.00189 |
+| square | 32 | 0.02443 ± 0.00085 | 0.04688 ± 0.00126 | 0.09047 ± 0.00183 | 0.17308 ± 0.00244 |
+| square | 64 | 0.02356 ± 0.00084 | 0.04528 ± 0.00169 | 0.08800 ± 0.00265 | 0.16934 ± 0.00300 |
+| square | 128 | 0.02288 ± 0.00091 | 0.04576 ± 0.00110 | 0.09013 ± 0.00152 | 0.17388 ± 0.00168 |
+| triangular | 16 | 0.02644 ± 0.00065 | 0.05008 ± 0.00107 | 0.09582 ± 0.00161 | 0.18157 ± 0.00169 |
+| triangular | 32 | 0.02482 ± 0.00047 | 0.04852 ± 0.00085 | 0.09434 ± 0.00151 | 0.18052 ± 0.00249 |
+| triangular | 64 | 0.02409 ± 0.00101 | 0.04722 ± 0.00147 | 0.09170 ± 0.00146 | 0.17676 ± 0.00159 |
+| triangular | 128 | 0.02444 ± 0.00128 | 0.04770 ± 0.00162 | 0.09254 ± 0.00209 | 0.17648 ± 0.00231 |
+
+## Soft exponent-free Z(δ), G=(N+1)(T2−T1)/W
+
+Z(δ)=E[(1−G/δ)+]; same samples and covariance, not additional independent evidence.
+
+| Lattice | L | δ=.025 | δ=.05 | δ=.1 | δ=.2 |
+|---|---:|---:|---:|---:|---:|
+| square | 16 | 0.01815 ± 0.00123 | 0.02996 ± 0.00217 | 0.05294 ± 0.00402 | 0.09694 ± 0.00744 |
+| square | 32 | 0.01389 ± 0.00055 | 0.02616 ± 0.00089 | 0.05027 ± 0.00131 | 0.09699 ± 0.00191 |
+| square | 64 | 0.01402 ± 0.00052 | 0.02684 ± 0.00111 | 0.05168 ± 0.00219 | 0.10049 ± 0.00322 |
+| square | 128 | 0.01365 ± 0.00054 | 0.02718 ± 0.00080 | 0.05440 ± 0.00102 | 0.10634 ± 0.00186 |
+| triangular | 16 | 0.01439 ± 0.00043 | 0.02634 ± 0.00065 | 0.04990 ± 0.00107 | 0.09547 ± 0.00160 |
+| triangular | 32 | 0.01372 ± 0.00031 | 0.02665 ± 0.00049 | 0.05209 ± 0.00090 | 0.10119 ± 0.00160 |
+| triangular | 64 | 0.01360 ± 0.00079 | 0.02709 ± 0.00127 | 0.05294 ± 0.00181 | 0.10274 ± 0.00223 |
+| triangular | 128 | 0.01435 ± 0.00103 | 0.02813 ± 0.00137 | 0.05487 ± 0.00174 | 0.10610 ± 0.00236 |
+
+## Exact control and uncertainty
+
+L=3 was enumerated once per lattice, all 9! permutations. Square: P(D=0)=3/35, E[D]=3/2, E[D²]=43/14. Triangular: 2/35, 3/2, 81/28. Every exact identity passed before the pilot.
+
+JSON preserves full delete-one covariance, all leave-one estimates, atom Wilson intervals and an exact one-sided 95% upper bound if an atom count is zero. At n=16,000, zero observations would imply only P(atom)<0.0001872 at one-sided 95%; not zero probability. In the actual L=128 runs each lattice had one direct atom.
+
+The Beta CDF uses P[Beta(d,N+1−d)≤x]=P[Bin(N,x)≥d]. Backward binomial-tail sums avoid subtraction near zero; d=0 is retained as an atom. This integrates over labels without adding Monte Carlo noise or evidence.
+
+## What this does and does not decide
+
+Read the accompanying research note for the interpretation. This pilot distinguishes shrinking single-step double births from finite near-diagonal occupancy. It does not establish absence of a diagonal atom in any scaling limit.
diff --git a/analysis/birth-gap-20260929/analyze.py b/analysis/birth-gap-20260929/analyze.py
new file mode 100644
index 000000000..bb0f472fc
--- /dev/null
+++ b/analysis/birth-gap-20260929/analyze.py
@@ -0,0 +1,260 @@
+#!/usr/bin/env python3
+"""Joint-birth finite-size readout; stdlib, no fitting and no new random samples.
+
+Conditional on D=J2-J1, the uniform-label gap has Beta(D,N+1-D)
+law (a point mass at zero for D=0). Integer-parameter CDFs are evaluated
+by the exact binomial-tail identity, using floating-point recurrences.
+"""
+import argparse
+import collections
+import gzip
+import json
+import math
+import pathlib
+
+HERE = pathlib.Path(__file__).resolve().parent
+EPSILONS = (0.025, 0.05, 0.1, 0.2)
+
+
+def read_batch(path):
+    with gzip.open(path, "rt") as f:
+        return json.load(f)
+
+
+def combine(batches):
+    hist = collections.Counter()
+    for b in batches:
+        for j1, j2, count in b["histogram"]:
+            hist[j1, j2] += count
+    return hist
+
+
+def quantile(hist, u):
+    target = u * sum(hist.values())
+    running = 0
+    for value, count in sorted(hist.items()):
+        running += count
+        if running >= target:
+            return value
+    raise ValueError("empty quantile histogram")
+
+
+def beta_gap_cdfs(n, x):
+    """Return P[Beta(d,n+1-d)<=x]; omitted right tail underflows to zero.
+
+    Bin(n,x) survival, accumulated backwards to avoid cancellation near zero.
+    This recurrence is not a general extreme-tail special-function library.
+    Refuse numerical underflow at start. Terminating the already-underflowed
+    right tail makes L=512 inexpensive without a scientific approximation.
+    """
+    if x <= 0:
+        return [1.0] + [0.0] * n
+    if x >= 1:
+        return [1.0] * (n + 1)
+    pmf = [math.exp(n * math.log1p(-x))]
+    if pmf[0] == 0:
+        raise ValueError("binomial recurrence start underflow; out of pilot range")
+    ratio = x / (1 - x)
+    for k in range(n):
+        next_value = pmf[k] * ((n - k) / (k + 1)) * ratio
+        if next_value == 0:
+            break
+        pmf.append(next_value)
+    total = math.fsum(pmf)
+    if abs(total - 1) > 1e-9:
+        raise ArithmeticError(f"binomial recurrence normalization {total}")
+    survival = [0.0] * len(pmf)
+    running = 0.0
+    for k in range(len(pmf) - 1, 0, -1):
+        running += pmf[k]
+        survival[k] = running / total
+    survival[0] = 1.0
+    return survival
+
+
+def metrics(hist, L):
+    n, count = L * L, sum(hist.values())
+    pooled = collections.Counter()
+    gaps = collections.Counter()
+    s1 = s2 = s11 = s22 = s12 = 0
+    for (j1, j2), c in hist.items():
+        pooled[j1] += c
+        pooled[j2] += c
+        gaps[j2 - j1] += c
+        s1 += c * j1; s2 += c * j2
+        s11 += c * j1 * j1; s22 += c * j2 * j2; s12 += c * j1 * j2
+    width = quantile(pooled, .75) - quantile(pooled, .25)
+    if width <= 0:
+        raise ValueError("birth-mixture IQR is zero")
+    scale = L ** 1.25
+    mean_d = sum(d * c for d, c in gaps.items()) / count
+    atom = gaps[0] / count
+    var1 = s11 / count - (s1 / count) ** 2
+    var2 = s22 / count - (s2 / count) ** 2
+    cov12 = s12 / count - s1 * s2 / count ** 2
+    result = {
+        "direct_atom": atom,
+        "mean_D": mean_d,
+        "mean_D_over_L54": mean_d / scale,
+        "pooled_birth_IQR_J": float(width),
+        "pooled_birth_IQR_J_over_L54": width / scale,
+        "mean_D_over_IQR": mean_d / width,
+        "mean_label_gap": mean_d / (n + 1),
+        "birth_J_correlation": cov12 / math.sqrt(var1 * var2),
+    }
+    for eps in EPSILONS:
+        tag = f"{eps:g}"
+        result[f"P_D_L54_le_{tag}"] = sum(c for d, c in gaps.items() if d <= eps * scale) / count
+        result[f"P_D_IQR_le_{tag}"] = sum(c for d, c in gaps.items() if d <= eps * width) / count
+        result[f"Z_D_L54_{tag}"] = math.fsum(c * max(0.0, 1 - d / (eps * scale)) for d, c in gaps.items()) / count
+        result[f"Z_D_IQR_{tag}"] = math.fsum(c * max(0.0, 1 - d / (eps * width)) for d, c in gaps.items()) / count
+        for key, x in (("label_L34", eps / L ** .75),
+                       ("label_IQR", eps * width / (n + 1))):
+            probabilities = beta_gap_cdfs(n, x)
+            result[f"P_{key}_le_{tag}"] = math.fsum(c * probabilities[d] for d, c in gaps.items() if d < len(probabilities)) / count
+            tilted = beta_gap_cdfs(n + 1, x)
+            # E[(1-B/x)+], B~Beta(d,n+1-d), exact truncated first
+            # moment. Tilted beta has parameters d+1,n+1-d (sum n+2).
+            result[f"Z_{key}_{tag}"] = math.fsum(
+                c * max(0.0, (probabilities[d] if d < len(probabilities) else 0.0)
+                        - d / ((n + 1) * x) * (tilted[d + 1] if d + 1 < len(tilted) else 0.0))
+                for d, c in gaps.items()) / count
+    return result
+
+
+def wilson(count, n):
+    z = 1.959963984540054
+    p = count / n
+    center = (p + z * z / (2 * n)) / (1 + z * z / n)
+    half = z * math.sqrt(p * (1 - p) / n + z * z / (4 * n * n)) / (1 + z * z / n)
+    return [max(0.0, center - half), min(1.0, center + half)]
+
+
+def summarize(batch_data, L):
+    hist = combine(batch_data)
+    full = metrics(hist, L)
+    names = list(full)
+    b = len(batch_data)
+    leave = [metrics(combine(batch_data[:k] + batch_data[k+1:]), L) for k in range(b)]
+    means = {key: sum(x[key] for x in leave) / b for key in names}
+    covariance = [[(b - 1) / b * sum((x[a] - means[a]) * (x[c] - means[c]) for x in leave)
+                   for c in names] for a in names]
+    count = sum(hist.values())
+    atom = sum(c for (j1, j2), c in hist.items() if j1 == j2)
+    return {
+        "L": L, "N": L*L, "samples": count, "batch_count": b,
+        "point": full,
+        "jackknife_se": {key: math.sqrt(max(0, covariance[i][i])) for i, key in enumerate(names)},
+        "direct_atom_count": atom,
+        "direct_atom_wilson_two_sided_95": wilson(atom, count),
+        "direct_atom_zero_count_one_sided_95_upper": -math.expm1(math.log(.05)/count) if atom == 0 else None,
+        "metric_order": names,
+        "delete_one_batch_covariance": covariance,
+        "delete_one_batch_metrics": leave,
+        "uncertainty": f"paired complete-batch delete-one; re-estimates mixture IQR in each delete-one; {b} batches, exploratory uncertainty only",
+        "positive_near_diagonal_rule": "subtract direct_atom from any P_* (zero is otherwise included)",
+    }
+
+
+def main():
+    global EPSILONS
+    ap = argparse.ArgumentParser()
+    ap.add_argument("--data", type=pathlib.Path, default=HERE / "data")
+    ap.add_argument("--output-json", type=pathlib.Path, default=HERE / "summary.json")
+    ap.add_argument("--output-md", type=pathlib.Path, default=HERE / "SUMMARY.md")
+    ap.add_argument("--epsilons", type=float, nargs="+", default=list(EPSILONS),
+                    help="fixed dimensionless CDF/Z grid, choose before reading the new block")
+    args = ap.parse_args()
+    if any(not math.isfinite(e) or e <= 0 for e in args.epsilons):
+        ap.error("epsilons must be finite and positive")
+    EPSILONS = tuple(sorted(set(args.epsilons)))
+    manifest = json.loads((args.data / "run.json").read_text())
+    if manifest["status"] != "completed":
+        raise ValueError("run is incomplete")
+    groups = collections.defaultdict(list)
+    controls = []
+    for ref in manifest["controls"]:
+        b = read_batch(args.data / ref["file"])
+        controls.append({"lattice": b["lattice"], "L": 3, "permutations": b["samples"],
+                         "direct_atom_count": b["direct_atom_count"],
+                         "mean_D": b["sum_D"] / b["samples"],
+                         "mean_D2": b["sum_D2"] / b["samples"]})
+    for ref in manifest["batches"]:
+        b = read_batch(args.data / ref["file"])
+        groups[b["lattice"], b["L"]].append(b)
+    cells = []
+    for (lattice, L), batches in sorted(groups.items()):
+        cell = summarize(batches, L)
+        cell["lattice"] = lattice
+        cells.append(cell)
+    report = {
+        "schema": "matching-one.joint-birth-pilot.v1",
+        "epsilons": EPSILONS,
+        "definitions": {
+            "J1_J2": "first occupied-site counts at global ambient homology rank >=1 and =2, SAME uniform permutation",
+            "D": "J2-J1",
+            "L54": "D/L^(5/4), prescribed nearcritical benchmark, no exponent fit; not proved square-site scaling",
+            "label_L34": "L^(3/4)*(T2-T1), conditional Beta(D,N+1-D) mixing of uniform labels",
+            "pooled_birth_IQR_J": "Q75-Q25 of equal-weight mixture of J1 and J2; empirical lower quantiles",
+            "label_IQR": "(N+1)*(T2-T1)/(pooled_birth_IQR_J); exponent-free self-normalized readout",
+            "Z": "E[(1-G/delta)+] with delta on the same epsilon grid; exact histogram or conditional-Beta integration, added on theory-agent request after acquisition",
+            "triangular_geometry": "nearest neighbours (+/-1,0),(0,+/-1),(1,1),(-1,-1); LxL primitive-coordinate rhombus, not square conformal modulus",
+        },
+        "controls": controls, "exact_control_reference": manifest.get("exact_control_reference"), "cells": cells,
+        "production_role": manifest.get("production_role", "pilot"),
+        "total_production_samples": sum(c["samples"] for c in cells),
+        "production_seconds_sum": manifest["aggregate_engine_seconds"],
+        "acquisition_wall_seconds_including_build_control_benchmark": manifest["wall_seconds"],
+        "limitations": [
+            "Finite-size pilot, not a noncoalescence proof or arm-exponent estimate.",
+            "A finite epsilon grid does not implement epsilon->0 after L->infinity.",
+            "No cross-lattice equality asserted: primitive square and triangular tori have different conformal moduli.",
+            "All readouts within one cell use the SAME sample block; Beta reconstruction adds no independent evidence.",
+            "Finite batch counts support exploratory uncertainty only; no p-values, model selection or data-chosen exponent.",
+            "A stable typical gap does not exclude an additional limiting diagonal atom.",
+        ]}
+    args.output_json.write_text(json.dumps(report, indent=2) + "\n")
+    batch_counts = sorted(set(c["batch_count"] for c in cells))
+    sample_counts = sorted(set(c["samples"] for c in cells))
+    platform_label = f"{manifest['machine']} / Python {manifest['python']}"
+    lines = ["# Executed joint-birth block — 2026-09-29", "",
+             f"{report['total_production_samples']:,} independent-filtration samples; per-cell batch counts {batch_counts}, sample counts {sample_counts}.",
+             f"Acquisition ({platform_label}) including compilation, requested controls and benchmarks: {manifest['wall_seconds']:.2f} s, at most {manifest['workers']} concurrent jobs.", "",
+             f"Production seed namespace: `{report['production_role']}`.", "",
+             "## Scale and direct jump", "",
+             "Errors after ± are complete-batch jackknife SE, not confidence limits.", "",
+             "| Lattice | L | Direct 0→2 / samples | E[D]/L^(5/4) | E[D]/mixture IQR |", "|---|---:|---:|---:|---:|"]
+    for c in cells:
+        p, e = c["point"], c["jackknife_se"]
+        lines.append(f"| {c['lattice']} | {c['L']} | {c['direct_atom_count']} / {c['samples']} | {p['mean_D_over_L54']:.4f} ± {e['mean_D_over_L54']:.4f} | {p['mean_D_over_IQR']:.4f} ± {e['mean_D_over_IQR']:.4f} |")
+    grid_header = "| Lattice | L | " + " | ".join(f"ε/δ={eps:g}" for eps in EPSILONS) + " |"
+    grid_separator = "|---|---:|" + "---:|" * len(EPSILONS)
+    for key, title in (("label_L34", "Continuous-label near diagonal: P[L^(3/4)(T2−T1) ≤ ε]"),
+                       ("label_IQR", "Exponent-free near diagonal: P[(N+1)(T2−T1)/W ≤ ε]"),
+                       ("D_L54", "Discrete near diagonal: P[D/L^(5/4) ≤ ε]"),
+                       ("D_IQR", "Discrete exponent-free near diagonal: P[D/W ≤ ε]")):
+        lines += ["", "## " + title, "", "W is the pooled J1/J2 interquartile width; zero atom included. Subtract the preceding direct-atom frequency for strictly positive mass.", "",
+                  grid_header, grid_separator]
+        for c in cells:
+            values = [f"{c['point'][f'P_{key}_le_{eps:g}']:.5f} ± {c['jackknife_se'][f'P_{key}_le_{eps:g}']:.5f}" for eps in EPSILONS]
+            lines.append(f"| {c['lattice']} | {c['L']} | " + " | ".join(values) + " |")
+    for key, title in (("label_L34", "Soft near-diagonal Z(δ), G=L^(3/4)(T2−T1)"),
+                       ("label_IQR", "Soft exponent-free Z(δ), G=(N+1)(T2−T1)/W")):
+        lines += ["", "## " + title, "", "Z(δ)=E[(1−G/δ)+]; same samples and covariance, not additional independent evidence.", "",
+                  grid_header, grid_separator]
+        for c in cells:
+            values = [f"{c['point'][f'Z_{key}_{eps:g}']:.5f} ± {c['jackknife_se'][f'Z_{key}_{eps:g}']:.5f}" for eps in EPSILONS]
+            lines.append(f"| {c['lattice']} | {c['L']} | " + " | ".join(values) + " |")
+    control_text = ("L=3 was enumerated once per lattice, all 9! permutations. Square: P(D=0)=3/35, E[D]=3/2, E[D²]=43/14. Triangular: 2/35, 3/2, 81/28. Every exact identity passed before this block."
+                    if controls else "This block did not repeat exact enumeration; supplied prior-control reference: " + str(manifest.get("exact_control_reference")))
+    lines += ["", "## Exact control and uncertainty", "", control_text, "",
+              "JSON preserves full delete-one covariance, all leave-one estimates, atom Wilson intervals and an exact one-sided 95% upper bound if an atom count is zero. Zero observations give only the finite-sample upper bound 1−0.05^(1/n), never zero probability.", "",
+              "The Beta CDF uses P[Beta(d,N+1−d)≤x]=P[Bin(N,x)≥d]. Backward binomial-tail sums avoid subtraction near zero; d=0 is retained as an atom. This integrates over labels without adding Monte Carlo noise or evidence.", "",
+              "## What this does and does not decide", "",
+              "Read the accompanying research note for the interpretation. This pilot distinguishes shrinking single-step double births from finite near-diagonal occupancy. It does not establish absence of a diagonal atom in any scaling limit.", ""]
+    args.output_md.write_text("\n".join(lines))
+    print(args.output_md.read_text())
+
+
+if __name__ == "__main__":
+    main()
diff --git a/analysis/birth-gap-20260929/cloud-SUMMARY.md b/analysis/birth-gap-20260929/cloud-SUMMARY.md
new file mode 100644
index 000000000..763806352
--- /dev/null
+++ b/analysis/birth-gap-20260929/cloud-SUMMARY.md
@@ -0,0 +1,123 @@
+# Executed joint-birth block — 2026-09-29
+
+560,000 independent-filtration samples; per-cell batch counts [14], sample counts [70000].
+Acquisition (aarch64 / Python 3.9.9) including compilation, requested controls and benchmarks: 329.71 s, at most 14 concurrent jobs.
+
+Production seed namespace: `cloud-independent-20260929`.
+
+## Scale and direct jump
+
+Errors after ± are complete-batch jackknife SE, not confidence limits.
+
+| Lattice | L | Direct 0→2 / samples | E[D]/L^(5/4) | E[D]/mixture IQR |
+|---|---:|---:|---:|---:|
+| square | 64 | 50 / 70000 | 0.4251 ± 0.0014 | 0.7398 ± 0.0025 |
+| square | 128 | 8 / 70000 | 0.4257 ± 0.0019 | 0.7244 ± 0.0041 |
+| square | 256 | 4 / 70000 | 0.4247 ± 0.0011 | 0.7107 ± 0.0025 |
+| square | 512 | 1 / 70000 | 0.4264 ± 0.0013 | 0.7046 ± 0.0023 |
+| triangular | 64 | 34 / 70000 | 0.4077 ± 0.0011 | 0.7236 ± 0.0128 |
+| triangular | 128 | 5 / 70000 | 0.4065 ± 0.0013 | 0.7028 ± 0.0055 |
+| triangular | 256 | 2 / 70000 | 0.4067 ± 0.0008 | 0.6907 ± 0.0032 |
+| triangular | 512 | 0 / 70000 | 0.4081 ± 0.0014 | 0.6855 ± 0.0028 |
+
+## Continuous-label near diagonal: P[L^(3/4)(T2−T1) ≤ ε]
+
+W is the pooled J1/J2 interquartile width; zero atom included. Subtract the preceding direct-atom frequency for strictly positive mass.
+
+| Lattice | L | ε/δ=0.00625 | ε/δ=0.0125 | ε/δ=0.025 | ε/δ=0.05 | ε/δ=0.1 | ε/δ=0.2 |
+|---|---:|---:|---:|---:|---:|---:|---:|
+| square | 64 | 0.01205 ± 0.00023 | 0.02360 ± 0.00043 | 0.04691 ± 0.00072 | 0.09247 ± 0.00091 | 0.17651 ± 0.00130 | 0.33010 ± 0.00166 |
+| square | 128 | 0.01184 ± 0.00045 | 0.02365 ± 0.00059 | 0.04701 ± 0.00064 | 0.09176 ± 0.00080 | 0.17716 ± 0.00147 | 0.32902 ± 0.00231 |
+| square | 256 | 0.01181 ± 0.00033 | 0.02384 ± 0.00049 | 0.04659 ± 0.00069 | 0.09206 ± 0.00117 | 0.17921 ± 0.00153 | 0.33283 ± 0.00200 |
+| square | 512 | 0.01136 ± 0.00038 | 0.02341 ± 0.00058 | 0.04589 ± 0.00062 | 0.09117 ± 0.00105 | 0.17519 ± 0.00148 | 0.32862 ± 0.00202 |
+| triangular | 64 | 0.01302 ± 0.00020 | 0.02520 ± 0.00027 | 0.04879 ± 0.00044 | 0.09491 ± 0.00079 | 0.18319 ± 0.00115 | 0.34109 ± 0.00167 |
+| triangular | 128 | 0.01260 ± 0.00033 | 0.02472 ± 0.00055 | 0.04869 ± 0.00086 | 0.09501 ± 0.00130 | 0.18273 ± 0.00139 | 0.34193 ± 0.00164 |
+| triangular | 256 | 0.01278 ± 0.00033 | 0.02472 ± 0.00031 | 0.04840 ± 0.00063 | 0.09443 ± 0.00098 | 0.18373 ± 0.00087 | 0.34152 ± 0.00159 |
+| triangular | 512 | 0.01237 ± 0.00035 | 0.02415 ± 0.00065 | 0.04735 ± 0.00109 | 0.09467 ± 0.00170 | 0.18254 ± 0.00236 | 0.34036 ± 0.00234 |
+
+## Exponent-free near diagonal: P[(N+1)(T2−T1)/W ≤ ε]
+
+W is the pooled J1/J2 interquartile width; zero atom included. Subtract the preceding direct-atom frequency for strictly positive mass.
+
+| Lattice | L | ε/δ=0.00625 | ε/δ=0.0125 | ε/δ=0.025 | ε/δ=0.05 | ε/δ=0.1 | ε/δ=0.2 |
+|---|---:|---:|---:|---:|---:|---:|---:|
+| square | 64 | 0.00720 ± 0.00016 | 0.01376 ± 0.00026 | 0.02706 ± 0.00048 | 0.05383 ± 0.00077 | 0.10548 ± 0.00095 | 0.20059 ± 0.00140 |
+| square | 128 | 0.00693 ± 0.00030 | 0.01392 ± 0.00048 | 0.02778 ± 0.00059 | 0.05497 ± 0.00065 | 0.10718 ± 0.00089 | 0.20564 ± 0.00172 |
+| square | 256 | 0.00704 ± 0.00026 | 0.01418 ± 0.00037 | 0.02840 ± 0.00060 | 0.05542 ± 0.00079 | 0.10929 ± 0.00133 | 0.21091 ± 0.00168 |
+| square | 512 | 0.00682 ± 0.00028 | 0.01383 ± 0.00043 | 0.02843 ± 0.00057 | 0.05512 ± 0.00068 | 0.10958 ± 0.00094 | 0.20945 ± 0.00155 |
+| triangular | 64 | 0.00759 ± 0.00015 | 0.01459 ± 0.00025 | 0.02824 ± 0.00044 | 0.05469 ± 0.00099 | 0.10634 ± 0.00209 | 0.20483 ± 0.00360 |
+| triangular | 128 | 0.00736 ± 0.00023 | 0.01452 ± 0.00038 | 0.02849 ± 0.00066 | 0.05609 ± 0.00105 | 0.10940 ± 0.00163 | 0.20891 ± 0.00189 |
+| triangular | 256 | 0.00771 ± 0.00025 | 0.01492 ± 0.00034 | 0.02896 ± 0.00036 | 0.05652 ± 0.00077 | 0.11062 ± 0.00103 | 0.21368 ± 0.00135 |
+| triangular | 512 | 0.00747 ± 0.00024 | 0.01469 ± 0.00041 | 0.02847 ± 0.00080 | 0.05643 ± 0.00117 | 0.11217 ± 0.00189 | 0.21455 ± 0.00213 |
+
+## Discrete near diagonal: P[D/L^(5/4) ≤ ε]
+
+W is the pooled J1/J2 interquartile width; zero atom included. Subtract the preceding direct-atom frequency for strictly positive mass.
+
+| Lattice | L | ε/δ=0.00625 | ε/δ=0.0125 | ε/δ=0.025 | ε/δ=0.05 | ε/δ=0.1 | ε/δ=0.2 |
+|---|---:|---:|---:|---:|---:|---:|---:|
+| square | 64 | 0.01064 ± 0.00022 | 0.02063 ± 0.00045 | 0.04116 ± 0.00098 | 0.09260 ± 0.00100 | 0.17597 ± 0.00147 | 0.32983 ± 0.00191 |
+| square | 128 | 0.00867 ± 0.00045 | 0.02190 ± 0.00064 | 0.04410 ± 0.00057 | 0.09006 ± 0.00087 | 0.17669 ± 0.00161 | 0.32849 ± 0.00238 |
+| square | 256 | 0.01093 ± 0.00031 | 0.02226 ± 0.00051 | 0.04534 ± 0.00065 | 0.09206 ± 0.00127 | 0.17863 ± 0.00154 | 0.33189 ± 0.00206 |
+| square | 512 | 0.01123 ± 0.00037 | 0.02294 ± 0.00064 | 0.04534 ± 0.00062 | 0.09059 ± 0.00116 | 0.17487 ± 0.00152 | 0.32896 ± 0.00198 |
+| triangular | 64 | 0.01171 ± 0.00031 | 0.02270 ± 0.00042 | 0.04343 ± 0.00045 | 0.09516 ± 0.00104 | 0.18276 ± 0.00135 | 0.34063 ± 0.00185 |
+| triangular | 128 | 0.00939 ± 0.00030 | 0.02296 ± 0.00058 | 0.04527 ± 0.00092 | 0.09261 ± 0.00138 | 0.18237 ± 0.00137 | 0.34184 ± 0.00179 |
+| triangular | 256 | 0.01206 ± 0.00036 | 0.02310 ± 0.00031 | 0.04773 ± 0.00066 | 0.09394 ± 0.00103 | 0.18326 ± 0.00100 | 0.34021 ± 0.00171 |
+| triangular | 512 | 0.01229 ± 0.00039 | 0.02383 ± 0.00065 | 0.04670 ± 0.00112 | 0.09373 ± 0.00172 | 0.18246 ± 0.00258 | 0.34034 ± 0.00228 |
+
+## Discrete exponent-free near diagonal: P[D/W ≤ ε]
+
+W is the pooled J1/J2 interquartile width; zero atom included. Subtract the preceding direct-atom frequency for strictly positive mass.
+
+| Lattice | L | ε/δ=0.00625 | ε/δ=0.0125 | ε/δ=0.025 | ε/δ=0.05 | ε/δ=0.1 | ε/δ=0.2 |
+|---|---:|---:|---:|---:|---:|---:|---:|
+| square | 64 | 0.00071 ± 0.00011 | 0.01064 ± 0.00022 | 0.02063 ± 0.00045 | 0.05191 ± 0.00088 | 0.10260 ± 0.00105 | 0.19351 ± 0.00155 |
+| square | 128 | 0.00421 ± 0.00023 | 0.01349 ± 0.00065 | 0.02611 ± 0.00069 | 0.05263 ± 0.00065 | 0.10601 ± 0.00093 | 0.20389 ± 0.00160 |
+| square | 256 | 0.00560 ± 0.00030 | 0.01270 ± 0.00037 | 0.02816 ± 0.00059 | 0.05426 ± 0.00079 | 0.10899 ± 0.00127 | 0.21044 ± 0.00169 |
+| square | 512 | 0.00664 ± 0.00033 | 0.01327 ± 0.00047 | 0.02791 ± 0.00061 | 0.05447 ± 0.00070 | 0.10954 ± 0.00108 | 0.20889 ± 0.00206 |
+| triangular | 64 | 0.00049 ± 0.00007 | 0.01171 ± 0.00031 | 0.02270 ± 0.00042 | 0.05409 ± 0.00064 | 0.10547 ± 0.00102 | 0.20239 ± 0.00127 |
+| triangular | 128 | 0.00480 ± 0.00027 | 0.01426 ± 0.00044 | 0.02736 ± 0.00065 | 0.05429 ± 0.00095 | 0.10591 ± 0.00705 | 0.20581 ± 0.00653 |
+| triangular | 256 | 0.00633 ± 0.00025 | 0.01411 ± 0.00040 | 0.02883 ± 0.00037 | 0.05627 ± 0.00074 | 0.11030 ± 0.00084 | 0.21306 ± 0.00104 |
+| triangular | 512 | 0.00739 ± 0.00027 | 0.01469 ± 0.00044 | 0.02836 ± 0.00087 | 0.05596 ± 0.00121 | 0.11230 ± 0.00199 | 0.21454 ± 0.00195 |
+
+## Soft near-diagonal Z(δ), G=L^(3/4)(T2−T1)
+
+Z(δ)=E[(1−G/δ)+]; same samples and covariance, not additional independent evidence.
+
+| Lattice | L | ε/δ=0.00625 | ε/δ=0.0125 | ε/δ=0.025 | ε/δ=0.05 | ε/δ=0.1 | ε/δ=0.2 |
+|---|---:|---:|---:|---:|---:|---:|---:|
+| square | 64 | 0.00636 ± 0.00015 | 0.01209 ± 0.00023 | 0.02367 ± 0.00040 | 0.04679 ± 0.00060 | 0.09092 ± 0.00078 | 0.17300 ± 0.00106 |
+| square | 128 | 0.00591 ± 0.00025 | 0.01183 ± 0.00039 | 0.02361 ± 0.00049 | 0.04655 ± 0.00056 | 0.09079 ± 0.00073 | 0.17293 ± 0.00115 |
+| square | 256 | 0.00591 ± 0.00020 | 0.01189 ± 0.00029 | 0.02360 ± 0.00043 | 0.04646 ± 0.00064 | 0.09123 ± 0.00090 | 0.17467 ± 0.00123 |
+| square | 512 | 0.00562 ± 0.00022 | 0.01148 ± 0.00034 | 0.02317 ± 0.00046 | 0.04576 ± 0.00064 | 0.08974 ± 0.00087 | 0.17182 ± 0.00124 |
+| triangular | 64 | 0.00678 ± 0.00012 | 0.01296 ± 0.00017 | 0.02501 ± 0.00023 | 0.04848 ± 0.00039 | 0.09393 ± 0.00058 | 0.17923 ± 0.00088 |
+| triangular | 128 | 0.00637 ± 0.00019 | 0.01253 ± 0.00031 | 0.02463 ± 0.00050 | 0.04830 ± 0.00076 | 0.09399 ± 0.00105 | 0.17899 ± 0.00112 |
+| triangular | 256 | 0.00653 ± 0.00020 | 0.01264 ± 0.00025 | 0.02463 ± 0.00032 | 0.04799 ± 0.00054 | 0.09377 ± 0.00058 | 0.17923 ± 0.00077 |
+| triangular | 512 | 0.00626 ± 0.00019 | 0.01229 ± 0.00032 | 0.02396 ± 0.00061 | 0.04752 ± 0.00093 | 0.09336 ± 0.00148 | 0.17855 ± 0.00180 |
+
+## Soft exponent-free Z(δ), G=(N+1)(T2−T1)/W
+
+Z(δ)=E[(1−G/δ)+]; same samples and covariance, not additional independent evidence.
+
+| Lattice | L | ε/δ=0.00625 | ε/δ=0.0125 | ε/δ=0.025 | ε/δ=0.05 | ε/δ=0.1 | ε/δ=0.2 |
+|---|---:|---:|---:|---:|---:|---:|---:|
+| square | 64 | 0.00395 ± 0.00012 | 0.00721 ± 0.00016 | 0.01380 ± 0.00026 | 0.02713 ± 0.00045 | 0.05355 ± 0.00063 | 0.10357 ± 0.00083 |
+| square | 128 | 0.00348 ± 0.00016 | 0.00695 ± 0.00028 | 0.01390 ± 0.00041 | 0.02769 ± 0.00049 | 0.05444 ± 0.00057 | 0.10580 ± 0.00082 |
+| square | 256 | 0.00356 ± 0.00015 | 0.00707 ± 0.00022 | 0.01421 ± 0.00033 | 0.02807 ± 0.00049 | 0.05533 ± 0.00075 | 0.10821 ± 0.00103 |
+| square | 512 | 0.00336 ± 0.00015 | 0.00683 ± 0.00024 | 0.01399 ± 0.00036 | 0.02792 ± 0.00047 | 0.05528 ± 0.00064 | 0.10758 ± 0.00087 |
+| triangular | 64 | 0.00405 ± 0.00009 | 0.00757 ± 0.00014 | 0.01451 ± 0.00023 | 0.02802 ± 0.00045 | 0.05435 ± 0.00100 | 0.10518 ± 0.00191 |
+| triangular | 128 | 0.00373 ± 0.00014 | 0.00734 ± 0.00022 | 0.01443 ± 0.00036 | 0.02839 ± 0.00060 | 0.05560 ± 0.00094 | 0.10778 ± 0.00133 |
+| triangular | 256 | 0.00391 ± 0.00015 | 0.00763 ± 0.00022 | 0.01479 ± 0.00026 | 0.02883 ± 0.00037 | 0.05622 ± 0.00061 | 0.10961 ± 0.00076 |
+| triangular | 512 | 0.00378 ± 0.00014 | 0.00742 ± 0.00022 | 0.01454 ± 0.00038 | 0.02843 ± 0.00068 | 0.05650 ± 0.00105 | 0.11022 ± 0.00155 |
+
+## Exact control and uncertainty
+
+This block did not repeat exact enumeration; supplied prior-control reference: abc78c6a:analysis/birth-gap-20260929/data/run.json
+
+JSON preserves full delete-one covariance, all leave-one estimates, atom Wilson intervals and an exact one-sided 95% upper bound if an atom count is zero. Zero observations give only the finite-sample upper bound 1−0.05^(1/n), never zero probability.
+
+The Beta CDF uses P[Beta(d,N+1−d)≤x]=P[Bin(N,x)≥d]. Backward binomial-tail sums avoid subtraction near zero; d=0 is retained as an atom. This integrates over labels without adding Monte Carlo noise or evidence.
+
+## What this does and does not decide
+
+Read the accompanying research note for the interpretation. This pilot distinguishes shrinking single-step double births from finite near-diagonal occupancy. It does not establish absence of a diagonal atom in any scaling limit.
diff --git a/analysis/birth-gap-20260929/cloud-analysis-time.txt b/analysis/birth-gap-20260929/cloud-analysis-time.txt
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+++ b/analysis/birth-gap-20260929/cloud-analysis-time.txt
@@ -0,0 +1,3 @@
+real 17.02
+user 16.96
+sys 0.03
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+{
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+  "utc_started": "2026-09-29T11:42:47.765899+00:00",
+  "command": "python3 /workspace/Matching-One-TV2N0X/birth-gap-20260929/run_pilot.py --compiler g++ --role cloud-independent-20260929 --sizes 64 128 256 512 --batches 14 --samples 5000 --workers 14 --budget-seconds 300 --skip-exact-control --exact-control-reference abc78c6a:analysis/birth-gap-20260929/data/run.json --git-base-commit abc78c6a1c69e83c9a2daf4d92911833bd6f3937 --output cloud-data-5k",
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+  "workers": 14,
+  "compiler": "g++ (GCC) 10.3.1\nCopyright (C) 2020 Free Software Foundation, Inc.\nThis is free software; see the source for copying conditions.  There is NO\nwarranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.",
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+  "production_role": "cloud-independent-20260929",
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+  },
+  "seed_rule": "first 64 SHA256 bits of matching-one/birth-gap/20260929/{role}/{lattice}/{L}/{batch}",
+  "random_streams": "separate mt19937_64 per (role,lattice,L,batch); no paired/cross-size common random numbers",
+  "sampling": "unbiased Fisher-Yates of all N sites; stop after same-filtration global rank reaches 2",
+  "controls": [],
+  "benchmarks": [
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+  ],
+  "build_command": "g++ -O3 -std=c++17 -DNDEBUG /workspace/Matching-One-TV2N0X/birth-gap-20260929/engine.cpp -o /workspace/Matching-One-TV2N0X/birth-gap-20260929/tmp/matching-birth-gap-build-ylhk2zq_/birth-gap",
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+}
diff --git a/analysis/birth-gap-20260929/cloud-data/benchmark-cloud-independent-20260929-square-L512-b00.json.gz b/analysis/birth-gap-20260929/cloud-data/benchmark-cloud-independent-20260929-square-L512-b00.json.gz
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index 000000000..86758bf63
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diff --git a/analysis/birth-gap-20260929/cloud-data/benchmark-cloud-independent-20260929-triangular-L512-b00.json.gz b/analysis/birth-gap-20260929/cloud-data/benchmark-cloud-independent-20260929-triangular-L512-b00.json.gz
new file mode 100644
index 000000000..3eccc2a98
Binary files /dev/null and b/analysis/birth-gap-20260929/cloud-data/benchmark-cloud-independent-20260929-triangular-L512-b00.json.gz differ
diff --git a/analysis/birth-gap-20260929/cloud-data/run.json b/analysis/birth-gap-20260929/cloud-data/run.json
new file mode 100644
index 000000000..fe74fc2fe
--- /dev/null
+++ b/analysis/birth-gap-20260929/cloud-data/run.json
@@ -0,0 +1,55 @@
+{
+  "schema": "matching-one.joint-birth-run.v1",
+  "utc_started": "2026-09-29T11:41:54.042025+00:00",
+  "command": "python3 /workspace/Matching-One-TV2N0X/birth-gap-20260929/run_pilot.py --compiler g++ --role cloud-independent-20260929 --sizes 64 128 256 512 --batches 14 --samples 10000 --workers 14 --budget-seconds 300 --skip-exact-control --exact-control-reference abc78c6a:analysis/birth-gap-20260929/data/run.json --git-base-commit abc78c6a1c69e83c9a2daf4d92911833bd6f3937 --output cloud-data",
+  "platform": "Linux-5.10.0-182.0.0.95.r3582_286.hce2.aarch64-aarch64-with-glibc2.34",
+  "machine": "aarch64",
+  "python": "3.9.9",
+  "workers": 14,
+  "compiler": "g++ (GCC) 10.3.1\nCopyright (C) 2020 Free Software Foundation, Inc.\nThis is free software; see the source for copying conditions.  There is NO\nwarranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.",
+  "git_base_commit": "abc78c6a1c69e83c9a2daf4d92911833bd6f3937",
+  "production_role": "cloud-independent-20260929",
+  "exact_control_reference": "abc78c6a:analysis/birth-gap-20260929/data/run.json",
+  "exact_control_skipped": true,
+  "estimated_wall_budget_seconds": 300.0,
+  "source_sha256": {
+    "engine.cpp": "72997c726bc734da7695d104fc8fb51ed5672506c967c2a88312bdc092925506",
+    "run_pilot.py": "8338502301e34d719e3729d435d794246f80a8fb2954ffb8617450f6f0de747c"
+  },
+  "seed_rule": "first 64 SHA256 bits of matching-one/birth-gap/20260929/{role}/{lattice}/{L}/{batch}",
+  "random_streams": "separate mt19937_64 per (role,lattice,L,batch); no paired/cross-size common random numbers",
+  "sampling": "unbiased Fisher-Yates of all N sites; stop after same-filtration global rank reaches 2",
+  "controls": [],
+  "benchmarks": [
+    {
+      "file": "benchmark-cloud-independent-20260929-square-L512-b00.json.gz",
+      "lattice": "square",
+      "L": 512,
+      "batch": 0,
+      "role": "benchmark-cloud-independent-20260929",
+      "seed": "11773303504918451570",
+      "samples": 200,
+      "runtime_seconds": 3.187666129,
+      "command": "/workspace/Matching-One-TV2N0X/birth-gap-20260929/tmp/matching-birth-gap-build-j4ke5sm6/birth-gap square 512 200 11773303504918451570 /workspace/Matching-One-TV2N0X/birth-gap-20260929/tmp/matching-birth-gap-build-j4ke5sm6/benchmark-cloud-independent-20260929-square-L512-b00.json sample",
+      "sha256": "2f761e78f28ef629496217661f5ea9ce5ff67757343ca4d0100ffe6453f0d824",
+      "dependency_group": "benchmark-cloud-independent-20260929/square/L512/batch0"
+    },
+    {
+      "file": "benchmark-cloud-independent-20260929-triangular-L512-b00.json.gz",
+      "lattice": "triangular",
+      "L": 512,
+      "batch": 0,
+      "role": "benchmark-cloud-independent-20260929",
+      "seed": "8466267847465560210",
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+      "dependency_group": "benchmark-cloud-independent-20260929/triangular/L512/batch0"
+    }
+  ],
+  "batches": [],
+  "build_command": "g++ -O3 -std=c++17 -DNDEBUG /workspace/Matching-One-TV2N0X/birth-gap-20260929/engine.cpp -o /workspace/Matching-One-TV2N0X/birth-gap-20260929/tmp/matching-birth-gap-build-j4ke5sm6/birth-gap",
+  "estimated_production_cpu_seconds": 6031.443064499999,
+  "status": "benchmark-budget-stop"
+}
diff --git a/analysis/birth-gap-20260929/cloud-summary.json b/analysis/birth-gap-20260929/cloud-summary.json
new file mode 100644
index 000000000..55bedf888
--- /dev/null
+++ b/analysis/birth-gap-20260929/cloud-summary.json
@@ -0,0 +1,34061 @@
+{
+  "schema": "matching-one.joint-birth-pilot.v1",
+  "epsilons": [
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+    0.0125,
+    0.025,
+    0.05,
+    0.1,
+    0.2
+  ],
+  "definitions": {
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+    "D": "J2-J1",
+    "L54": "D/L^(5/4), prescribed nearcritical benchmark, no exponent fit; not proved square-site scaling",
+    "label_L34": "L^(3/4)*(T2-T1), conditional Beta(D,N+1-D) mixing of uniform labels",
+    "pooled_birth_IQR_J": "Q75-Q25 of equal-weight mixture of J1 and J2; empirical lower quantiles",
+    "label_IQR": "(N+1)*(T2-T1)/(pooled_birth_IQR_J); exponent-free self-normalized readout",
+    "Z": "E[(1-G/delta)+] with delta on the same epsilon grid; exact histogram or conditional-Beta integration, added on theory-agent request after acquisition",
+    "triangular_geometry": "nearest neighbours (+/-1,0),(0,+/-1),(1,1),(-1,-1); LxL primitive-coordinate rhombus, not square conformal modulus"
+  },
+  "controls": [],
+  "exact_control_reference": "abc78c6a:analysis/birth-gap-20260929/data/run.json",
+  "cells": [
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+      "N": 4096,
+      "samples": 70000,
+      "batch_count": 14,
+      "point": {
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+        "mean_D": 76.94288571428571,
+        "mean_D_over_L54": 0.425053408223695,
+        "pooled_birth_IQR_J": 104.0,
+        "pooled_birth_IQR_J_over_L54": 0.5745242597140698,
+        "mean_D_over_IQR": 0.7398354395604395,
+        "mean_label_gap": 0.01878029917361135,
+        "birth_J_correlation": 0.5651986205372764,
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+        "Z_D_L54_0.00625": 0.0018671569120598138,
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+        "pooled_birth_IQR_J",
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+        "mean_D_over_IQR",
+        "mean_label_gap",
+        "birth_J_correlation",
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+      "uncertainty": "paired complete-batch delete-one; re-estimates mixture IQR in each delete-one; 14 batches, exploratory uncertainty only",
+      "positive_near_diagonal_rule": "subtract direct_atom from any P_* (zero is otherwise included)",
+      "lattice": "triangular"
+    }
+  ],
+  "production_role": "cloud-independent-20260929",
+  "total_production_samples": 560000,
+  "production_seconds_sum": 4491.686453404,
+  "acquisition_wall_seconds_including_build_control_benchmark": 329.7057283539325,
+  "limitations": [
+    "Finite-size pilot, not a noncoalescence proof or arm-exponent estimate.",
+    "A finite epsilon grid does not implement epsilon->0 after L->infinity.",
+    "No cross-lattice equality asserted: primitive square and triangular tori have different conformal moduli.",
+    "All readouts within one cell use the SAME sample block; Beta reconstruction adds no independent evidence.",
+    "Finite batch counts support exploratory uncertainty only; no p-values, model selection or data-chosen exponent.",
+    "A stable typical gap does not exclude an additional limiting diagonal atom."
+  ]
+}
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+++ b/analysis/birth-gap-20260929/data/run.json
@@ -0,0 +1,914 @@
+{
+  "schema": "matching-one.joint-birth-run.v1",
+  "utc_started": "2026-09-29T11:30:39.694830+00:00",
+  "command": "python3 /private/tmp/matching-frontier-20260929-Jk39nH/analysis/birth-gap-20260929/run_pilot.py --sizes 16 32 64 128 --batches 8 --samples 2000 --workers 4",
+  "platform": "macOS-27.0-arm64-arm-64bit-Mach-O",
+  "machine": "arm64",
+  "python": "3.13.7",
+  "workers": 4,
+  "compiler": "Apple clang version 21.0.0 (clang-2100.3.34.2)\nTarget: arm64-apple-darwin27.0.0\nThread model: posix\nInstalledDir: /Library/Developer/CommandLineTools/usr/bin",
+  "git_base_commit": "d31fa5fe9584b77595e4a78b9560aa2ff77cf0b3",
+  "source_sha256": {
+    "engine.cpp": "72997c726bc734da7695d104fc8fb51ed5672506c967c2a88312bdc092925506",
+    "run_pilot.py": "da256dc5a9500b543b67f0a2d89735472f2e9e17b20629c104248156ab1245a5"
+  },
+  "seed_rule": "first 64 SHA256 bits of matching-one/birth-gap/20260929/{role}/{lattice}/{L}/{batch}",
+  "random_streams": "separate mt19937_64 per (role,lattice,L,batch); no paired/cross-size common random numbers",
+  "sampling": "unbiased Fisher-Yates of all N sites; stop after same-filtration global rank reaches 2",
+  "controls": [
+    {
+      "file": "exact-control-square-L3-b00.json.gz",
+      "lattice": "square",
+      "L": 3,
+      "batch": 0,
+      "role": "exact-control",
+      "seed": "16372218219864572294",
+      "samples": 362880,
+      "runtime_seconds": 0.061204958,
+      "command": "/var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/birth-gap square 3 0 16372218219864572294 /var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/exact-control-square-L3-b00.json exact",
+      "sha256": "dccb0e132d7b2100bd39c6414f385366354c9099facbecbbaa1e3a5fd8f37a78",
+      "dependency_group": "exact-control/square/L3/batch0"
+    },
+    {
+      "file": "exact-control-triangular-L3-b00.json.gz",
+      "lattice": "triangular",
+      "L": 3,
+      "batch": 0,
+      "role": "exact-control",
+      "seed": "12970221764285794744",
+      "samples": 362880,
+      "runtime_seconds": 0.036185875,
+      "command": "/var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/birth-gap triangular 3 0 12970221764285794744 /var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/exact-control-triangular-L3-b00.json exact",
+      "sha256": "eed9f4d3f0e996bfaba8e6b0ab3928ba26776bef0b2b1a18d86e08b051d30391",
+      "dependency_group": "exact-control/triangular/L3/batch0"
+    }
+  ],
+  "benchmarks": [
+    {
+      "file": "benchmark-square-L128-b00.json.gz",
+      "lattice": "square",
+      "L": 128,
+      "batch": 0,
+      "role": "benchmark",
+      "seed": "17618861968012045904",
+      "samples": 200,
+      "runtime_seconds": 0.063210917,
+      "command": "/var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/birth-gap square 128 200 17618861968012045904 /var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/benchmark-square-L128-b00.json sample",
+      "sha256": "daba37605a1cf8f6956c8753b63f19ba5385713558c314c7d45ea39799ec1681",
+      "dependency_group": "benchmark/square/L128/batch0"
+    },
+    {
+      "file": "benchmark-triangular-L128-b00.json.gz",
+      "lattice": "triangular",
+      "L": 128,
+      "batch": 0,
+      "role": "benchmark",
+      "seed": "7859738520867152032",
+      "samples": 200,
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+      "runtime_seconds": 0.183019208,
+      "command": "/var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/birth-gap triangular 64 2000 958813401741228500 /var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/pilot-triangular-L64-b03.json sample",
+      "sha256": "79d33f84c0dd182eee25f74ed7e34ccf3f9630102e9298cac3382f7dc3e87a97",
+      "dependency_group": "pilot/triangular/L64/batch3"
+    },
+    {
+      "file": "pilot-triangular-L64-b04.json.gz",
+      "lattice": "triangular",
+      "L": 64,
+      "batch": 4,
+      "role": "pilot",
+      "seed": "2169422698097871123",
+      "samples": 2000,
+      "runtime_seconds": 0.189221917,
+      "command": "/var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/birth-gap triangular 64 2000 2169422698097871123 /var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/pilot-triangular-L64-b04.json sample",
+      "sha256": "aa3dc386c99008cb500932acee0b05e0d46c7b0b7350afb21e25126059eddc14",
+      "dependency_group": "pilot/triangular/L64/batch4"
+    },
+    {
+      "file": "pilot-triangular-L64-b05.json.gz",
+      "lattice": "triangular",
+      "L": 64,
+      "batch": 5,
+      "role": "pilot",
+      "seed": "5927921256005987723",
+      "samples": 2000,
+      "runtime_seconds": 0.185478,
+      "command": "/var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/birth-gap triangular 64 2000 5927921256005987723 /var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/pilot-triangular-L64-b05.json sample",
+      "sha256": "0b2ea77ab678c82c4a72553ca4438b5dd0180831dab645f236155d2ba198c75e",
+      "dependency_group": "pilot/triangular/L64/batch5"
+    },
+    {
+      "file": "pilot-triangular-L64-b06.json.gz",
+      "lattice": "triangular",
+      "L": 64,
+      "batch": 6,
+      "role": "pilot",
+      "seed": "9976285594350850911",
+      "samples": 2000,
+      "runtime_seconds": 0.184383083,
+      "command": "/var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/birth-gap triangular 64 2000 9976285594350850911 /var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/pilot-triangular-L64-b06.json sample",
+      "sha256": "0c6119d67f752abb342d6c557ff137d6b7a032b2c43028e3fed73282038237b9",
+      "dependency_group": "pilot/triangular/L64/batch6"
+    },
+    {
+      "file": "pilot-triangular-L64-b07.json.gz",
+      "lattice": "triangular",
+      "L": 64,
+      "batch": 7,
+      "role": "pilot",
+      "seed": "733749475041577561",
+      "samples": 2000,
+      "runtime_seconds": 0.185996667,
+      "command": "/var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/birth-gap triangular 64 2000 733749475041577561 /var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/pilot-triangular-L64-b07.json sample",
+      "sha256": "c0f6914fb6912399b561480674e8e665f9ef953459e46ff3b17824ca77327654",
+      "dependency_group": "pilot/triangular/L64/batch7"
+    },
+    {
+      "file": "pilot-triangular-L128-b00.json.gz",
+      "lattice": "triangular",
+      "L": 128,
+      "batch": 0,
+      "role": "pilot",
+      "seed": "9872378506023293213",
+      "samples": 2000,
+      "runtime_seconds": 0.7272865,
+      "command": "/var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/birth-gap triangular 128 2000 9872378506023293213 /var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/pilot-triangular-L128-b00.json sample",
+      "sha256": "767f47696c8db1fa1115bdcbe5a79562d486f38da0fc3d88cd73ee9aab4fd165",
+      "dependency_group": "pilot/triangular/L128/batch0"
+    },
+    {
+      "file": "pilot-triangular-L128-b01.json.gz",
+      "lattice": "triangular",
+      "L": 128,
+      "batch": 1,
+      "role": "pilot",
+      "seed": "15516249391103618402",
+      "samples": 2000,
+      "runtime_seconds": 0.718490041,
+      "command": "/var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/birth-gap triangular 128 2000 15516249391103618402 /var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/pilot-triangular-L128-b01.json sample",
+      "sha256": "84c3381dc4874962a921f28d1f95230326a9c75bd1e102c9d5b8776d0db7717f",
+      "dependency_group": "pilot/triangular/L128/batch1"
+    },
+    {
+      "file": "pilot-triangular-L128-b02.json.gz",
+      "lattice": "triangular",
+      "L": 128,
+      "batch": 2,
+      "role": "pilot",
+      "seed": "6293947287737208001",
+      "samples": 2000,
+      "runtime_seconds": 0.719656333,
+      "command": "/var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/birth-gap triangular 128 2000 6293947287737208001 /var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/pilot-triangular-L128-b02.json sample",
+      "sha256": "171948215f2aef2d808d52657eb4490e6c394bddd2c93caca55217edf7581f07",
+      "dependency_group": "pilot/triangular/L128/batch2"
+    },
+    {
+      "file": "pilot-triangular-L128-b03.json.gz",
+      "lattice": "triangular",
+      "L": 128,
+      "batch": 3,
+      "role": "pilot",
+      "seed": "6926748883488171353",
+      "samples": 2000,
+      "runtime_seconds": 0.72070525,
+      "command": "/var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/birth-gap triangular 128 2000 6926748883488171353 /var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/pilot-triangular-L128-b03.json sample",
+      "sha256": "8cfaf0dc7acdb9c7e23407fbd3bf4bb296e74916a70cf7d2133d4667130a4bb8",
+      "dependency_group": "pilot/triangular/L128/batch3"
+    },
+    {
+      "file": "pilot-triangular-L128-b04.json.gz",
+      "lattice": "triangular",
+      "L": 128,
+      "batch": 4,
+      "role": "pilot",
+      "seed": "13460684323953900354",
+      "samples": 2000,
+      "runtime_seconds": 0.714876958,
+      "command": "/var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/birth-gap triangular 128 2000 13460684323953900354 /var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/pilot-triangular-L128-b04.json sample",
+      "sha256": "dbab1b659d16306dee59b65c86ea2a523418924bc5eab70dfeb54fc5a10ce60b",
+      "dependency_group": "pilot/triangular/L128/batch4"
+    },
+    {
+      "file": "pilot-triangular-L128-b05.json.gz",
+      "lattice": "triangular",
+      "L": 128,
+      "batch": 5,
+      "role": "pilot",
+      "seed": "9139919503906188317",
+      "samples": 2000,
+      "runtime_seconds": 0.714073292,
+      "command": "/var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/birth-gap triangular 128 2000 9139919503906188317 /var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/pilot-triangular-L128-b05.json sample",
+      "sha256": "60ca66ed403b4da52bebcfbe557ed18f1ce7bacdd1d7f1aa6237f071dab1b15d",
+      "dependency_group": "pilot/triangular/L128/batch5"
+    },
+    {
+      "file": "pilot-triangular-L128-b06.json.gz",
+      "lattice": "triangular",
+      "L": 128,
+      "batch": 6,
+      "role": "pilot",
+      "seed": "2341140374670824118",
+      "samples": 2000,
+      "runtime_seconds": 0.715843167,
+      "command": "/var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/birth-gap triangular 128 2000 2341140374670824118 /var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/pilot-triangular-L128-b06.json sample",
+      "sha256": "288342f91859f7936fdd7a40095175afc7d53fe288174f49bf8746539a4bfe82",
+      "dependency_group": "pilot/triangular/L128/batch6"
+    },
+    {
+      "file": "pilot-triangular-L128-b07.json.gz",
+      "lattice": "triangular",
+      "L": 128,
+      "batch": 7,
+      "role": "pilot",
+      "seed": "16137988657530107339",
+      "samples": 2000,
+      "runtime_seconds": 0.715420125,
+      "command": "/var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/birth-gap triangular 128 2000 16137988657530107339 /var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/pilot-triangular-L128-b07.json sample",
+      "sha256": "6509fe53ed643a8082c7b8ca5c124e59a6dbb5512d25a3f32cb9f078506d3dcf",
+      "dependency_group": "pilot/triangular/L128/batch7"
+    }
+  ],
+  "build_command": "clang++ -O3 -std=c++17 -DNDEBUG /private/tmp/matching-frontier-20260929-Jk39nH/analysis/birth-gap-20260929/engine.cpp -o /var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-birth-gap-build-x15vhgzl/birth-gap",
+  "estimated_production_cpu_seconds": 13.53837510625,
+  "wall_seconds": 6.891348666977137,
+  "aggregate_engine_seconds": 17.432192414,
+  "status": "completed",
+  "utc_finished": "2026-09-29T11:30:46.697045+00:00"
+}
diff --git a/analysis/birth-gap-20260929/engine.cpp b/analysis/birth-gap-20260929/engine.cpp
new file mode 100644
index 000000000..d2be130fd
--- /dev/null
+++ b/analysis/birth-gap-20260929/engine.cpp
@@ -0,0 +1,147 @@
+// Paired rank births in one site-permutation filtration. C++17, no libraries.
+// Lifted union-find convention adapted from src/threshold_rank_axis_mc.cpp.
+// Global image rank is the span of ALL cycle windings, not Boolean wraps and
+// not the maximum of component ranks. Coordinates are integer torus periods.
+#include <algorithm>
+#include <array>
+#include <chrono>
+#include <cmath>
+#include <cstdint>
+#include <fstream>
+#include <iostream>
+#include <map>
+#include <numeric>
+#include <random>
+#include <stdexcept>
+#include <string>
+#include <vector>
+
+struct Neighbor { int vertex, dx, dy; };
+
+class BirthEngine {
+    int L, N, degree;
+    std::vector<std::array<Neighbor,6>> neighbor;
+    std::vector<int> parent, size;
+    std::vector<std::int64_t> dx, dy;
+    std::vector<unsigned char> active;
+    int rank = 0;
+    std::int64_t basis_x = 0, basis_y = 0;
+    struct Root { int vertex; std::int64_t x, y; };
+    Root find(int v) {
+        if (parent[v] == v) return {v, 0, 0};
+        Root r = find(parent[v]);
+        dx[v] += r.x; dy[v] += r.y; parent[v] = r.vertex;
+        return {r.vertex, dx[v], dy[v]};
+    }
+    void edge(int u, Neighbor e) {
+        Root a = find(u), b = find(e.vertex);
+        std::int64_t x = a.x + e.dx - b.x;
+        std::int64_t y = a.y + e.dy - b.y;
+        if (a.vertex == b.vertex) {
+            if (x % L || y % L) throw std::logic_error("nonperiodic cycle");
+            x /= L; y /= L;
+            if (!(x || y) || rank == 2) return;
+            if (rank == 0) { basis_x=x; basis_y=y; rank=1; }
+            else if (basis_x*y != basis_y*x) rank=2;
+            return;
+        }
+        if (size[a.vertex] < size[b.vertex]) {
+            std::swap(a,b); x=-x; y=-y;
+        }
+        parent[b.vertex]=a.vertex; dx[b.vertex]=x; dy[b.vertex]=y;
+        size[a.vertex]+=size[b.vertex];
+    }
+public:
+    BirthEngine(int length, bool triangular): L(length), N(L*L),
+        degree(triangular?6:4), neighbor(N), parent(N), size(N), dx(N),
+        dy(N), active(N) {
+        if (L < 3 || L > 1024) throw std::invalid_argument("3 <= L <= 1024");
+        const int sx[6]={1,-1,0,0,1,-1}, sy[6]={0,0,1,-1,1,-1};
+        for (int y=0;y<L;++y) for (int x=0;x<L;++x)
+            for (int j=0;j<degree;++j)
+                neighbor[x+L*y][j]={((x+sx[j]+L)%L)+L*((y+sy[j]+L)%L),sx[j],sy[j]};
+    }
+    std::pair<int,int> births(const std::vector<int>& order) {
+        std::fill(active.begin(),active.end(),0);
+        rank=0; int j1=0;
+        for (int k=0;k<N;++k) {
+            int v=order[k]; active[v]=1; parent[v]=v; size[v]=1; dx[v]=dy[v]=0;
+            for (int j=0;j<degree;++j) {
+                Neighbor e=neighbor[v][j];
+                if (active[e.vertex]) edge(v,e);
+            }
+            // Inspect only after every incident edge of the new site is added:
+            // a genuine 0->2 insertion has J1=J2, regardless of edge order.
+            if (rank >= 1 && j1 == 0) j1=k+1;
+            if (rank == 2) return {j1,k+1};
+        }
+        throw std::logic_error("full torus did not reach rank two");
+    }
+};
+
+// Unbiased bounded integer, unlike rng()%bound. Explicit Fisher-Yates makes
+// reproduction independent of the implementation of std::shuffle.
+std::uint64_t bounded(std::mt19937_64& rng, std::uint64_t bound) {
+    const std::uint64_t threshold=(-bound)%bound;
+    while (true) { auto r=rng(); if (r>=threshold) return r%bound; }
+}
+
+int main(int argc,char** argv) {
+    try {
+        if (argc != 7) {
+            std::cerr << "engine square|triangular L samples seed output.json sample|exact\n";
+            return 2;
+        }
+        std::string lattice=argv[1], mode=argv[6];
+        if (lattice!="square" && lattice!="triangular") throw std::invalid_argument("lattice");
+        int L=std::stoi(argv[2]), N=L*L;
+        std::uint64_t samples=std::stoull(argv[3]), seed=std::stoull(argv[4]);
+        if (mode!="sample" && mode!="exact") throw std::invalid_argument("mode");
+        if (mode=="exact" && L!=3) throw std::invalid_argument("exact limited to L=3");
+        if (samples==0 && mode=="sample") throw std::invalid_argument("samples must be positive");
+        BirthEngine engine(L,lattice=="triangular");
+        std::vector<int> order(N); std::iota(order.begin(),order.end(),0);
+        std::mt19937_64 rng(seed);
+        std::map<std::pair<int,int>,std::uint64_t> histogram;
+        std::uint64_t count=0, atom=0, sum_d=0, sum_d2=0;
+        auto start=std::chrono::steady_clock::now();
+        do {
+            if (mode=="sample") {
+                std::iota(order.begin(),order.end(),0);
+                for (int i=N-1;i>0;--i) std::swap(order[i],order[bounded(rng,i+1)]);
+            }
+            auto birth=engine.births(order); ++histogram[birth];
+            std::uint64_t d=birth.second-birth.first;
+            ++count; atom+=(d==0); sum_d+=d; sum_d2+=d*d;
+            if (mode=="sample" && count==samples) break;
+        } while (mode=="sample" || std::next_permutation(order.begin(),order.end()));
+        if (mode=="exact") {
+            const std::uint64_t numerator=lattice=="square"?3:2;
+            const std::uint64_t d2_num=lattice=="square"?43:81;
+            const std::uint64_t d2_den=lattice=="square"?14:28;
+            if (count!=362880 || atom*35!=count*numerator || sum_d*2!=3*count ||
+                sum_d2*d2_den!=d2_num*count)
+                throw std::logic_error("L=3 exact paired-birth control failed");
+        }
+        double seconds=std::chrono::duration<double>(std::chrono::steady_clock::now()-start).count();
+        std::ofstream out(argv[5]); if (!out) throw std::runtime_error("cannot open output");
+        out.precision(17);
+        out << "{\"schema\":\"matching-one.joint-birth-batch.v1\",\"lattice\":\"" << lattice
+            << "\",\"L\":" << L << ",\"N\":" << N << ",\"mode\":\"" << mode
+            << "\",\"seed\":\"" << seed << "\",\"samples\":" << count
+            << ",\"runtime_seconds\":" << seconds << ",\"direct_atom_count\":" << atom
+            << ",\"sum_D\":" << sum_d << ",\"sum_D2\":" << sum_d2
+            << ",\"histogram_columns\":[\"J1\",\"J2\",\"count\"],\"histogram\":[";
+        bool comma=false;
+        for (const auto& h:histogram) {
+            if (comma) out << ','; comma=true;
+            out << '[' << h.first.first << ',' << h.first.second << ',' << h.second << ']';
+        }
+        out << "]}\n";
+        if (!out) throw std::runtime_error("output write failed");
+        std::cout << lattice << " L=" << L << " " << mode << " samples=" << count
+                  << " seconds=" << seconds << " atom=" << atom << " mean_D="
+                  << static_cast<double>(sum_d)/count << '\n';
+        return 0;
+    } catch (const std::exception& e) { std::cerr << e.what() << '\n'; return 1; }
+}
diff --git a/analysis/birth-gap-20260929/run_pilot.py b/analysis/birth-gap-20260929/run_pilot.py
new file mode 100644
index 000000000..88591ed97
--- /dev/null
+++ b/analysis/birth-gap-20260929/run_pilot.py
@@ -0,0 +1,142 @@
+#!/usr/bin/env python3
+"""Compile, exact-control, benchmark, then bounded independent joint-birth pilot.
+
+Only Python stdlib and a C++17 compiler are required. No cloud or package install.
+Each batch is a fresh mt19937_64 stream; lattice/size/batch streams are distinct.
+"""
+import argparse
+import concurrent.futures
+import datetime
+import gzip
+import hashlib
+import json
+import os
+import pathlib
+import platform
+import re
+import shlex
+import subprocess
+import tempfile
+import time
+
+HERE = pathlib.Path(__file__).resolve().parent
+
+
+def seed_for(lattice, length, batch, role):
+    tag = f"matching-one/birth-gap/20260929/{role}/{lattice}/{length}/{batch}"
+    return int.from_bytes(hashlib.sha256(tag.encode()).digest()[:8], "big")
+
+
+def main():
+    ap = argparse.ArgumentParser()
+    ap.add_argument("--sizes", type=int, nargs="+", default=[16, 32, 64, 128])
+    ap.add_argument("--batches", type=int, default=8)
+    ap.add_argument("--samples", type=int, default=2000)
+    ap.add_argument("--workers", type=int, default=4)
+    ap.add_argument("--compiler", default="clang++", help="C++17 compiler command, e.g. g++")
+    ap.add_argument("--role", default="pilot", help="production seed namespace; use a NEW value for independent blocks")
+    ap.add_argument("--skip-exact-control", action="store_true")
+    ap.add_argument("--exact-control-reference", help="prior exact-run reference required when skipping local exhaustive control")
+    ap.add_argument("--git-base-commit", help="record source base when remote directory is not a Git checkout")
+    ap.add_argument("--budget-seconds", type=float, default=240,
+                    help="stop before production if benchmark-estimated wall time exceeds this bound")
+    ap.add_argument("--output", type=pathlib.Path, default=HERE / "data")
+    args = ap.parse_args()
+    if args.workers > 14 or args.workers < 1:
+        ap.error("bounded pilot permits 1..14 workers; choose from measured local/remote quota")
+    if not re.fullmatch(r"[A-Za-z0-9][A-Za-z0-9_-]*", args.role):
+        ap.error("--role must be a filename-safe nonempty seed namespace")
+    if args.skip_exact_control and not args.exact_control_reference:
+        ap.error("--skip-exact-control requires --exact-control-reference")
+    if min(args.sizes) < 3 or max(args.sizes) > 1024 or args.batches < 2 or args.samples < 1 or args.budget_seconds <= 0:
+        ap.error("require sizes 3..1024, >=2 batches, positive samples and budget")
+    args.output.mkdir(parents=True, exist_ok=True)
+    manifest_path = args.output / "run.json"
+    if manifest_path.exists():
+        raise SystemExit("Refusing to overwrite an existing run; choose another --output")
+    compiler = shlex.split(args.compiler)
+    if not compiler:
+        ap.error("empty compiler command")
+    base_commit = args.git_base_commit
+    if base_commit is None:
+        try:
+            base_commit = subprocess.check_output(["git", "rev-parse", "HEAD"], cwd=HERE,
+                                                 text=True, stderr=subprocess.DEVNULL).strip()
+        except (subprocess.CalledProcessError, FileNotFoundError):
+            base_commit = "unavailable-not-a-git-checkout"
+    run = {"schema": "matching-one.joint-birth-run.v1",
+           "utc_started": datetime.datetime.now(datetime.timezone.utc).isoformat(),
+           "command": shlex.join(["python3", str(pathlib.Path(__file__).resolve()), *os.sys.argv[1:]]),
+           "platform": platform.platform(), "machine": platform.machine(),
+           "python": platform.python_version(), "workers": args.workers,
+           "compiler": subprocess.check_output([*compiler, "--version"], text=True).strip(),
+           "git_base_commit": base_commit,
+           "production_role": args.role,
+           "exact_control_reference": args.exact_control_reference,
+           "exact_control_skipped": args.skip_exact_control,
+           "estimated_wall_budget_seconds": args.budget_seconds,
+           "source_sha256": {p.name: hashlib.sha256(p.read_bytes()).hexdigest()
+                             for p in (HERE / "engine.cpp", pathlib.Path(__file__).resolve())},
+           "seed_rule": "first 64 SHA256 bits of matching-one/birth-gap/20260929/{role}/{lattice}/{L}/{batch}",
+           "random_streams": "separate mt19937_64 per (role,lattice,L,batch); no paired/cross-size common random numbers",
+           "sampling": "unbiased Fisher-Yates of all N sites; stop after same-filtration global rank reaches 2",
+           "controls": [], "benchmarks": [], "batches": []}
+    start = time.monotonic()
+    with tempfile.TemporaryDirectory(prefix="matching-birth-gap-build-") as tmp:
+        binary = pathlib.Path(tmp) / "birth-gap"
+        build = [*compiler, "-O3", "-std=c++17", "-DNDEBUG", str(HERE / "engine.cpp"), "-o", str(binary)]
+        subprocess.run(build, check=True)
+        run["build_command"] = shlex.join(build)
+
+        def job(lattice, length, batch, role, n, mode="sample"):
+            name = f"{role}-{lattice}-L{length}-b{batch:02d}.json"
+            raw = pathlib.Path(tmp) / name
+            seed = seed_for(lattice, length, batch, role)
+            command = [str(binary), lattice, str(length), str(n), str(seed), str(raw), mode]
+            log = subprocess.check_output(command, text=True).strip()
+            content = raw.read_bytes()
+            result = json.loads(content)
+            target = args.output / (name + ".gz")
+            with target.open("wb") as f:
+                with gzip.GzipFile(fileobj=f, mode="wb", mtime=0) as gz:
+                    gz.write(content)
+            print(log, flush=True)
+            return {"file": target.name, "lattice": lattice, "L": length, "batch": batch,
+                    "role": role, "seed": str(seed), "samples": result["samples"],
+                    "runtime_seconds": result["runtime_seconds"], "command": shlex.join(command),
+                    "sha256": hashlib.sha256(target.read_bytes()).hexdigest(),
+                    "dependency_group": f"{role}/{lattice}/L{length}/batch{batch}"}
+
+        if not args.skip_exact_control:
+            for lattice in ("square", "triangular"):
+                run["controls"].append(job(lattice, 3, 0, "exact-control", 0, "exact"))
+        # Benchmark the largest requested cell sequentially BEFORE production.
+        # These samples have separate seeds and are excluded from every estimate.
+        for lattice in ("square", "triangular"):
+            benchmark_role = "benchmark" if args.role == "pilot" else f"benchmark-{args.role}"
+            run["benchmarks"].append(job(lattice, max(args.sizes), 0, benchmark_role, 200))
+        estimated_cpu = sum(b["runtime_seconds"] / b["samples"] for b in run["benchmarks"]) * args.samples * args.batches * sum((L / max(args.sizes)) ** 2 for L in args.sizes)
+        run["estimated_production_cpu_seconds"] = estimated_cpu
+        print(f"largest-cell benchmark predicts roughly {estimated_cpu:.1f} aggregate CPU-seconds", flush=True)
+        if estimated_cpu / args.workers > args.budget_seconds:
+            run["status"] = "benchmark-budget-stop"
+            manifest_path.write_text(json.dumps(run, indent=2) + "\n")
+            raise SystemExit("Benchmark exceeds bounded pilot; no production started")
+        jobs = [(lattice, L, b) for lattice in ("square", "triangular")
+                for L in args.sizes for b in range(args.batches)]
+        with concurrent.futures.ThreadPoolExecutor(max_workers=args.workers) as pool:
+            futures = [pool.submit(job, lattice, L, b, args.role, args.samples) for lattice, L, b in jobs]
+            for future in concurrent.futures.as_completed(futures):
+                run["batches"].append(future.result())
+                run["batches"].sort(key=lambda x: (x["lattice"], x["L"], x["batch"]))
+                manifest_path.write_text(json.dumps(run, indent=2) + "\n")
+    run["wall_seconds"] = time.monotonic() - start
+    run["aggregate_engine_seconds"] = sum(b["runtime_seconds"] for b in run["batches"])
+    run["status"] = "completed"
+    run["utc_finished"] = datetime.datetime.now(datetime.timezone.utc).isoformat()
+    manifest_path.write_text(json.dumps(run, indent=2) + "\n")
+    print(f"Completed in {run['wall_seconds']:.2f}s; manifest {manifest_path}")
+
+
+if __name__ == "__main__":
+    main()
diff --git a/analysis/birth-gap-20260929/summary.json b/analysis/birth-gap-20260929/summary.json
new file mode 100644
index 000000000..2f812844c
--- /dev/null
+++ b/analysis/birth-gap-20260929/summary.json
@@ -0,0 +1,17338 @@
+{
+  "schema": "matching-one.joint-birth-pilot.v1",
+  "epsilons": [
+    0.025,
+    0.05,
+    0.1,
+    0.2
+  ],
+  "definitions": {
+    "J1_J2": "first occupied-site counts at global ambient homology rank >=1 and =2, SAME uniform permutation",
+    "D": "J2-J1",
+    "L54": "D/L^(5/4), prescribed nearcritical benchmark, no exponent fit; not proved square-site scaling",
+    "label_L34": "L^(3/4)*(T2-T1), conditional Beta(D,N+1-D) mixing of uniform labels",
+    "pooled_birth_IQR_J": "Q75-Q25 of equal-weight mixture of J1 and J2; empirical lower quantiles",
+    "label_IQR": "(N+1)*(T2-T1)/(pooled_birth_IQR_J); exponent-free self-normalized readout",
+    "Z": "E[(1-G/delta)+] with delta on the same epsilon grid; exact histogram or conditional-Beta integration, added on theory-agent request after acquisition",
+    "triangular_geometry": "nearest neighbours (+/-1,0),(0,+/-1),(1,1),(-1,-1); LxL primitive-coordinate rhombus, not square conformal modulus"
+  },
+  "controls": [
+    {
+      "lattice": "square",
+      "L": 3,
+      "permutations": 362880,
+      "direct_atom_count": 31104,
+      "mean_D": 1.5,
+      "mean_D2": 3.0714285714285716
+    },
+    {
+      "lattice": "triangular",
+      "L": 3,
+      "permutations": 362880,
+      "direct_atom_count": 20736,
+      "mean_D": 1.5,
+      "mean_D2": 2.892857142857143
+    }
+  ],
+  "cells": [
+    {
+      "L": 16,
+      "N": 256,
+      "samples": 16000,
+      "batch_count": 8,
+      "point": {
+        "direct_atom": 0.0060625,
+        "mean_D": 13.535625,
+        "mean_D_over_L54": 0.42298828125,
+        "pooled_birth_IQR_J": 16.0,
+        "pooled_birth_IQR_J_over_L54": 0.5,
+        "mean_D_over_IQR": 0.8459765625,
+        "mean_label_gap": 0.05266780155642023,
+        "birth_J_correlation": 0.4762002529264091,
+        "P_D_L54_le_0.025": 0.0060625,
+        "P_D_IQR_le_0.025": 0.0060625,
+        "Z_D_L54_0.025": 0.0060625,
+        "Z_D_IQR_0.025": 0.0060625,
+        "P_label_L34_le_0.025": 0.053519352944065304,
+        "Z_label_L34_0.025": 0.03005358760618102,
+        "P_label_IQR_le_0.025": 0.030091061348945774,
+        "Z_label_IQR_0.025": 0.018154405720284265,
+        "P_D_L54_le_0.05": 0.067625,
+        "P_D_IQR_le_0.05": 0.0060625,
+        "Z_D_L54_0.05": 0.0291484375,
+        "Z_D_IQR_0.05": 0.0060625,
+        "P_label_L34_le_0.05": 0.09852641080757128,
+        "Z_label_L34_0.05": 0.053120627224607135,
+        "P_label_IQR_le_0.05": 0.053340179786129115,
+        "Z_label_IQR_0.05": 0.029962274396509756,
+        "P_D_L54_le_0.1": 0.177625,
+        "P_D_IQR_le_0.1": 0.067625,
+        "Z_D_L54_0.1": 0.07221484375,
+        "Z_D_IQR_0.1": 0.0291484375,
+        "P_label_L34_le_0.1": 0.18322683754076557,
+        "Z_label_L34_0.1": 0.09727573324481398,
+        "P_label_IQR_le_0.1": 0.09818353175271727,
+        "Z_label_IQR_0.1": 0.052943930532670554,
+        "P_D_L54_le_0.2": 0.32125,
+        "P_D_IQR_le_0.2": 0.177625,
+        "Z_D_L54_0.2": 0.157275390625,
+        "Z_D_IQR_0.2": 0.07221484375,
+        "P_label_L34_le_0.2": 0.3348577986087651,
+        "Z_label_L34_0.2": 0.17905574301130733,
+        "P_label_IQR_le_0.2": 0.1825928075187743,
+        "Z_label_IQR_0.2": 0.0969412249611066
+      },
+      "jackknife_se": {
+        "direct_atom": 0.0005781551633057882,
+        "mean_D": 0.10177284536442635,
+        "mean_D_over_L54": 0.0031804014176383235,
+        "pooled_birth_IQR_J": 1.2808688457449497,
+        "pooled_birth_IQR_J_over_L54": 0.04002715142952968,
+        "mean_D_over_IQR": 0.06352721784254237,
+        "mean_label_gap": 0.00039600328935574864,
+        "birth_J_correlation": 0.009553582750114309,
+        "P_D_L54_le_0.025": 0.0005781551633057882,
+        "P_D_IQR_le_0.025": 0.0005781551633057882,
+        "Z_D_L54_0.025": 0.0005781551633057882,
+        "Z_D_IQR_0.025": 0.0005781551633057882,
+        "P_label_L34_le_0.025": 0.0009443597321754387,
+        "Z_label_L34_0.025": 0.000695453748390335,
+        "P_label_IQR_le_0.025": 0.0022035978866232426,
+        "Z_label_IQR_0.025": 0.0012319069702050624,
+        "P_D_L54_le_0.05": 0.0016250000000000038,
+        "P_D_IQR_le_0.05": 0.0005781551633057882,
+        "Z_D_L54_0.05": 0.0007891508967458763,
+        "Z_D_IQR_0.05": 0.0005781551633057882,
+        "P_label_L34_le_0.05": 0.0013885850709170141,
+        "Z_label_L34_0.05": 0.0009054268730168807,
+        "P_label_IQR_le_0.05": 0.004091273482195937,
+        "Z_label_IQR_0.05": 0.002171570177072862,
+        "P_D_L54_le_0.1": 0.003033665274877895,
+        "P_D_IQR_le_0.1": 0.0016250000000000038,
+        "Z_D_L54_0.1": 0.0010810880204202686,
+        "Z_D_IQR_0.1": 0.0032513313144247754,
+        "P_label_L34_le_0.1": 0.0021814558801377414,
+        "Z_label_L34_0.1": 0.0013233207335418072,
+        "P_label_IQR_le_0.1": 0.007641106176502777,
+        "Z_label_IQR_0.1": 0.004017277459732897,
+        "P_D_L54_le_0.2": 0.003827765106989411,
+        "P_D_IQR_le_0.2": 0.003033665274877895,
+        "Z_D_L54_0.2": 0.0019635435014980955,
+        "Z_D_IQR_0.2": 0.008370155221891063,
+        "P_label_L34_le_0.2": 0.0029031810868561216,
+        "Z_label_L34_0.2": 0.0018942161276261568,
+        "P_label_IQR_le_0.2": 0.013925108657570897,
+        "Z_label_IQR_0.2": 0.007439101000003368
+      },
+      "direct_atom_count": 97,
+      "direct_atom_wilson_two_sided_95": [
+        0.00497257571959138,
+        0.007389547420015931
+      ],
+      "direct_atom_zero_count_one_sided_95_upper": null,
+      "metric_order": [
+        "direct_atom",
+        "mean_D",
+        "mean_D_over_L54",
+        "pooled_birth_IQR_J",
+        "pooled_birth_IQR_J_over_L54",
+        "mean_D_over_IQR",
+        "mean_label_gap",
+        "birth_J_correlation",
+        "P_D_L54_le_0.025",
+        "P_D_IQR_le_0.025",
+        "Z_D_L54_0.025",
+        "Z_D_IQR_0.025",
+        "P_label_L34_le_0.025",
+        "Z_label_L34_0.025",
+        "P_label_IQR_le_0.025",
+        "Z_label_IQR_0.025",
+        "P_D_L54_le_0.05",
+        "P_D_IQR_le_0.05",
+        "Z_D_L54_0.05",
+        "Z_D_IQR_0.05",
+        "P_label_L34_le_0.05",
+        "Z_label_L34_0.05",
+        "P_label_IQR_le_0.05",
+        "Z_label_IQR_0.05",
+        "P_D_L54_le_0.1",
+        "P_D_IQR_le_0.1",
+        "Z_D_L54_0.1",
+        "Z_D_IQR_0.1",
+        "P_label_L34_le_0.1",
+        "Z_label_L34_0.1",
+        "P_label_IQR_le_0.1",
+        "Z_label_IQR_0.1",
+        "P_D_L54_le_0.2",
+        "P_D_IQR_le_0.2",
+        "Z_D_L54_0.2",
+        "Z_D_IQR_0.2",
+        "P_label_L34_le_0.2",
+        "Z_label_L34_0.2",
+        "P_label_IQR_le_0.2",
+        "Z_label_IQR_0.2"
+      ],
+      "delete_one_batch_covariance": [
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+          "P_D_IQR_le_0.05": 0.052571428571428575,
+          "Z_D_L54_0.05": 0.04467614194067525,
+          "Z_D_IQR_0.05": 0.025100230414746547,
+          "P_label_L34_le_0.05": 0.09254244104258474,
+          "Z_label_L34_0.05": 0.04672949621934114,
+          "P_label_IQR_le_0.05": 0.05405633013311486,
+          "Z_label_IQR_0.05": 0.027156151326630025,
+          "P_D_L54_le_0.1": 0.17814285714285713,
+          "P_D_IQR_le_0.1": 0.103,
+          "Z_D_L54_0.1": 0.08978184856997082,
+          "Z_D_IQR_0.1": 0.05158006912442396,
+          "P_label_L34_le_0.1": 0.1790531994180597,
+          "Z_label_L34_0.1": 0.09166745209444985,
+          "P_label_IQR_le_0.1": 0.10629613152590718,
+          "Z_label_IQR_0.1": 0.05368122858661917,
+          "P_D_L54_le_0.2": 0.3392142857142857,
+          "P_D_IQR_le_0.2": 0.20235714285714285,
+          "Z_D_L54_0.2": 0.17414045935315886,
+          "Z_D_IQR_0.2": 0.10292453917050691,
+          "P_label_L34_le_0.2": 0.3380641914943523,
+          "Z_label_L34_0.2": 0.17584693083390948,
+          "P_label_IQR_le_0.2": 0.20429108117427558,
+          "Z_label_IQR_0.2": 0.104860831266393
+        },
+        {
+          "direct_atom": 7.142857142857143e-05,
+          "mean_D": 174.3715,
+          "mean_D_over_L54": 0.4050074774486269,
+          "pooled_birth_IQR_J": 247.0,
+          "pooled_birth_IQR_J_over_L54": 0.5736995261829534,
+          "mean_D_over_IQR": 0.7059574898785425,
+          "mean_label_gap": 0.010642142203234665,
+          "birth_J_correlation": 0.6143390136214282,
+          "P_D_L54_le_0.025": 0.04528571428571428,
+          "P_D_IQR_le_0.025": 0.02692857142857143,
+          "Z_D_L54_0.025": 0.022185100628769692,
+          "Z_D_IQR_0.025": 0.011948814343551188,
+          "P_label_L34_le_0.025": 0.048080477067990714,
+          "Z_label_L34_0.025": 0.024318606611659854,
+          "P_label_IQR_le_0.025": 0.02797270961203281,
+          "Z_label_IQR_0.025": 0.014090613595928543,
+          "P_D_L54_le_0.05": 0.09042857142857143,
+          "P_D_IQR_le_0.05": 0.05392857142857143,
+          "Z_D_L54_0.05": 0.0454782678935774,
+          "Z_D_IQR_0.05": 0.02573308270676692,
+          "P_label_L34_le_0.05": 0.09297935528317751,
+          "Z_label_L34_0.05": 0.047496079543255215,
+          "P_label_IQR_le_0.05": 0.054846951752000994,
+          "Z_label_IQR_0.05": 0.027805377611737264,
+          "P_D_L54_le_0.1": 0.17814285714285713,
+          "P_D_IQR_le_0.1": 0.10307142857142858,
+          "Z_D_L54_0.1": 0.09005559183723567,
+          "Z_D_IQR_0.1": 0.05212579525737421,
+          "P_label_L34_le_0.1": 0.1788679655063224,
+          "Z_label_L34_0.1": 0.09195751016859643,
+          "P_label_IQR_le_0.1": 0.10601785492286311,
+          "Z_label_IQR_0.1": 0.054174564316932884,
+          "P_D_L54_le_0.2": 0.3399285714285714,
+          "P_D_IQR_le_0.2": 0.20214285714285715,
+          "Z_D_L54_0.2": 0.1745287417218837,
+          "Z_D_IQR_0.2": 0.10276315789473685,
+          "P_label_L34_le_0.2": 0.33905943095183444,
+          "Z_label_L34_0.2": 0.17624235948357805,
+          "P_label_IQR_le_0.2": 0.20351705478068235,
+          "Z_label_IQR_0.2": 0.10470031739975051
+        },
+        {
+          "direct_atom": 0.0,
+          "mean_D": 174.83142857142857,
+          "mean_D_over_L54": 0.4060757398110017,
+          "pooled_birth_IQR_J": 248.0,
+          "pooled_birth_IQR_J_over_L54": 0.5760221963294431,
+          "mean_D_over_IQR": 0.7049654377880185,
+          "mean_label_gap": 0.010670212302192773,
+          "birth_J_correlation": 0.6092193279089649,
+          "P_D_L54_le_0.025": 0.045714285714285714,
+          "P_D_IQR_le_0.025": 0.027357142857142858,
+          "Z_D_L54_0.025": 0.022461039447714656,
+          "Z_D_IQR_0.025": 0.012172811059907836,
+          "P_label_L34_le_0.025": 0.048563225971069185,
+          "Z_label_L34_0.025": 0.02461275435566503,
+          "P_label_IQR_le_0.025": 0.02839173394917915,
+          "Z_label_IQR_0.025": 0.014363992419657794,
+          "P_D_L54_le_0.05": 0.09192857142857143,
+          "P_D_IQR_le_0.05": 0.054214285714285715,
+          "Z_D_L54_0.05": 0.046025973133516626,
+          "Z_D_IQR_0.05": 0.026167050691244243,
+          "P_label_L34_le_0.05": 0.09432540130137733,
+          "Z_label_L34_0.05": 0.048078901455075936,
+          "P_label_IQR_le_0.05": 0.05561697198840646,
+          "Z_label_IQR_0.05": 0.028238548088462538,
+          "P_D_L54_le_0.1": 0.17942857142857144,
+          "P_D_IQR_le_0.1": 0.10471428571428572,
+          "Z_D_L54_0.1": 0.09121521831499767,
+          "Z_D_IQR_0.1": 0.05300057603686636,
+          "P_label_L34_le_0.1": 0.1802885542361496,
+          "Z_label_L34_0.1": 0.09308535699850261,
+          "P_label_IQR_le_0.1": 0.10801234484648349,
+          "Z_label_IQR_0.1": 0.055084342548715796,
+          "P_D_L54_le_0.2": 0.339,
+          "P_D_IQR_le_0.2": 0.20314285714285715,
+          "Z_D_L54_0.2": 0.1752567318227908,
+          "Z_D_IQR_0.2": 0.104338133640553,
+          "P_label_L34_le_0.2": 0.3383510312022169,
+          "Z_label_L34_0.2": 0.17693837633060994,
+          "P_label_IQR_le_0.2": 0.20524850450816756,
+          "Z_label_IQR_0.2": 0.10623722882318418
+        }
+      ],
+      "uncertainty": "paired complete-batch delete-one; re-estimates mixture IQR in each delete-one; eight batches, exploratory uncertainty only",
+      "positive_near_diagonal_rule": "subtract direct_atom from any P_* (zero is otherwise included)",
+      "lattice": "triangular"
+    }
+  ],
+  "total_production_samples": 128000,
+  "production_seconds_sum": 17.432192414,
+  "acquisition_wall_seconds_including_build_control_benchmark": 6.891348666977137,
+  "limitations": [
+    "Finite-size pilot, not a noncoalescence proof or arm-exponent estimate.",
+    "Four fixed epsilons do not implement epsilon->0 after L->infinity.",
+    "No cross-lattice equality asserted: primitive square and triangular tori have different conformal moduli.",
+    "All readouts within one cell use the SAME sample block; Beta reconstruction adds no independent evidence.",
+    "Eight batches support only pilot uncertainty; no p-values, model selection or data-chosen exponent.",
+    "A stable typical gap does not exclude an additional limiting diagonal atom."
+  ]
+}
diff --git a/analysis/birth-markov-kernel-20260929/RESULT.md b/analysis/birth-markov-kernel-20260929/RESULT.md
new file mode 100644
index 000000000..037d97f30
--- /dev/null
+++ b/analysis/birth-markov-kernel-20260929/RESULT.md
@@ -0,0 +1,57 @@
+# Does rank one retain birth-history memory?
+
+2026-09-29; retrospective reuse of the completed 560k cloud block. No new samples.
+
+One contrast, selected before this scoring: cutoffs a,b,c are the .25/.50/.75 lower quantiles of the pooled J1/J2 births.
+Among paths with J1<=b<J2, compare P(J2>c) for early (J1<=a) versus late (a<J1<=b) first birth.
+Every current-rank-only time-inhomogeneous Markov process predicts difference zero.
+
+## Result
+
+L512 is the primary available-size readout; smaller sizes supply context. Errors are 14-batch delete-one SE, not confidence intervals.
+All thresholds are re-estimated in every deletion; the JSON retains their values and the complete metric covariance.
+
+| Lattice | L | Rank-one risk count | Early survival | Late survival | Difference (percentage points) |
+|---|---:|---:|---:|---:|---:|
+| square | 64 | 29042/70000 | 0.4649 | 0.4524 | +1.248 ± 0.553 |
+| square | 128 | 28218/70000 | 0.4516 | 0.4437 | +0.785 ± 0.480 |
+| square | 256 | 27775/70000 | 0.4578 | 0.4438 | +1.395 ± 0.623 |
+| square | 512 | 27537/70000 | 0.4491 | 0.4349 | +1.422 ± 0.511 |
+| triangular | 64 | 28622/70000 | 0.4493 | 0.4430 | +0.623 ± 0.521 |
+| triangular | 128 | 27429/70000 | 0.4427 | 0.4391 | +0.352 ± 0.712 |
+| triangular | 256 | 26832/70000 | 0.4428 | 0.4323 | +1.054 ± 0.476 |
+| triangular | 512 | 26681/70000 | 0.4325 | 0.4255 | +0.702 ± 0.515 |
+
+## Existing exact L3 archive, a different evidential object
+
+No 9! enumeration rerun: the stored complete tables are evaluated with integer/Fraction arithmetic.
+All 120 count-time triples 0<=a<b<c<=9 are checked here. This finite exhaustive identity check is separate from the ONE fixed production triple; it is not a production cutoff search.
+
+- square L3: 0 violating triples out of 120; rank-only insertion process Markov under the full-kernel criterion: True.
+  - Exact iid-label bridge at p=(1/3,1/2,2/3): contrast 0 (+0.0000 percentage points); Markov identity violated: False.
+- triangular L3: 2 violating triples out of 120; rank-only insertion process Markov under the full-kernel criterion: False.
+  - [3, 4, 5]: early/late survival 3/5 / 13/25; difference 2/25; conditional covariance 1/90.
+  - [4, 5, 6]: early/late survival 3/16 / 1/7; difference 5/112; conditional covariance 1/90.
+  - Exact iid-label bridge at p=(1/3,1/2,2/3): contrast -1156/716639 (-0.1613 percentage points); Markov identity violated: True.
+
+The exact clock bridge is a multinomial mixture, not a replacement of count k by its expectation Np. Insertion and label clocks have separately evaluated properties; neither finite result is extrapolated to larger sizes or a continuum limit.
+
+## Scope
+
+- Not prospective or independent model-elimination certification; this block already supplied gap readouts.
+- Finite-size conditional dependence does not prove non-Markovianity of a continuum limit.
+- Conditioning on rank one tests the Markov property; it is not a causal effect of earlier birth.
+- A zero contrast at one triple would not prove Markovianity; the theorem requires all triples.
+- Square and triangular primitive-coordinate tori have different moduli; no universality comparison.
+- No fitted exponent, pooled cross-lattice significance, adaptive cutoff search or hidden-state identification.
+
+Source: `../birth-gap-20260929/cloud-data-5k/run.json`; all 112 production batches only.
+No local-pilot pooling or benchmark rows. Grain: one uniform-permutation filtration, represented by weighted (J1,J2) histogram rows.
+One run took 2.74 s on Python 3.11.15 (arm64).
+One exact four-cell control is recorded in result.json; old physical enumeration was not rerun.
+
+Reproduce from the repository root:
+
+```sh
+python3 analysis/birth-markov-kernel-20260929/analyze.py
+```
diff --git a/analysis/birth-markov-kernel-20260929/analyze.py b/analysis/birth-markov-kernel-20260929/analyze.py
new file mode 100644
index 000000000..d3105751f
--- /dev/null
+++ b/analysis/birth-markov-kernel-20260929/analyze.py
@@ -0,0 +1,328 @@
+#!/usr/bin/env python3
+"""One preselected three-time rank-memory contrast on existing paired births.
+
+Retrospective archive analysis; no resampling, exponent fit, cutoff scan,
+new percolation simulation, or independent-validation claim. Standard library.
+The source cloud block has already been used for gap statistics.
+"""
+import argparse
+from collections import Counter, defaultdict
+from fractions import Fraction
+import gzip
+import hashlib
+from itertools import combinations
+import json
+import math
+from pathlib import Path
+import platform
+import sys
+import time
+
+
+HERE = Path(__file__).resolve().parent
+QUANTILES = (0.25, 0.50, 0.75)
+
+
+def lower_quantile(hist, q):
+    target = q * sum(hist.values())
+    total = 0
+    for k, count in sorted(hist.items()):
+        total += count
+        if total >= target:
+            return k
+    raise ValueError("empty birth distribution")
+
+
+def combine(batches):
+    hist = Counter()
+    for b in batches:
+        hist.update({(j1, j2): count for j1, j2, count in b["histogram"]})
+    return hist
+
+
+def table_metrics(table, samples):
+    # Rows: early/late first birth. Columns: survives past c / exits by c.
+    n11, n10 = table[0]
+    n01, n00 = table[1]
+    early, late = n11 + n10, n01 + n00
+    risk = early + late
+    if min(early, late) <= 0:
+        raise ValueError("empty early/late rank-one risk group")
+    pe, pl = n11 / early, n01 / late
+    covariance = (n11 * n00 - n10 * n01) / risk**2
+    contrast = pe - pl
+    # Same quantity by its conditional-covariance identity.
+    assert math.isclose(covariance, early * late / risk**2 * contrast,
+                        rel_tol=1e-12, abs_tol=1e-15)
+    return {
+        "rank_one_risk_fraction": risk / samples,
+        "early_fraction_within_risk": early / risk,
+        "survival_early": pe,
+        "survival_late": pl,
+        "survival_difference": contrast,
+        "conditional_covariance": covariance,
+    }
+
+
+def summarize_hist(hist):
+    marginal = Counter()
+    for (j1, j2), count in hist.items():
+        marginal[j1] += count
+        marginal[j2] += count
+    a, b, c = [lower_quantile(marginal, q) for q in QUANTILES]
+    if not a < b < c:
+        raise ValueError("distinct three-time cutoffs not available")
+    table = [[0, 0], [0, 0]]
+    for (j1, j2), count in hist.items():
+        if j1 <= b < j2:
+            table[0 if j1 <= a else 1][0 if j2 > c else 1] += count
+    samples = sum(hist.values())
+    point = table_metrics(table, samples)
+    n11, n10 = table[0]
+    n01, n00 = table[1]
+    # These are empirical kernel numerators, all over the same sample count.
+    kernel_counts = {
+        "H_a_b": n11 + n10,
+        "H_a_c": n11,
+        "H_b_c": n11 + n01,
+        "H_b_b": n11 + n10 + n01 + n00,
+    }
+    return {
+        "cutoffs_insertion_counts": [a, b, c],
+        "risk_table_early_late_by_survive_exit": table,
+        "kernel_counts": kernel_counts,
+        "point": point,
+    }
+
+
+def summarize_cell(batches, refs, lattice, L):
+    b = len(batches)
+    if b < 3 or len({x["samples"] for x in batches}) != 1:
+        raise ValueError("this delete-one contract requires equal complete batches")
+    full = summarize_hist(combine(batches))
+    leave = [summarize_hist(combine(batches[:i] + batches[i+1:]))
+             for i in range(b)]
+    names = list(full["point"])
+    centers = {n: math.fsum(r["point"][n] for r in leave)/b for n in names}
+    cov = [[(b-1)/b * math.fsum((r["point"][a]-centers[a]) *
+                              (r["point"][c]-centers[c]) for r in leave)
+            for c in names] for a in names]
+    return {
+        "lattice": lattice, "L": L, "N": L*L,
+        "samples": sum(x["samples"] for x in batches), "batches": b,
+        "role": "primary_largest_available_size" if L == 512 else "size_context",
+        **full,
+        "metric_order": names,
+        "jackknife_se": {n: math.sqrt(max(0, cov[i][i])) for i, n in enumerate(names)},
+        "delete_one_batch_covariance": cov,
+        "delete_one": [{"omitted_batch": refs[i]["batch"], **r}
+                       for i, r in enumerate(leave)],
+        "source_batches": [{k: r[k] for k in
+                            ("file", "batch", "samples", "sha256", "dependency_group")}
+                           for r in refs],
+    }
+
+
+def exact_control():
+    # One four-cell independence example and one dependence example;
+    # not a physical simulation or a rerun of any earlier census.
+    independent, dependent = [[1, 1], [1, 1]], [[3, 1], [1, 3]]
+    assert table_metrics(independent, 4)["survival_difference"] == 0
+    got = table_metrics(dependent, 8)
+    assert Fraction(got["survival_difference"]) == Fraction(1, 2)
+    assert Fraction(got["conditional_covariance"]) == Fraction(1, 8)
+    # Birth-pair cells exactly realize the four risk-table categories at 2,3,4.
+    pairs = {(1, 5): 3, (1, 4): 1, (3, 5): 1, (3, 4): 3}
+    table = [[0, 0], [0, 0]]
+    for (j1, j2), count in pairs.items():
+        if j1 <= 3 < j2:
+            table[int(j1 > 2)][int(j2 <= 4)] += count
+    assert table == dependent
+    return {"independent_table": independent, "dependent_table": dependent,
+            "dependent_contrast": "1/2", "dependent_covariance": "1/8",
+            "scope": "one exact four-cell arithmetic and inequality-boundary control"}
+
+
+def exact_archive_analysis(data_path):
+    """Analyze all triples of the existing 9! tables; never regenerate them."""
+    source = json.loads((data_path / "run.json").read_text())
+    results = []
+    for ref in source["controls"]:
+        raw = (data_path / ref["file"]).read_bytes()
+        assert hashlib.sha256(raw).hexdigest() == ref["sha256"]
+        b = json.loads(gzip.decompress(raw))
+        assert b["L"] == 3 and b["N"] == 9 and b["mode"] == "exact"
+        assert sum(row[2] for row in b["histogram"]) == b["samples"] == math.factorial(9)
+        def h(s, t):
+            return sum(n for j1, j2, n in b["histogram"] if j1 <= s and j2 > t)
+        count = 0
+        violations = []
+        for a, middle, c in combinations(range(b["N"]+1), 3):
+            count += 1
+            A, B, C, D = h(a, middle), h(a, c), h(middle, c), h(middle, middle)
+            determinant = B*D-A*C
+            if determinant:
+                assert D > A > 0
+                violations.append({
+                    "cutoffs": [a, middle, c],
+                    "risk_table": [[B, A-B], [C-B, D-A-C+B]],
+                    "conditional_covariance": str(Fraction(determinant, D**2)),
+                    "survival_early": str(Fraction(B, A)),
+                    "survival_late": str(Fraction(C-B, D-A)),
+                    "survival_difference": str(Fraction(B, A)-Fraction(C-B, D-A)),
+                })
+        # Exact uniform-label bridge: counts at p<=q are multinomial,
+        # independent of the permutation. Fixed rational times, no search.
+        def label_h(p, q):
+            total = Fraction(0)
+            n = b["N"]
+            for k in range(n+1):
+                for ell in range(k, n+1):
+                    mass = (math.comb(n, k)*math.comb(n-k, ell-k) *
+                            p**k * (q-p)**(ell-k) * (1-q)**(n-ell))
+                    total += mass * Fraction(h(k, ell), b["samples"])
+            return total
+        a, middle, c = Fraction(1, 3), Fraction(1, 2), Fraction(2, 3)
+        A, B, C, D = label_h(a, middle), label_h(a, c), label_h(middle, c), label_h(middle, middle)
+        label_covariance = (B*D-A*C)/D**2
+        label_difference = B/A-(C-B)/(D-A)
+        results.append({
+            "lattice": b["lattice"], "L": 3, "permutations": b["samples"],
+            "source_file": str(data_path / ref["file"]), "source_sha256": ref["sha256"],
+            "triples_checked": count, "violations": violations,
+            "rank_insertion_process_is_markov": not violations,
+            "exact_uniform_label_bridge": {
+                "times": [str(a), str(middle), str(c)],
+                "H_a_b": str(A), "H_a_c": str(B), "H_b_c": str(C), "H_b_b": str(D),
+                "conditional_covariance": str(label_covariance),
+                "survival_difference": str(label_difference),
+                "survival_difference_decimal": float(label_difference),
+                "markov_identity_violated_at_this_triple": label_covariance != 0,
+                "method": "exact multinomial clock mixture of stored complete-permutation H_J; Fraction arithmetic, no random labels drawn",
+            },
+            "boundary": "exact finite archived histogram under its engine convention; not a label-clock, large-L or continuum claim",
+        })
+    return results
+
+
+def main():
+    ap = argparse.ArgumentParser()
+    ap.add_argument("--data", type=Path,
+                    default=HERE.parent / "birth-gap-20260929/cloud-data-5k")
+    ap.add_argument("--output-json", type=Path, default=HERE / "result.json")
+    ap.add_argument("--output-md", type=Path, default=HERE / "RESULT.md")
+    ap.add_argument("--exact-data", type=Path, default=HERE.parent / "birth-gap-20260929/data")
+    args = ap.parse_args()
+    started = time.perf_counter()
+    control = exact_control()
+    exact_archives = exact_archive_analysis(args.exact_data)
+    manifest_path = args.data / "run.json"
+    manifest = json.loads(manifest_path.read_text())
+    if manifest["status"] != "completed":
+        raise ValueError("incomplete source block")
+    groups = defaultdict(list)
+    seeds, dependencies = set(), set()
+    for ref in manifest["batches"]:  # controls and timing benchmarks never enter
+        raw = (args.data / ref["file"]).read_bytes()
+        if hashlib.sha256(raw).hexdigest() != ref["sha256"]:
+            raise ValueError("source hash mismatch: " + ref["file"])
+        batch = json.loads(gzip.decompress(raw))
+        assert batch["histogram_columns"] == ["J1", "J2", "count"]
+        assert batch["samples"] == ref["samples"]
+        assert (batch["lattice"], batch["L"]) == (ref["lattice"], ref["L"])
+        assert batch["seed"] == ref["seed"]
+        assert ref["seed"] not in seeds and ref["dependency_group"] not in dependencies
+        seeds.add(ref["seed"]); dependencies.add(ref["dependency_group"])
+        rows = batch["histogram"]
+        assert len({(j1, j2) for j1, j2, count in rows}) == len(rows)
+        assert all(1 <= j1 <= j2 <= batch["N"] and count > 0 for j1, j2, count in rows)
+        assert sum(count for j1, j2, count in rows) == batch["samples"]
+        groups[batch["lattice"], batch["L"]].append((ref, batch))
+    cells = []
+    for (lattice, L), items in sorted(groups.items()):
+        items.sort(key=lambda x: x[0]["batch"])
+        refs, batches = zip(*items)
+        cells.append(summarize_cell(list(batches), list(refs), lattice, L))
+    result = {
+        "schema": "matching-one.birth-markov-kernel.v1",
+        "as_of": "2026-09-29",
+        "question": "Does entry history still predict completion after conditioning on present rank one?",
+        "contract": {
+            "time": "occupied-site insertion count, not iid-label time at finite N",
+            "cutoffs": "lower quantiles .25/.50/.75 of the equal-weight J1/J2 mixture, by cell",
+            "quantiles": QUANTILES,
+            "risk": "J1<=b<J2",
+            "early": "J1<=a", "late": "a<J1<=b",
+            "future": "J2>c",
+            "contrast": "P(future|risk,early)-P(future|risk,late)",
+            "null": "any time-inhomogeneous Markov description using current rank alone predicts zero",
+            "selection": "one triple selected before this scoring, primary L512; lower sizes are context, no cutoff scan",
+            "status": "retrospective exploratory reanalysis of an already-inspected random block",
+            "uncertainty": "complete equal-batch delete-one; recompute all three cutoffs in every replicate; SE not confidence bounds",
+        },
+        "source_manifest": str(manifest_path),
+        "source_manifest_sha256": hashlib.sha256(manifest_path.read_bytes()).hexdigest(),
+        "source_production_role": manifest["production_role"],
+        "source_engine_commit": manifest["git_base_commit"],
+        "source_samples": sum(c["samples"] for c in cells),
+        "no_new_samples": True,
+        "environment": {"python": sys.version, "platform": platform.platform()},
+        "exact_control": control,
+        "existing_exact_archives": exact_archives,
+        "cells": cells,
+        "limitations": [
+            "Not prospective or independent model-elimination certification; this block already supplied gap readouts.",
+            "Finite-size conditional dependence does not prove non-Markovianity of a continuum limit.",
+            "Conditioning on rank one tests the Markov property; it is not a causal effect of earlier birth.",
+            "A zero contrast at one triple would not prove Markovianity; the theorem requires all triples.",
+            "Square and triangular primitive-coordinate tori have different moduli; no universality comparison.",
+            "No fitted exponent, pooled cross-lattice significance, adaptive cutoff search or hidden-state identification.",
+        ],
+        "wall_seconds": time.perf_counter()-started,
+    }
+    args.output_json.write_text(json.dumps(result, indent=2, allow_nan=False)+"\n")
+    lines = ["# Does rank one retain birth-history memory?", "",
+             "2026-09-29; retrospective reuse of the completed 560k cloud block. No new samples.", "",
+             "One contrast, selected before this scoring: cutoffs a,b,c are the .25/.50/.75 lower quantiles of the pooled J1/J2 births.",
+             "Among paths with J1<=b<J2, compare P(J2>c) for early (J1<=a) versus late (a<J1<=b) first birth.",
+             "Every current-rank-only time-inhomogeneous Markov process predicts difference zero.", "",
+             "## Result", "",
+             "L512 is the primary available-size readout; smaller sizes supply context. Errors are 14-batch delete-one SE, not confidence intervals.",
+             "All thresholds are re-estimated in every deletion; the JSON retains their values and the complete metric covariance.", "",
+             "| Lattice | L | Rank-one risk count | Early survival | Late survival | Difference (percentage points) |",
+             "|---|---:|---:|---:|---:|---:|"]
+    for cell in cells:
+        p, e = cell["point"], cell["jackknife_se"]
+        risk = sum(map(sum, cell["risk_table_early_late_by_survive_exit"]))
+        lines.append(f"| {cell['lattice']} | {cell['L']} | {risk}/{cell['samples']} | "
+                     f"{p['survival_early']:.4f} | {p['survival_late']:.4f} | "
+                     f"{100*p['survival_difference']:+.3f} ± {100*e['survival_difference']:.3f} |")
+    lines += ["", "## Existing exact L3 archive, a different evidential object", "",
+              "No 9! enumeration rerun: the stored complete tables are evaluated with integer/Fraction arithmetic.",
+              "All 120 count-time triples 0<=a<b<c<=9 are checked here. This finite exhaustive identity check is separate from the ONE fixed production triple; it is not a production cutoff search.", ""]
+    for r in exact_archives:
+        lines.append(f"- {r['lattice']} L3: {len(r['violations'])} violating triples out of {r['triples_checked']}; "
+                     f"rank-only insertion process Markov under the full-kernel criterion: {r['rank_insertion_process_is_markov']}.")
+        for v in r["violations"]:
+            lines.append(f"  - {v['cutoffs']}: early/late survival {v['survival_early']} / {v['survival_late']}; "
+                         f"difference {v['survival_difference']}; conditional covariance {v['conditional_covariance']}.")
+        bridge = r["exact_uniform_label_bridge"]
+        lines.append(f"  - Exact iid-label bridge at p=(1/3,1/2,2/3): contrast {bridge['survival_difference']} "
+                     f"({100*bridge['survival_difference_decimal']:+.4f} percentage points); "
+                     f"Markov identity violated: {bridge['markov_identity_violated_at_this_triple']}.")
+    lines += ["", "The exact clock bridge is a multinomial mixture, not a replacement of count k by its expectation Np. Insertion and label clocks have separately evaluated properties; neither finite result is extrapolated to larger sizes or a continuum limit.",
+              "", "## Scope", ""] + ["- "+x for x in result["limitations"]]
+    lines += ["", "Source: `../birth-gap-20260929/cloud-data-5k/run.json`; all 112 production batches only.",
+              "No local-pilot pooling or benchmark rows. Grain: one uniform-permutation filtration, represented by weighted (J1,J2) histogram rows.",
+              f"One run took {result['wall_seconds']:.2f} s on Python {platform.python_version()} ({platform.machine()}).",
+              "One exact four-cell control is recorded in result.json; old physical enumeration was not rerun.", "",
+              "Reproduce from the repository root:", "", "```sh",
+              "python3 analysis/birth-markov-kernel-20260929/analyze.py", "```", ""]
+    args.output_md.write_text("\n".join(lines))
+    print(json.dumps({"seconds": result["wall_seconds"], "samples": result["source_samples"],
+                      "primary": [{"lattice": c["lattice"], "point": c["point"], "se": c["jackknife_se"]}
+                                  for c in cells if c["L"] == 512]}, indent=2))
+
+
+if __name__ == "__main__":
+    main()
diff --git a/analysis/birth-markov-kernel-20260929/exact-clock.json b/analysis/birth-markov-kernel-20260929/exact-clock.json
new file mode 100644
index 000000000..41ca8b76e
--- /dev/null
+++ b/analysis/birth-markov-kernel-20260929/exact-clock.json
@@ -0,0 +1,56 @@
+{
+  "schema": "matching-one.birth-kernel-exact-clock.v1",
+  "as_of": "2026-09-29",
+  "definition": "H(p,q)=P(T1<=p,T2>q), 0<=p<=q<=1, iid Uniform site labels",
+  "clock_transform": "sum_(k<=ell) Multinomial(N;k,ell-k,N-ell;p,q-p,1-q) H_J(k,ell)",
+  "determinant": "H(a,c)H(b,b)-H(a,b)H(b,c), a<b<c",
+  "triangular_A": "2*p**3 - 6*p**2 + 3*p",
+  "triangular_B": "-2*q**3 + 3*q + 1",
+  "cases": [
+    {
+      "lattice": "square",
+      "L": 3,
+      "N": 9,
+      "source_path": "analysis/birth-gap-20260929/data/exact-control-square-L3-b00.json.gz",
+      "source_sha256": "dccb0e132d7b2100bd39c6414f385366354c9099facbecbbaa1e3a5fd8f37a78",
+      "permutations_in_archive": 362880,
+      "kernel_closed_form": "6*p**3*(1 - q**2)**3",
+      "kernel_polynomial": "-6*p**3*q**6 + 18*p**3*q**4 - 18*p**3*q**2 + 6*p**3",
+      "coefficient_matrix_rank": 1,
+      "markov_determinant_closed_form": "0",
+      "markov_identity_holds_for_all_label_triples": true,
+      "specified_label_triple": [
+        "1/3",
+        "1/2",
+        "2/3"
+      ],
+      "conditional_survival_difference_at_triple": "0",
+      "exact_polynomial_identity": true
+    },
+    {
+      "lattice": "triangular",
+      "L": 3,
+      "N": 9,
+      "source_path": "analysis/birth-gap-20260929/data/exact-control-triangular-L3-b00.json.gz",
+      "source_sha256": "eed9f4d3f0e996bfaba8e6b0ab3928ba26776bef0b2b1a18d86e08b051d30391",
+      "permutations_in_archive": 362880,
+      "kernel_closed_form": "9*p**3*(1 - q)**3*(2*p**3 - 6*p**2 + 3*p - 2*q**3 + 3*q + 1)",
+      "kernel_polynomial": "-18*p**6*q**3 + 54*p**6*q**2 - 54*p**6*q + 18*p**6 + 54*p**5*q**3 - 162*p**5*q**2 + 162*p**5*q - 54*p**5 - 27*p**4*q**3 + 81*p**4*q**2 - 81*p**4*q + 27*p**4 + 18*p**3*q**6 - 54*p**3*q**5 + 27*p**3*q**4 + 54*p**3*q**3 - 54*p**3*q**2 + 9*p**3",
+      "coefficient_matrix_rank": 2,
+      "markov_determinant_closed_form": "81*a**3*b**3*(1 - b)**3*(1 - c)**3*(2*b**3 - 3*b - 2*c**3 + 3*c)*(-2*a**3 + 6*a**2 - 3*a + 2*b**3 - 6*b**2 + 3*b)",
+      "markov_identity_holds_for_all_label_triples": false,
+      "specified_label_triple": [
+        "1/3",
+        "1/2",
+        "2/3"
+      ],
+      "conditional_survival_difference_at_triple": "-1156/716639",
+      "exact_polynomial_identity": true
+    }
+  ],
+  "scope": "Exact archived L3 engine convention; rank natural filtration, ordinary deterministic-time Markov property; no continuum extrapolation or hidden-positive-state dimension claim.",
+  "no_new_permutations_or_random_labels": true,
+  "python": "3.11.15",
+  "sympy": "1.14.0",
+  "wall_seconds": 0.0988064159755595
+}
diff --git a/analysis/birth-markov-kernel-20260929/exact_clock.py b/analysis/birth-markov-kernel-20260929/exact_clock.py
new file mode 100644
index 000000000..d1c05773e
--- /dev/null
+++ b/analysis/birth-markov-kernel-20260929/exact_clock.py
@@ -0,0 +1,88 @@
+#!/usr/bin/env python3
+"""Exact L3 two-time uniform-label kernels from archived 9! permutation tables.
+
+Uses SymPy for polynomial arithmetic, not new enumeration or simulations.
+"""
+from pathlib import Path
+import gzip
+import hashlib
+import json
+import math
+import platform
+import time
+import sympy as S
+
+
+HERE = Path(__file__).resolve().parent
+DATA = HERE.parent / "birth-gap-20260929/data"
+
+
+def main():
+    started = time.perf_counter()
+    source = json.loads((DATA / "run.json").read_text())
+    p, q = S.symbols("p q")
+    a, b, c = S.symbols("a b c")
+    A = lambda x: 2*x**3-6*x**2+3*x
+    B = lambda x: -2*x**3+3*x+1
+    records = []
+    for ref in source["controls"]:
+        raw = (DATA / ref["file"]).read_bytes()
+        assert hashlib.sha256(raw).hexdigest() == ref["sha256"]
+        batch = json.loads(gzip.decompress(raw))
+        n, total = batch["N"], batch["samples"]
+        assert n == 9 and total == math.factorial(n) and batch["mode"] == "exact"
+        assert sum(row[2] for row in batch["histogram"]) == total
+        def h(k, ell):
+            return sum(count for i, j, count in batch["histogram"] if i <= k and j > ell)
+        kernel = S.expand(sum(
+            S.Rational(h(k, ell), total)*math.comb(n, k)*math.comb(n-k, ell-k) *
+            p**k*(q-p)**(ell-k)*(1-q)**(n-ell)
+            for k in range(n+1) for ell in range(k, n+1)))
+        if batch["lattice"] == "square":
+            closed_form = 6*p**3*(1-q**2)**3
+            determinant_form = S.Integer(0)
+        else:
+            assert batch["lattice"] == "triangular"
+            closed_form = 9*p**3*(1-q)**3*(A(p)+B(q))
+            determinant_form = 81*a**3*b**3*(1-b)**3*(1-c)**3*(A(b)-A(a))*(B(c)-B(b))
+        assert S.expand(kernel-closed_form) == 0
+        substitute = lambda x, y: kernel.subs({p: x, q: y}, simultaneous=True)
+        determinant = substitute(a, c)*substitute(b, b)-substitute(a, b)*substitute(b, c)
+        assert S.expand(determinant-determinant_form) == 0
+        coefficients = S.Matrix(n+1, n+1, lambda i, j: kernel.coeff(p, i).coeff(q, j))
+        times = (S.Rational(1, 3), S.Rational(1, 2), S.Rational(2, 3))
+        x, y, z = times
+        Hab, Hac, Hbc, Hbb = substitute(x, y), substitute(x, z), substitute(y, z), substitute(y, y)
+        difference = S.cancel(Hac/Hab-(Hbc-Hac)/(Hbb-Hab))
+        records.append({
+            "lattice": batch["lattice"], "L": 3, "N": n,
+            "source_path": "analysis/birth-gap-20260929/data/"+ref["file"],
+            "source_sha256": ref["sha256"], "permutations_in_archive": total,
+            "kernel_closed_form": str(closed_form),
+            "kernel_polynomial": str(kernel),
+            "coefficient_matrix_rank": int(coefficients.rank()),
+            "markov_determinant_closed_form": str(determinant_form),
+            "markov_identity_holds_for_all_label_triples": determinant_form == 0,
+            "specified_label_triple": [str(x) for x in times],
+            "conditional_survival_difference_at_triple": str(difference),
+            "exact_polynomial_identity": True,
+        })
+    result = {
+        "schema": "matching-one.birth-kernel-exact-clock.v1",
+        "as_of": "2026-09-29",
+        "definition": "H(p,q)=P(T1<=p,T2>q), 0<=p<=q<=1, iid Uniform site labels",
+        "clock_transform": "sum_(k<=ell) Multinomial(N;k,ell-k,N-ell;p,q-p,1-q) H_J(k,ell)",
+        "determinant": "H(a,c)H(b,b)-H(a,b)H(b,c), a<b<c",
+        "triangular_A": str(A(p)), "triangular_B": str(B(q)),
+        "cases": records,
+        "scope": "Exact archived L3 engine convention; rank natural filtration, ordinary deterministic-time Markov property; no continuum extrapolation or hidden-positive-state dimension claim.",
+        "no_new_permutations_or_random_labels": True,
+        "python": platform.python_version(), "sympy": S.__version__,
+        "wall_seconds": time.perf_counter()-started,
+    }
+    (HERE / "exact-clock.json").write_text(json.dumps(result, indent=2)+"\n")
+    print(json.dumps(result, indent=2))
+
+
+if __name__ == "__main__":
+    main()
diff --git a/analysis/birth-markov-kernel-20260929/result.json b/analysis/birth-markov-kernel-20260929/result.json
new file mode 100644
index 000000000..88c2066b4
--- /dev/null
+++ b/analysis/birth-markov-kernel-20260929/result.json
@@ -0,0 +1,5392 @@
+{
+  "schema": "matching-one.birth-markov-kernel.v1",
+  "as_of": "2026-09-29",
+  "question": "Does entry history still predict completion after conditioning on present rank one?",
+  "contract": {
+    "time": "occupied-site insertion count, not iid-label time at finite N",
+    "cutoffs": "lower quantiles .25/.50/.75 of the equal-weight J1/J2 mixture, by cell",
+    "quantiles": [
+      0.25,
+      0.5,
+      0.75
+    ],
+    "risk": "J1<=b<J2",
+    "early": "J1<=a",
+    "late": "a<J1<=b",
+    "future": "J2>c",
+    "contrast": "P(future|risk,early)-P(future|risk,late)",
+    "null": "any time-inhomogeneous Markov description using current rank alone predicts zero",
+    "selection": "one triple selected before this scoring, primary L512; lower sizes are context, no cutoff scan",
+    "status": "retrospective exploratory reanalysis of an already-inspected random block",
+    "uncertainty": "complete equal-batch delete-one; recompute all three cutoffs in every replicate; SE not confidence bounds"
+  },
+  "source_manifest": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/birth-gap-20260929/cloud-data-5k/run.json",
+  "source_manifest_sha256": "50b519b94da0bad9093b7accb2c6e7e5bee44d85930d446f8a74be3f9766470a",
+  "source_production_role": "cloud-independent-20260929",
+  "source_engine_commit": "abc78c6a1c69e83c9a2daf4d92911833bd6f3937",
+  "source_samples": 560000,
+  "no_new_samples": true,
+  "environment": {
+    "python": "3.11.15 (main, Jun 23 2026, 15:46:51) [Clang 22.1.3 ]",
+    "platform": "macOS-27.0-arm64-arm-64bit"
+  },
+  "exact_control": {
+    "independent_table": [
+      [
+        1,
+        1
+      ],
+      [
+        1,
+        1
+      ]
+    ],
+    "dependent_table": [
+      [
+        3,
+        1
+      ],
+      [
+        1,
+        3
+      ]
+    ],
+    "dependent_contrast": "1/2",
+    "dependent_covariance": "1/8",
+    "scope": "one exact four-cell arithmetic and inequality-boundary control"
+  },
+  "existing_exact_archives": [
+    {
+      "lattice": "square",
+      "L": 3,
+      "permutations": 362880,
+      "source_file": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/birth-gap-20260929/data/exact-control-square-L3-b00.json.gz",
+      "source_sha256": "dccb0e132d7b2100bd39c6414f385366354c9099facbecbbaa1e3a5fd8f37a78",
+      "triples_checked": 120,
+      "violations": [],
+      "rank_insertion_process_is_markov": true,
+      "exact_uniform_label_bridge": {
+        "times": [
+          "1/3",
+          "1/2",
+          "2/3"
+        ],
+        "H_a_b": "3/32",
+        "H_a_c": "250/6561",
+        "H_b_c": "125/972",
+        "H_b_b": "81/256",
+        "conditional_covariance": "0",
+        "survival_difference": "0",
+        "survival_difference_decimal": 0.0,
+        "markov_identity_violated_at_this_triple": false,
+        "method": "exact multinomial clock mixture of stored complete-permutation H_J; Fraction arithmetic, no random labels drawn"
+      },
+      "boundary": "exact finite archived histogram under its engine convention; not a label-clock, large-L or continuum claim"
+    },
+    {
+      "lattice": "triangular",
+      "L": 3,
+      "permutations": 362880,
+      "source_file": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/birth-gap-20260929/data/exact-control-triangular-L3-b00.json.gz",
+      "source_sha256": "eed9f4d3f0e996bfaba8e6b0ab3928ba26776bef0b2b1a18d86e08b051d30391",
+      "triples_checked": 120,
+      "violations": [
+        {
+          "cutoffs": [
+            3,
+            4,
+            5
+          ],
+          "risk_table": [
+            [
+              23328,
+              15552
+            ],
+            [
+              101088,
+              93312
+            ]
+          ],
+          "conditional_covariance": "1/90",
+          "survival_early": "3/5",
+          "survival_late": "13/25",
+          "survival_difference": "2/25"
+        },
+        {
+          "cutoffs": [
+            4,
+            5,
+            6
+          ],
+          "risk_table": [
+            [
+              23328,
+              101088
+            ],
+            [
+              15552,
+              93312
+            ]
+          ],
+          "conditional_covariance": "1/90",
+          "survival_early": "3/16",
+          "survival_late": "1/7",
+          "survival_difference": "5/112"
+        }
+      ],
+      "rank_insertion_process_is_markov": false,
+      "exact_uniform_label_bridge": {
+        "times": [
+          "1/3",
+          "1/2",
+          "2/3"
+        ],
+        "H_a_b": "287/2592",
+        "H_a_c": "76/2187",
+        "H_b_c": "287/2592",
+        "H_b_b": "45/128",
+        "conditional_covariance": "-4624/13286025",
+        "survival_difference": "-1156/716639",
+        "survival_difference_decimal": -0.0016130855284180738,
+        "markov_identity_violated_at_this_triple": true,
+        "method": "exact multinomial clock mixture of stored complete-permutation H_J; Fraction arithmetic, no random labels drawn"
+      },
+      "boundary": "exact finite archived histogram under its engine convention; not a label-clock, large-L or continuum claim"
+    }
+  ],
+  "cells": [
+    {
+      "lattice": "square",
+      "L": 64,
+      "N": 4096,
+      "samples": 70000,
+      "batches": 14,
+      "role": "size_context",
+      "cutoffs_insertion_counts": [
+        2376,
+        2428,
+        2480
+      ],
+      "risk_table_early_late_by_survive_exit": [
+        [
+          6201,
+          7138
+        ],
+        [
+          7104,
+          8599
+        ]
+      ],
+      "kernel_counts": {
+        "H_a_b": 13339,
+        "H_a_c": 6201,
+        "H_b_c": 13305,
+        "H_b_b": 29042
+      },
+      "point": {
+        "rank_one_risk_fraction": 0.4148857142857143,
+        "early_fraction_within_risk": 0.4593003236691688,
+        "survival_early": 0.4648774270934853,
+        "survival_late": 0.4523976310259186,
+        "survival_difference": 0.012479796067566662,
+        "conditional_covariance": 0.0030992766883034656
+      },
+      "metric_order": [
+        "rank_one_risk_fraction",
+        "early_fraction_within_risk",
+        "survival_early",
+        "survival_late",
+        "survival_difference",
+        "conditional_covariance"
+      ],
+      "jackknife_se": {
+        "rank_one_risk_fraction": 0.0013569057051326747,
+        "early_fraction_within_risk": 0.003247458471242331,
+        "survival_early": 0.004835451081033996,
+        "survival_late": 0.0037305149771411444,
+        "survival_difference": 0.0055289270963451995,
+        "conditional_covariance": 0.0013737335485123454
+      },
+      "delete_one_batch_covariance": [
+        [
+          1.841193092621601e-06,
+          2.0320565639726226e-06,
+          2.2836084440189043e-06,
+          1.1944120168628762e-06,
+          1.0891964271560285e-06,
+          2.7232932314563015e-07
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+          "batch": 11,
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+          "sha256": "18dd1eb5c4a4939a2eb60ac6015c15fba57c53b905a3dcee5e7429be9466b239",
+          "dependency_group": "cloud-independent-20260929/triangular/L512/batch11"
+        },
+        {
+          "file": "cloud-independent-20260929-triangular-L512-b12.json.gz",
+          "batch": 12,
+          "samples": 5000,
+          "sha256": "fa65fedbe181bca3a4020f357afe7638bac127d4aeeb798d9f86b0c6c858be74",
+          "dependency_group": "cloud-independent-20260929/triangular/L512/batch12"
+        },
+        {
+          "file": "cloud-independent-20260929-triangular-L512-b13.json.gz",
+          "batch": 13,
+          "samples": 5000,
+          "sha256": "111a16337cce62804352e90de94e68e6918e56e73ad8f20de7fe66724a56b0bf",
+          "dependency_group": "cloud-independent-20260929/triangular/L512/batch13"
+        }
+      ]
+    }
+  ],
+  "limitations": [
+    "Not prospective or independent model-elimination certification; this block already supplied gap readouts.",
+    "Finite-size conditional dependence does not prove non-Markovianity of a continuum limit.",
+    "Conditioning on rank one tests the Markov property; it is not a causal effect of earlier birth.",
+    "A zero contrast at one triple would not prove Markovianity; the theorem requires all triples.",
+    "Square and triangular primitive-coordinate tori have different moduli; no universality comparison.",
+    "No fitted exponent, pooled cross-lattice significance, adaptive cutoff search or hidden-state identification."
+  ],
+  "wall_seconds": 2.7442724999855272
+}
diff --git a/analysis/birth-selection-mechanism-20260930/RESULT.md b/analysis/birth-selection-mechanism-20260930/RESULT.md
new file mode 100644
index 000000000..7f6b88d8f
--- /dev/null
+++ b/analysis/birth-selection-mechanism-20260930/RESULT.md
@@ -0,0 +1,33 @@
+# Three physical mechanism controls
+
+Exact fractions on three specified square-L4 preparations; no census or production.
+
+| Set | c | e | P(J1<=4 given A6) | Var_safe(degree) | S2(0) | S2'(0) | S2(log 2) |
+|---|---:|---:|---:|---:|---:|---:|---:|
+| A | 0 | 2 | 1/15 | 6/25 | 43/45 | -2/75 | 59/63 |
+| B | 0 | 3 | 1/15 | 16/25 | 14/15 | -16/225 | 71/81 |
+| C | 0 | 2 | 0 | 6/25 | 43/45 | -2/75 | 59/63 |
+
+A/B have the same entire conditional birth law but different futures and source response.
+A/C have the same c,e and two-step response but different conditional birth laws.
+For iid thinning, their homology-retention polynomials are u^4,u^4,u^6.
+The source kick is an explicitly modified source, not the original Bernoulli experiment.
+
+## Actual longer-window response, not extrapolated curvature
+
+| Insertions after kick | A derivative | B derivative | C derivative |
+|---:|---:|---:|---:|
+| 1 | 0 | 0 | 0 |
+| 2 | -2/75 | -16/225 | -2/75 |
+| 3 | -7/180 | -173/1800 | -17/450 |
+| 4 | -17/525 | -307/4200 | -6/175 |
+| 5 | -2/105 | -47/1260 | -2/105 |
+| 6 | -1/150 | -67/6300 | -2/315 |
+| 7 | 0 | 0 | 0 |
+| 8 | 0 | 0 | 0 |
+| 9 | 0 | 0 | 0 |
+| 10 | 0 | 0 | 0 |
+
+Mean remaining steps-to-completion derivatives: A=-779/6300; B=-454/1575; C=-391/3150.
+
+Only subsets reachable from the three specified starts are visited; no uniform-start full census.
diff --git a/analysis/birth-selection-mechanism-20260930/derive.py b/analysis/birth-selection-mechanism-20260930/derive.py
new file mode 100644
index 000000000..4c9d589a7
--- /dev/null
+++ b/analysis/birth-selection-mechanism-20260930/derive.py
@@ -0,0 +1,131 @@
+#!/usr/bin/env python3
+"""Three specified geometries and their mechanism responses; not a census.
+
+Reuse only the lifted topology function. All outputs are exact fractions.
+No parameter search, permutation enumeration, sampling, or old-data rescore.
+"""
+from fractions import Fraction as F
+from functools import lru_cache
+import importlib.util
+from itertools import combinations
+import json
+from math import comb
+from pathlib import Path
+
+HERE = Path(__file__).resolve().parent
+SOURCE = HERE.parent / "birth-completion-geometry-20260929/analyze.py"
+SPEC = importlib.util.spec_from_file_location("topology_only", SOURCE)
+TOPO = importlib.util.module_from_spec(SPEC)
+SPEC.loader.exec_module(TOPO)
+TOPO.L, TOPO.N = 4, 16
+NEIGHBORS = [[((v % 4 + dx) % 4 + 4*((v//4 + dy) % 4), dx, dy)
+              for dx, dy in [(1, 0), (-1, 0), (0, 1), (0, -1)]]
+             for v in range(16)]
+
+
+@lru_cache(None)
+def topology(mask):
+    return TOPO.topology(mask, NEIGHBORS)
+
+
+@lru_cache(None)
+def survival(mask, h):
+    if topology(mask)[0] != 1:
+        return F(0)
+    if h == 0:
+        return F(1)
+    vacant = [v for v in range(16) if not mask >> v & 1]
+    return sum((survival(mask | (1 << v), h-1) for v in vacant), F(0))/len(vacant)
+
+
+def geometry(name, vertices):
+    mask = sum(1 << v for v in vertices)
+    rank, direction = topology(mask)
+    vacancies = [v for v in range(16) if not mask >> v & 1]
+    completing = [v for v in vacancies if topology(mask | (1 << v))[0] == 2]
+    safe = [v for v in vacancies if v not in completing]
+    edges = [(v, w) for v, w in combinations(safe, 2)
+             if topology(mask | (1 << v) | (1 << w))[0] == 2]
+    degrees = [sum(v in pair for pair in edges) for v in safe]
+    m, s, c, e = len(vacancies), len(safe), len(completing), len(edges)
+    mean_d = F(sum(degrees), s)
+    variance_d = F(sum(d*d for d in degrees), s)-mean_d**2
+    retention_counts = [sum(topology(sum(1 << v for v in subset))[0] >= 1
+                            for subset in combinations(vertices, a))
+                        for a in range(len(vertices)+1)]
+    # exp(theta)=2: a finite, fully normalized source kick, not a derivative
+    # inferred by numerical differencing. Only the first insertion is tilted.
+    weights = [2**d for d in degrees]
+    tilted_mean_d = F(sum(w*d for w, d in zip(weights, degrees)), sum(weights))
+    lag_rows = []
+    for h in range(1, m+1):
+        tails = [survival(mask | (1 << v), h-1) for v in safe]
+        lag_rows.append({
+            "lag_insertions": h,
+            "uniform_survival": str(survival(mask, h)),
+            "first_kick_derivative_at_zero": str(sum(((d-mean_d)*f for d, f in zip(degrees, tails)), F(0))/m),
+            "first_kick_exp_theta_2_survival": str(F(s, m)*sum((w*f for w, f in zip(weights, tails)), F(0))/sum(weights)),
+        })
+    return {
+        "name": name, "occupied": vertices, "rank": rank,
+        "direction": direction, "c": c, "e": e, "synergy_edges": edges,
+        "safe_degree_counts": {str(d): degrees.count(d) for d in sorted(set(degrees))},
+        "safe_degree_mean": str(mean_d), "safe_degree_variance": str(variance_d),
+        "retaining_subsets_by_size": retention_counts,
+        "birth_cdf_at_count_a": [str(F(n, comb(len(vertices), a)))
+                                  for a, n in enumerate(retention_counts)],
+        "survival_two_uniform": str(F(s*(s-1)-2*e, m*(m-1))),
+        "survival_two_first_kick_derivative_at_zero": str(-F(s, m*(m-1))*variance_d),
+        "survival_two_first_kick_exp_theta_2": str(F(s, m)*(1-F(c, m-1)-tilted_mean_d/(m-1))),
+        "all_lags_single_kick": lag_rows,
+        "mean_remaining_steps_to_completion_derivative": str(sum((F(r["first_kick_derivative_at_zero"]) for r in lag_rows), F(0))),
+    }
+
+
+def main():
+    rows = [geometry("A", [0, 1, 4, 5, 8, 12]),
+            geometry("B", [0, 2, 4, 5, 8, 12]),
+            geometry("C", [0, 1, 5, 8, 9, 12])]
+    # A and B are the already published L4 geometric witness. C is one
+    # specified six-site bent loop, not a best-effect search over subsets.
+    assert all((r["rank"], r["direction"], r["c"]) == (1, (0, 1), 0) for r in rows)
+    assert rows[0]["birth_cdf_at_count_a"] == rows[1]["birth_cdf_at_count_a"]
+    assert rows[0]["e"] == rows[2]["e"] == 2 and rows[1]["e"] == 3
+    result = {
+        "schema": "matching-one.birth-selection-intervention.v1",
+        "date": "2026-09-30", "lattice": "square NN occupied-site L4 torus",
+        "clock": "insertion count; k=6; one tilted insertion then original uniform continuation at every later count",
+        "source": "P_theta(v|A,safe) proportional to exp(theta*synergy_degree(v)); exit mass c/m unchanged",
+        "scope": "exact specified-preparation controls, not uniform-start population inference or L512 evidence",
+        "topology_source": str(SOURCE.relative_to(HERE.parent.parent)),
+        "topology_source_commit": "8b37c20e5f2b1946cefbed98b09280a0cad25ed2",
+        "rows": rows,
+        "equal_A_B_entry_ensemble_e_tilt_survival_derivative": str(-F(1, 180)),
+    }
+    (HERE / "result.json").write_text(json.dumps(result, indent=2)+"\n")
+    lines = ["# Three physical mechanism controls", "",
+             "Exact fractions on three specified square-L4 preparations; no census or production.", "",
+             "| Set | c | e | P(J1<=4 given A6) | Var_safe(degree) | S2(0) | S2'(0) | S2(log 2) |",
+             "|---|---:|---:|---:|---:|---:|---:|---:|"]
+    for r in rows:
+        lines.append(f"| {r['name']} | {r['c']} | {r['e']} | {r['birth_cdf_at_count_a'][4]} | "
+                     f"{r['safe_degree_variance']} | {r['survival_two_uniform']} | "
+                     f"{r['survival_two_first_kick_derivative_at_zero']} | {r['survival_two_first_kick_exp_theta_2']} |")
+    lines += ["", "A/B have the same entire conditional birth law but different futures and source response.",
+              "A/C have the same c,e and two-step response but different conditional birth laws.",
+              "For iid thinning, their homology-retention polynomials are u^4,u^4,u^6.",
+              "The source kick is an explicitly modified source, not the original Bernoulli experiment.", ""]
+    lines += ["## Actual longer-window response, not extrapolated curvature", "",
+              "| Insertions after kick | A derivative | B derivative | C derivative |",
+              "|---:|---:|---:|---:|"]
+    for h in range(10):
+        lines.append("| " + str(h+1) + " | " + " | ".join(r["all_lags_single_kick"][h]["first_kick_derivative_at_zero"] for r in rows) + " |")
+    lines += ["", "Mean remaining steps-to-completion derivatives: " + "; ".join(
+        f"{r['name']}={r['mean_remaining_steps_to_completion_derivative']}" for r in rows) + ".", "",
+        "Only subsets reachable from the three specified starts are visited; no uniform-start full census.", ""]
+    (HERE / "RESULT.md").write_text("\n".join(lines))
+    print("\n".join(lines))
+
+
+if __name__ == "__main__":
+    main()
diff --git a/analysis/birth-selection-mechanism-20260930/result.json b/analysis/birth-selection-mechanism-20260930/result.json
new file mode 100644
index 000000000..5553cb0b0
--- /dev/null
+++ b/analysis/birth-selection-mechanism-20260930/result.json
@@ -0,0 +1,372 @@
+{
+  "schema": "matching-one.birth-selection-intervention.v1",
+  "date": "2026-09-30",
+  "lattice": "square NN occupied-site L4 torus",
+  "clock": "insertion count; k=6; one tilted insertion then original uniform continuation at every later count",
+  "source": "P_theta(v|A,safe) proportional to exp(theta*synergy_degree(v)); exit mass c/m unchanged",
+  "scope": "exact specified-preparation controls, not uniform-start population inference or L512 evidence",
+  "topology_source": "analysis/birth-completion-geometry-20260929/analyze.py",
+  "topology_source_commit": "8b37c20e5f2b1946cefbed98b09280a0cad25ed2",
+  "rows": [
+    {
+      "name": "A",
+      "occupied": [
+        0,
+        1,
+        4,
+        5,
+        8,
+        12
+      ],
+      "rank": 1,
+      "direction": [
+        0,
+        1
+      ],
+      "c": 0,
+      "e": 2,
+      "synergy_edges": [
+        [
+          2,
+          3
+        ],
+        [
+          6,
+          7
+        ]
+      ],
+      "safe_degree_counts": {
+        "0": 6,
+        "1": 4
+      },
+      "safe_degree_mean": "2/5",
+      "safe_degree_variance": "6/25",
+      "retaining_subsets_by_size": [
+        0,
+        0,
+        0,
+        0,
+        1,
+        2,
+        1
+      ],
+      "birth_cdf_at_count_a": [
+        "0",
+        "0",
+        "0",
+        "0",
+        "1/15",
+        "1/3",
+        "1"
+      ],
+      "survival_two_uniform": "43/45",
+      "survival_two_first_kick_derivative_at_zero": "-2/75",
+      "survival_two_first_kick_exp_theta_2": "59/63",
+      "all_lags_single_kick": [
+        {
+          "lag_insertions": 1,
+          "uniform_survival": "1",
+          "first_kick_derivative_at_zero": "0",
+          "first_kick_exp_theta_2_survival": "1"
+        },
+        {
+          "lag_insertions": 2,
+          "uniform_survival": "43/45",
+          "first_kick_derivative_at_zero": "-2/75",
+          "first_kick_exp_theta_2_survival": "59/63"
+        },
+        {
+          "lag_insertions": 3,
+          "uniform_survival": "5/6",
+          "first_kick_derivative_at_zero": "-7/180",
+          "first_kick_exp_theta_2_survival": "29/36"
+        },
+        {
+          "lag_insertions": 4,
+          "uniform_survival": "127/210",
+          "first_kick_derivative_at_zero": "-17/525",
+          "first_kick_exp_theta_2_survival": "57/98"
+        },
+        {
+          "lag_insertions": 5,
+          "uniform_survival": "41/126",
+          "first_kick_derivative_at_zero": "-2/105",
+          "first_kick_exp_theta_2_survival": "275/882"
+        },
+        {
+          "lag_insertions": 6,
+          "uniform_survival": "1/10",
+          "first_kick_derivative_at_zero": "-1/150",
+          "first_kick_exp_theta_2_survival": "2/21"
+        },
+        {
+          "lag_insertions": 7,
+          "uniform_survival": "0",
+          "first_kick_derivative_at_zero": "0",
+          "first_kick_exp_theta_2_survival": "0"
+        },
+        {
+          "lag_insertions": 8,
+          "uniform_survival": "0",
+          "first_kick_derivative_at_zero": "0",
+          "first_kick_exp_theta_2_survival": "0"
+        },
+        {
+          "lag_insertions": 9,
+          "uniform_survival": "0",
+          "first_kick_derivative_at_zero": "0",
+          "first_kick_exp_theta_2_survival": "0"
+        },
+        {
+          "lag_insertions": 10,
+          "uniform_survival": "0",
+          "first_kick_derivative_at_zero": "0",
+          "first_kick_exp_theta_2_survival": "0"
+        }
+      ],
+      "mean_remaining_steps_to_completion_derivative": "-779/6300"
+    },
+    {
+      "name": "B",
+      "occupied": [
+        0,
+        2,
+        4,
+        5,
+        8,
+        12
+      ],
+      "rank": 1,
+      "direction": [
+        0,
+        1
+      ],
+      "c": 0,
+      "e": 3,
+      "synergy_edges": [
+        [
+          1,
+          3
+        ],
+        [
+          3,
+          6
+        ],
+        [
+          6,
+          7
+        ]
+      ],
+      "safe_degree_counts": {
+        "0": 6,
+        "1": 2,
+        "2": 2
+      },
+      "safe_degree_mean": "3/5",
+      "safe_degree_variance": "16/25",
+      "retaining_subsets_by_size": [
+        0,
+        0,
+        0,
+        0,
+        1,
+        2,
+        1
+      ],
+      "birth_cdf_at_count_a": [
+        "0",
+        "0",
+        "0",
+        "0",
+        "1/15",
+        "1/3",
+        "1"
+      ],
+      "survival_two_uniform": "14/15",
+      "survival_two_first_kick_derivative_at_zero": "-16/225",
+      "survival_two_first_kick_exp_theta_2": "71/81",
+      "all_lags_single_kick": [
+        {
+          "lag_insertions": 1,
+          "uniform_survival": "1",
+          "first_kick_derivative_at_zero": "0",
+          "first_kick_exp_theta_2_survival": "1"
+        },
+        {
+          "lag_insertions": 2,
+          "uniform_survival": "14/15",
+          "first_kick_derivative_at_zero": "-16/225",
+          "first_kick_exp_theta_2_survival": "71/81"
+        },
+        {
+          "lag_insertions": 3,
+          "uniform_survival": "23/30",
+          "first_kick_derivative_at_zero": "-173/1800",
+          "first_kick_exp_theta_2_survival": "223/324"
+        },
+        {
+          "lag_insertions": 4,
+          "uniform_survival": "53/105",
+          "first_kick_derivative_at_zero": "-307/4200",
+          "first_kick_exp_theta_2_survival": "335/756"
+        },
+        {
+          "lag_insertions": 5,
+          "uniform_survival": "61/252",
+          "first_kick_derivative_at_zero": "-47/1260",
+          "first_kick_exp_theta_2_survival": "17/81"
+        },
+        {
+          "lag_insertions": 6,
+          "uniform_survival": "1/15",
+          "first_kick_derivative_at_zero": "-67/6300",
+          "first_kick_exp_theta_2_survival": "65/1134"
+        },
+        {
+          "lag_insertions": 7,
+          "uniform_survival": "0",
+          "first_kick_derivative_at_zero": "0",
+          "first_kick_exp_theta_2_survival": "0"
+        },
+        {
+          "lag_insertions": 8,
+          "uniform_survival": "0",
+          "first_kick_derivative_at_zero": "0",
+          "first_kick_exp_theta_2_survival": "0"
+        },
+        {
+          "lag_insertions": 9,
+          "uniform_survival": "0",
+          "first_kick_derivative_at_zero": "0",
+          "first_kick_exp_theta_2_survival": "0"
+        },
+        {
+          "lag_insertions": 10,
+          "uniform_survival": "0",
+          "first_kick_derivative_at_zero": "0",
+          "first_kick_exp_theta_2_survival": "0"
+        }
+      ],
+      "mean_remaining_steps_to_completion_derivative": "-454/1575"
+    },
+    {
+      "name": "C",
+      "occupied": [
+        0,
+        1,
+        5,
+        8,
+        9,
+        12
+      ],
+      "rank": 1,
+      "direction": [
+        0,
+        1
+      ],
+      "c": 0,
+      "e": 2,
+      "synergy_edges": [
+        [
+          2,
+          3
+        ],
+        [
+          10,
+          11
+        ]
+      ],
+      "safe_degree_counts": {
+        "0": 6,
+        "1": 4
+      },
+      "safe_degree_mean": "2/5",
+      "safe_degree_variance": "6/25",
+      "retaining_subsets_by_size": [
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        1
+      ],
+      "birth_cdf_at_count_a": [
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "1"
+      ],
+      "survival_two_uniform": "43/45",
+      "survival_two_first_kick_derivative_at_zero": "-2/75",
+      "survival_two_first_kick_exp_theta_2": "59/63",
+      "all_lags_single_kick": [
+        {
+          "lag_insertions": 1,
+          "uniform_survival": "1",
+          "first_kick_derivative_at_zero": "0",
+          "first_kick_exp_theta_2_survival": "1"
+        },
+        {
+          "lag_insertions": 2,
+          "uniform_survival": "43/45",
+          "first_kick_derivative_at_zero": "-2/75",
+          "first_kick_exp_theta_2_survival": "59/63"
+        },
+        {
+          "lag_insertions": 3,
+          "uniform_survival": "49/60",
+          "first_kick_derivative_at_zero": "-17/450",
+          "first_kick_exp_theta_2_survival": "199/252"
+        },
+        {
+          "lag_insertions": 4,
+          "uniform_survival": "41/70",
+          "first_kick_derivative_at_zero": "-6/175",
+          "first_kick_exp_theta_2_survival": "55/98"
+        },
+        {
+          "lag_insertions": 5,
+          "uniform_survival": "13/42",
+          "first_kick_derivative_at_zero": "-2/105",
+          "first_kick_exp_theta_2_survival": "29/98"
+        },
+        {
+          "lag_insertions": 6,
+          "uniform_survival": "2/21",
+          "first_kick_derivative_at_zero": "-2/315",
+          "first_kick_exp_theta_2_survival": "40/441"
+        },
+        {
+          "lag_insertions": 7,
+          "uniform_survival": "0",
+          "first_kick_derivative_at_zero": "0",
+          "first_kick_exp_theta_2_survival": "0"
+        },
+        {
+          "lag_insertions": 8,
+          "uniform_survival": "0",
+          "first_kick_derivative_at_zero": "0",
+          "first_kick_exp_theta_2_survival": "0"
+        },
+        {
+          "lag_insertions": 9,
+          "uniform_survival": "0",
+          "first_kick_derivative_at_zero": "0",
+          "first_kick_exp_theta_2_survival": "0"
+        },
+        {
+          "lag_insertions": 10,
+          "uniform_survival": "0",
+          "first_kick_derivative_at_zero": "0",
+          "first_kick_exp_theta_2_survival": "0"
+        }
+      ],
+      "mean_remaining_steps_to_completion_derivative": "-391/3150"
+    }
+  ],
+  "equal_A_B_entry_ensemble_e_tilt_survival_derivative": "-1/180"
+}
diff --git a/analysis/completion-hazard-production-20260929/ESTIMANDS.md b/analysis/completion-hazard-production-20260929/ESTIMANDS.md
new file mode 100644
index 000000000..0f8803e6f
--- /dev/null
+++ b/analysis/completion-hazard-production-20260929/ESTIMANDS.md
@@ -0,0 +1,204 @@
+# Completion-hazard production: fixed estimands and interpretation
+
+This scorer uses the newly generated independent square block only. It does
+not pool older random data or count derived readouts as independent evidence.
+The initial design is L=512, N=262144, 14 equal batches of 10000 filtrations,
+with fixed insertion cutoffs a=154646, b=155385, c=156120. The scorer reads
+these parameters from the generating contract rather than embedding them in
+the estimator. The clock is uniform site-permutation **insertion count**,
+not an iid-label time or a fitted near-critical clock.
+
+## Inputs and invocation
+
+```sh
+python3 analysis/completion-hazard-production-20260929/analyze.py \
+  --data-dir analysis/completion-hazard-production-20260929/data \
+  --output-dir analysis/completion-hazard-production-20260929/results
+```
+
+`contract.json` must be in the parent of `--data-dir`. Named integer fields
+`a,b,c,L` are required, either at the top level or nested inside objects;
+repeated occurrences must agree. N is read if present, otherwise inferred
+as L squared. The square geometry and `0<=a<b<c<=N` are checked. Matching
+named fields in `data/run.json` are cross-checked, and absent metadata names
+are reported, not presented as verified matches.
+
+`data/run.json["batches"]` is a nonempty list of objects with `file`,
+`samples`, `seed`, `sha256`. File names are relative to data-dir (or absolute
+inside it). Distinct files/seeds/content hashes are required; duplicated
+blocks are not independent replicates. Each listed CSV or `.csv.gz` is read
+**once**; the same raw bytes are hashed, decompressed and parsed. Its
+expected number of samples and every required row field are checked.
+
+**The fixed production block must be complete before any scoring.** Require
+`run.status == "completed"`, `len(run.batches) == contract.batches`, and
+each `batch.samples == contract.samples_per_batch`. The latter two contract
+fields are required top-level integers. For the current design this means
+all 14 batches, each with 10000 rows; the CSV row counts are also checked
+against those declarations. An in-progress run or its subset of finished
+batches is rejected, not described as a complete analysis. All contracted
+batches are included; there is no data-dependent batch selection.
+
+Required CSV columns are
+
+    J1,J2,dx_b,dy_b,nu_b,tau,nu_tau.
+
+For rank one at b, D=(dx_b,dy_b) is a nonzero primitive unoriented direction.
+Only its sign is canonicalized: the first nonzero coordinate is positive.
+Nonprimitive directions are errors, not silently rounded or merged.
+`nu_b=-1` outside rank one. At tau the analogous rule holds for `nu_tau`.
+Require `b<=tau<c`, and `nu_tau=nu_b` if tau=b. These are semantic checks,
+not a new validation of the engine's topological count. Hashes and range
+checks cannot establish that tau is independent of the filtration or that
+the seeds generate independent blocks; those remain generation assumptions.
+
+## Fixed initial cohorts
+
+The two initial cohorts partition all rank-one risk at b:
+
+    E: J1<=a, J2>b;
+    L: a<J1<=b, J2>b.
+
+Here L used as a cohort label means **late**, not the linear size. These
+cohorts are fixed using the history by b. They are not redefined by rank
+at tau or by survival to c. Rows outside rank one at b are excluded from
+these cohort estimands, with their count reported.
+
+For each member of either cohort define
+
+    Z = 1{J2>c},
+    I = (c-b) * 1{J2>tau} * nu_tau/(N-tau).
+
+When J2<=tau, I is zero; the sentinel -1 is not multiplied into it.
+Otherwise the initial cohort condition ensures rank one at tau. I is
+**not** truncated to [0,1]; its individual value can exceed one.
+
+## Why the random-time compensator is unbiased
+
+Let G_k be the revealed permutation history. A rank-one set A_k completes
+at the next insertion with conditional probability nu_2(A_k)/(N-k).
+For either initial cohort g in G_b and every k>=b,
+
+    E[1_g 1{J2>k} nu_2(A_k)/(N-k)] = P(g,J2=k+1).
+
+Independence and uniformity of tau on `{b,...,c-1}` therefore give
+
+    E[I | g]
+      = sum_(k=b)^(c-1) E[1{J2>k}nu_2(A_k)/(N-k) | g]
+      = P(b<J2<=c | g),
+    E[Z+I | g] = 1.                                  (1)
+
+Conditioning on the entire sampled filtration instead of g gives
+`E_tau[I | filtration]` equal to its sum of conditional one-step hazards,
+not generally to its realized completion indicator. There is no pointwise
+identity `I=1-Z`. Completion changes the risk set along the path, which is
+why retaining `1{J2>tau}` is essential. Fresh entrants after b are not part
+of this compensator cohort.
+
+## Primary scalar outputs and signs
+
+The JSON uses these ordered primary scalars; all enter the joint aligned
+batch covariance:
+
+| Key | Definition |
+|---|---|
+| `delta_survival` | mean Z in E minus mean Z in L |
+| `delta_integrated_hazard` | minus mean I in E plus mean I in L |
+| `closure_residual` | delta_survival minus delta_integrated_hazard |
+| `delta_nu_instantaneous_scaled` | (c-b)/(N-b) times (mean nu_b in E minus mean nu_b in L) |
+| `delta_overlap_D` | overlap-weighted survival difference within exact D |
+| `delta_overlap_D_nu` | overlap-weighted survival difference within exact (D, integer nu_b) |
+| `closure_early` | mean Z in E plus mean I in E minus one |
+| `closure_late` | mean Z in L plus mean I in L minus one |
+
+Equation (1) makes the expected pooled survival and integrated-hazard
+differences equal. The groupwise residuals additionally expose common
+closure bias that could cancel in their difference. The per-cohort means
+of Z, I and nu_b are also reported as descriptive components.
+
+The instantaneous scaled nu difference has the **hazard sign**: a larger
+early nu_b predicts lower *next-step* early survival. Multiplication by
+c-b is only an explicitly labelled window scaling. It is not integration
+over the evolving completion intensity and does not assume geometry or
+hazard stays constant during the interval.
+
+Closing the random-time identity on the new block supports the implemented
+geometry-to-exit map and compensator. It does **not** demonstrate that nu_b
+is a sufficient recursive state: I observes geometry at a later random
+time, whereas a model based on the state at b must predict its evolution.
+
+## Two fixed common-support conditional comparisons
+
+For either fixed stratification S=D or S=(D,nu_b), let a cell contain early
+count e_s and late count l_s, with survival counts z_Es and z_Ls. Only
+cells with both e_s>0 and l_s>0 contribute. Define
+
+    w_s = e_s*l_s/(e_s+l_s),
+    Delta_S = sum_s w_s*(z_Es/e_s-z_Ls/l_s) / sum_s w_s.       (2)
+
+D is the exact primitive unoriented integer direction; nu_b is its exact
+integer value. There are no bins, regressions, fitted cutoffs, new features,
+or post-hoc subgroup searches. For each stratification report the number
+of shared and one-sided cells, supported counts, and excluded fractions
+in E, L and the combined initial risk set. Empty or entirely non-overlapping
+groups are `not_scoreable`, not zero. The overlap-weight sum is a weight
+sum, not an asserted effective sample size. Exact cell counts/survivors
+are retained in JSON; every direction has descriptive cohort means.
+
+These overlap-weighted targets differ from the pooled risk difference and
+from each other. Therefore their reduction relative to the pooled estimate
+is **not** a mediated fraction, explained fraction, or causal adjustment
+effect. No such ratio is reported.
+
+If the state (time b, rank one, D,nu_b) suffices to predict future survival,
+then early and late entry must have equal survival probability within
+every such population cell on common support. A nonzero population
+Delta_(D,nu) is a genuine violation of this necessary prediction. A zero
+weighted difference can hide opposite cell differences; it establishes
+neither cellwise equality nor a recursively Markov compressed process.
+Sampling variation must be distinguished from a population violation.
+The D-only statistic addresses the analogous weaker conditioning claim.
+
+## Aligned whole-batch delete-one uncertainty
+
+The point estimate pools all rows through per-batch sufficient statistics.
+For each of B batches, remove that entire batch simultaneously for all
+cohorts, all directions and all scalar outputs, then recompute (2), including
+which cells still share support and their weights. No direction is deleted
+independently and no original full-data stratum weights are held fixed.
+
+For the vector theta_(-j) and its delete-one mean theta_bar,
+
+    Cov_JK = (B-1)/B * sum_j
+               (theta_(-j)-theta_bar)(theta_(-j)-theta_bar)^T.
+
+The JSON contains the complete ordered covariance matrix, all B aligned
+replicate vectors and marginal SEs. The estimates shown are pooled plug-in
+estimates, not jackknife bias-corrected estimates. Exact identities such
+as `closure_residual=closure_early-closure_late` make a singular covariance
+expected; the scorer neither inverts it nor invents independent tests.
+
+This formula assumes the planned equal-size independent batches. The CLI
+rejects partial or unequal-size production blocks using the fixed contract
+above. Internally the covariance routine additionally returns `not_scoreable`
+for fewer than two batches or unequal batch sizes. If one scalar loses its necessary groups
+or common support in any delete-one replicate, its SE and covariance row/
+column are null with an explicit reason. Other fully scoreable scalars still
+use all the same B deletions; no pairwise subset covariance is substituted.
+
+## Files produced and boundaries
+
+The command writes `result.json` and `RESULT.md` under --output-dir. Structural
+input errors (hash mismatch, invalid rows, missing files, inconsistent
+parameters, duplicated batch identities) terminate the run rather than
+silently omitting data. A failed scientific comparison due to absent groups
+instead remains in the report as `not_scoreable`.
+
+This is one fixed finite-lag comparison from one new random block. Survival,
+compensator, directions and completion counts share its randomness. Old
+archives are not independently pooled. The scorer makes no statement about
+causal intervention, original-U operator identity, iid-label-clock closure,
+or a limiting non-Markov theorem. The finite model distinction here is
+whether the specified b-state leaves predictive entry-history dependence,
+not whether adding ever more descriptive coordinates yields a smaller
+retrospective residual.
diff --git a/analysis/completion-hazard-production-20260929/analyze.py b/analysis/completion-hazard-production-20260929/analyze.py
new file mode 100644
index 000000000..13424008b
--- /dev/null
+++ b/analysis/completion-hazard-production-20260929/analyze.py
@@ -0,0 +1,543 @@
+#!/usr/bin/env python3
+"""Score the independent completion-hazard block; Python standard library only.
+
+Read contract.json beside --data-dir, and batches from --data-dir/run.json.
+Each compressed batch is read once for SHA256 and CSV parsing. Jackknife
+replicates reuse batch sufficient statistics, with stratum weights recomputed.
+"""
+
+import argparse
+import csv
+from dataclasses import dataclass, field
+import gzip
+import hashlib
+import io
+import json
+import math
+from pathlib import Path
+import sys
+
+
+REQUIRED_COLUMNS = ("J1", "J2", "dx_b", "dy_b", "nu_b", "tau", "nu_tau")
+PRIMARY = (
+    "delta_survival",
+    "delta_integrated_hazard",
+    "closure_residual",
+    "delta_nu_instantaneous_scaled",
+    "delta_overlap_D",
+    "delta_overlap_D_nu",
+    "closure_early",
+    "closure_late",
+)
+LABELS = {
+    "delta_survival": "Pooled survival: early minus late",
+    "delta_integrated_hazard": "Integrated-hazard prediction: -I early + I late",
+    "closure_residual": "Survival difference minus integrated prediction",
+    "delta_nu_instantaneous_scaled": "Instantaneous nu difference, window-scaled (not integrated)",
+    "delta_overlap_D": "Exact-D overlap-weighted survival difference",
+    "delta_overlap_D_nu": "Exact-(D,nu_b) overlap-weighted survival difference",
+    "closure_early": "Early map residual: mean Z + mean I - 1",
+    "closure_late": "Late map residual: mean Z + mean I - 1",
+}
+
+
+def integer(value, label):
+    if isinstance(value, bool):
+        raise ValueError(f"{label}: Boolean is not an integer parameter")
+    if isinstance(value, int):
+        return value
+    if isinstance(value, str):
+        try:
+            return int(value)
+        except ValueError:
+            pass
+    raise ValueError(f"{label}: expected an integer, got {value!r}")
+
+
+def named_parameter(document, name, required=False):
+    """Allow top-level or nested named parameters, never conflicting values."""
+    matches = []
+
+    def visit(value, path):
+        if isinstance(value, dict):
+            for key, item in value.items():
+                location = f"{path}.{key}" if path else key
+                if key == name and not isinstance(item, (dict, list)):
+                    matches.append((location, integer(item, location)))
+                if isinstance(item, (dict, list)):
+                    visit(item, location)
+        elif isinstance(value, list):
+            for index, item in enumerate(value):
+                if isinstance(item, (dict, list)):
+                    visit(item, f"{path}[{index}]")
+
+    visit(document, "")
+    values = {value for _, value in matches}
+    if len(values) > 1:
+        raise ValueError(f"Conflicting {name} fields: {matches}")
+    if not values:
+        if required:
+            raise ValueError(f"contract.json must contain named integer {name}")
+        return None
+    return values.pop()
+
+
+@dataclass
+class Moments:
+    n: int = 0
+    z: int = 0
+    integrated: float = 0.0
+    nu: int = 0
+
+    def observe(self, z, integrated, nu):
+        self.n += 1
+        self.z += z
+        self.integrated += integrated
+        self.nu += nu
+
+    def merge(self, other):
+        self.n += other.n
+        self.z += other.z
+        self.integrated += other.integrated
+        self.nu += other.nu
+
+    def report(self):
+        return {
+            "count": self.n,
+            "survivors": self.z,
+            "sum_I": self.integrated,
+            "sum_nu_b": self.nu,
+            "mean_Z": self.z/self.n if self.n else None,
+            "mean_I": self.integrated/self.n if self.n else None,
+            "mean_nu_b": self.nu/self.n if self.n else None,
+        }
+
+
+def pair():
+    return [Moments(), Moments()]
+
+
+@dataclass
+class BatchStats:
+    samples: int = 0
+    sign_canonicalizations: int = 0
+    groups: list = field(default_factory=pair)
+    directions: dict = field(default_factory=dict)
+    direction_nu: dict = field(default_factory=dict)
+
+    def observe(self, group, direction, z, integrated, nu):
+        self.groups[group].observe(z, integrated, nu)
+        for cells, key in ((self.directions, direction),
+                           (self.direction_nu, (*direction, nu))):
+            if key not in cells:
+                cells[key] = pair()
+            cells[key][group].observe(z, integrated, nu)
+
+    def merge(self, other):
+        self.samples += other.samples
+        self.sign_canonicalizations += other.sign_canonicalizations
+        for mine, theirs in zip(self.groups, other.groups):
+            mine.merge(theirs)
+        for mine, theirs in ((self.directions, other.directions),
+                             (self.direction_nu, other.direction_nu)):
+            for key, values in theirs.items():
+                if key not in mine:
+                    mine[key] = pair()
+                for target, source in zip(mine[key], values):
+                    target.merge(source)
+
+
+def aggregate(blocks, omit=None):
+    result = BatchStats()
+    for index, block in enumerate(blocks):
+        if index != omit:
+            result.merge(block)
+    return result
+
+
+def read_batch(data_dir, entry, parameters):
+    if not isinstance(entry, dict):
+        raise ValueError("run.json batches must be objects")
+    for name in ("file", "samples", "seed", "sha256"):
+        if name not in entry:
+            raise ValueError(f"Batch entry is missing {name}")
+    path = (data_dir / entry["file"]).resolve()
+    if not path.is_relative_to(data_dir):
+        raise ValueError(f"Batch file is outside data-dir: {entry['file']}")
+    expected_samples = integer(entry["samples"], "batch.samples")
+    if expected_samples <= 0:
+        raise ValueError("Every listed batch must have positive samples")
+    seed = str(integer(entry["seed"], "batch.seed"))
+    expected_hash = str(entry["sha256"]).strip().lower()
+    if len(expected_hash) != 64 or any(ch not in "0123456789abcdef" for ch in expected_hash):
+        raise ValueError(f"Invalid batch SHA256: {entry['file']}")
+
+    # Exactly one filesystem read of the batch; both checks use these bytes.
+    raw = path.read_bytes()
+    digest = hashlib.sha256(raw).hexdigest()
+    if digest != expected_hash:
+        raise ValueError(f"SHA256 mismatch for {entry['file']}")
+    payload = gzip.decompress(raw) if path.suffix == ".gz" else raw
+    reader = csv.DictReader(io.StringIO(payload.decode("utf-8-sig"), newline=""))
+    if reader.fieldnames is None or len(set(reader.fieldnames)) != len(reader.fieldnames):
+        raise ValueError(f"Missing or duplicate CSV header: {entry['file']}")
+    missing = set(REQUIRED_COLUMNS)-set(reader.fieldnames)
+    if missing:
+        raise ValueError(f"Missing CSV columns {sorted(missing)} in {entry['file']}")
+
+    a, b, c, n = (parameters[key] for key in ("a", "b", "c", "N"))
+    stats = BatchStats()
+    for line_number, row in enumerate(reader, start=2):
+        try:
+            if None in row:
+                raise ValueError("Extra CSV fields")
+            j1, j2, dx, dy, nu, tau, nu_tau = (
+                integer(row[key], key) for key in REQUIRED_COLUMNS)
+            if not 1 <= j1 <= j2 <= n:
+                raise ValueError("Require 1 <= J1 <= J2 <= N")
+            if not b <= tau < c:
+                raise ValueError("tau is outside [b,c-1]")
+            at_b = j1 <= b < j2
+            at_tau = j1 <= tau < j2
+            if at_b:
+                if not 0 <= nu <= n-b:
+                    raise ValueError("nu_b is invalid on rank-one risk")
+                if math.gcd(abs(dx), abs(dy)) != 1:
+                    raise ValueError("Rank-one direction must be nonzero and primitive")
+            elif nu != -1:
+                raise ValueError("nu_b must be -1 outside rank-one risk")
+            if at_tau:
+                if not 0 <= nu_tau <= n-tau:
+                    raise ValueError("nu_tau is invalid on rank-one risk")
+            elif nu_tau != -1:
+                raise ValueError("nu_tau must be -1 outside rank-one risk")
+            if tau == b and nu_tau != nu:
+                raise ValueError("nu_tau must equal nu_b when tau=b")
+        except (ValueError, KeyError, TypeError) as error:
+            raise ValueError(f"{entry['file']} CSV line {line_number}: {error}") from error
+        stats.samples += 1
+        if not at_b:
+            continue
+        if dx < 0 or (dx == 0 and dy < 0):
+            dx, dy = -dx, -dy
+            stats.sign_canonicalizations += 1
+        group = 0 if j1 <= a else 1
+        integrated = (c-b)*nu_tau/(n-tau) if j2 > tau else 0.0
+        stats.observe(group, (dx, dy), int(j2 > c), integrated, nu)
+    if stats.samples != expected_samples:
+        raise ValueError(f"Sample count mismatch in {entry['file']}: "
+                         f"expected {expected_samples}, found {stats.samples}")
+    provenance = {
+        "file": str(path.relative_to(data_dir)), "samples": stats.samples,
+        "seed": seed, "sha256": digest, "sha256_verified": True,
+    }
+    return stats, provenance
+
+
+def overlap(cells, totals, detailed=False):
+    weights, early, late = [], [], []
+    supported = [0, 0]
+    common = early_only = late_only = 0
+    rows = []
+    for key, (e, l) in sorted(cells.items()):
+        weight = e.n*l.n/(e.n+l.n) if e.n and l.n else 0.0
+        if weight:
+            common += 1
+            supported[0] += e.n
+            supported[1] += l.n
+            weights.append(weight)
+            early.append(weight*e.z/e.n)
+            late.append(weight*l.z/l.n)
+        elif e.n:
+            early_only += 1
+        elif l.n:
+            late_only += 1
+        if detailed:
+            record = {
+                "direction": list(key[:2]), "early_count": e.n, "late_count": l.n,
+                "early_survivors": e.z, "late_survivors": l.z,
+                "overlap_weight": weight,
+            }
+            if len(key) == 3:
+                record["nu_b"] = key[2]
+            rows.append(record)
+    total_weight = math.fsum(weights)
+    early_mean = math.fsum(early)/total_weight if total_weight else None
+    late_mean = math.fsum(late)/total_weight if total_weight else None
+    risk = sum(group.n for group in totals)
+    return {
+        "status": "scoreable" if total_weight else "not_scoreable",
+        "reason": None if total_weight else "No cell has both early and late risk",
+        "difference": early_mean-late_mean if total_weight else None,
+        "weighted_survival_early": early_mean,
+        "weighted_survival_late": late_mean,
+        "overlap_weight_sum": total_weight,
+        "common_cells": common,
+        "early_only_cells": early_only,
+        "late_only_cells": late_only,
+        "supported_early_count": supported[0],
+        "supported_late_count": supported[1],
+        "drop_fraction_early": 1-supported[0]/totals[0].n if totals[0].n else None,
+        "drop_fraction_late": 1-supported[1]/totals[1].n if totals[1].n else None,
+        "drop_fraction_all_risk": 1-sum(supported)/risk if risk else None,
+        **({"cells": rows} if detailed else {}),
+    }
+
+
+def score(stats, parameters, detailed=False):
+    e, l = (group.report() for group in stats.groups)
+    by_d = overlap(stats.directions, stats.groups, detailed)
+    by_dn = overlap(stats.direction_nu, stats.groups, detailed)
+    values = {name: None for name in PRIMARY}
+    reasons = {name: None for name in PRIMARY}
+    if e["count"]:
+        values["closure_early"] = e["mean_Z"]+e["mean_I"]-1
+    else:
+        reasons["closure_early"] = "Early cohort is empty"
+    if l["count"]:
+        values["closure_late"] = l["mean_Z"]+l["mean_I"]-1
+    else:
+        reasons["closure_late"] = "Late cohort is empty"
+    if e["count"] and l["count"]:
+        values["delta_survival"] = e["mean_Z"]-l["mean_Z"]
+        values["delta_integrated_hazard"] = -e["mean_I"]+l["mean_I"]
+        values["closure_residual"] = values["delta_survival"]-values["delta_integrated_hazard"]
+        values["delta_nu_instantaneous_scaled"] = (
+            (parameters["c"]-parameters["b"])/(parameters["N"]-parameters["b"])
+            * (e["mean_nu_b"]-l["mean_nu_b"]))
+    else:
+        for name in PRIMARY[:4]:
+            reasons[name] = "Both initial cohorts must be nonempty"
+    for name, table in (("delta_overlap_D", by_d), ("delta_overlap_D_nu", by_dn)):
+        values[name] = table["difference"]
+        reasons[name] = table["reason"]
+    result = {
+        "samples": stats.samples,
+        "rank1_at_b_count": e["count"]+l["count"],
+        "excluded_non_rank1_at_b": stats.samples-e["count"]-l["count"],
+        "cohorts": {"early": e, "late": l},
+        "scalars": {
+            name: {"estimate": values[name],
+                   "status": "scoreable" if values[name] is not None else "not_scoreable",
+                   "reason": reasons[name]}
+            for name in PRIMARY
+        },
+        "conditional_D": by_d,
+        "conditional_D_nu": by_dn,
+    }
+    if detailed:
+        result["per_direction"] = [
+            {"direction": list(direction), "early": groups[0].report(),
+             "late": groups[1].report()}
+            for direction, groups in sorted(stats.directions.items())
+        ]
+    return result
+
+
+def jackknife(blocks, parameters, point):
+    b_count = len(blocks)
+    replicates = [score(aggregate(blocks, omit=i), parameters) for i in range(b_count)]
+    matrix = [[replicate["scalars"][name]["estimate"] for name in PRIMARY]
+              for replicate in replicates]
+    covariance = [[None for _ in PRIMARY] for _ in PRIMARY]
+    means, eligible, reasons = {}, {}, {}
+    global_problem = ("At least two batches are required" if b_count < 2 else
+                      "Unequal batch sizes: equal-size delete-one covariance is not reported"
+                      if len({batch.samples for batch in blocks}) != 1 else None)
+    for j, name in enumerate(PRIMARY):
+        missing = [i for i, row in enumerate(matrix) if row[j] is None]
+        problem = global_problem
+        if point["scalars"][name]["estimate"] is None:
+            problem = "Full-data point estimate is not scoreable"
+        elif missing:
+            problem = f"Undefined delete-one estimate after omitting batch indices {missing}"
+        eligible[name] = problem is None
+        reasons[name] = problem
+        means[name] = math.fsum(row[j] for row in matrix)/b_count if problem is None else None
+    for j, left in enumerate(PRIMARY):
+        for k, right in enumerate(PRIMARY):
+            if eligible[left] and eligible[right]:
+                covariance[j][k] = (b_count-1)/b_count * math.fsum(
+                    (row[j]-means[left])*(row[k]-means[right]) for row in matrix)
+    uncertainty = {}
+    for j, name in enumerate(PRIMARY):
+        uncertainty[name] = {
+            "status": "scoreable" if eligible[name] else "not_scoreable",
+            "reason": reasons[name],
+            "standard_error": math.sqrt(max(0.0, covariance[j][j])) if eligible[name] else None,
+        }
+    return {
+        "method": "aligned equal-size whole-batch delete-one; weights and support recomputed",
+        "batch_count": b_count,
+        "point_estimator": "pooled plug-in, not jackknife-bias-corrected",
+        "scalar_order": list(PRIMARY),
+        "leave_one_out_estimates": matrix,
+        "leave_one_out_mean": means,
+        "covariance": covariance,
+        "uncertainty": uncertainty,
+        "no_partial_batch_subsets_used_for_covariance": True,
+        "exact_linear_relations_can_make_covariance_singular": True,
+    }
+
+
+def fmt(value, digits=7):
+    return "not_scoreable" if value is None else f"{value:.{digits}g}"
+
+
+def markdown(result):
+    p, point, jk = result["parameters"], result["point"], result["jackknife"]
+    lines = [
+        "# Completion-hazard production: fixed-cohort readout", "",
+        f"Square L={p['L']}; N={p['N']}; fixed insertion cutoffs "
+        f"a={p['a']}, b={p['b']}, c={p['c']}.", "",
+        f"Scored {point['samples']:,} filtrations in {len(result['batches'])} batches. "
+        "This is one shared random block, not independent evidence for each column.", "",
+        "## Primary estimates", "",
+        "| Estimand | Estimate | Aligned batch SE | Status |",
+        "|---|---:|---:|---|",
+    ]
+    for name in PRIMARY:
+        value, uncertainty = point["scalars"][name], jk["uncertainty"][name]
+        status = value["status"] if value["estimate"] is None else uncertainty["status"]
+        reason = value["reason"] or uncertainty["reason"]
+        if reason:
+            status += ": " + reason
+        lines.append(f"| {LABELS[name]} | {fmt(value['estimate'])} | "
+                     f"{fmt(uncertainty['standard_error'])} | {status} |")
+    lines += ["", "The nu contrast has the hazard sign (early minus late) and is only "
+              "instantaneous. It is not an integrated survival prediction.", "",
+              "## Initial cohort means", "",
+              "| Cohort | Count | mean Z | mean I | mean nu_b |",
+              "|---|---:|---:|---:|---:|"]
+    for name, group in point["cohorts"].items():
+        lines.append(f"| {name} | {group['count']} | {fmt(group['mean_Z'])} | "
+                     f"{fmt(group['mean_I'])} | {fmt(group['mean_nu_b'])} |")
+    lines += ["", "## Exact common-support conditional estimands", "",
+              "| Stratification | Common cells | Supported E / L | Dropped E | Dropped L | Dropped total |",
+              "|---|---:|---:|---:|---:|---:|"]
+    for label, key in (("D", "conditional_D"), ("(D, integer nu_b)", "conditional_D_nu")):
+        table = point[key]
+        lines.append(f"| {label} | {table['common_cells']} | "
+                     f"{table['supported_early_count']} / {table['supported_late_count']} | "
+                     f"{fmt(table['drop_fraction_early'])} | {fmt(table['drop_fraction_late'])} | "
+                     f"{fmt(table['drop_fraction_all_risk'])} |")
+    lines += ["", "Weights are e*l/(e+l) in each exact shared cell. These targets differ "
+              "from the pooled difference and from each other. No mediated/explained "
+              "fraction is defined. Missing support is reported, not imputed or binned.", "",
+              "## Direction-resolved descriptive means", "",
+              "| Primitive unoriented D | E n | L n | E Z | L Z | E I | L I | E nu_b | L nu_b |",
+              "|---|---:|---:|---:|---:|---:|---:|---:|---:|"]
+    for row in point["per_direction"]:
+        e, l = row["early"], row["late"]
+        lines.append(f"| {tuple(row['direction'])} | {e['count']} | {l['count']} | "
+                     + " | ".join(fmt(group[field]) for field in ("mean_Z", "mean_I", "mean_nu_b")
+                                  for group in (e, l)) + " |")
+    lines += ["", "## Interpretation and uncertainty", "",
+              "- E[Z+I | initial cohort]=1 is an expectation identity, not a pathwise equality. "
+              "Closure assesses the implemented geometry-to-exit map; it does not establish nu_b closure.",
+              "- The (D,nu_b) finite-lag contrast is a distinct necessary prediction of a "
+              "state-sufficiency model. A zero weighted contrast can hide opposite cell effects "
+              "and does not prove recursive Markovness.",
+              "- All delete-one replicates remove the same whole batch for every metric. "
+              "All stratum weights and shared-cell support are recomputed. Full covariance and "
+              "replicate vectors are in result.json; no pairwise subset covariance is used.",
+              "- Per-direction rows are descriptive. No claim about causal mediation, the iid-label "
+              "clock, or a large-size non-Markov limit is made.", "",
+              "## Input provenance", "",
+              "Each listed batch was read once, checked against its SHA256 and expected sample count, "
+              "then retained only as sufficient statistics for resampling. Checksums do not prove "
+              "tau independence or independent RNG design; those are generator-contract assumptions.", "",
+              "| Batch | Samples | Seed | SHA256 |", "|---|---:|---|---|"]
+    for row in result["batches"]:
+        lines.append(f"| {row['file']} | {row['samples']} | {row['seed']} | {row['sha256']} |")
+    lines += ["", "No old archive was pooled into these estimates.", ""]
+    return "\n".join(lines)
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument("--data-dir", type=Path, required=True)
+    parser.add_argument("--output-dir", type=Path, required=True)
+    args = parser.parse_args()
+    data_dir = args.data_dir.resolve()
+    contract_path = data_dir.parent / "contract.json"
+    contract = json.loads(contract_path.read_text())
+    run = json.loads((data_dir / "run.json").read_text())
+    if run.get("status") != "completed":
+        raise ValueError("Only run.status='completed' may be scored; partial production blocks are not accepted")
+    expected_batches = integer(contract["batches"], "contract.batches")
+    expected_samples = integer(contract["samples_per_batch"], "contract.samples_per_batch")
+    parameters = {name: named_parameter(contract, name, required=True) for name in ("a", "b", "c", "L")}
+    parameters["N"] = named_parameter(contract, "N")
+    if parameters["N"] is None:
+        parameters["N"] = parameters["L"]**2
+    a, b, c, n, size = (parameters[name] for name in ("a", "b", "c", "N", "L"))
+    if size < 2 or n != size**2 or not 0 <= a < b < c <= n:
+        raise ValueError("Require square N=L^2 and 0 <= a < b < c <= N")
+    metadata_checks = {}
+    for name, expected in parameters.items():
+        observed = named_parameter(run, name)
+        if observed is not None and observed != expected:
+            raise ValueError(f"run.json {name}={observed} differs from contract {expected}")
+        metadata_checks[name] = "matches" if observed is not None else "not_recorded_by_name"
+    entries = run.get("batches")
+    if not isinstance(entries, list) or not entries:
+        raise ValueError("run.json must contain a nonempty batches list")
+    if len(entries) != expected_batches:
+        raise ValueError(f"Incomplete fixed block: contract requires {expected_batches} batches, "
+                         f"run lists {len(entries)}")
+    for index, entry in enumerate(entries):
+        if integer(entry["samples"], f"batch[{index}].samples") != expected_samples:
+            raise ValueError(f"Batch {index} does not have contract.samples_per_batch={expected_samples}; "
+                             "partial batches are not accepted")
+    blocks, provenance = [], []
+    seen_files, seen_seeds, seen_hashes = set(), set(), set()
+    for entry in entries:
+        block, info = read_batch(data_dir, entry, parameters)
+        for name, seen in (("file", seen_files), ("seed", seen_seeds), ("sha256", seen_hashes)):
+            if info[name] in seen:
+                raise ValueError(f"Repeated batch {name}; cannot count as an independent batch: {info[name]}")
+            seen.add(info[name])
+        blocks.append(block)
+        provenance.append(info)
+    total = aggregate(blocks)
+    point = score(total, parameters, detailed=True)
+    result = {
+        "schema": "matching-one.completion-hazard-production.v1",
+        "parameters": parameters,
+        "fixed_block": {
+            "run_status": run["status"], "required_batches": expected_batches,
+            "required_samples_per_batch": expected_samples,
+            "required_total_samples": expected_batches*expected_samples,
+        },
+        "contract_file": str(contract_path),
+        "run_metadata_parameter_checks": metadata_checks,
+        "clock": "finite uniform-permutation insertion count",
+        "data_policy": "only listed new-block files; no old-archive pooling",
+        "tau_design_assumption": "independent uniform integer in [b,c-1] for each full filtration",
+        "direction_convention": "primitive, first nonzero coordinate positive; sign normalized only",
+        "direction_sign_canonicalizations": total.sign_canonicalizations,
+        "batches": provenance,
+        "point": point,
+        "jackknife": jackknife(blocks, parameters, point),
+        "batch_cohort_sufficient_statistics": [
+            {"batch_index": i, "early": block.groups[0].report(), "late": block.groups[1].report()}
+            for i, block in enumerate(blocks)
+        ],
+    }
+    args.output_dir.mkdir(parents=True, exist_ok=True)
+    output_json = args.output_dir / "result.json"
+    output_md = args.output_dir / "RESULT.md"
+    output_json.write_text(json.dumps(result, ensure_ascii=False, indent=2, sort_keys=True,
+                                      allow_nan=False)+"\n")
+    output_md.write_text(markdown(result))
+    print(json.dumps({"result_json": str(output_json), "report": str(output_md),
+                      "batches": len(blocks), "samples": total.samples}))
+
+
+if __name__ == "__main__":
+    try:
+        main()
+    except (ValueError, OSError, KeyError, TypeError, csv.Error, EOFError) as error:
+        print(f"analysis failed: {error}", file=sys.stderr)
+        sys.exit(2)
diff --git a/analysis/completion-hazard-production-20260929/contract.json b/analysis/completion-hazard-production-20260929/contract.json
new file mode 100644
index 000000000..86c8447fd
--- /dev/null
+++ b/analysis/completion-hazard-production-20260929/contract.json
@@ -0,0 +1,34 @@
+{
+  "schema": "matching-one.completion-hazard-contract.v1",
+  "namespace": "completion-independent-20260929",
+  "frozen_before_new_data": true,
+  "parent_commit": "e772037d09678a2dda299ca99f24dc6f6a765af9",
+  "lattice": "square",
+  "L": 512,
+  "N": 262144,
+  "a": 154646,
+  "b": 155385,
+  "c": 156120,
+  "cutoff_source": "analysis/birth-markov-kernel-20260929/result.json; old cloud square L512 pooled .25/.50/.75 birth-mixture quantiles, held fixed in this new block",
+  "batches": 14,
+  "samples_per_batch": 10000,
+  "workers": 14,
+  "sampling": "uniform Fisher-Yates permutation; independent mt19937_64 stream for uniform integer tau in [b,c-1]",
+  "early": "J1<=a and J2>b",
+  "late": "a<J1<=b and J2>b",
+  "primary": [
+    "early-minus-late survival P(J2>c)",
+    "overlap-weighted early-minus-late survival within exact direction",
+    "overlap-weighted early-minus-late survival within exact (direction,integer nu_b)"
+  ],
+  "mechanistic_readouts": [
+    "immediate mean nu_b contrast scaled by (c-b)/(N-b), not a finite-lag explanation",
+    "I=(c-b)*1{J2>tau}*nu_tau/(N-tau); hazard-predicted survival contrast mean(I_late)-mean(I_early)",
+    "actual survival contrast minus hazard-predicted contrast with same-batch covariance"
+  ],
+  "strata_weights": "w=e*l/(e+l), only strata with both groups; recompute on every batch deletion; not a mediated fraction of the pooled contrast",
+  "prospective_model_contrast": "rank-only, rank+direction, and rank+direction+instantaneous completion-count predictive sufficiency each imply their corresponding fixed finite-lag contrast is zero; a zero contrast does not establish full Markovness",
+  "analysis": "aligned delete-one batch estimates and full covariance; fixed sample count; no p-value stopping, cutoff scan, free exponent, regression feature search or pooling with old archives",
+  "benchmark": "32 independent samples, excluded from scientific block; runtime alone decides resource feasibility",
+  "scope": "one existing large size, one lattice and insertion clock; not an original-U identification or continuum/universality certificate"
+}
diff --git a/analysis/completion-hazard-production-20260929/data/benchmark-square-L512-b00.csv.gz b/analysis/completion-hazard-production-20260929/data/benchmark-square-L512-b00.csv.gz
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diff --git a/analysis/completion-hazard-production-20260929/data/production-square-L512-b13.csv.gz b/analysis/completion-hazard-production-20260929/data/production-square-L512-b13.csv.gz
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diff --git a/analysis/completion-hazard-production-20260929/data/run.json b/analysis/completion-hazard-production-20260929/data/run.json
new file mode 100644
index 000000000..505b75208
--- /dev/null
+++ b/analysis/completion-hazard-production-20260929/data/run.json
@@ -0,0 +1,263 @@
+{
+  "schema": "matching-one.completion-hazard-run.v1",
+  "status": "completed",
+  "pid": 410,
+  "utc_started": "2026-09-29T14:43:58.518550+00:00",
+  "source_commit": "9520c9bdf702a62435f1054e64184b1df4fc0ab3",
+  "source_sha256": {
+    "engine.cpp": "42ac8a013cb731f4da4e3121ca429fd475555063e876638ad5d375d195448115",
+    "run.py": "9a9a06dec6d698e2827c0ac04034479d197c408e679b48deac9ee088d3c6ad70",
+    "contract.json": "6cde2b4b3adff1500ac02cf4b6b8e679259fd36a1e5676855fc314e486596964"
+  },
+  "contract": {
+    "schema": "matching-one.completion-hazard-contract.v1",
+    "namespace": "completion-independent-20260929",
+    "frozen_before_new_data": true,
+    "parent_commit": "e772037d09678a2dda299ca99f24dc6f6a765af9",
+    "lattice": "square",
+    "L": 512,
+    "N": 262144,
+    "a": 154646,
+    "b": 155385,
+    "c": 156120,
+    "cutoff_source": "analysis/birth-markov-kernel-20260929/result.json; old cloud square L512 pooled .25/.50/.75 birth-mixture quantiles, held fixed in this new block",
+    "batches": 14,
+    "samples_per_batch": 10000,
+    "workers": 14,
+    "sampling": "uniform Fisher-Yates permutation; independent mt19937_64 stream for uniform integer tau in [b,c-1]",
+    "early": "J1<=a and J2>b",
+    "late": "a<J1<=b and J2>b",
+    "primary": [
+      "early-minus-late survival P(J2>c)",
+      "overlap-weighted early-minus-late survival within exact direction",
+      "overlap-weighted early-minus-late survival within exact (direction,integer nu_b)"
+    ],
+    "mechanistic_readouts": [
+      "immediate mean nu_b contrast scaled by (c-b)/(N-b), not a finite-lag explanation",
+      "I=(c-b)*1{J2>tau}*nu_tau/(N-tau); hazard-predicted survival contrast mean(I_late)-mean(I_early)",
+      "actual survival contrast minus hazard-predicted contrast with same-batch covariance"
+    ],
+    "strata_weights": "w=e*l/(e+l), only strata with both groups; recompute on every batch deletion; not a mediated fraction of the pooled contrast",
+    "prospective_model_contrast": "rank-only, rank+direction, and rank+direction+instantaneous completion-count predictive sufficiency each imply their corresponding fixed finite-lag contrast is zero; a zero contrast does not establish full Markovness",
+    "analysis": "aligned delete-one batch estimates and full covariance; fixed sample count; no p-value stopping, cutoff scan, free exponent, regression feature search or pooling with old archives",
+    "benchmark": "32 independent samples, excluded from scientific block; runtime alone decides resource feasibility",
+    "scope": "one existing large size, one lattice and insertion clock; not an original-U identification or continuum/universality certificate"
+  },
+  "command": "python3 /workspace/Matching-One-TV2N0X/completion-hazard-20260929/run.py --source-commit 9520c9bdf702a62435f1054e64184b1df4fc0ab3",
+  "platform": "Linux-5.10.0-182.0.0.95.r3582_286.hce2.aarch64-aarch64-with-glibc2.34",
+  "machine": "aarch64",
+  "python": "3.9.9",
+  "compiler": "g++ (GCC) 10.3.1\nCopyright (C) 2020 Free Software Foundation, Inc.\nThis is free software; see the source for copying conditions.  There is NO\nwarranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.",
+  "workers": 14,
+  "budget_seconds": 900,
+  "seed_rule": "first 64 SHA256 bits of matching-one/completion-independent-20260929/{role}/square/L512/{batch}/{stream}; stream=permutation or audit-time",
+  "batches": [
+    {
+      "file": "production-square-L512-b00.csv.gz",
+      "batch": 0,
+      "samples": 10000,
+      "seed": "18419746559907930925",
+      "time_seed": "12723381226631050454",
+      "role": "production",
+      "runtime_seconds": 308.89561684895307,
+      "engine_log": "samples=10000 seconds=308.887",
+      "command": "/workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/engine sample square 512 10000 18419746559907930925 12723381226631050454 155385 156120 /workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/production-square-L512-b00.csv",
+      "sha256": "32f16cc37728d67fdbe3446a48da1bb4d36c48fc4f051648ac8e36e2595d4026",
+      "dependency_group": "completion-independent-20260929/square/L512/batch0"
+    },
+    {
+      "file": "production-square-L512-b01.csv.gz",
+      "batch": 1,
+      "samples": 10000,
+      "seed": "13442365983088886094",
+      "time_seed": "8591617387320025580",
+      "role": "production",
+      "runtime_seconds": 312.38477401598357,
+      "engine_log": "samples=10000 seconds=312.374",
+      "command": "/workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/engine sample square 512 10000 13442365983088886094 8591617387320025580 155385 156120 /workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/production-square-L512-b01.csv",
+      "sha256": "fdc708228c1e0535787aa3426ea725b276ecef0b8ba193215b1bf93512518892",
+      "dependency_group": "completion-independent-20260929/square/L512/batch1"
+    },
+    {
+      "file": "production-square-L512-b02.csv.gz",
+      "batch": 2,
+      "samples": 10000,
+      "seed": "12192384056273429425",
+      "time_seed": "13433160703667021117",
+      "role": "production",
+      "runtime_seconds": 316.13183840503916,
+      "engine_log": "samples=10000 seconds=316.117",
+      "command": "/workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/engine sample square 512 10000 12192384056273429425 13433160703667021117 155385 156120 /workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/production-square-L512-b02.csv",
+      "sha256": "57b5f9ef9bcc6f21a5a14f5e7b6c24fccb94388f349943738916cc1700ab4897",
+      "dependency_group": "completion-independent-20260929/square/L512/batch2"
+    },
+    {
+      "file": "production-square-L512-b03.csv.gz",
+      "batch": 3,
+      "samples": 10000,
+      "seed": "16795263932832913938",
+      "time_seed": "786699782798800568",
+      "role": "production",
+      "runtime_seconds": 316.3825669710059,
+      "engine_log": "samples=10000 seconds=316.348",
+      "command": "/workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/engine sample square 512 10000 16795263932832913938 786699782798800568 155385 156120 /workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/production-square-L512-b03.csv",
+      "sha256": "65ec5cee09f76a03c69b702a21bb6f27dd70930b48e3b7f047de33103032eaa4",
+      "dependency_group": "completion-independent-20260929/square/L512/batch3"
+    },
+    {
+      "file": "production-square-L512-b04.csv.gz",
+      "batch": 4,
+      "samples": 10000,
+      "seed": "5606540329541986450",
+      "time_seed": "4353823182260169812",
+      "role": "production",
+      "runtime_seconds": 321.3780810958706,
+      "engine_log": "samples=10000 seconds=321.35",
+      "command": "/workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/engine sample square 512 10000 5606540329541986450 4353823182260169812 155385 156120 /workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/production-square-L512-b04.csv",
+      "sha256": "033189fab591caa248c66c9710d63104036dc3d26b88e495d965615a84ecc4fb",
+      "dependency_group": "completion-independent-20260929/square/L512/batch4"
+    },
+    {
+      "file": "production-square-L512-b05.csv.gz",
+      "batch": 5,
+      "samples": 10000,
+      "seed": "176734671850227651",
+      "time_seed": "12631360215171416686",
+      "role": "production",
+      "runtime_seconds": 322.3972083879635,
+      "engine_log": "samples=10000 seconds=322.373",
+      "command": "/workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/engine sample square 512 10000 176734671850227651 12631360215171416686 155385 156120 /workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/production-square-L512-b05.csv",
+      "sha256": "b38b95839aab1c6f6575a9b94702c41a032a2cc00430c6eb904840472488fbd4",
+      "dependency_group": "completion-independent-20260929/square/L512/batch5"
+    },
+    {
+      "file": "production-square-L512-b06.csv.gz",
+      "batch": 6,
+      "samples": 10000,
+      "seed": "14922363405798405830",
+      "time_seed": "6274897957855625106",
+      "role": "production",
+      "runtime_seconds": 316.1888132409658,
+      "engine_log": "samples=10000 seconds=316.161",
+      "command": "/workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/engine sample square 512 10000 14922363405798405830 6274897957855625106 155385 156120 /workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/production-square-L512-b06.csv",
+      "sha256": "330829d82ec4dad029f53dc07fe8bda5252bb35118aa9b507012c73df88e5a35",
+      "dependency_group": "completion-independent-20260929/square/L512/batch6"
+    },
+    {
+      "file": "production-square-L512-b07.csv.gz",
+      "batch": 7,
+      "samples": 10000,
+      "seed": "3540279280158270918",
+      "time_seed": "5624797256931927200",
+      "role": "production",
+      "runtime_seconds": 317.39242523605935,
+      "engine_log": "samples=10000 seconds=317.357",
+      "command": "/workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/engine sample square 512 10000 3540279280158270918 5624797256931927200 155385 156120 /workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/production-square-L512-b07.csv",
+      "sha256": "a292a7a8be0fb5f79f2449745709c51942fc4261fd50e542569d33d04005ee30",
+      "dependency_group": "completion-independent-20260929/square/L512/batch7"
+    },
+    {
+      "file": "production-square-L512-b08.csv.gz",
+      "batch": 8,
+      "samples": 10000,
+      "seed": "182843979474712324",
+      "time_seed": "14595628627417457844",
+      "role": "production",
+      "runtime_seconds": 321.9213431030512,
+      "engine_log": "samples=10000 seconds=321.885",
+      "command": "/workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/engine sample square 512 10000 182843979474712324 14595628627417457844 155385 156120 /workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/production-square-L512-b08.csv",
+      "sha256": "113325911eb95469467bfe73496d5b57448413125dcea383b55b0ff7875db4ac",
+      "dependency_group": "completion-independent-20260929/square/L512/batch8"
+    },
+    {
+      "file": "production-square-L512-b09.csv.gz",
+      "batch": 9,
+      "samples": 10000,
+      "seed": "4164285986391024372",
+      "time_seed": "15552832912891450204",
+      "role": "production",
+      "runtime_seconds": 320.87680300488137,
+      "engine_log": "samples=10000 seconds=320.843",
+      "command": "/workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/engine sample square 512 10000 4164285986391024372 15552832912891450204 155385 156120 /workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/production-square-L512-b09.csv",
+      "sha256": "b5712154670f8179ddf32487f35a794ea16b5ef4f0cdad2025c3f8e447f321da",
+      "dependency_group": "completion-independent-20260929/square/L512/batch9"
+    },
+    {
+      "file": "production-square-L512-b10.csv.gz",
+      "batch": 10,
+      "samples": 10000,
+      "seed": "15500924042285991404",
+      "time_seed": "1741135588251435614",
+      "role": "production",
+      "runtime_seconds": 317.8622832458932,
+      "engine_log": "samples=10000 seconds=317.826",
+      "command": "/workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/engine sample square 512 10000 15500924042285991404 1741135588251435614 155385 156120 /workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/production-square-L512-b10.csv",
+      "sha256": "98bb914abc5188216cd5b9b4584ba46539bd4c16c1b4d745b6a8033748243668",
+      "dependency_group": "completion-independent-20260929/square/L512/batch10"
+    },
+    {
+      "file": "production-square-L512-b11.csv.gz",
+      "batch": 11,
+      "samples": 10000,
+      "seed": "17186215622830953477",
+      "time_seed": "3349485547084033433",
+      "role": "production",
+      "runtime_seconds": 316.24242771603167,
+      "engine_log": "samples=10000 seconds=316.207",
+      "command": "/workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/engine sample square 512 10000 17186215622830953477 3349485547084033433 155385 156120 /workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/production-square-L512-b11.csv",
+      "sha256": "73aaa04f05ee279532f7c781a97acda5191840cf79fc2cee72f2a6c9caaed7f1",
+      "dependency_group": "completion-independent-20260929/square/L512/batch11"
+    },
+    {
+      "file": "production-square-L512-b12.csv.gz",
+      "batch": 12,
+      "samples": 10000,
+      "seed": "11279241988091952515",
+      "time_seed": "16543947972952860182",
+      "role": "production",
+      "runtime_seconds": 323.19172599795274,
+      "engine_log": "samples=10000 seconds=323.156",
+      "command": "/workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/engine sample square 512 10000 11279241988091952515 16543947972952860182 155385 156120 /workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/production-square-L512-b12.csv",
+      "sha256": "75f556befdd4986e26f6594f7d2ec44fa6517b9b19021b9cbaa612116471875c",
+      "dependency_group": "completion-independent-20260929/square/L512/batch12"
+    },
+    {
+      "file": "production-square-L512-b13.csv.gz",
+      "batch": 13,
+      "samples": 10000,
+      "seed": "10329155733035115099",
+      "time_seed": "12897823005956447827",
+      "role": "production",
+      "runtime_seconds": 316.4162373950239,
+      "engine_log": "samples=10000 seconds=316.381",
+      "command": "/workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/engine sample square 512 10000 10329155733035115099 12897823005956447827 155385 156120 /workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/production-square-L512-b13.csv",
+      "sha256": "d5ffad750a152bd62325ae22375496b861da96f8687ded8ad7c0a8220d06a202",
+      "dependency_group": "completion-independent-20260929/square/L512/batch13"
+    }
+  ],
+  "benchmarks": [
+    {
+      "file": "benchmark-square-L512-b00.csv.gz",
+      "batch": 0,
+      "samples": 32,
+      "seed": "14042891078363688929",
+      "time_seed": "15354029970648824631",
+      "role": "benchmark",
+      "runtime_seconds": 0.6683522029779851,
+      "engine_log": "samples=32 seconds=0.661251",
+      "command": "/workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/engine sample square 512 32 14042891078363688929 15354029970648824631 155385 156120 /workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/benchmark-square-L512-b00.csv",
+      "sha256": "b073852b02522a07fc8b94588578ab96b7b57d12ee9267b1c5f4355e4056a3c6",
+      "dependency_group": "excluded-benchmark"
+    }
+  ],
+  "cgroup": {
+    "cpu/cpu.cfs_quota_us": "1450000",
+    "cpu/cpu.cfs_period_us": "100000",
+    "memory/memory.limit_in_bytes": "26843545600"
+  },
+  "build_command": "g++ -O3 -std=c++17 /workspace/Matching-One-TV2N0X/completion-hazard-20260929/engine.cpp -o /workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/engine",
+  "binary_sha256": "1d6befbb44bd72afcea4691af7c58cb0c0a458d1e933e1bc28f289482c7d1d5a",
+  "estimated_production_wall_seconds": 208.86006343062036,
+  "wall_seconds": 325.5115620449651,
+  "utc_finished": "2026-09-29T14:49:24.030113+00:00"
+}
diff --git a/analysis/completion-hazard-production-20260929/engine.cpp b/analysis/completion-hazard-production-20260929/engine.cpp
new file mode 100644
index 000000000..6870c5fe8
--- /dev/null
+++ b/analysis/completion-hazard-production-20260929/engine.cpp
@@ -0,0 +1,181 @@
+// Completion geometry on independent uniform-permutation filtrations.
+// Lifted union-find convention retained from ../birth-gap-20260929/engine.cpp.
+// The old engine and archives are unchanged. C++17, standard library only.
+#include <algorithm>
+#include <array>
+#include <chrono>
+#include <cstdint>
+#include <fstream>
+#include <iostream>
+#include <numeric>
+#include <random>
+#include <stdexcept>
+#include <string>
+#include <vector>
+
+struct Neighbor { int vertex, dx, dy; };
+struct Observation { int j1=0, j2=0, dx_b=0, dy_b=0, nu_b=-1, tau=0, nu_tau=-1; };
+
+class Geometry {
+    int L, N, degree;
+    std::vector<std::array<Neighbor,6>> neighbor;
+    std::vector<int> parent, size;
+    std::vector<std::int64_t> dx, dy;
+    std::vector<unsigned char> active;
+    int rank=0;
+    std::int64_t bx=0, by=0;
+    struct Root { int vertex; std::int64_t x,y; };
+    Root find(int v) {
+        if (parent[v]==v) return {v,0,0};
+        Root r=find(parent[v]);
+        dx[v]+=r.x; dy[v]+=r.y; parent[v]=r.vertex;
+        return {r.vertex,dx[v],dy[v]};
+    }
+    void edge(int v, Neighbor e) {
+        Root a=find(v), b=find(e.vertex);
+        std::int64_t x=a.x+e.dx-b.x, y=a.y+e.dy-b.y;
+        if (a.vertex==b.vertex) {
+            if (x%L || y%L) throw std::logic_error("nonperiodic cycle");
+            x/=L; y/=L;
+            if (!(x||y) || rank==2) return;
+            if (!rank) {bx=x;by=y;rank=1;}
+            else if (bx*y!=by*x) rank=2;
+        } else {
+            if (size[a.vertex]<size[b.vertex]) {std::swap(a,b);x=-x;y=-y;}
+            parent[b.vertex]=a.vertex;dx[b.vertex]=x;dy[b.vertex]=y;
+            size[a.vertex]+=size[b.vertex];
+        }
+    }
+public:
+    Geometry(int length,bool triangular):L(length),N(L*L),degree(triangular?6:4),
+        neighbor(N),parent(N),size(N),dx(N),dy(N),active(N) {
+        if (L<3 || L>1024) throw std::invalid_argument("L outside [3,1024]");
+        const int sx[]={1,-1,0,0,1,-1},sy[]={0,0,1,-1,1,-1};
+        for(int y=0;y<L;++y)for(int x=0;x<L;++x)for(int j=0;j<degree;++j)
+            neighbor[x+L*y][j]={((x+sx[j]+L)%L)+L*((y+sy[j]+L)%L),sx[j],sy[j]};
+    }
+    void reset(){std::fill(active.begin(),active.end(),0);rank=0;bx=by=0;}
+    int insert(int v) {
+        if(active[v])throw std::logic_error("double insertion");
+        active[v]=1;parent[v]=v;size[v]=1;dx[v]=dy[v]=0;
+        for(int j=0;j<degree;++j)if(active[neighbor[v][j].vertex])edge(v,neighbor[v][j]);
+        return rank;
+    }
+    std::pair<int,int> direction()const{
+        if(rank!=1)return {0,0};
+        auto g=std::gcd(bx,by);auto x=bx/g,y=by/g;
+        if(x<0 || (x==0 && y<0)){x=-x;y=-y;}
+        return {static_cast<int>(x),static_cast<int>(y)};
+    }
+    int completion_count() {
+        if(rank!=1)return -1;
+        int result=0;
+        // A new cycle through v arises precisely from two neighbours in
+        // the same old component. Cycles between different old components
+        // cannot arise from a single star. Existing windings are all along b.
+        for(int v=0;v<N;++v)if(!active[v]){
+            std::array<Root,6> roots; int used=0; bool completes=false;
+            for(int j=0;j<degree && !completes;++j){
+                Neighbor e=neighbor[v][j];if(!active[e.vertex])continue;
+                Root r=find(e.vertex);r.x=e.dx-r.x;r.y=e.dy-r.y;
+                for(int h=0;h<used;++h)if(roots[h].vertex==r.vertex){
+                    auto x=r.x-roots[h].x,y=r.y-roots[h].y;
+                    if(x%L || y%L)throw std::logic_error("hypothetical nonperiodic cycle");
+                    if(bx*y!=by*x){completes=true;break;}
+                }
+                roots[used++]=r;
+            }
+            result+=completes;
+        }
+        return result;
+    }
+    // Query only: no insertion. The lifted-potential criterion is the same
+    // one used by completion_count; old sampling/observations are unchanged.
+    bool completes_if_inserted(int v) {
+        if(rank!=1 || active[v])throw std::logic_error("rank-one vacancy required");
+        std::array<Root,6> roots;int used=0;
+        for(int j=0;j<degree;++j){
+            Neighbor e=neighbor[v][j];if(!active[e.vertex])continue;
+            Root r=find(e.vertex);r.x=e.dx-r.x;r.y=e.dy-r.y;
+            for(int h=0;h<used;++h)if(roots[h].vertex==r.vertex){
+                auto x=r.x-roots[h].x,y=r.y-roots[h].y;
+                if(x%L || y%L)throw std::logic_error("hypothetical nonperiodic cycle");
+                if(bx*y!=by*x)return true;
+            }
+            roots[used++]=r;
+        }
+        return false;
+    }
+    Observation observe(const std::vector<int>&order,int b,int tau){
+        reset();Observation o;o.tau=tau;
+        for(int k=1;k<=N;++k){
+            insert(order[k-1]);
+            if(rank>=1 && !o.j1)o.j1=k;
+            if(rank==2){o.j2=k;return o;}
+            if(k==b){
+                auto d=direction();o.dx_b=d.first;o.dy_b=d.second;
+                o.nu_b=completion_count();
+            }
+            if(k==tau)o.nu_tau=(tau==b?o.nu_b:completion_count());
+        }
+        throw std::logic_error("full torus not rank two");
+    }
+    int control(std::ostream&out){
+        if(L!=3)throw std::logic_error("control L3 only");
+        out<<"mask,rank,dx,dy,nu2\n";int controls=0;
+        for(int mask=0;mask<(1<<N);++mask){
+            reset();for(int v=0;v<N;++v)if(mask>>v&1)insert(v);
+            auto d=direction();int nu=completion_count();
+            if(rank==1){
+                int brute=0;
+                for(int v=0;v<N;++v)if(!(mask>>v&1)){
+                    Geometry copy=*this;brute+=(copy.insert(v)==2);++controls;
+                }
+                if(nu!=brute)throw std::logic_error("completion probe mismatch");
+            }
+            out<<mask<<','<<rank<<','<<d.first<<','<<d.second<<','<<nu<<'\n';
+        }
+        return controls;
+    }
+};
+
+std::uint64_t bounded(std::mt19937_64&rng,std::uint64_t bound){
+    std::uint64_t threshold=(-bound)%bound;
+    while(true){auto r=rng();if(r>=threshold)return r%bound;}
+}
+
+int main(int argc,char**argv){
+    try{
+        if(argc<3)throw std::invalid_argument("mode lattice ...");
+        std::string mode=argv[1],lattice=argv[2];
+        if(lattice!="square" && lattice!="triangular")throw std::invalid_argument("lattice");
+        if(mode=="control"){
+            if(argc!=4)throw std::invalid_argument("control square|triangular output.csv");
+            Geometry g(3,lattice=="triangular");std::ofstream out(argv[3]);
+            if(!out)throw std::runtime_error("cannot open control output");
+            int checks=g.control(out);out.close();
+            if(!out)throw std::runtime_error("control write failed");
+            std::cout<<"512 subsets; "<<checks<<" single-site completion probes agree\n";return 0;
+        }
+        if(mode!="sample" || argc!=10)
+            throw std::invalid_argument("sample square|triangular L samples permutation_seed time_seed b c output.csv");
+        int L=std::stoi(argv[3]),N=L*L,b=std::stoi(argv[7]),c=std::stoi(argv[8]);
+        auto samples=std::stoull(argv[4]),seed=std::stoull(argv[5]),time_seed=std::stoull(argv[6]);
+        if(!samples || b<1 || b>=c || c>N)throw std::invalid_argument("sample/time bounds");
+        Geometry g(L,lattice=="triangular");std::vector<int>order(N);
+        std::mt19937_64 rng(seed),time_rng(time_seed);
+        std::ofstream out(argv[9]);if(!out)throw std::runtime_error("cannot open output");
+        out<<"J1,J2,dx_b,dy_b,nu_b,tau,nu_tau\n";
+        auto start=std::chrono::steady_clock::now();
+        for(std::uint64_t i=0;i<samples;++i){
+            std::iota(order.begin(),order.end(),0);
+            for(int v=N-1;v>0;--v)std::swap(order[v],order[bounded(rng,v+1)]);
+            int tau=b+bounded(time_rng,c-b);
+            auto o=g.observe(order,b,tau);
+            out<<o.j1<<','<<o.j2<<','<<o.dx_b<<','<<o.dy_b<<','<<o.nu_b<<','<<o.tau<<','<<o.nu_tau<<'\n';
+        }
+        out.close();if(!out)throw std::runtime_error("output write failed");
+        double sec=std::chrono::duration<double>(std::chrono::steady_clock::now()-start).count();
+        std::cout<<"samples="<<samples<<" seconds="<<sec<<'\n';return 0;
+    }catch(const std::exception&e){std::cerr<<e.what()<<'\n';return 1;}
+}
diff --git a/analysis/completion-hazard-production-20260929/execution-receipt.json b/analysis/completion-hazard-production-20260929/execution-receipt.json
new file mode 100644
index 000000000..d21d26d65
--- /dev/null
+++ b/analysis/completion-hazard-production-20260929/execution-receipt.json
@@ -0,0 +1,31 @@
+{
+  "schema": "matching-one.completion-hazard-execution.v1",
+  "date": "2026-09-29",
+  "instance": "DevEnvC_TV2N0X",
+  "instance_id": "4a8d1d443419434889e49148ed0a7ba6",
+  "remote_directory": "/workspace/Matching-One-TV2N0X/completion-hazard-20260929",
+  "source_commit_before_acquisition": "9520c9bdf702a62435f1054e64184b1df4fc0ab3",
+  "runner_pid": 410,
+  "samples": 140000,
+  "excluded_benchmark_samples": 32,
+  "wall_seconds": 325.5115620449651,
+  "workers": 14,
+  "observed_cpu_quota_cores": 14.5,
+  "observed_memory_limit_GiB": 25,
+  "target_ssh_key_rotations_after_start": 1,
+  "key_mode": "600",
+  "other_environment_operations": "none; no jobs inspected on other environments",
+  "packages_installed_on_target": "gcc-c++ 10.3.1 and dependencies; compiler was absent after container restart",
+  "remote_python": "3.9.9",
+  "analysis_python": "3.11.15 existing local research environment; standard library only",
+  "scoring_runs": 1,
+  "run_manifest_sha256": "7c63af9d9c7dcf5eff4b0f14644fe83560d73190532d8528f3e0cf0c5f0af2df",
+  "analysis_source_sha256": "83222cbd9e73d6fe7c452f32a12e67da5ae331d27aa47707460bd4a323181365",
+  "result_json_sha256": "35cfa9ae64e97b4ef81e4fb339e4c37fa1f6615c8968510e32711c79c3f88abf",
+  "batch_hashes_verified_during_scoring": true,
+  "results_downloaded_before_shutdown": true,
+  "postrun_process_check": "no remaining research process on this target",
+  "tunnel": "own managed tunnel stopped",
+  "terminal_cloud_state": "Ready, confirmed by live selected-instance status after power-off",
+  "remote_data_deleted": false
+}
diff --git a/analysis/completion-hazard-production-20260929/preproduction-control.json b/analysis/completion-hazard-production-20260929/preproduction-control.json
new file mode 100644
index 000000000..8fa409e02
--- /dev/null
+++ b/analysis/completion-hazard-production-20260929/preproduction-control.json
@@ -0,0 +1,12 @@
+{
+  "schema": "matching-one.completion-geometry-control.v1",
+  "date": "2026-09-29",
+  "method": "One new C++ completion-site scan control against explicit copied-state single-site insertion; all 512 L3 subsets per lattice. Independent stored lifted-DFS direction/nu2 census also compared.",
+  "source_reference": "analysis/birth-completion-geometry-20260929/result.json at e772037d09678a2dda299ca99f24dc6f6a765af9",
+  "square": {"subsets": 512, "rank_one_states": 162, "single_site_probes": 648, "mismatches": 0},
+  "triangular": {"subsets": 512, "rank_one_states": 180, "single_site_probes": 810, "mismatches": 0},
+  "compile": "clang++ -O3 -std=c++17 analysis/completion-hazard-production-20260929/engine.cpp -o engine",
+  "reproduce": ["engine control square control-square.csv", "engine control triangular control-triangular.csv"],
+  "no_permutation_reenumeration": true,
+  "scope": "Hypothetical cycle and direction convention, not a full statistical calibration or continuum check"
+}
diff --git a/analysis/completion-hazard-production-20260929/results/RESULT.md b/analysis/completion-hazard-production-20260929/results/RESULT.md
new file mode 100644
index 000000000..9b735587b
--- /dev/null
+++ b/analysis/completion-hazard-production-20260929/results/RESULT.md
@@ -0,0 +1,79 @@
+# Completion-hazard production: fixed-cohort readout
+
+Square L=512; N=262144; fixed insertion cutoffs a=154646, b=155385, c=156120.
+
+Scored 140,000 filtrations in 14 batches. This is one shared random block, not independent evidence for each column.
+
+## Primary estimates
+
+| Estimand | Estimate | Aligned batch SE | Status |
+|---|---:|---:|---|
+| Pooled survival: early minus late | 0.006857066 | 0.004444348 | scoreable |
+| Integrated-hazard prediction: -I early + I late | 0.01550219 | 0.003505008 | scoreable |
+| Survival difference minus integrated prediction | -0.008645121 | 0.006389162 | scoreable |
+| Instantaneous nu difference, window-scaled (not integrated) | -0.01504115 | 0.003301171 | scoreable |
+| Exact-D overlap-weighted survival difference | -0.004149993 | 0.004511942 | scoreable |
+| Exact-(D,nu_b) overlap-weighted survival difference | 0.001216127 | 0.004763563 | scoreable |
+| Early map residual: mean Z + mean I - 1 | -0.009410927 | 0.006314693 | scoreable |
+| Late map residual: mean Z + mean I - 1 | -0.0007658063 | 0.004128975 | scoreable |
+
+The nu contrast has the hazard sign (early minus late) and is only instantaneous. It is not an integrated survival prediction.
+
+## Initial cohort means
+
+| Cohort | Count | mean Z | mean I | mean nu_b |
+|---|---:|---:|---:|---:|
+| early | 24218 | 0.4426873 | 0.5479018 | 99.40206 |
+| late | 30505 | 0.4358302 | 0.563404 | 101.5868 |
+
+## Exact common-support conditional estimands
+
+| Stratification | Common cells | Supported E / L | Dropped E | Dropped L | Dropped total |
+|---|---:|---:|---:|---:|---:|
+| D | 6 | 24218 / 30490 | 0 | 0.0004917227 | 0.0002741078 |
+| (D, integer nu_b) | 1202 | 24105 / 30015 | 0.004665951 | 0.01606294 | 0.01101913 |
+
+Weights are e*l/(e+l) in each exact shared cell. These targets differ from the pooled difference and from each other. No mediated/explained fraction is defined. Missing support is reported, not imputed or binned.
+
+## Direction-resolved descriptive means
+
+| Primitive unoriented D | E n | L n | E Z | L Z | E I | L I | E nu_b | L nu_b |
+|---|---:|---:|---:|---:|---:|---:|---:|---:|
+| (0, 1) | 11168 | 13113 | 0.4595272 | 0.4607641 | 0.5356463 | 0.5455196 | 94.1918 | 93.52048 |
+| (1, -2) | 2 | 4 | 0 | 0 | 0.5654377 | 0.6576087 | 197 | 353.75 |
+| (1, -1) | 925 | 2072 | 0.2648649 | 0.2890927 | 0.6603045 | 0.6938002 | 155.0865 | 155.2375 |
+| (1, 0) | 11184 | 13150 | 0.4544886 | 0.4587072 | 0.5390584 | 0.5393133 | 95.34505 | 92.65224 |
+| (1, 1) | 938 | 2148 | 0.2782516 | 0.2886406 | 0.6889624 | 0.6927524 | 154.4733 | 151.9046 |
+| (1, 2) | 0 | 10 | not_scoreable | 0.1 | not_scoreable | 0.5775421 | not_scoreable | 302.4 |
+| (2, -1) | 1 | 3 | 0 | 0.3333333 | 0 | 0.3531556 | 301 | 186.3333 |
+| (2, 1) | 0 | 5 | not_scoreable | 0 | not_scoreable | 1.243709 | not_scoreable | 250.6 |
+
+## Interpretation and uncertainty
+
+- E[Z+I | initial cohort]=1 is an expectation identity, not a pathwise equality. Closure assesses the implemented geometry-to-exit map; it does not establish nu_b closure.
+- The (D,nu_b) finite-lag contrast is a distinct necessary prediction of a state-sufficiency model. A zero weighted contrast can hide opposite cell effects and does not prove recursive Markovness.
+- All delete-one replicates remove the same whole batch for every metric. All stratum weights and shared-cell support are recomputed. Full covariance and replicate vectors are in result.json; no pairwise subset covariance is used.
+- Per-direction rows are descriptive. No claim about causal mediation, the iid-label clock, or a large-size non-Markov limit is made.
+
+## Input provenance
+
+Each listed batch was read once, checked against its SHA256 and expected sample count, then retained only as sufficient statistics for resampling. Checksums do not prove tau independence or independent RNG design; those are generator-contract assumptions.
+
+| Batch | Samples | Seed | SHA256 |
+|---|---:|---|---|
+| production-square-L512-b00.csv.gz | 10000 | 18419746559907930925 | 32f16cc37728d67fdbe3446a48da1bb4d36c48fc4f051648ac8e36e2595d4026 |
+| production-square-L512-b01.csv.gz | 10000 | 13442365983088886094 | fdc708228c1e0535787aa3426ea725b276ecef0b8ba193215b1bf93512518892 |
+| production-square-L512-b02.csv.gz | 10000 | 12192384056273429425 | 57b5f9ef9bcc6f21a5a14f5e7b6c24fccb94388f349943738916cc1700ab4897 |
+| production-square-L512-b03.csv.gz | 10000 | 16795263932832913938 | 65ec5cee09f76a03c69b702a21bb6f27dd70930b48e3b7f047de33103032eaa4 |
+| production-square-L512-b04.csv.gz | 10000 | 5606540329541986450 | 033189fab591caa248c66c9710d63104036dc3d26b88e495d965615a84ecc4fb |
+| production-square-L512-b05.csv.gz | 10000 | 176734671850227651 | b38b95839aab1c6f6575a9b94702c41a032a2cc00430c6eb904840472488fbd4 |
+| production-square-L512-b06.csv.gz | 10000 | 14922363405798405830 | 330829d82ec4dad029f53dc07fe8bda5252bb35118aa9b507012c73df88e5a35 |
+| production-square-L512-b07.csv.gz | 10000 | 3540279280158270918 | a292a7a8be0fb5f79f2449745709c51942fc4261fd50e542569d33d04005ee30 |
+| production-square-L512-b08.csv.gz | 10000 | 182843979474712324 | 113325911eb95469467bfe73496d5b57448413125dcea383b55b0ff7875db4ac |
+| production-square-L512-b09.csv.gz | 10000 | 4164285986391024372 | b5712154670f8179ddf32487f35a794ea16b5ef4f0cdad2025c3f8e447f321da |
+| production-square-L512-b10.csv.gz | 10000 | 15500924042285991404 | 98bb914abc5188216cd5b9b4584ba46539bd4c16c1b4d745b6a8033748243668 |
+| production-square-L512-b11.csv.gz | 10000 | 17186215622830953477 | 73aaa04f05ee279532f7c781a97acda5191840cf79fc2cee72f2a6c9caaed7f1 |
+| production-square-L512-b12.csv.gz | 10000 | 11279241988091952515 | 75f556befdd4986e26f6594f7d2ec44fa6517b9b19021b9cbaa612116471875c |
+| production-square-L512-b13.csv.gz | 10000 | 10329155733035115099 | d5ffad750a152bd62325ae22375496b861da96f8687ded8ad7c0a8220d06a202 |
+
+No old archive was pooled into these estimates.
diff --git a/analysis/completion-hazard-production-20260929/results/result.json b/analysis/completion-hazard-production-20260929/results/result.json
new file mode 100644
index 000000000..a9d2cc80d
--- /dev/null
+++ b/analysis/completion-hazard-production-20260929/results/result.json
@@ -0,0 +1,19331 @@
+{
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+  "clock": "finite uniform-permutation insertion count",
+  "contract_file": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/completion-hazard-production-20260929/contract.json",
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+          "sum_nu_b": 326291,
+          "survivors": 620
+        }
+      },
+      {
+        "direction": [
+          1,
+          2
+        ],
+        "early": {
+          "count": 0,
+          "mean_I": null,
+          "mean_Z": null,
+          "mean_nu_b": null,
+          "sum_I": 0.0,
+          "sum_nu_b": 0,
+          "survivors": 0
+        },
+        "late": {
+          "count": 10,
+          "mean_I": 0.5775421020756097,
+          "mean_Z": 0.1,
+          "mean_nu_b": 302.4,
+          "sum_I": 5.775421020756097,
+          "sum_nu_b": 3024,
+          "survivors": 1
+        }
+      },
+      {
+        "direction": [
+          2,
+          -1
+        ],
+        "early": {
+          "count": 1,
+          "mean_I": 0.0,
+          "mean_Z": 0.0,
+          "mean_nu_b": 301.0,
+          "sum_I": 0.0,
+          "sum_nu_b": 301,
+          "survivors": 0
+        },
+        "late": {
+          "count": 3,
+          "mean_I": 0.35315564851191317,
+          "mean_Z": 0.3333333333333333,
+          "mean_nu_b": 186.33333333333334,
+          "sum_I": 1.0594669455357395,
+          "sum_nu_b": 559,
+          "survivors": 1
+        }
+      },
+      {
+        "direction": [
+          2,
+          1
+        ],
+        "early": {
+          "count": 0,
+          "mean_I": null,
+          "mean_Z": null,
+          "mean_nu_b": null,
+          "sum_I": 0.0,
+          "sum_nu_b": 0,
+          "survivors": 0
+        },
+        "late": {
+          "count": 5,
+          "mean_I": 1.2437086699929425,
+          "mean_Z": 0.0,
+          "mean_nu_b": 250.6,
+          "sum_I": 6.218543349964713,
+          "sum_nu_b": 1253,
+          "survivors": 0
+        }
+      }
+    ],
+    "rank1_at_b_count": 54723,
+    "samples": 140000,
+    "scalars": {
+      "closure_early": {
+        "estimate": -0.009410927210762643,
+        "reason": null,
+        "status": "scoreable"
+      },
+      "closure_late": {
+        "estimate": -0.0007658063318577435,
+        "reason": null,
+        "status": "scoreable"
+      },
+      "closure_residual": {
+        "estimate": -0.0086451208789049,
+        "reason": null,
+        "status": "scoreable"
+      },
+      "delta_integrated_hazard": {
+        "estimate": 0.015502186518906669,
+        "reason": null,
+        "status": "scoreable"
+      },
+      "delta_nu_instantaneous_scaled": {
+        "estimate": -0.015041153963458436,
+        "reason": null,
+        "status": "scoreable"
+      },
+      "delta_overlap_D": {
+        "estimate": -0.004149992812322534,
+        "reason": null,
+        "status": "scoreable"
+      },
+      "delta_overlap_D_nu": {
+        "estimate": 0.001216126747729529,
+        "reason": null,
+        "status": "scoreable"
+      },
+      "delta_survival": {
+        "estimate": 0.006857065640001769,
+        "reason": null,
+        "status": "scoreable"
+      }
+    }
+  },
+  "run_metadata_parameter_checks": {
+    "L": "matches",
+    "N": "matches",
+    "a": "matches",
+    "b": "matches",
+    "c": "matches"
+  },
+  "schema": "matching-one.completion-hazard-production.v1",
+  "tau_design_assumption": "independent uniform integer in [b,c-1] for each full filtration"
+}
diff --git a/analysis/completion-hazard-production-20260929/run.log b/analysis/completion-hazard-production-20260929/run.log
new file mode 100644
index 000000000..3577e0c3a
--- /dev/null
+++ b/analysis/completion-hazard-production-20260929/run.log
@@ -0,0 +1,17 @@
+benchmark batch=0: samples=32 seconds=0.661251
+Estimated ideal-parallel production wall: 208.86s
+production batch=0: samples=10000 seconds=308.887
+production batch=1: samples=10000 seconds=312.374
+production batch=2: samples=10000 seconds=316.117
+production batch=6: samples=10000 seconds=316.161
+production batch=11: samples=10000 seconds=316.207
+production batch=3: samples=10000 seconds=316.348
+production batch=13: samples=10000 seconds=316.381
+production batch=7: samples=10000 seconds=317.357
+production batch=10: samples=10000 seconds=317.826
+production batch=9: samples=10000 seconds=320.843
+production batch=4: samples=10000 seconds=321.35
+production batch=8: samples=10000 seconds=321.885
+production batch=5: samples=10000 seconds=322.373
+production batch=12: samples=10000 seconds=323.156
+Completed 14 batches in 325.51s
diff --git a/analysis/completion-hazard-production-20260929/run.py b/analysis/completion-hazard-production-20260929/run.py
new file mode 100644
index 000000000..6b090db6f
--- /dev/null
+++ b/analysis/completion-hazard-production-20260929/run.py
@@ -0,0 +1,127 @@
+#!/usr/bin/env python3
+"""Run one predeclared geometry-marked production block; Python stdlib only."""
+import argparse
+from concurrent.futures import ThreadPoolExecutor, as_completed
+from datetime import datetime, timezone
+import gzip
+import hashlib
+import json
+import os
+from pathlib import Path
+import platform
+import shlex
+import subprocess
+import tempfile
+import time
+
+HERE = Path(__file__).resolve().parent
+
+
+def sha(path):
+    return hashlib.sha256(path.read_bytes()).hexdigest()
+
+
+def seed(role, batch, stream):
+    name = f"matching-one/completion-independent-20260929/{role}/square/L512/{batch}/{stream}"
+    return int.from_bytes(hashlib.sha256(name.encode()).digest()[:8], "big")
+
+
+def main():
+    ap = argparse.ArgumentParser()
+    ap.add_argument("--source-commit", required=True)
+    ap.add_argument("--compiler", default="g++")
+    ap.add_argument("--output", type=Path, default=HERE / "data")
+    ap.add_argument("--budget-seconds", type=float, default=900)
+    args = ap.parse_args()
+    contract = json.loads((HERE / "contract.json").read_text())
+    args.output.mkdir(parents=True, exist_ok=True)
+    manifest = args.output / "run.json"
+    if manifest.exists():
+        raise SystemExit("Existing run is not overwritten or silently resumed")
+    workers = contract["workers"]
+    started = time.monotonic()
+    run = {
+        "schema": "matching-one.completion-hazard-run.v1",
+        "status": "building", "pid": os.getpid(),
+        "utc_started": datetime.now(timezone.utc).isoformat(),
+        "source_commit": args.source_commit,
+        "source_sha256": {name: sha(HERE / name) for name in ("engine.cpp", "run.py", "contract.json")},
+        "contract": contract,
+        "command": shlex.join(["python3", str(Path(__file__).resolve()), *os.sys.argv[1:]]),
+        "platform": platform.platform(), "machine": platform.machine(),
+        "python": platform.python_version(),
+        "compiler": subprocess.check_output([args.compiler, "--version"], text=True).strip(),
+        "workers": workers, "budget_seconds": args.budget_seconds,
+        "seed_rule": "first 64 SHA256 bits of matching-one/completion-independent-20260929/{role}/square/L512/{batch}/{stream}; stream=permutation or audit-time",
+        "batches": [], "benchmarks": [],
+    }
+    def save():
+        manifest.write_text(json.dumps(run, indent=2)+"\n")
+    for name in ("cpu/cpu.cfs_quota_us", "cpu/cpu.cfs_period_us", "memory/memory.limit_in_bytes", "cpu.max", "memory.max"):
+        path = Path("/sys/fs/cgroup") / name
+        if path.exists():
+            run.setdefault("cgroup", {})[name] = path.read_text().strip()
+    save()
+    try:
+        with tempfile.TemporaryDirectory(prefix="completion-build-", dir=HERE) as tmp:
+            tmp = Path(tmp)
+            binary = tmp / "engine"
+            build = [args.compiler, "-O3", "-std=c++17", str(HERE / "engine.cpp"), "-o", str(binary)]
+            subprocess.run(build, check=True)
+            run["build_command"] = shlex.join(build)
+            run["binary_sha256"] = sha(binary)
+
+            def job(role, batch, samples):
+                stem = f"{role}-square-L512-b{batch:02d}"
+                raw = tmp / (stem+".csv")
+                permutation_seed, audit_seed = seed(role, batch, "permutation"), seed(role, batch, "audit-time")
+                command = [str(binary), "sample", "square", str(contract["L"]), str(samples),
+                           str(permutation_seed), str(audit_seed), str(contract["b"]), str(contract["c"]), str(raw)]
+                begin = time.monotonic()
+                log = subprocess.check_output(command, text=True).strip()
+                seconds = time.monotonic()-begin
+                target = args.output / (stem+".csv.gz")
+                content = raw.read_bytes()
+                if content.count(b"\n") != samples+1:
+                    raise RuntimeError("incomplete sample output")
+                with target.open("wb") as output:
+                    with gzip.GzipFile(fileobj=output, mode="wb", mtime=0) as gz:
+                        gz.write(content)
+                print(f"{role} batch={batch}: {log}", flush=True)
+                return {"file": target.name, "batch": batch, "samples": samples,
+                        "seed": str(permutation_seed), "time_seed": str(audit_seed),
+                        "role": role, "runtime_seconds": seconds, "engine_log": log,
+                        "command": shlex.join(command), "sha256": sha(target),
+                        "dependency_group": f"completion-independent-20260929/square/L512/batch{batch}" if role=="production" else "excluded-benchmark"}
+
+            run["status"] = "benchmarking"; save()
+            benchmark = job("benchmark", 0, 32)
+            run["benchmarks"].append(benchmark)
+            estimate = benchmark["runtime_seconds"]/32 * contract["samples_per_batch"]*contract["batches"]/workers
+            run["estimated_production_wall_seconds"] = estimate
+            print(f"Estimated ideal-parallel production wall: {estimate:.2f}s", flush=True)
+            if estimate > args.budget_seconds:
+                run["status"] = "benchmark-budget-stop"; save()
+                return
+            run["status"] = "running"; save()
+            with ThreadPoolExecutor(max_workers=workers) as pool:
+                futures = [pool.submit(job, "production", b, contract["samples_per_batch"])
+                           for b in range(contract["batches"])]
+                for future in as_completed(futures):
+                    run["batches"].append(future.result())
+                    run["batches"].sort(key=lambda x: x["batch"])
+                    save()
+        run["status"] = "completed"
+    except BaseException as error:
+        run["status"] = "failed"
+        run["error"] = str(error)
+        raise
+    finally:
+        run["wall_seconds"] = time.monotonic()-started
+        run["utc_finished"] = datetime.now(timezone.utc).isoformat()
+        save()
+    print(f"Completed {len(run['batches'])} batches in {run['wall_seconds']:.2f}s", flush=True)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/analysis/completion-pair-synergy-20260929/RESULT.md b/analysis/completion-pair-synergy-20260929/RESULT.md
new file mode 100644
index 000000000..690a79eb2
--- /dev/null
+++ b/analysis/completion-pair-synergy-20260929/RESULT.md
@@ -0,0 +1,30 @@
+# Completion-pair synergy: exact L4 successor calculation
+
+Uniform-permutation count clock. Exhaustive occupied subsets and integer prefix DP, not 16! paths.
+
+## square
+
+19932 rank-one configurations; 72 (k,D,nu2) cells.
+34 cells have different pair counts; 34 different successor laws.
+
+First weak-history witness: k=6, a=4, D=[0, 1], nu=0.
+Identical one-step survival: 1.
+Two-step early-minus-late survival: -4/4131.
+Mean synergy-edge early-minus-late: 20/459.
+
+Wall time: 0.540s.
+
+## triangular
+
+23502 rank-one configurations; 90 (k,D,nu2) cells.
+48 cells have different pair counts; 48 different successor laws.
+
+First weak-history witness: k=5, a=4, D=[0, 1], nu=0.
+Identical one-step survival: 1.
+Two-step early-minus-late survival: 2/121.
+Mean synergy-edge early-minus-late: -10/11.
+
+Wall time: 0.693s.
+
+Complete per-cell integer birth masses, edge sums and successor counts are in result.json.
+Existence witnesses are selected lexicographically from an exact finite census, not significance-ranked samples.
diff --git a/analysis/completion-pair-synergy-20260929/analyze.py b/analysis/completion-pair-synergy-20260929/analyze.py
new file mode 100644
index 000000000..dc88c4ad8
--- /dev/null
+++ b/analysis/completion-pair-synergy-20260929/analyze.py
@@ -0,0 +1,211 @@
+#!/usr/bin/env python3
+"""Exact L4 successor geometry, not a new stochastic production block.
+
+Reuse the existing lifted topology routine. Enumerate 2^16 subsets per
+lattice, count prefix entry histories by integer DP (not 16! permutations),
+and resolve the two-insertion law at matched direction/completion count.
+"""
+from collections import Counter, defaultdict
+from fractions import Fraction
+import importlib.util
+from itertools import combinations
+import json
+import math
+from pathlib import Path
+import platform
+import time
+
+
+HERE = Path(__file__).resolve().parent
+SOURCE = HERE.parent / "birth-completion-geometry-20260929/analyze.py"
+SPEC = importlib.util.spec_from_file_location("existing_topology", SOURCE)
+TOPO = importlib.util.module_from_spec(SPEC)
+SPEC.loader.exec_module(TOPO)
+# Only topology() is called; its original L3 acquisition/DP is not run.
+L, N = 4, 16
+TOPO.L, TOPO.N = L, N
+FACT = [math.factorial(k) for k in range(N+1)]
+
+
+def occupied(mask):
+    return [v for v in range(N) if mask >> v & 1]
+
+
+def solve(lattice):
+    started = time.perf_counter()
+    steps = [(1, 0), (-1, 0), (0, 1), (0, -1)]
+    if lattice == "triangular":
+        steps += [(1, 1), (-1, -1)]
+    neighbors = [[(((v % L+dx) % L)+L*((v//L+dy) % L), dx, dy)
+                  for dx, dy in steps] for v in range(N)]
+    topo = [TOPO.topology(mask, neighbors) for mask in range(1 << N)]
+    rank_counts = Counter((mask.bit_count(), r) for mask, (r, _) in enumerate(topo))
+    # This is new L4 geometry, not a rerun of the old L3 complete table.
+    assert topo[0] == (0, None) and topo[-1] == (2, None)
+    assert topo[(1 << L)-1] == (1, (1, 0))  # one complete horizontal row
+    completion = {}
+    for mask, (rank, direction) in enumerate(topo):
+        if rank == 1:
+            completion[mask] = tuple(v for v in range(N) if not mask >> v & 1
+                                     and topo[mask | (1 << v)][0] == 2)
+    ways = {}
+    cells = {}
+    geometry = {}
+    checked = 0
+    for mask, complete_sites in completion.items():
+        k, m = mask.bit_count(), N-mask.bit_count()
+        direction = topo[mask][1]
+        birth = Counter()
+        for v in occupied(mask):
+            old = mask ^ (1 << v)
+            if topo[old][0] == 0:
+                birth[k] += FACT[k-1]
+            else:
+                assert topo[old] == (1, direction)
+                birth.update(ways[old])
+        assert sum(birth.values()) == FACT[k]
+        ways[mask] = birth
+        safe = [v for v in range(N) if not mask >> v & 1 and v not in complete_sites]
+        edges = [(v, w) for v, w in combinations(safe, 2)
+                 if topo[mask | (1 << v) | (1 << w)][0] == 2]
+        degrees = Counter(v for edge in edges for v in edge)
+        c, e = len(complete_sites), len(edges)
+        next_counts = Counter()
+        for v in safe:
+            new = mask | (1 << v)
+            assert topo[new] == (1, direction)
+            assert len(completion[new]) == c+degrees[v]
+            next_counts[len(completion[new])] += 1
+        # Direct unordered-pair outcome versus graph formula, one arithmetic
+        # check inside the new enumeration, not a second topology census.
+        vacancies = [v for v in range(N) if not mask >> v & 1]
+        surviving_ordered_pairs = 2*sum(topo[mask | (1 << v) | (1 << w)][0] == 1
+                                      for v, w in combinations(vacancies, 2))
+        assert surviving_ordered_pairs == (m-c)*(m-c-1)-2*e
+        checked += 1
+        signature = tuple(sorted(next_counts.items()))
+        key = k, direction, c
+        if key not in cells:
+            cells[key] = {"masks": 0, "edge_counts": Counter(), "signatures": {},
+                          "birth": defaultdict(lambda: {"prefix_mass": 0, "edges_sum": 0,
+                                                        "survival2_numerator_sum": 0,
+                                                        "successor_nu_counts": Counter()})}
+        cell = cells[key]
+        cell["masks"] += 1
+        cell["edge_counts"][e] += 1
+        cell["signatures"].setdefault(signature, mask)
+        for j, count in birth.items():
+            entry = cell["birth"][j]
+            entry["prefix_mass"] += count
+            entry["edges_sum"] += count*e
+            entry["survival2_numerator_sum"] += count*surviving_ordered_pairs
+            for nu, number in next_counts.items():
+                entry["successor_nu_counts"][nu] += count*number
+        geometry[mask] = {"mask": mask, "occupied": occupied(mask), "k": k,
+                          "direction": list(direction), "nu2": c,
+                          "completion_sites": list(complete_sites), "safe_sites": safe,
+                          "synergy_edges": edges, "edge_count": e,
+                          "successor_nu_counts": dict(sorted(next_counts.items())),
+                          "birth_prefix_counts": dict(sorted(birth.items())),
+                          "survival2": str(Fraction(surviving_ordered_pairs, m*(m-1))) if m >= 2 else None}
+    strong_witness = None
+    weak_witness = None
+    summaries = []
+    variable_edge_cells = variable_transition_cells = 0
+    for (k, direction, c), cell in sorted(cells.items()):
+        m = N-k
+        edge_variation = len(cell["edge_counts"]) > 1
+        transition_variation = len(cell["signatures"]) > 1
+        variable_edge_cells += edge_variation
+        variable_transition_cells += transition_variation
+        if edge_variation and strong_witness is None:
+            candidate_masks = [mask for mask in cell["signatures"].values()]
+            first = min(candidate_masks)
+            second = min(mask for mask in candidate_masks
+                         if geometry[mask]["edge_count"] != geometry[first]["edge_count"])
+            strong_witness = {"selection": "first lexicographic cell with distinct edge counts",
+                              "preparations": [geometry[first], geometry[second]]}
+        birth_rows = []
+        for j, stats in sorted(cell["birth"].items()):
+            birth_rows.append({"j1": j, **stats,
+                               "successor_nu_counts": dict(sorted(stats["successor_nu_counts"].items()))})
+        summaries.append({"k": k, "direction": list(direction), "nu2": c,
+                          "configurations": cell["masks"], "edge_count_histogram": dict(sorted(cell["edge_counts"].items())),
+                          "distinct_successor_signatures": len(cell["signatures"]),
+                          "birth_sufficient_statistics": birth_rows})
+        if weak_witness is not None or m < 2:
+            continue
+        for a in range(1, k):
+            early = [r for r in birth_rows if r["j1"] <= a]
+            late = [r for r in birth_rows if r["j1"] > a]
+            if not early or not late:
+                continue
+            def group(rows):
+                mass = sum(r["prefix_mass"] for r in rows)
+                edges_sum = sum(r["edges_sum"] for r in rows)
+                num = sum(r["survival2_numerator_sum"] for r in rows)
+                next_counts = Counter()
+                for r in rows:
+                    next_counts.update(r["successor_nu_counts"])
+                return {"prefix_mass": mass, "edges_sum": edges_sum,
+                        "mean_edges": str(Fraction(edges_sum, mass)),
+                        "survival2": str(Fraction(num, mass*m*(m-1))),
+                        "survivor_successor_law": {nu: str(Fraction(count, mass*(m-c)))
+                                                   for nu, count in sorted(next_counts.items())} if m > c else {},
+                        "mean_nu_next_given_survival": str(Fraction(sum(nu*count for nu, count in next_counts.items()), mass*(m-c))) if m > c else None}
+            eg, lg = group(early), group(late)
+            delta_e = Fraction(eg["mean_edges"])-Fraction(lg["mean_edges"])
+            if not delta_e:
+                continue
+            delta_z = Fraction(eg["survival2"])-Fraction(lg["survival2"])
+            assert delta_z == -2*delta_e/(m*(m-1))
+            weak_witness = {"selection": "first lexicographic (k,direction,nu2,a) with nonzero exact mean-edge difference; structural existence, no p-value",
+                            "k": k, "a": a, "direction": list(direction), "nu2": c,
+                            "early": eg, "late": lg,
+                            "mean_edge_difference": str(delta_e), "survival2_difference": str(delta_z),
+                            "one_step_survival_both": str(Fraction(m-c, m))}
+            break
+    return {"lattice": lattice, "L": L, "N": N, "occupied_subsets": 1 << N,
+            "rank_counts": [[k, rank, count] for (k, rank), count in sorted(rank_counts.items())],
+            "rank_one_configurations": len(completion), "coarse_cells": len(cells),
+            "cells_with_edge_variation": variable_edge_cells,
+            "cells_with_successor_variation": variable_transition_cells,
+            "pair_formula_rows": checked,
+            "strong_preparation_witness": strong_witness,
+            "uniform_start_birth_history_witness": weak_witness,
+            "exact_cell_sufficient_statistics": summaries,
+            "wall_seconds": time.perf_counter()-started}
+
+
+def main():
+    result = {"schema": "matching-one.completion-pair-synergy.v1", "date": "2026-09-29",
+              "source": "uniform site permutation; square NN or triangular +diagonal occupied graph on L4 torus",
+              "clock": "insertion count, not iid-label time", "python": platform.python_version(),
+              "topology_source": "analysis/birth-completion-geometry-20260929/analyze.py:topology",
+              "topology_source_commit": "8b37c20e5f2b1946cefbed98b09280a0cad25ed2",
+              "scope": "Exact finite L4 computation, no Monte Carlo, 16! enumeration, cloud or continuum inference",
+              "models": [solve(lattice) for lattice in ("square", "triangular")]}
+    (HERE / "result.json").write_text(json.dumps(result, indent=2)+"\n")
+    lines = ["# Completion-pair synergy: exact L4 successor calculation", "",
+             "Uniform-permutation count clock. Exhaustive occupied subsets and integer prefix DP, not 16! paths.", ""]
+    for model in result["models"]:
+        lines += [f"## {model['lattice']}", "",
+                  f"{model['rank_one_configurations']} rank-one configurations; {model['coarse_cells']} (k,D,nu2) cells.",
+                  f"{model['cells_with_edge_variation']} cells have different pair counts; {model['cells_with_successor_variation']} different successor laws.", ""]
+        witness = model["uniform_start_birth_history_witness"]
+        if witness:
+            lines += [f"First weak-history witness: k={witness['k']}, a={witness['a']}, D={witness['direction']}, nu={witness['nu2']}.",
+                      f"Identical one-step survival: {witness['one_step_survival_both']}.",
+                      f"Two-step early-minus-late survival: {witness['survival2_difference']}.",
+                      f"Mean synergy-edge early-minus-late: {witness['mean_edge_difference']}.", ""]
+        else:
+            lines += ["No birth-cutoff mean-pair witness found; not a full weak-Markov proof.", ""]
+        lines += [f"Wall time: {model['wall_seconds']:.3f}s.", ""]
+    lines += ["Complete per-cell integer birth masses, edge sums and successor counts are in result.json.",
+              "Existence witnesses are selected lexicographically from an exact finite census, not significance-ranked samples.", ""]
+    (HERE / "RESULT.md").write_text("\n".join(lines))
+    print("\n".join(lines))
+
+
+if __name__ == "__main__":
+    main()
diff --git a/analysis/completion-pair-synergy-20260929/result.json b/analysis/completion-pair-synergy-20260929/result.json
new file mode 100644
index 000000000..180c38105
--- /dev/null
+++ b/analysis/completion-pair-synergy-20260929/result.json
@@ -0,0 +1,10387 @@
+{
+  "schema": "matching-one.completion-pair-synergy.v1",
+  "date": "2026-09-29",
+  "source": "uniform site permutation; square NN or triangular +diagonal occupied graph on L4 torus",
+  "clock": "insertion count, not iid-label time",
+  "python": "3.11.15",
+  "topology_source": "analysis/birth-completion-geometry-20260929/analyze.py:topology",
+  "topology_source_commit": "8b37c20e5f2b1946cefbed98b09280a0cad25ed2",
+  "scope": "Exact finite L4 computation, no Monte Carlo, 16! enumeration, cloud or continuum inference",
+  "models": [
+    {
+      "lattice": "square",
+      "L": 4,
+      "N": 16,
+      "occupied_subsets": 65536,
+      "rank_counts": [
+        [
+          0,
+          0,
+          1
+        ],
+        [
+          1,
+          0,
+          16
+        ],
+        [
+          2,
+          0,
+          120
+        ],
+        [
+          3,
+          0,
+          560
+        ],
+        [
+          4,
+          0,
+          1812
+        ],
+        [
+          4,
+          1,
+          8
+        ],
+        [
+          5,
+          0,
+          4272
+        ],
+        [
+          5,
+          1,
+          96
+        ],
+        [
+          6,
+          0,
+          7448
+        ],
+        [
+          6,
+          1,
+          560
+        ],
+        [
+          7,
+          0,
+          9440
+        ],
+        [
+          7,
+          1,
+          1984
+        ],
+        [
+          7,
+          2,
+          16
+        ],
+        [
+          8,
+          0,
+          8082
+        ],
+        [
+          8,
+          1,
+          4580
+        ],
+        [
+          8,
+          2,
+          208
+        ],
+        [
+          9,
+          0,
+          3984
+        ],
+        [
+          9,
+          1,
+          6368
+        ],
+        [
+          9,
+          2,
+          1088
+        ],
+        [
+          10,
+          0,
+          792
+        ],
+        [
+          10,
+          1,
+          4704
+        ],
+        [
+          10,
+          2,
+          2512
+        ],
+        [
+          11,
+          0,
+          32
+        ],
+        [
+          11,
+          1,
+          1472
+        ],
+        [
+          11,
+          2,
+          2864
+        ],
+        [
+          12,
+          1,
+          160
+        ],
+        [
+          12,
+          2,
+          1660
+        ],
+        [
+          13,
+          2,
+          560
+        ],
+        [
+          14,
+          2,
+          120
+        ],
+        [
+          15,
+          2,
+          16
+        ],
+        [
+          16,
+          2,
+          1
+        ]
+      ],
+      "rank_one_configurations": 19932,
+      "coarse_cells": 72,
+      "cells_with_edge_variation": 34,
+      "cells_with_successor_variation": 34,
+      "pair_formula_rows": 19932,
+      "strong_preparation_witness": {
+        "selection": "first lexicographic cell with distinct edge counts",
+        "preparations": [
+          {
+            "mask": 4403,
+            "occupied": [
+              0,
+              1,
+              4,
+              5,
+              8,
+              12
+            ],
+            "k": 6,
+            "direction": [
+              0,
+              1
+            ],
+            "nu2": 0,
+            "completion_sites": [],
+            "safe_sites": [
+              2,
+              3,
+              6,
+              7,
+              9,
+              10,
+              11,
+              13,
+              14,
+              15
+            ],
+            "synergy_edges": [
+              [
+                2,
+                3
+              ],
+              [
+                6,
+                7
+              ]
+            ],
+            "edge_count": 2,
+            "successor_nu_counts": {
+              "0": 6,
+              "1": 4
+            },
+            "birth_prefix_counts": {
+              "4": 48,
+              "5": 192,
+              "6": 480
+            },
+            "survival2": "43/45"
+          },
+          {
+            "mask": 4405,
+            "occupied": [
+              0,
+              2,
+              4,
+              5,
+              8,
+              12
+            ],
+            "k": 6,
+            "direction": [
+              0,
+              1
+            ],
+            "nu2": 0,
+            "completion_sites": [],
+            "safe_sites": [
+              1,
+              3,
+              6,
+              7,
+              9,
+              10,
+              11,
+              13,
+              14,
+              15
+            ],
+            "synergy_edges": [
+              [
+                1,
+                3
+              ],
+              [
+                3,
+                6
+              ],
+              [
+                6,
+                7
+              ]
+            ],
+            "edge_count": 3,
+            "successor_nu_counts": {
+              "0": 6,
+              "1": 2,
+              "2": 2
+            },
+            "birth_prefix_counts": {
+              "4": 48,
+              "5": 192,
+              "6": 480
+            },
+            "survival2": "14/15"
+          }
+        ]
+      },
+      "uniform_start_birth_history_witness": {
+        "selection": "first lexicographic (k,direction,nu2,a) with nonzero exact mean-edge difference; structural existence, no p-value",
+        "k": 6,
+        "a": 4,
+        "direction": [
+          0,
+          1
+        ],
+        "nu2": 0,
+        "early": {
+          "prefix_mass": 10368,
+          "edges_sum": 26880,
+          "mean_edges": "70/27",
+          "survival2": "229/243",
+          "survivor_successor_law": {
+            "0": "3/5",
+            "1": "38/135",
+            "2": "16/135"
+          },
+          "mean_nu_next_given_survival": "14/27"
+        },
+        "late": {
+          "prefix_mass": 156672,
+          "edges_sum": 399360,
+          "mean_edges": "130/51",
+          "survival2": "433/459",
+          "survivor_successor_law": {
+            "0": "3/5",
+            "1": "74/255",
+            "2": "28/255"
+          },
+          "mean_nu_next_given_survival": "26/51"
+        },
+        "mean_edge_difference": "20/459",
+        "survival2_difference": "-4/4131",
+        "one_step_survival_both": "1"
+      },
+      "exact_cell_sufficient_statistics": [
+        {
+          "k": 4,
+          "direction": [
+            0,
+            1
+          ],
+          "nu2": 0,
+          "configurations": 4,
+          "edge_count_histogram": {
+            "0": 4
+          },
+          "distinct_successor_signatures": 1,
+          "birth_sufficient_statistics": [
+            {
+              "j1": 4,
+              "prefix_mass": 96,
+              "edges_sum": 0,
+              "survival2_numerator_sum": 12672,
+              "successor_nu_counts": {
+                "0": 1152
+              }
+            }
+          ]
+        },
+        {
+          "k": 4,
+          "direction": [
+            1,
+            0
+          ],
+          "nu2": 0,
+          "configurations": 4,
+          "edge_count_histogram": {
+            "0": 4
+          },
+          "distinct_successor_signatures": 1,
+          "birth_sufficient_statistics": [
+            {
+              "j1": 4,
+              "prefix_mass": 96,
+              "edges_sum": 0,
+              "survival2_numerator_sum": 12672,
+              "successor_nu_counts": {
+                "0": 1152
+              }
+            }
+          ]
+        },
+        {
+          "k": 5,
+          "direction": [
+            0,
+            1
+          ],
+          "nu2": 0,
+          "configurations": 48,
+          "edge_count_histogram": {
+            "1": 48
+          },
+          "distinct_successor_signatures": 1,
+          "birth_sufficient_statistics": [
+            {
+              "j1": 4,
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+              "edges_sum": 1152,
+              "survival2_numerator_sum": 124416,
+              "successor_nu_counts": {
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+                "1": 2304
+              }
+            },
+            {
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+              "edges_sum": 4608,
+              "survival2_numerator_sum": 497664,
+              "successor_nu_counts": {
+                "0": 41472,
+                "1": 9216
+              }
+            }
+          ]
+        },
+        {
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+            1,
+            0
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+          "configurations": 48,
+          "edge_count_histogram": {
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+          ]
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+          ]
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+              "successor_nu_counts": {
+                "4": 371589120
+              }
+            },
+            {
+              "j1": 10,
+              "prefix_mass": 139345920,
+              "edges_sum": 0,
+              "survival2_numerator_sum": 0,
+              "successor_nu_counts": {
+                "4": 139345920
+              }
+            }
+          ]
+        },
+        {
+          "k": 11,
+          "direction": [
+            1,
+            1
+          ],
+          "nu2": 5,
+          "configurations": 32,
+          "edge_count_histogram": {
+            "0": 32
+          },
+          "distinct_successor_signatures": 1,
+          "birth_sufficient_statistics": [
+            {
+              "j1": 4,
+              "prefix_mass": 7741440,
+              "edges_sum": 0,
+              "survival2_numerator_sum": 0,
+              "successor_nu_counts": {}
+            },
+            {
+              "j1": 5,
+              "prefix_mass": 55848960,
+              "edges_sum": 0,
+              "survival2_numerator_sum": 0,
+              "successor_nu_counts": {}
+            },
+            {
+              "j1": 6,
+              "prefix_mass": 157593600,
+              "edges_sum": 0,
+              "survival2_numerator_sum": 0,
+              "successor_nu_counts": {}
+            },
+            {
+              "j1": 7,
+              "prefix_mass": 282009600,
+              "edges_sum": 0,
+              "survival2_numerator_sum": 0,
+              "successor_nu_counts": {}
+            },
+            {
+              "j1": 8,
+              "prefix_mass": 348364800,
+              "edges_sum": 0,
+              "survival2_numerator_sum": 0,
+              "successor_nu_counts": {}
+            },
+            {
+              "j1": 9,
+              "prefix_mass": 286433280,
+              "edges_sum": 0,
+              "survival2_numerator_sum": 0,
+              "successor_nu_counts": {}
+            },
+            {
+              "j1": 10,
+              "prefix_mass": 139345920,
+              "edges_sum": 0,
+              "survival2_numerator_sum": 0,
+              "successor_nu_counts": {}
+            }
+          ]
+        },
+        {
+          "k": 12,
+          "direction": [
+            0,
+            1
+          ],
+          "nu2": 4,
+          "configurations": 4,
+          "edge_count_histogram": {
+            "0": 4
+          },
+          "distinct_successor_signatures": 1,
+          "birth_sufficient_statistics": [
+            {
+              "j1": 4,
+              "prefix_mass": 11612160,
+              "edges_sum": 0,
+              "survival2_numerator_sum": 0,
+              "successor_nu_counts": {}
+            },
+            {
+              "j1": 5,
+              "prefix_mass": 85155840,
+              "edges_sum": 0,
+              "survival2_numerator_sum": 0,
+              "successor_nu_counts": {}
+            },
+            {
+              "j1": 6,
+              "prefix_mass": 251596800,
+              "edges_sum": 0,
+              "survival2_numerator_sum": 0,
+              "successor_nu_counts": {}
+            },
+            {
+              "j1": 7,
+              "prefix_mass": 474163200,
+              "edges_sum": 0,
+              "survival2_numerator_sum": 0,
+              "successor_nu_counts": {}
+            },
+            {
+              "j1": 8,
+              "prefix_mass": 582543360,
+              "edges_sum": 0,
+              "survival2_numerator_sum": 0,
+              "successor_nu_counts": {}
+            },
+            {
+              "j1": 9,
+              "prefix_mass": 371589120,
+              "edges_sum": 0,
+              "survival2_numerator_sum": 0,
+              "successor_nu_counts": {}
+            },
+            {
+              "j1": 10,
+              "prefix_mass": 139345920,
+              "edges_sum": 0,
+              "survival2_numerator_sum": 0,
+              "successor_nu_counts": {}
+            }
+          ]
+        },
+        {
+          "k": 12,
+          "direction": [
+            1,
+            0
+          ],
+          "nu2": 4,
+          "configurations": 4,
+          "edge_count_histogram": {
+            "0": 4
+          },
+          "distinct_successor_signatures": 1,
+          "birth_sufficient_statistics": [
+            {
+              "j1": 4,
+              "prefix_mass": 11612160,
+              "edges_sum": 0,
+              "survival2_numerator_sum": 0,
+              "successor_nu_counts": {}
+            },
+            {
+              "j1": 5,
+              "prefix_mass": 85155840,
+              "edges_sum": 0,
+              "survival2_numerator_sum": 0,
+              "successor_nu_counts": {}
+            },
+            {
+              "j1": 6,
+              "prefix_mass": 251596800,
+              "edges_sum": 0,
+              "survival2_numerator_sum": 0,
+              "successor_nu_counts": {}
+            },
+            {
+              "j1": 7,
+              "prefix_mass": 474163200,
+              "edges_sum": 0,
+              "survival2_numerator_sum": 0,
+              "successor_nu_counts": {}
+            },
+            {
+              "j1": 8,
+              "prefix_mass": 582543360,
+              "edges_sum": 0,
+              "survival2_numerator_sum": 0,
+              "successor_nu_counts": {}
+            },
+            {
+              "j1": 9,
+              "prefix_mass": 371589120,
+              "edges_sum": 0,
+              "survival2_numerator_sum": 0,
+              "successor_nu_counts": {}
+            },
+            {
+              "j1": 10,
+              "prefix_mass": 139345920,
+              "edges_sum": 0,
+              "survival2_numerator_sum": 0,
+              "successor_nu_counts": {}
+            }
+          ]
+        },
+        {
+          "k": 12,
+          "direction": [
+            1,
+            1
+          ],
+          "nu2": 4,
+          "configurations": 4,
+          "edge_count_histogram": {
+            "0": 4
+          },
+          "distinct_successor_signatures": 1,
+          "birth_sufficient_statistics": [
+            {
+              "j1": 4,
+              "prefix_mass": 11612160,
+              "edges_sum": 0,
+              "survival2_numerator_sum": 0,
+              "successor_nu_counts": {}
+            },
+            {
+              "j1": 5,
+              "prefix_mass": 85155840,
+              "edges_sum": 0,
+              "survival2_numerator_sum": 0,
+              "successor_nu_counts": {}
+            },
+            {
+              "j1": 6,
+              "prefix_mass": 251596800,
+              "edges_sum": 0,
+              "survival2_numerator_sum": 0,
+              "successor_nu_counts": {}
+            },
+            {
+              "j1": 7,
+              "prefix_mass": 474163200,
+              "edges_sum": 0,
+              "survival2_numerator_sum": 0,
+              "successor_nu_counts": {}
+            },
+            {
+              "j1": 8,
+              "prefix_mass": 582543360,
+              "edges_sum": 0,
+              "survival2_numerator_sum": 0,
+              "successor_nu_counts": {}
+            },
+            {
+              "j1": 9,
+              "prefix_mass": 371589120,
+              "edges_sum": 0,
+              "survival2_numerator_sum": 0,
+              "successor_nu_counts": {}
+            },
+            {
+              "j1": 10,
+              "prefix_mass": 139345920,
+              "edges_sum": 0,
+              "survival2_numerator_sum": 0,
+              "successor_nu_counts": {}
+            }
+          ]
+        }
+      ],
+      "wall_seconds": 0.6931376670254394
+    }
+  ]
+}
diff --git a/analysis/deterministic-block-encoding-20260929/check.py b/analysis/deterministic-block-encoding-20260929/check.py
new file mode 100644
index 000000000..962816940
--- /dev/null
+++ b/analysis/deterministic-block-encoding-20260929/check.py
@@ -0,0 +1,153 @@
+#!/usr/bin/env python3
+"""One exact census of four labelled states encoded into two corner blocks.
+
+Print the deterministic result JSON. This checks only the stated full tensor
+query algebra, not reachability of those columns by a physical rank experiment.
+No dependencies, simulation, floating-point rank, or whole-machine enumeration.
+"""
+
+from collections import Counter
+from itertools import product
+import json
+from math import comb, factorial, gcd
+
+
+CORNERS = ((0, 0), (1, 0), (0, 1), (1, 1))
+
+
+def integer_rank(rows):
+    """Gaussian elimination over Q using nonzero integer row operations.
+
+    Eliminate with b[pivot]*row - row[pivot]*b. Primitive normalization
+    controls coefficient growth; no reduction modulo a prime is used.
+    """
+    basis = []
+    for source in rows:
+        row = list(source)
+        for pivot, b in basis:
+            if row[pivot]:
+                factor, leading = row[pivot], b[pivot]
+                row = [leading * x - factor * y for x, y in zip(row, b)]
+                divisor = 0
+                for x in row:
+                    divisor = gcd(divisor, x)
+                if divisor:
+                    row = [x // divisor for x in row]
+        pivot = next((i for i, x in enumerate(row) if x), None)
+        if pivot is not None:
+            if row[pivot] < 0:
+                row = [-x for x in row]
+            basis.append((pivot, row))
+    return len(basis)
+
+
+def tensor_row(word):
+    first, second = CORNERS[word % 4], CORNERS[word // 4]
+    return tuple(x * y for x in (1, *first) for y in (1, *second))
+
+
+def main():
+    rows = tuple(tensor_row(word) for word in range(16))
+    rank_counts = Counter()
+    image_rank_counts = Counter()
+    injective_cardinality_counts = Counter()
+    deficient_informative_block_counts = Counter()
+    examples = {
+        (0, 1, 2, 3): "one_informative_block",
+        (0, 5, 10, 15): "direct_copy",
+        (0, 1, 4, 5): "split_bits",
+        (0, 5, 6, 3): "parity_second_block",
+    }
+    example_results = {}
+    checked = 0
+    for code in product(range(16), repeat=4):
+        actual = integer_rank(rows[word] for word in code)
+        image_size = len(set(code))
+        sizes = (len({word % 4 for word in code}),
+                 len({word // 4 for word in code}))
+        exception = image_size == 4 and sizes in ((4, 1), (1, 4))
+        predicted = 3 if exception else image_size
+        if actual != predicted:
+            raise AssertionError((code, sizes, actual, predicted))
+        rank_counts[actual] += 1
+        image_rank_counts[image_size, actual] += 1
+        if image_size == 4:
+            injective_cardinality_counts[sizes, actual] += 1
+            if exception:
+                deficient_informative_block_counts[sizes.index(4)] += 1
+        if code in examples:
+            example_results[examples[code]] = {
+                "corner_word_ids": list(code), "exact_rank": actual,
+                "local_image_cardinalities": list(sizes),
+            }
+        checked += 1
+
+    # Independent combinatorial count: choose the distinct words, then a
+    # surjection from the four labelled states; deficient injections have
+    # a unique informative block, a constant other corner, and 4! labels.
+    surjections = {1: 1, 2: 14, 3: 36, 4: 24}
+    distinct_word_counts = {m: comb(16, m) * n for m, n in surjections.items()}
+    deficient_injections = 2 * 4 * factorial(4)
+    expected_ranks = {
+        1: distinct_word_counts[1],
+        2: distinct_word_counts[2],
+        3: distinct_word_counts[3] + deficient_injections,
+        4: distinct_word_counts[4] - deficient_injections,
+    }
+    assert checked == 16 ** 4
+    assert dict(rank_counts) == expected_ranks
+    assert sum(deficient_informative_block_counts.values()) == deficient_injections
+
+    result = {
+        "schema": "matching-one.deterministic-block-encoding.v1",
+        "contract": {
+            "logical_states": 4,
+            "physical_blocks": 2,
+            "corner_order": [list(corner) for corner in CORNERS],
+            "word_id": "first_corner + 4*second_corner",
+            "response": "(1,u0,v0) tensor (1,u1,v1)",
+            "columns": ["1", "u1", "v1", "u0", "u0*u1", "u0*v1",
+                        "v0", "v0*u1", "v0*v1"],
+            "query_language": "full algebraic tensor span",
+            "physical_product_query_reachability_verified_here": False,
+        },
+        "arithmetic": "exact integer Gaussian elimination over Q",
+        "assignments_checked": checked,
+        "rank_counts": {str(k): v for k, v in sorted(rank_counts.items())},
+        "by_distinct_codewords_and_rank": [
+            {"distinct_codewords": m, "rank": r, "assignments": n}
+            for (m, r), n in sorted(image_rank_counts.items())
+        ],
+        "injective_assignments": distinct_word_counts[4],
+        "injective_rank3": deficient_injections,
+        "injective_rank4": expected_ranks[4],
+        "injective_by_local_cardinality": [
+            {"local_image_cardinalities": list(sizes), "rank": r,
+             "assignments": n}
+            for (sizes, r), n in sorted(injective_cardinality_counts.items())
+        ],
+        "rank3_informative_block_counts": {
+            str(k): v for k, v in sorted(deficient_informative_block_counts.items())
+        },
+        "unlabelled_injective_codebooks": {
+            "total": comb(16, 4), "rank3": deficient_injections // factorial(4),
+            "rank4": expected_ranks[4] // factorial(4),
+        },
+        "classification_mismatches": 0,
+        "combinatorial_count_crosscheck": "passed",
+        "examples": example_results,
+        "general_k_rank3_injective_count": "24*k*4^(k-1), k>=1",
+        "scope": "finite algebraic census supporting the all-k proof in the note",
+        "not_claimed": [
+            "all tensor columns are physically reachable in every source contract",
+            "a stochastic-encoder obstruction",
+            "a full-machine positive versus linear dimension theorem",
+            "optimal noisy error probability",
+            "novelty certification",
+        ],
+    }
+    print(json.dumps(result, ensure_ascii=False, indent=2, sort_keys=True))
+
+
+if __name__ == "__main__":
+    main()
diff --git a/analysis/deterministic-block-encoding-20260929/result.json b/analysis/deterministic-block-encoding-20260929/result.json
new file mode 100644
index 000000000..042eb8871
--- /dev/null
+++ b/analysis/deterministic-block-encoding-20260929/result.json
@@ -0,0 +1,58 @@
+{
+  "arithmetic": "exact integer Gaussian elimination over Q",
+  "assignments_checked": 65536,
+  "by_distinct_codewords_and_rank": [
+    {"assignments": 16, "distinct_codewords": 1, "rank": 1},
+    {"assignments": 1680, "distinct_codewords": 2, "rank": 2},
+    {"assignments": 20160, "distinct_codewords": 3, "rank": 3},
+    {"assignments": 192, "distinct_codewords": 4, "rank": 3},
+    {"assignments": 43488, "distinct_codewords": 4, "rank": 4}
+  ],
+  "classification_mismatches": 0,
+  "combinatorial_count_crosscheck": "passed",
+  "contract": {
+    "columns": ["1", "u1", "v1", "u0", "u0*u1", "u0*v1", "v0", "v0*u1", "v0*v1"],
+    "corner_order": [[0, 0], [1, 0], [0, 1], [1, 1]],
+    "logical_states": 4,
+    "physical_blocks": 2,
+    "physical_product_query_reachability_verified_here": false,
+    "query_language": "full algebraic tensor span",
+    "response": "(1,u0,v0) tensor (1,u1,v1)",
+    "word_id": "first_corner + 4*second_corner"
+  },
+  "examples": {
+    "direct_copy": {"corner_word_ids": [0, 5, 10, 15], "exact_rank": 4, "local_image_cardinalities": [4, 4]},
+    "one_informative_block": {"corner_word_ids": [0, 1, 2, 3], "exact_rank": 3, "local_image_cardinalities": [4, 1]},
+    "parity_second_block": {"corner_word_ids": [0, 5, 6, 3], "exact_rank": 4, "local_image_cardinalities": [4, 2]},
+    "split_bits": {"corner_word_ids": [0, 1, 4, 5], "exact_rank": 4, "local_image_cardinalities": [2, 2]}
+  },
+  "general_k_rank3_injective_count": "24*k*4^(k-1), k>=1",
+  "injective_assignments": 43680,
+  "injective_by_local_cardinality": [
+    {"assignments": 96, "local_image_cardinalities": [1, 4], "rank": 3},
+    {"assignments": 864, "local_image_cardinalities": [2, 2], "rank": 4},
+    {"assignments": 6912, "local_image_cardinalities": [2, 3], "rank": 4},
+    {"assignments": 2016, "local_image_cardinalities": [2, 4], "rank": 4},
+    {"assignments": 6912, "local_image_cardinalities": [3, 2], "rank": 4},
+    {"assignments": 17280, "local_image_cardinalities": [3, 3], "rank": 4},
+    {"assignments": 3456, "local_image_cardinalities": [3, 4], "rank": 4},
+    {"assignments": 96, "local_image_cardinalities": [4, 1], "rank": 3},
+    {"assignments": 2016, "local_image_cardinalities": [4, 2], "rank": 4},
+    {"assignments": 3456, "local_image_cardinalities": [4, 3], "rank": 4},
+    {"assignments": 576, "local_image_cardinalities": [4, 4], "rank": 4}
+  ],
+  "injective_rank3": 192,
+  "injective_rank4": 43488,
+  "not_claimed": [
+    "all tensor columns are physically reachable in every source contract",
+    "a stochastic-encoder obstruction",
+    "a full-machine positive versus linear dimension theorem",
+    "optimal noisy error probability",
+    "novelty certification"
+  ],
+  "rank3_informative_block_counts": {"0": 96, "1": 96},
+  "rank_counts": {"1": 16, "2": 1680, "3": 20352, "4": 43488},
+  "schema": "matching-one.deterministic-block-encoding.v1",
+  "scope": "finite algebraic census supporting the all-k proof in the note",
+  "unlabelled_injective_codebooks": {"rank3": 8, "rank4": 1812, "total": 1820}
+}
diff --git a/analysis/directional-hazard-contrast-20260929/RESULT.md b/analysis/directional-hazard-contrast-20260929/RESULT.md
new file mode 100644
index 000000000..b81e33c4a
--- /dev/null
+++ b/analysis/directional-hazard-contrast-20260929/RESULT.md
@@ -0,0 +1,17 @@
+# Directional next-insertion hazard: one retrospective check
+
+Same completed 140k block, unchanged cutoffs, cohorts and exact-D overlap weights.
+
+| Quantity | Estimate | Aligned batch SE |
+|---|---:|---:|
+| pooled_nu_early_minus_late | -2.184732729 | 0.4794961673 |
+| within_D_nu_early_minus_late | 1.629168057 | 0.3909761647 |
+| within_D_survival_early_minus_late | -0.004149992812 | 0.004511941724 |
+| within_D_next_insertion_exit_difference | 1.526024089e-05 | 3.662231425e-06 |
+| within_D_window_scaled_exit_difference | 0.01121627705 | 0.002691740097 |
+
+Positive nu contrast means greater early-cohort next-insertion exit probability.
+Window scaling is not a finite-window exit estimate. The last survival contrast is the unchanged original result.
+
+This necessity check was selected after the original report; it is not independent validation.
+No new observations, subgroup searches, thresholds, cutoffs or samples are added.
diff --git a/analysis/directional-hazard-contrast-20260929/analyze.py b/analysis/directional-hazard-contrast-20260929/analyze.py
new file mode 100644
index 000000000..88fa3772c
--- /dev/null
+++ b/analysis/directional-hazard-contrast-20260929/analyze.py
@@ -0,0 +1,122 @@
+#!/usr/bin/env python3
+"""One retrospective directional Markov-necessity check; no new samples.
+
+Reuse the saved nu_b, unchanged cohorts/cutoffs, exact direction and overlap
+weights from the completed block. Do not mutate its scorer or results.
+"""
+import hashlib
+import importlib.util
+import json
+import math
+from pathlib import Path
+import platform
+import sys
+import time
+
+
+HERE = Path(__file__).resolve().parent
+PARENT = HERE.parent / "completion-hazard-production-20260929"
+SPEC = importlib.util.spec_from_file_location("completion_parent", PARENT / "analyze.py")
+BASE = importlib.util.module_from_spec(SPEC)
+sys.modules[SPEC.name] = BASE
+SPEC.loader.exec_module(BASE)
+NAMES = ["pooled_nu_early_minus_late", "within_D_nu_early_minus_late", "within_D_survival_early_minus_late"]
+
+
+def score(stats):
+    early, late = stats.groups
+    if not early.n or not late.n:
+        raise ValueError("Both risk cohorts are required")
+    denominator, numerator, survival = 0.0, 0.0, 0.0
+    support = [0, 0]
+    rows = []
+    for direction, (e, l) in sorted(stats.directions.items()):
+        weight = e.n*l.n/(e.n+l.n) if e.n and l.n else 0.0
+        if weight:
+            denominator += weight
+            numerator += weight*(e.nu/e.n-l.nu/l.n)
+            survival += weight*(e.z/e.n-l.z/l.n)
+            support[0] += e.n
+            support[1] += l.n
+        rows.append({"direction": list(direction), "early": e.report(), "late": l.report(),
+                     "weight": weight,
+                     "mean_nu_difference": e.nu/e.n-l.nu/l.n if weight else None})
+    if not denominator:
+        raise ValueError("No common direction support")
+    return [early.nu/early.n-late.nu/late.n, numerator/denominator, survival/denominator], {
+        "supported_early": support[0], "supported_late": support[1],
+        "early_total": early.n, "late_total": late.n, "overlap_weight_sum": denominator,
+        "direction_rows": rows,
+    }
+
+
+def main():
+    started = time.perf_counter()
+    contract = json.loads((PARENT / "contract.json").read_text())
+    run = json.loads((PARENT / "data/run.json").read_text())
+    assert run["status"] == "completed"
+    assert len(run["batches"]) == contract["batches"]
+    assert all(b["samples"] == contract["samples_per_batch"] for b in run["batches"])
+    blocks, provenance = [], []
+    for entry in run["batches"]:
+        stats, ref = BASE.read_batch((PARENT / "data").resolve(), entry, contract)
+        blocks.append(stats); provenance.append(ref)
+    pooled, detail = score(BASE.aggregate(blocks))
+    deletes = [score(BASE.aggregate(blocks, omit=i))[0] for i in range(len(blocks))]
+    B = len(blocks)
+    center = [math.fsum(row[j] for row in deletes)/B for j in range(len(NAMES))]
+    covariance = [[(B-1)/B*math.fsum((row[j]-center[j])*(row[k]-center[k]) for row in deletes)
+                   for k in range(len(NAMES))] for j in range(len(NAMES))]
+    se = [math.sqrt(covariance[j][j]) for j in range(len(NAMES))]
+    # The already-published overlap survival statistic must retain its value:
+    # we are replacing only its outcome by nu_b for the new necessity check.
+    original = json.loads((PARENT / "results/result.json").read_text())
+    assert abs(pooled[2]-original["point"]["scalars"]["delta_overlap_D"]["estimate"]) < 1e-14
+    assert abs(se[2]-original["jackknife"]["uncertainty"]["delta_overlap_D"]["standard_error"]) < 1e-14
+    step_factor = 1/(contract["N"]-contract["b"])
+    window_factor = (contract["c"]-contract["b"])*step_factor
+    derived = {
+        "within_D_next_insertion_exit_difference": {"estimate": pooled[1]*step_factor, "se": se[1]*step_factor},
+        "within_D_window_scaled_exit_difference": {"estimate": pooled[1]*window_factor, "se": se[1]*window_factor},
+    }
+    result = {
+        "schema": "matching-one.directional-hazard-contrast.v1", "date": "2026-09-29",
+        "status": "retrospective_single_necessary_condition_not_new_validation",
+        "source_commit": "87f6644d3b57c7dbd67c44c63bcceffb6b7693ac",
+        "source_manifest": "analysis/completion-hazard-production-20260929/data/run.json",
+        "source_manifest_sha256": hashlib.sha256((PARENT / "data/run.json").read_bytes()).hexdigest(),
+        "source_reader_sha256": hashlib.sha256((PARENT / "analyze.py").read_bytes()).hexdigest(),
+        "samples": sum(block.samples for block in blocks),
+        "cutoffs": [contract[k] for k in ("a", "b", "c")], "N": contract["N"],
+        "question": "Does direction-only Markov closure pass its exact next-insertion hazard necessity?",
+        "null": "For every direction d, E[nu_b|early,d]=E[nu_b|late,d] under marked-rank Markovness",
+        "selection": "Question chosen after seeing the published finite-lag readout; no time, threshold, direction subset or feature scan",
+        "weights": "same exact-D overlap weights e*l/(e+l), recomputed in every whole-batch deletion",
+        "scalar_order": NAMES, "point": dict(zip(NAMES, pooled)), "se": dict(zip(NAMES, se)),
+        "covariance": covariance, "delete_one_vectors": deletes, "derived": derived,
+        "support": detail, "batches": provenance,
+        "boundaries": [
+            "Raw counts, next-insertion probability and window-scaled intensity have different units.",
+            "Within-(D,nu_b) immediate nu equality is tautological and is not tested as closure evidence.",
+            "Same 140k block, not independent corroboration; original prospective estimands and results unchanged.",
+            "One aggregate zero does not prove sectorwise equality; nonzero population weighted contrast contradicts the necessary null.",
+            "Batch SE is approximate; no multiplicity-adjusted certification or continuum extrapolation.",
+        ],
+        "python": platform.python_version(), "wall_seconds": time.perf_counter()-started,
+    }
+    (HERE / "result.json").write_text(json.dumps(result, indent=2, allow_nan=False)+"\n")
+    lines = ["# Directional next-insertion hazard: one retrospective check", "",
+             "Same completed 140k block, unchanged cutoffs, cohorts and exact-D overlap weights.", "",
+             "| Quantity | Estimate | Aligned batch SE |", "|---|---:|---:|"]
+    lines += [f"| {name} | {pooled[j]:.10g} | {se[j]:.10g} |" for j, name in enumerate(NAMES)]
+    lines += [f"| {name} | {value['estimate']:.10g} | {value['se']:.10g} |" for name, value in derived.items()]
+    lines += ["", "Positive nu contrast means greater early-cohort next-insertion exit probability.",
+              "Window scaling is not a finite-window exit estimate. The last survival contrast is the unchanged original result.",
+              "", "This necessity check was selected after the original report; it is not independent validation.",
+              "No new observations, subgroup searches, thresholds, cutoffs or samples are added.", ""]
+    (HERE / "RESULT.md").write_text("\n".join(lines))
+    print("\n".join(lines))
+
+
+if __name__ == "__main__":
+    main()
diff --git a/analysis/directional-hazard-contrast-20260929/result.json b/analysis/directional-hazard-contrast-20260929/result.json
new file mode 100644
index 000000000..432d96295
--- /dev/null
+++ b/analysis/directional-hazard-contrast-20260929/result.json
@@ -0,0 +1,460 @@
+{
+  "schema": "matching-one.directional-hazard-contrast.v1",
+  "date": "2026-09-29",
+  "status": "retrospective_single_necessary_condition_not_new_validation",
+  "source_commit": "87f6644d3b57c7dbd67c44c63bcceffb6b7693ac",
+  "source_manifest": "analysis/completion-hazard-production-20260929/data/run.json",
+  "source_manifest_sha256": "7c63af9d9c7dcf5eff4b0f14644fe83560d73190532d8528f3e0cf0c5f0af2df",
+  "source_reader_sha256": "83222cbd9e73d6fe7c452f32a12e67da5ae331d27aa47707460bd4a323181365",
+  "samples": 140000,
+  "cutoffs": [
+    154646,
+    155385,
+    156120
+  ],
+  "N": 262144,
+  "question": "Does direction-only Markov closure pass its exact next-insertion hazard necessity?",
+  "null": "For every direction d, E[nu_b|early,d]=E[nu_b|late,d] under marked-rank Markovness",
+  "selection": "Question chosen after seeing the published finite-lag readout; no time, threshold, direction subset or feature scan",
+  "weights": "same exact-D overlap weights e*l/(e+l), recomputed in every whole-batch deletion",
+  "scalar_order": [
+    "pooled_nu_early_minus_late",
+    "within_D_nu_early_minus_late",
+    "within_D_survival_early_minus_late"
+  ],
+  "point": {
+    "pooled_nu_early_minus_late": -2.1847327292311007,
+    "within_D_nu_early_minus_late": 1.6291680567969193,
+    "within_D_survival_early_minus_late": -0.0041499928123224636
+  },
+  "se": {
+    "pooled_nu_early_minus_late": 0.4794961673090409,
+    "within_D_nu_early_minus_late": 0.39097616465913365,
+    "within_D_survival_early_minus_late": 0.004511941723660848
+  },
+  "covariance": [
+    [
+      0.22991657446405975,
+      0.1775224479955025,
+      -0.00016932981133567867
+    ],
+    [
+      0.1775224479955025,
+      0.15286236133156597,
+      -0.00027094610841093555
+    ],
+    [
+      -0.00016932981133567867,
+      -0.00027094610841093555,
+      2.035761811771162e-05
+    ]
+  ],
+  "delete_one_vectors": [
+    [
+      -2.018000629567865,
+      1.783977958199084,
+      -0.002919145291324874
+    ],
+    [
+      -2.325420226578686,
+      1.556267956392056,
+      -0.00477357738437172
+    ],
+    [
+      -1.9892709822265857,
+      1.745199967518535,
+      -0.004934137714091619
+    ],
+    [
+      -2.090701930202556,
+      1.6792616516650039,
+      -0.0026017718607748838
+    ],
+    [
+      -2.2475318645691686,
+      1.5771929977455288,
+      -0.003950169891317702
+    ],
+    [
+      -2.2448636873931918,
+      1.6477781532691969,
+      -0.004951711375779151
+    ],
+    [
+      -2.492703518103994,
+      1.3825079611013626,
+      -0.004385325139063321
+    ],
+    [
+      -2.344017881267618,
+      1.4361283568998433,
+      -0.003003390681550195
+    ],
+    [
+      -2.068444329682549,
+      1.6948946913620369,
+      -0.00554643231550222
+    ],
+    [
+      -2.116659397318955,
+      1.7040425471394742,
+      -0.004327132560620712
+    ],
+    [
+      -2.204421520887692,
+      1.5834555641416685,
+      -0.0015369172452727713
+    ],
+    [
+      -2.1374625763463655,
+      1.6589565083934306,
+      -0.00642420751014362
+    ],
+    [
+      -2.1183765338579974,
+      1.6878822494583414,
+      -0.0050376866339166505
+    ],
+    [
+      -2.1866233444075363,
+      1.672343042596936,
+      -0.0037193135551129575
+    ]
+  ],
+  "derived": {
+    "within_D_next_insertion_exit_difference": {
+      "estimate": 1.5260240886453782e-05,
+      "se": 3.6622314246024563e-06
+    },
+    "within_D_window_scaled_exit_difference": {
+      "estimate": 0.01121627705154353,
+      "se": 0.0026917400970828055
+    }
+  },
+  "support": {
+    "supported_early": 24218,
+    "supported_late": 30490,
+    "early_total": 24218,
+    "late_total": 30505,
+    "overlap_weight_sum": 13369.571391148975,
+    "direction_rows": [
+      {
+        "direction": [
+          0,
+          1
+        ],
+        "early": {
+          "count": 11168,
+          "survivors": 5132,
+          "sum_I": 5982.097617860022,
+          "sum_nu_b": 1051934,
+          "mean_Z": 0.4595272206303725,
+          "mean_I": 0.5356462766708473,
+          "mean_nu_b": 94.19179799426934
+        },
+        "late": {
+          "count": 13113,
+          "survivors": 6042,
+          "sum_I": 7153.399168479315,
+          "sum_nu_b": 1226334,
+          "mean_Z": 0.46076412720201326,
+          "mean_I": 0.5455196498497152,
+          "mean_nu_b": 93.52047586364677
+        },
+        "weight": 6031.299534615543,
+        "mean_nu_difference": 0.6713221306225705
+      },
+      {
+        "direction": [
+          1,
+          -2
+        ],
+        "early": {
+          "count": 2,
+          "survivors": 0,
+          "sum_I": 1.1308753166338306,
+          "sum_nu_b": 394,
+          "mean_Z": 0.0,
+          "mean_I": 0.5654376583169153,
+          "mean_nu_b": 197.0
+        },
+        "late": {
+          "count": 4,
+          "survivors": 0,
+          "sum_I": 2.630434986274932,
+          "sum_nu_b": 1415,
+          "mean_Z": 0.0,
+          "mean_I": 0.657608746568733,
+          "mean_nu_b": 353.75
+        },
+        "weight": 1.3333333333333333,
+        "mean_nu_difference": -156.75
+      },
+      {
+        "direction": [
+          1,
+          -1
+        ],
+        "early": {
+          "count": 925,
+          "survivors": 245,
+          "sum_I": 610.7816934992491,
+          "sum_nu_b": 143455,
+          "mean_Z": 0.2648648648648649,
+          "mean_I": 0.6603045335127017,
+          "mean_nu_b": 155.08648648648648
+        },
+        "late": {
+          "count": 2072,
+          "survivors": 599,
+          "sum_I": 1437.5539123693968,
+          "sum_nu_b": 321652,
+          "mean_Z": 0.2890926640926641,
+          "mean_I": 0.6938001507574308,
+          "mean_nu_b": 155.23745173745175
+        },
+        "weight": 639.5061728395061,
+        "mean_nu_difference": -0.15096525096527103
+      },
+      {
+        "direction": [
+          1,
+          0
+        ],
+        "early": {
+          "count": 11184,
+          "survivors": 5083,
+          "sum_I": 6028.8292146761405,
+          "sum_nu_b": 1066339,
+          "mean_Z": 0.45448855507868385,
+          "mean_I": 0.5390584061763359,
+          "mean_nu_b": 95.34504649499284
+        },
+        "late": {
+          "count": 13150,
+          "survivors": 6032,
+          "sum_I": 7091.969947987316,
+          "sum_nu_b": 1218377,
+          "mean_Z": 0.45870722433460076,
+          "mean_I": 0.5393133040294537,
+          "mean_nu_b": 92.65224334600761
+        },
+        "weight": 6043.79058107997,
+        "mean_nu_difference": 2.6928031489852344
+      },
+      {
+        "direction": [
+          1,
+          1
+        ],
+        "early": {
+          "count": 938,
+          "survivors": 261,
+          "sum_I": 646.2467634577035,
+          "sum_nu_b": 144896,
+          "mean_Z": 0.27825159914712155,
+          "mean_I": 0.6889624343898758,
+          "mean_nu_b": 154.4733475479744
+        },
+        "late": {
+          "count": 2148,
+          "survivors": 620,
+          "sum_I": 1488.0321827081254,
+          "sum_nu_b": 326291,
+          "mean_Z": 0.2886405959031657,
+          "mean_I": 0.6927524128063899,
+          "mean_nu_b": 151.90456238361267
+        },
+        "weight": 652.8917692806222,
+        "mean_nu_difference": 2.568785164361742
+      },
+      {
+        "direction": [
+          1,
+          2
+        ],
+        "early": {
+          "count": 0,
+          "survivors": 0,
+          "sum_I": 0.0,
+          "sum_nu_b": 0,
+          "mean_Z": null,
+          "mean_I": null,
+          "mean_nu_b": null
+        },
+        "late": {
+          "count": 10,
+          "survivors": 1,
+          "sum_I": 5.775421020756097,
+          "sum_nu_b": 3024,
+          "mean_Z": 0.1,
+          "mean_I": 0.5775421020756097,
+          "mean_nu_b": 302.4
+        },
+        "weight": 0.0,
+        "mean_nu_difference": null
+      },
+      {
+        "direction": [
+          2,
+          -1
+        ],
+        "early": {
+          "count": 1,
+          "survivors": 0,
+          "sum_I": 0.0,
+          "sum_nu_b": 301,
+          "mean_Z": 0.0,
+          "mean_I": 0.0,
+          "mean_nu_b": 301.0
+        },
+        "late": {
+          "count": 3,
+          "survivors": 1,
+          "sum_I": 1.0594669455357395,
+          "sum_nu_b": 559,
+          "mean_Z": 0.3333333333333333,
+          "mean_I": 0.35315564851191317,
+          "mean_nu_b": 186.33333333333334
+        },
+        "weight": 0.75,
+        "mean_nu_difference": 114.66666666666666
+      },
+      {
+        "direction": [
+          2,
+          1
+        ],
+        "early": {
+          "count": 0,
+          "survivors": 0,
+          "sum_I": 0.0,
+          "sum_nu_b": 0,
+          "mean_Z": null,
+          "mean_I": null,
+          "mean_nu_b": null
+        },
+        "late": {
+          "count": 5,
+          "survivors": 0,
+          "sum_I": 6.218543349964713,
+          "sum_nu_b": 1253,
+          "mean_Z": 0.0,
+          "mean_I": 1.2437086699929425,
+          "mean_nu_b": 250.6
+        },
+        "weight": 0.0,
+        "mean_nu_difference": null
+      }
+    ]
+  },
+  "batches": [
+    {
+      "file": "production-square-L512-b00.csv.gz",
+      "samples": 10000,
+      "seed": "18419746559907930925",
+      "sha256": "32f16cc37728d67fdbe3446a48da1bb4d36c48fc4f051648ac8e36e2595d4026",
+      "sha256_verified": true
+    },
+    {
+      "file": "production-square-L512-b01.csv.gz",
+      "samples": 10000,
+      "seed": "13442365983088886094",
+      "sha256": "fdc708228c1e0535787aa3426ea725b276ecef0b8ba193215b1bf93512518892",
+      "sha256_verified": true
+    },
+    {
+      "file": "production-square-L512-b02.csv.gz",
+      "samples": 10000,
+      "seed": "12192384056273429425",
+      "sha256": "57b5f9ef9bcc6f21a5a14f5e7b6c24fccb94388f349943738916cc1700ab4897",
+      "sha256_verified": true
+    },
+    {
+      "file": "production-square-L512-b03.csv.gz",
+      "samples": 10000,
+      "seed": "16795263932832913938",
+      "sha256": "65ec5cee09f76a03c69b702a21bb6f27dd70930b48e3b7f047de33103032eaa4",
+      "sha256_verified": true
+    },
+    {
+      "file": "production-square-L512-b04.csv.gz",
+      "samples": 10000,
+      "seed": "5606540329541986450",
+      "sha256": "033189fab591caa248c66c9710d63104036dc3d26b88e495d965615a84ecc4fb",
+      "sha256_verified": true
+    },
+    {
+      "file": "production-square-L512-b05.csv.gz",
+      "samples": 10000,
+      "seed": "176734671850227651",
+      "sha256": "b38b95839aab1c6f6575a9b94702c41a032a2cc00430c6eb904840472488fbd4",
+      "sha256_verified": true
+    },
+    {
+      "file": "production-square-L512-b06.csv.gz",
+      "samples": 10000,
+      "seed": "14922363405798405830",
+      "sha256": "330829d82ec4dad029f53dc07fe8bda5252bb35118aa9b507012c73df88e5a35",
+      "sha256_verified": true
+    },
+    {
+      "file": "production-square-L512-b07.csv.gz",
+      "samples": 10000,
+      "seed": "3540279280158270918",
+      "sha256": "a292a7a8be0fb5f79f2449745709c51942fc4261fd50e542569d33d04005ee30",
+      "sha256_verified": true
+    },
+    {
+      "file": "production-square-L512-b08.csv.gz",
+      "samples": 10000,
+      "seed": "182843979474712324",
+      "sha256": "113325911eb95469467bfe73496d5b57448413125dcea383b55b0ff7875db4ac",
+      "sha256_verified": true
+    },
+    {
+      "file": "production-square-L512-b09.csv.gz",
+      "samples": 10000,
+      "seed": "4164285986391024372",
+      "sha256": "b5712154670f8179ddf32487f35a794ea16b5ef4f0cdad2025c3f8e447f321da",
+      "sha256_verified": true
+    },
+    {
+      "file": "production-square-L512-b10.csv.gz",
+      "samples": 10000,
+      "seed": "15500924042285991404",
+      "sha256": "98bb914abc5188216cd5b9b4584ba46539bd4c16c1b4d745b6a8033748243668",
+      "sha256_verified": true
+    },
+    {
+      "file": "production-square-L512-b11.csv.gz",
+      "samples": 10000,
+      "seed": "17186215622830953477",
+      "sha256": "73aaa04f05ee279532f7c781a97acda5191840cf79fc2cee72f2a6c9caaed7f1",
+      "sha256_verified": true
+    },
+    {
+      "file": "production-square-L512-b12.csv.gz",
+      "samples": 10000,
+      "seed": "11279241988091952515",
+      "sha256": "75f556befdd4986e26f6594f7d2ec44fa6517b9b19021b9cbaa612116471875c",
+      "sha256_verified": true
+    },
+    {
+      "file": "production-square-L512-b13.csv.gz",
+      "samples": 10000,
+      "seed": "10329155733035115099",
+      "sha256": "d5ffad750a152bd62325ae22375496b861da96f8687ded8ad7c0a8220d06a202",
+      "sha256_verified": true
+    }
+  ],
+  "boundaries": [
+    "Raw counts, next-insertion probability and window-scaled intensity have different units.",
+    "Within-(D,nu_b) immediate nu equality is tautological and is not tested as closure evidence.",
+    "Same 140k block, not independent corroboration; original prospective estimands and results unchanged.",
+    "One aggregate zero does not prove sectorwise equality; nonzero population weighted contrast contradicts the necessary null.",
+    "Batch SE is approximate; no multiplicity-adjusted certification or continuum extrapolation."
+  ],
+  "python": "3.11.15",
+  "wall_seconds": 0.32170062500517815
+}
diff --git a/analysis/exact-completion-pairs-20260929/README.md b/analysis/exact-completion-pairs-20260929/README.md
new file mode 100644
index 000000000..3c9491c91
--- /dev/null
+++ b/analysis/exact-completion-pairs-20260929/README.md
@@ -0,0 +1,89 @@
+# Exact conditional averaging of the existing safe-insertion block
+
+This is a post-readout estimator refinement of the fixed
+[safe-insertion target](../safe-insertion-production-20260929/contract.json),
+not a new acquisition, changed source, or independent confirmation.
+It replays exactly the same 14 x 10,000 permutations and replaces each
+four-probe average by its exact conditional expectation.
+
+## Geometry and target
+
+The [projected component-pair proof](../../notes/projected-component-pair-criterion-20260929.md)
+gives a necessary and sufficient criterion for two individually safe vacancies
+to complete ambient rank two together. Integer transverse potentials detect
+inconsistent shared-component or physical-edge constraints. Component-pair
+buckets generate only true synergy edges, then deduplicate them. No quadratic
+all-vacancy-pair scan is used in production. The four small physical controls
+do use direct insertion for their reference pair sets.
+
+For one rank-one prefix, let m=N-b, c=nu_b, s=m-c and e be the number of
+unordered safe synergy pairs. For s>0:
+
+    E[nu(A+V)-nu(A) | A, V uniform safe] = 2e/s.
+    P(survive two more insertions | A) = [s(s-1)-2e]/[m(m-1)].
+
+Conditioning on the full original prefix sample also fixes J1, directions,
+cell membership and overlap weights. Thus replacing the actual probe mean
+by 2e/s preserves the finite-sample conditional target and removes probe
+randomness; it does not remove configuration variation. This argument does
+not assume the current pair graph closes recursively.
+
+The proof's generic expected bound O(N*z^2+z^2*e) uses merged contact lists
+in its adjacency scan. The implementation instead uses bounded nested lists,
+O(N*z^3+z^2*e); z is fixed at four (square) or six (controls), giving the same
+expected O(N+e) fixed-lattice bound. Hash costs are expected, not worst-case;
+large e can still make explicit enumeration expensive.
+
+## Computation and inference
+
+`engine.cpp` reuses the unchanged lifted union-find, bounded integer RNG and
+full Fisher-Yates permutation stream. It builds transverse occupied-component
+potentials separately, computes c and e, and saves one row per old prefix.
+`run.py` takes all seeds and sample counts from the original manifest. It
+records source hashes, compiler, commands, timing and the original dependency
+groups. No new random prefix is generated; the first-128 timing sample occurs
+again in batch zero and is not extra data.
+
+`analyze.py` requires row-for-row equality of `(J1,rank_b,dx_b,dy_b,nu_b)` across
+all 140,000 old/replayed rows. It retains original early/late definitions,
+exact (D,c) support, weights n_early*n_late/(n_early+n_late), and the original
+direction-only secondary target. Every whole-batch deletion recomputes
+support and weights. Four actual old probes are averaged before inference.
+
+Eight jointly scored coordinates include the exact and old probe contrasts,
+their paired difference, the exact edge/two-step contrasts, cohort means,
+and the unchanged secondary. Derived consumer factors are applied inside
+cells; the two-step contrast is not another independent measurement.
+The JSON contains integer sufficient sums, all 14 deletion vectors and full
+8x8 covariance. The measured SE ratio is descriptive, not a guarantee that
+every empirical jackknife SE must shrink. No cutoff, feature, sign or size
+search is included.
+
+## Reproduction
+
+Owner routing preference updated 2026-09-30: medium/large replays go to
+Huawei first; local work is for small calculations, scoring and editing.
+This run began locally, was paused by the owner, then retained batches0..7
+and transferred only the missing batches8..13 to TV2N0X. `resume.py` records
+the first local restart; `finish_on_cloud.py` is this specific six-batch
+handoff, not a generic scheduler. No stopped partial batch is scored.
+
+To rescore the saved data without overwriting the report:
+
+```sh
+python3 analysis/exact-completion-pairs-20260929/analyze.py \
+  --output-dir analysis/exact-completion-pairs-20260929/reproduction
+```
+
+To reproduce geometry from seeds, use an isolated checkout of the replay
+source commit named in `data/run.json` (before replay outputs existed) on
+the selected compute host; use `--compiler g++` on the Huawei image:
+
+```sh
+python3 analysis/exact-completion-pairs-20260929/run.py \
+  --source-commit 8b5eee0d47b6c220c2ba46a5bae08d2d19b392a4 --workers 4
+```
+
+The runner deliberately does not overwrite an existing replay directory.
+This is output preservation, not a research permission gate. The original
+production and its published primary readout remain unchanged.
diff --git a/analysis/exact-completion-pairs-20260929/analyze.py b/analysis/exact-completion-pairs-20260929/analyze.py
new file mode 100644
index 000000000..67ad09488
--- /dev/null
+++ b/analysis/exact-completion-pairs-20260929/analyze.py
@@ -0,0 +1,249 @@
+#!/usr/bin/env python3
+"""Score the exact conditional mean on the SAME fixed 140k source prefixes.
+
+Reuse the committed original scorer for validation, cells and overlap weights;
+only actual original probes enter its Moments. Exact edge sums stay integers.
+"""
+import argparse
+import csv
+from datetime import datetime, timezone
+import gzip
+import hashlib
+import importlib.util
+from itertools import zip_longest
+import json
+import math
+from pathlib import Path
+import subprocess
+import sys
+
+ROOT = Path(__file__).resolve().parent
+REPO = ROOT.parent.parent
+OLD = ROOT.parent / "safe-insertion-production-20260929"
+OLD_COMMIT = "2a1677e1efe1137dffc07c3c4ae4ceebf15ecfd3"
+METRICS = ("RB_deltaY", "original_probe_deltaY", "paired_RB_minus_probe",
+           "RB_deltaE", "RB_deltaZ2", "RB_earlyY", "RB_lateY", "secondary_deltaNu")
+WARNINGS = [
+    "Estimator refinement after the original readout; same 140k prefixes and batch blocks, zero new samples.",
+    "No independent confirmation: reproducing old primary/secondary is source/estimand identity, not scientific replication.",
+    "Exact means conditional mean 2e/(m-c); finite-prefix uncertainty remains. Four actual old probes are averaged per prefix.",
+    "SE_RB/SE_probe is descriptive; variance reduction does not guarantee empirical delete-batch jackknife SE shrinkage.",
+    "No automatic Markov claim: one weighted moment can cancel across cells; finite L512 count clock only, no continuum claim.",
+]
+
+
+def require(condition, message):
+    if not condition:
+        raise ValueError(message)
+
+
+def git(*args):
+    return subprocess.check_output(["git", "-C", str(REPO), *args])
+
+
+def reference(path):
+    raw = path.read_bytes()
+    return {"path": str(path), "bytes": len(raw), "sha256": hashlib.sha256(raw).hexdigest()}
+
+
+def aligned_batch(old, original, replay):
+    identity, old_path, old_hash, old_entry = original
+    new_id, new_path, new_hash, new_entry = replay
+    require(identity == new_id and old_entry["seed"] == new_entry["seed"], "batch/seed mismatch")
+    require(new_entry["original_file"] == old_entry["file"] and
+            new_entry["dependency_group"] == old_entry["dependency_group"], "source block mismatch")
+    refs = [reference(p) for p in (old_path, new_path)]
+    require([r["sha256"] for r in refs] == [old_hash, new_hash], f"batch {identity}: hash mismatch")
+    stats, edges, candidates = old.BatchStats(), {}, 0
+    m = old.FIXED["N"] - old.FIXED["b"]
+    with gzip.open(old_path, "rt", newline="", encoding="utf-8-sig") as fo, \
+            gzip.open(new_path, "rt", newline="", encoding="utf-8-sig") as fn:
+        ro, rn = csv.reader(fo, strict=True), csv.reader(fn, strict=True)
+        require(next(ro, None) == list(old.COLUMNS), f"batch {identity}: old header mismatch")
+        require(next(rn, None) == list(old.COLUMNS[:5]) + ["synergy_edges", "nonlocal_candidates"],
+                f"batch {identity}: replay header mismatch")
+        for line, (o, n) in enumerate(zip_longest(ro, rn), 2):
+            label = f"batch {identity}, CSV line {line}"
+            require(o is not None and n is not None, f"{label}: unequal row counts")
+            require(len(o) == 9 and len(n) == 7 and o[:5] == n[:5], f"{label}: prefix mismatch")
+            values = [old.integer(v, label) for v in o]
+            stats.observe(values, old.FIXED)
+            j1, rank, dx, dy, c = values[:5]
+            e, candidate = [old.integer(v, label) for v in n[5:]]
+            if rank != 1:
+                require((e, candidate) == (-1, -1), f"{label}: outside-risk sentinel mismatch")
+                continue
+            s = m - c
+            require(0 <= e <= s * (s - 1) // 2 and candidate >= 0, f"{label}: invalid pair counts")
+            require(s > 1 or (e, candidate) == (0, 0), f"{label}: impossible pair counts")
+            candidates += candidate
+            if not s:
+                continue
+            if dx < 0 or (dx == 0 and dy < 0):
+                dx, dy = -dx, -dy
+            edges.setdefault((dx, dy, c), [0, 0])[int(j1 > old.FIXED["a"])] += e
+    require(stats.samples == 10000, f"batch {identity}: expected exactly 10000 aligned rows")
+    return (stats, edges), {"batch": identity, "rows": stats.samples, "all_first5_identical": True,
+                           "seed": old_entry["seed"], "dependency_group": old_entry["dependency_group"],
+                           "sum_nonlocal_candidates": candidates, "original": refs[0], "replay": refs[1]}
+
+
+def estimate(old, blocks, omit=None, detailed=False):
+    chosen = [block for i, block in enumerate(blocks) if i != omit]
+    total = old.aggregate([block[0] for block in chosen])
+    _, primary, secondary = old.estimate(total, old.FIXED, detailed=True)
+    require(primary["status"] == secondary["status"] == "scoreable", f"absent support, deletion={omit}")
+    edges = {}
+    for _, cells in chosen:
+        for key, sums in cells.items():
+            target = edges.setdefault(key, [0, 0])
+            for g in range(2):
+                target[g] += sums[g]
+    m = old.FIXED["N"] - old.FIXED["b"]
+    terms, rows = [[] for _ in range(7)], []
+    for cell in primary.pop("cells"):
+        key = (*cell["direction"], cell["nu_b"])
+        s, means, cohorts = m - key[2], [], {}
+        for g, name in enumerate(old.GROUPS):
+            moment, sum_e = cell[name], edges[key][g]
+            n, sum_probe = moment["prefixes"], moment["sum_probe_Y"]
+            mean_e = sum_e / n if n else None
+            mean_y = 2 * sum_e / (s * n) if n else None
+            means.append((mean_e, mean_y))
+            cohorts[name] = dict(n=n, sum_e=sum_e, sum_old_probe_Y=sum_probe,
+                                 mean_e=mean_e, meanY=mean_y, mean_old_prefixprobe_Y=moment["mean_Ybar"])
+        record = {k: cell[k] for k in ("direction", "nu_b", "common_support", "overlap_weight", "normalized_weight")}
+        record.update(safe_sites=s, **cohorts)
+        if cell["common_support"]:
+            de, dy = (means[0][j] - means[1][j] for j in range(2))
+            dp = cell["deltaY"]  # Actual four-probe prefix mean, unchanged original arithmetic.
+            vector = [dy, dp, dy - dp, de, -2 * de / (m * (m - 1)), means[0][1], means[1][1]]
+            for target, value in zip(terms, vector):
+                target.append(cell["overlap_weight"] * value)
+            record.update(zip(METRICS[:5], vector[:5]))
+        rows.append(record)
+    vector = [math.fsum(t) / primary["overlap_weight_sum"] for t in terms] + [secondary["delta"]]
+    vector[2], vector[4] = vector[0] - vector[1], -2 * vector[3] / (m * (m - 1))
+    primary.update(delta=vector[0], deltaE=vector[3], deltaZ2=vector[4], original_probe_deltaY=vector[1],
+                   weighted_early_mean=vector[5], weighted_late_mean=vector[6])
+    secondary_rows = secondary.pop("cells")
+    if detailed:
+        primary["cells"] = rows
+        secondary["cells"] = [{**{k: r[k] for k in ("direction", "common_support", "overlap_weight", "normalized_weight", "deltaNu")},
+                               **{g: dict(n=r[g]["prefixes"], sumNu=r[g]["sum_nu_b"], meanNu=r[g]["mean_nu_b"])
+                                  for g in old.GROUPS}} for r in secondary_rows]
+    return vector, primary, secondary, total.totals(4)
+
+
+def markdown(result):
+    lines = ["# Same-prefix exact completion-pair refinement", "",
+             "All 140000 rows aligned across 14 original batches; every first-five-field tuple matched.", "",
+             "Early minus late; fixed a,b and original overlap weights. No new random prefixes.", "",
+             "| Metric | Estimate | Batch jackknife SE |", "|---|---:|---:|"]
+    for key in METRICS:
+        lines.append(f"| {key} | {result['point_estimates'][key]:.15g} | {result['standard_errors'][key]:.15g} |")
+    lines += ["", f"Descriptive SE_RB/SE_probe: {result['refinement']['SE_RB_over_SE_probe']}. "
+              f"Variance ratio: {result['refinement']['variance_RB_over_probe']}.", "",
+              "| Support / cohort | Rank-one | Eligible | Supported |", "|---|---:|---:|---:|"]
+    for name in ("primary", "secondary"):
+        for group in ("early", "late", "combined"):
+            c = result[name]["coverage"][group]
+            lines.append(f"| {name} / {group} | {c['rank1_prefixes']} | {c['eligible_prefixes']} | {c['supported_prefixes']} |")
+    lines += ["", "Original committed primary and secondary point/SE reproduced (absolute tolerances 1e-14 / 1e-12).",
+              "This is a source/estimand identity check, not scientific replication.", "",
+              "JSON retains exact integer cell sums, mean_e, meanY, all 14 aligned deletion vectors/supports, "
+              "full 8x8 covariance, and input/source hashes and commits. First seven metrics share exact (D,c) cells; "
+              "the secondary uses all rank-one prefixes within D, including c=m.", "",
+              "Y=2e/(m-c); original prefix probe mean=(Y0+Y1+Y2+Y3)/4. "
+              "RB_deltaE averages cell mean-edge differences; RB_deltaZ2=-2*RB_deltaE/[m(m-1)]. "
+              "c=m has no conditional safe mean. Singular covariance is expected; no inverse is used.", ""]
+    return "\n".join(lines + [f"- {w}" for w in WARNINGS]) + "\n"
+
+
+def main():
+    ap = argparse.ArgumentParser(description=__doc__)
+    ap.add_argument("--data-dir", type=Path, default=ROOT / "data")
+    ap.add_argument("--output-dir", type=Path, default=ROOT / "results")
+    args = ap.parse_args()
+    output = args.output_dir.resolve()
+    require(ROOT in output.parents, "outputs must be inside this new analysis directory")
+    destinations = [output / "result.json", output / "RESULT.md"]
+    require(not any(p.exists() for p in destinations), "refusing to overwrite outputs; choose a fresh output directory")
+    for path in (OLD / "analyze.py", OLD / "results/result.json", OLD / "contract.json", OLD / "data/run.json"):
+        require(path.read_bytes() == git("show", f"{OLD_COMMIT}:{path.relative_to(REPO)}"), f"original changed: {path}")
+    sys.dont_write_bytecode = True
+    spec = importlib.util.spec_from_file_location("original_safe_insertion_score", OLD / "analyze.py")
+    old = importlib.util.module_from_spec(spec)
+    sys.modules[spec.name] = old
+    spec.loader.exec_module(old)
+    contract, contract_ref = old.read_json(OLD / "contract.json")
+    old.check_contract(contract)
+    original, original_ref = old.read_json(OLD / "data/run.json")
+    run, run_ref = old.read_json(args.data_dir.resolve() / "run.json")
+    old.check_contract(run["contract"], "replay.contract")
+    require(run["contract"] == original["contract"] == contract, "contract changed")
+    require(run["new_random_prefixes"] == 0 and run["original_manifest_sha256"] == original_ref["sha256"], "wrong source manifest")
+    original_entries = old.batch_entries(original, OLD / "data", old.FIXED)
+    replay_entries = old.batch_entries(run, args.data_dir.resolve(), old.FIXED)
+    require([v[0] for v in original_entries] == [v[0] for v in replay_entries] == list(range(14)), "require original 14 batch IDs")
+    blocks, manifests = [], []
+    for o, n in zip(original_entries, replay_entries):
+        block, manifest = aligned_batch(old, o, n)
+        blocks.append(block)
+        manifests.append(manifest)
+    point, primary, secondary, totals = estimate(old, blocks, detailed=True)
+    deletions = []
+    for i in range(14):
+        v, p, s, t = estimate(old, blocks, omit=i)
+        deletions.append(dict(deleted_batch=manifests[i]["batch"], vector=v, primary_support=p, secondary_support=s, totals=t))
+    vectors = [r["vector"] for r in deletions]
+    mean = [math.fsum(v[k] for v in vectors) / 14 for k in range(8)]
+    covariance = [[13 / 14 * math.fsum((v[j] - mean[j]) * (v[k] - mean[k]) for v in vectors)
+                   for k in range(8)] for j in range(8)]
+    se = [math.sqrt(covariance[k][k]) for k in range(8)]
+    committed, committed_ref = old.read_json(OLD / "results/result.json")
+    require(committed["status"] == "scored" and totals == committed["totals"], "original total identity failed")
+    checks = {}
+    for index, key in ((1, "deltaY"), (7, "secondary_deltaNu")):
+        dp, ds = point[index] - committed["point_estimates"][key], se[index] - committed["standard_errors"][key]
+        require(abs(dp) <= 1e-14 and abs(ds) <= 1e-12, f"original {key} point/SE identity failed: {dp}, {ds}")
+        checks[key] = dict(committed_point=committed["point_estimates"][key], committed_SE=committed["standard_errors"][key],
+                           reproduced_point=point[index], reproduced_SE=se[index], point_error=dp, SE_error=ds)
+    ratio = se[0] / se[1] if se[1] else None
+    sources = [ROOT / n for n in ("analyze.py", "engine.cpp", "run.py")]
+    sources += [OLD / n for n in ("analyze.py", "engine.cpp", "run.py")]
+    geometry = ROOT.parent / "completion-hazard-production-20260929/engine.cpp"
+    require(reference(geometry)["sha256"] == run["included_geometry_sha256"] == original["included_geometry_sha256"], "geometry changed")
+    for name, digest in run["source_sha256"].items():
+        require(reference(ROOT / name)["sha256"] == digest, f"replay source changed: {name}")
+    result = dict(schema="matching-one.exact-pair-score.v1", status="scored", parameters=old.FIXED,
+                  created_utc=datetime.now(timezone.utc).isoformat(), m=old.FIXED["N"] - old.FIXED["b"],
+                  metric_order=METRICS, point_vector=point, point_estimates=dict(zip(METRICS, point)),
+                  standard_errors=dict(zip(METRICS, se)), primary=primary, secondary=secondary, totals=totals,
+                  original_identity=dict(status="passed", point_tolerance=1e-14, SE_tolerance=1e-12, checks=checks),
+                  refinement=dict(SE_RB_over_SE_probe=ratio, variance_RB_over_probe=ratio ** 2 if ratio is not None else None),
+                  joint_uncertainty=dict(method="aligned whole-batch delete-one; support and weights recomputed",
+                      covariance_formula="(B-1)/B * sum_j (theta_(-j)-mean)(theta_(-j)-mean)^T", batch_count=14,
+                      metric_order=METRICS, covariance=covariance, delete_one_mean_vector=mean, delete_one=deletions,
+                      point_estimator="pooled plug-in; no jackknife bias correction"),
+                  input_manifest=dict(contract=contract_ref, original_run=original_ref, replay_run=run_ref,
+                                      committed_original_result=committed_ref, aligned_batches=manifests),
+                  source_manifest=dict(original_result_commit=OLD_COMMIT, original_acquisition_commit=original["source_commit"],
+                      replay_source_commit=run["source_commit"], scoring_head_commit=git("rev-parse", "HEAD").decode().strip(),
+                      files=[reference(p) for p in sources + [geometry, REPO / "notes/completion-pair-synergy-20260929.md"]],
+                      note="Commit references and working-file hashes are distinct; replay code may be uncommitted."),
+                  warnings=WARNINGS)
+    encoded = json.dumps(result, indent=2, allow_nan=False) + "\n"
+    prose = markdown(result)
+    output.mkdir(parents=True, exist_ok=True)
+    for path, content in zip(destinations, (encoded, prose)):
+        with path.open("x", encoding="utf-8") as handle:
+            handle.write(content)
+    print(json.dumps({"status": "scored", "outputs": [str(p) for p in destinations]}))
+
+
+if __name__ == "__main__":
+    try:
+        main()
+    except (OSError, EOFError, ValueError, KeyError, csv.Error, subprocess.CalledProcessError) as error:
+        sys.exit(f"exact-pair scoring error: {error}")
diff --git a/analysis/exact-completion-pairs-20260929/benchmark.json b/analysis/exact-completion-pairs-20260929/benchmark.json
new file mode 100644
index 000000000..bb7e52316
--- /dev/null
+++ b/analysis/exact-completion-pairs-20260929/benchmark.json
@@ -0,0 +1,21 @@
+{
+  "purpose": "One-time physical witness control and same-prefix runtime/source check before full replay",
+  "compiler_command": "clang++ -O3 -std=c++17 analysis/exact-completion-pairs-20260929/engine.cpp -o <temporary>/engine",
+  "physical_controls": [
+    {"lattice": "square", "L": 4, "mask": 4403, "nu": 0, "edges": 2},
+    {"lattice": "square", "L": 4, "mask": 4405, "nu": 0, "edges": 3},
+    {"lattice": "triangular", "L": 4, "mask": 4371, "nu": 0, "edges": 4},
+    {"lattice": "triangular", "L": 4, "mask": 4402, "nu": 0, "edges": 6}
+  ],
+  "control_comparison": "Full enumerated pair sets equalled direct copied-Geometry two-insertion sets for these four witnesses only",
+  "replay_command": "<temporary>/engine 512 128 9722445355900932119 155385 <temporary>/replay-b00-first128.csv",
+  "prefixes": 128,
+  "engine_seconds": 1.83744,
+  "rank_one_prefixes": 56,
+  "maximum_edges": 31070,
+  "row_alignment": "All first five fields equalled the existing safe-insertion batch 0 first 128 rows",
+  "nonlocal_candidate_check": "candidate count <= 6 * edge count in all rank-one rows",
+  "new_random_prefixes": 0,
+  "scientific_cohort_contrast_computed": false,
+  "dependency": "These rows recur in full replay batch 0; they are not extra evidence"
+}
diff --git a/analysis/exact-completion-pairs-20260929/cloud.log b/analysis/exact-completion-pairs-20260929/cloud.log
new file mode 100644
index 000000000..7f304d36d
--- /dev/null
+++ b/analysis/exact-completion-pairs-20260929/cloud.log
@@ -0,0 +1,7 @@
+batch=9: prefixes=10000 seconds=381.735
+batch=8: prefixes=10000 seconds=399.725
+batch=10: prefixes=10000 seconds=401.385
+batch=13: prefixes=10000 seconds=415.405
+batch=11: prefixes=10000 seconds=416.066
+batch=12: prefixes=10000 seconds=417.618
+Cloud continuation completed in 420.381s
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index 000000000..77fb5fe1b
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+}
diff --git a/analysis/exact-completion-pairs-20260929/data/run.json b/analysis/exact-completion-pairs-20260929/data/run.json
new file mode 100644
index 000000000..1191abe1a
--- /dev/null
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@@ -0,0 +1,297 @@
+{
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+    "weights": "w=n_early*n_late/(n_early+n_late) on common exact cells; normalize over cells and recompute all support and weights on aligned whole-batch deletion",
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+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch4",
+      "runtime_seconds": 233.1643263330334,
+      "engine_log": "prefixes=10000 seconds=232.437",
+      "command": "/var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-exact-pairs-resume-_2g_wuy3/engine 512 10000 13568940959106988201 155385 /var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-exact-pairs-resume-_2g_wuy3/exact-square-L512-b04.csv"
+    },
+    {
+      "batch": 5,
+      "samples": 10000,
+      "seed": "14221577344058561937",
+      "file": "exact-square-L512-b05.csv.gz",
+      "sha256": "5f275599206d6db5099bf138477821dfd4ea516f411fefe36c5de1f07477069a",
+      "original_file": "production-square-L512-b05.csv.gz",
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch5",
+      "runtime_seconds": 235.9306038749637,
+      "engine_log": "prefixes=10000 seconds=235.203",
+      "command": "/var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-exact-pairs-resume-_2g_wuy3/engine 512 10000 14221577344058561937 155385 /var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-exact-pairs-resume-_2g_wuy3/exact-square-L512-b05.csv"
+    },
+    {
+      "batch": 6,
+      "samples": 10000,
+      "seed": "2117620933003635365",
+      "file": "exact-square-L512-b06.csv.gz",
+      "sha256": "8a9ceb8fb776d7a318565864145efa3a4f8131b95c984c4a5d6a4c05b3e2ab84",
+      "original_file": "production-square-L512-b06.csv.gz",
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch6",
+      "runtime_seconds": 233.67998904100386,
+      "engine_log": "prefixes=10000 seconds=232.956",
+      "command": "/var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-exact-pairs-resume-_2g_wuy3/engine 512 10000 2117620933003635365 155385 /var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-exact-pairs-resume-_2g_wuy3/exact-square-L512-b06.csv"
+    },
+    {
+      "batch": 7,
+      "samples": 10000,
+      "seed": "10091810389689459151",
+      "file": "exact-square-L512-b07.csv.gz",
+      "sha256": "46f60c48a55d02559dc5a21f53d4a49eda3feb3f456837b322cc3a133d302b9e",
+      "original_file": "production-square-L512-b07.csv.gz",
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch7",
+      "runtime_seconds": 234.30471350002335,
+      "engine_log": "prefixes=10000 seconds=233.58",
+      "command": "/var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-exact-pairs-resume-_2g_wuy3/engine 512 10000 10091810389689459151 155385 /var/folders/qv/09841bwx4x9b1gqkht0f9nt40000gn/T/matching-exact-pairs-resume-_2g_wuy3/exact-square-L512-b07.csv"
+    },
+    {
+      "batch": 8,
+      "samples": 10000,
+      "seed": "11008454278899210265",
+      "file": "exact-square-L512-b08.csv.gz",
+      "sha256": "06a6924043ef47a1c664305490004c7cab7b137fa9a85e0735914a98921267e0",
+      "original_file": "production-square-L512-b08.csv.gz",
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch8",
+      "runtime_seconds": 399.7739146189997,
+      "engine_log": "prefixes=10000 seconds=399.725",
+      "command": "/workspace/Matching-One-TV2N0X/exact-pairs-20260930/analysis/exact-completion-pairs-20260929/exact-pair-cloud-q6j7j4co/engine 512 10000 11008454278899210265 155385 /workspace/Matching-One-TV2N0X/exact-pairs-20260930/analysis/exact-completion-pairs-20260929/exact-pair-cloud-q6j7j4co/exact-square-L512-b08.csv",
+      "execution": "TV2N0X"
+    },
+    {
+      "batch": 9,
+      "samples": 10000,
+      "seed": "17332517276911702520",
+      "file": "exact-square-L512-b09.csv.gz",
+      "sha256": "525d35bfc340547d0e0617ab1074b34818bcbd1762f438f6e9ffd4170305d17f",
+      "original_file": "production-square-L512-b09.csv.gz",
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch9",
+      "runtime_seconds": 381.7914042969933,
+      "engine_log": "prefixes=10000 seconds=381.735",
+      "command": "/workspace/Matching-One-TV2N0X/exact-pairs-20260930/analysis/exact-completion-pairs-20260929/exact-pair-cloud-q6j7j4co/engine 512 10000 17332517276911702520 155385 /workspace/Matching-One-TV2N0X/exact-pairs-20260930/analysis/exact-completion-pairs-20260929/exact-pair-cloud-q6j7j4co/exact-square-L512-b09.csv",
+      "execution": "TV2N0X"
+    },
+    {
+      "batch": 10,
+      "samples": 10000,
+      "seed": "17211728716206152366",
+      "file": "exact-square-L512-b10.csv.gz",
+      "sha256": "80d47a4fabc6a5419b7e5cfc7f1964fbe08c8080f015456012a9f775e2f8e4bf",
+      "original_file": "production-square-L512-b10.csv.gz",
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch10",
+      "runtime_seconds": 401.4404009709833,
+      "engine_log": "prefixes=10000 seconds=401.385",
+      "command": "/workspace/Matching-One-TV2N0X/exact-pairs-20260930/analysis/exact-completion-pairs-20260929/exact-pair-cloud-q6j7j4co/engine 512 10000 17211728716206152366 155385 /workspace/Matching-One-TV2N0X/exact-pairs-20260930/analysis/exact-completion-pairs-20260929/exact-pair-cloud-q6j7j4co/exact-square-L512-b10.csv",
+      "execution": "TV2N0X"
+    },
+    {
+      "batch": 11,
+      "samples": 10000,
+      "seed": "13292054708231777266",
+      "file": "exact-square-L512-b11.csv.gz",
+      "sha256": "5d0910c2cb54a063a0e685cacc121676adf0abecb9bbf559707cd95d60dc9d8f",
+      "original_file": "production-square-L512-b11.csv.gz",
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch11",
+      "runtime_seconds": 416.1216379549587,
+      "engine_log": "prefixes=10000 seconds=416.066",
+      "command": "/workspace/Matching-One-TV2N0X/exact-pairs-20260930/analysis/exact-completion-pairs-20260929/exact-pair-cloud-q6j7j4co/engine 512 10000 13292054708231777266 155385 /workspace/Matching-One-TV2N0X/exact-pairs-20260930/analysis/exact-completion-pairs-20260929/exact-pair-cloud-q6j7j4co/exact-square-L512-b11.csv",
+      "execution": "TV2N0X"
+    },
+    {
+      "batch": 12,
+      "samples": 10000,
+      "seed": "9255174391098419870",
+      "file": "exact-square-L512-b12.csv.gz",
+      "sha256": "02fc628489c24d33b6ca8a2bb9fc76bac2f61e9fa9fb1f0630b3660d8cff59db",
+      "original_file": "production-square-L512-b12.csv.gz",
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch12",
+      "runtime_seconds": 417.67217682197224,
+      "engine_log": "prefixes=10000 seconds=417.618",
+      "command": "/workspace/Matching-One-TV2N0X/exact-pairs-20260930/analysis/exact-completion-pairs-20260929/exact-pair-cloud-q6j7j4co/engine 512 10000 9255174391098419870 155385 /workspace/Matching-One-TV2N0X/exact-pairs-20260930/analysis/exact-completion-pairs-20260929/exact-pair-cloud-q6j7j4co/exact-square-L512-b12.csv",
+      "execution": "TV2N0X"
+    },
+    {
+      "batch": 13,
+      "samples": 10000,
+      "seed": "4111027655712514342",
+      "file": "exact-square-L512-b13.csv.gz",
+      "sha256": "cc846063ddacbe3a147d076b1979b5aadd44fa337dc4beb1967618b7f27871b2",
+      "original_file": "production-square-L512-b13.csv.gz",
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch13",
+      "runtime_seconds": 415.4595349589363,
+      "engine_log": "prefixes=10000 seconds=415.405",
+      "command": "/workspace/Matching-One-TV2N0X/exact-pairs-20260930/analysis/exact-completion-pairs-20260929/exact-pair-cloud-q6j7j4co/engine 512 10000 4111027655712514342 155385 /workspace/Matching-One-TV2N0X/exact-pairs-20260930/analysis/exact-completion-pairs-20260929/exact-pair-cloud-q6j7j4co/exact-square-L512-b13.csv",
+      "execution": "TV2N0X"
+    }
+  ],
+  "build_command": "clang++ -O3 -std=c++17 /private/tmp/matching-frontier-20260929-Jk39nH/analysis/exact-completion-pairs-20260929/engine.cpp -o /private/tmp/matching-frontier-20260929-Jk39nH/analysis/exact-completion-pairs-20260929/exact-pair-build-w2_bqml_/engine",
+  "binary_sha256": "2fa12e7a14075feda924271292c7a098be98a5848354322fda86043bc0ced36e",
+  "resumptions": [
+    {
+      "pid": 93063,
+      "utc_started": "2026-09-30T01:38:11.899918+00:00",
+      "reason": "Owner paused the task; stopped processes no longer existed on resume",
+      "retained_batches": [
+        0,
+        1,
+        2,
+        3
+      ],
+      "replayed_batches": [
+        4,
+        5,
+        6,
+        7,
+        8,
+        9,
+        10,
+        11,
+        12,
+        13
+      ],
+      "source_commit": "267f1ecfe50d4a3004185943bbc03034d51d6dc5",
+      "source_sha256": "544a0320c06aca1f0ceb22883bb0af53fc7c917c4f4a58c559710f8b2e3a7469",
+      "workers": 4,
+      "new_random_prefixes": 0,
+      "partial_files": "Original abandoned temporary CSVs retained; no partial batches scored",
+      "binary_sha256": "2fa12e7a14075feda924271292c7a098be98a5848354322fda86043bc0ced36e"
+    }
+  ],
+  "cloud_continuation": {
+    "instance": "DevEnvC_TV2N0X",
+    "instance_id": "4a8d1d443419434889e49148ed0a7ba6",
+    "source_commit": "0684c3b4eee0a647e643540ecfcdcd2918a70977",
+    "runner_sha256": "f119d53ea67c3c1b3ad5f40afa82e28ac676240064d40c91ebeba43b43471221",
+    "pid": 247,
+    "platform": "Linux-5.10.0-182.0.0.95.r3582_286.hce2.aarch64-aarch64-with-glibc2.34",
+    "python": "3.9.9",
+    "compiler": "g++ (GCC) 10.3.1\nCopyright (C) 2020 Free Software Foundation, Inc.\nThis is free software; see the source for copying conditions.  There is NO\nwarranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.",
+    "utc_started": "2026-09-30T01:46:19.133424+00:00",
+    "workers": 6,
+    "retained_local_batches": [
+      0,
+      1,
+      2,
+      3,
+      4,
+      5,
+      6,
+      7
+    ],
+    "completed_cloud_batches": [
+      9,
+      8,
+      10,
+      13,
+      11,
+      12
+    ],
+    "reason": "Owner requests medium/large computation on Huawei first; local incomplete workers stopped",
+    "new_random_prefixes": 0,
+    "binary_sha256": "72ddbdb1c2e26352ed8f62b820bfcdc741750bc5b50cf459f4b0adefe019fab7",
+    "benchmark_log": "prefixes=16 seconds=0.397452",
+    "wall_seconds": 420.380964801996
+  },
+  "utc_finished": "2026-09-30T01:53:19.514408+00:00",
+  "timing_note": "Mixed execution: local successful batches0..7, cloud8..13. Owner pause and abandoned partial work excluded from successful batch timings; cloud wall includes compile/benchmark."
+}
diff --git a/analysis/exact-completion-pairs-20260929/engine.cpp b/analysis/exact-completion-pairs-20260929/engine.cpp
new file mode 100644
index 000000000..2ea5915f2
--- /dev/null
+++ b/analysis/exact-completion-pairs-20260929/engine.cpp
@@ -0,0 +1,161 @@
+// Conditional mean of the existing safe-insertion target, with no safe probes.
+// Source permutations and lifted union-find are reused without modification.
+#define main original_completion_engine_main
+#include "../completion-hazard-production-20260929/engine.cpp"
+#undef main
+#include <unordered_map>
+#include <unordered_set>
+
+using EdgeKey = std::uint64_t;
+EdgeKey pair_key(int a,int b){if(a>b)std::swap(a,b);return (EdgeKey(a)<<32)|unsigned(b);}
+
+struct Contact {
+    std::array<int,6> component{};
+    std::array<std::int64_t,6> height{};
+    int n=0;
+};
+
+struct PairResult {
+    int nu=0;
+    std::uint64_t edges=0, candidates=0;
+};
+
+class ProjectedPairs {
+    int L,N,z;
+    std::vector<std::array<Neighbor,6>> neighbor;
+    std::vector<unsigned char> occupied,safe;
+    std::vector<int> component,stack;
+    std::vector<std::int64_t> potential;
+    std::vector<Contact> contact;
+public:
+    std::unordered_set<EdgeKey> edges;
+    ProjectedPairs(int length,bool triangular):L(length),N(L*L),z(triangular?6:4),
+        neighbor(N),occupied(N),safe(N),component(N),potential(N),contact(N){
+        const int sx[]={1,-1,0,0,1,-1},sy[]={0,0,1,-1,1,-1};
+        for(int y=0;y<L;++y)for(int x=0;x<L;++x)for(int j=0;j<z;++j)
+            neighbor[x+L*y][j]={((x+sx[j]+L)%L)+L*((y+sy[j]+L)%L),sx[j],sy[j]};
+        stack.reserve(N);
+    }
+    PairResult count(const std::vector<int>& order,int k,std::pair<int,int> d){
+        auto ell=[d](int x,int y){return std::int64_t(d.first)*y-std::int64_t(d.second)*x;};
+        std::fill(occupied.begin(),occupied.end(),0);
+        std::fill(safe.begin(),safe.end(),0);
+        std::fill(component.begin(),component.end(),-1);
+        for(int i=0;i<k;++i)occupied[order[i]]=1;
+        // Rank one guarantees a consistent transverse potential even when
+        // more than one occupied component carries the persistent direction.
+        for(int root=0;root<N;++root)if(occupied[root] && component[root]<0){
+            component[root]=root;potential[root]=0;stack.clear();stack.push_back(root);
+            while(!stack.empty()){
+                int u=stack.back();stack.pop_back();
+                for(int j=0;j<z;++j){auto e=neighbor[u][j];if(!occupied[e.vertex])continue;
+                    auto h=potential[u]+ell(e.dx,e.dy);
+                    if(component[e.vertex]<0){component[e.vertex]=root;potential[e.vertex]=h;stack.push_back(e.vertex);}
+                    else if(component[e.vertex]!=root || potential[e.vertex]!=h)
+                        throw std::logic_error("inconsistent rank-one transverse potential");
+                }
+            }
+        }
+        PairResult out;
+        for(int v=0;v<N;++v)if(!occupied[v]){
+            auto& row=contact[v];row.n=0;bool completes=false;
+            for(int j=0;j<z && !completes;++j){
+                auto e=neighbor[v][j];if(!occupied[e.vertex])continue;
+                int c=component[e.vertex];auto a=potential[e.vertex]-ell(e.dx,e.dy);
+                int h=0;while(h<row.n && row.component[h]!=c)++h;
+                if(h<row.n){if(row.height[h]!=a)completes=true;}
+                else {row.component[row.n]=c;row.height[row.n++]=a;}
+            }
+            if(completes)++out.nu;
+            else safe[v]=1;
+        }
+        // Each root pair has groups of vertices with equal relative height.
+        // Only cross-group pairs are enumerated; every one is a true edge.
+        using HeightGroups=std::unordered_map<std::int64_t,std::vector<int>>;
+        std::unordered_map<EdgeKey,HeightGroups> buckets;
+        for(int v=0;v<N;++v)if(safe[v]){
+            const auto& row=contact[v];
+            for(int i=0;i<row.n;++i)for(int j=i+1;j<row.n;++j){
+                int c=row.component[i],cc=row.component[j];
+                auto h=row.height[i]-row.height[j];
+                if(c>cc){std::swap(c,cc);h=-h;}
+                buckets[pair_key(c,cc)][h].push_back(v);
+            }
+        }
+        edges.clear();
+        for(const auto& bucket:buckets){
+            const auto& groups=bucket.second;
+            for(auto a=groups.begin();a!=groups.end();++a){
+                auto b=a;++b;
+                for(;b!=groups.end();++b)
+                    for(int v:a->second)for(int w:b->second){
+                        edges.insert(pair_key(v,w));++out.candidates;
+                    }
+            }
+        }
+        // A direct lattice edge provides one more relative-height constraint.
+        // One shared old component is enough for a conflict with that edge.
+        for(int v=0;v<N;++v)if(safe[v]){
+            const auto& a=contact[v];
+            for(int j=0;j<z;++j){
+                auto e=neighbor[v][j];int w=e.vertex;
+                if(w<=v || !safe[w])continue;
+                const auto& b=contact[w];bool mismatch=false;
+                for(int p=0;p<a.n && !mismatch;++p)for(int q=0;q<b.n;++q)
+                    if(a.component[p]==b.component[q] && a.height[p]-b.height[q]+ell(e.dx,e.dy)!=0){mismatch=true;break;}
+                if(mismatch)edges.insert(pair_key(v,w));
+            }
+        }
+        out.edges=edges.size();return out;
+    }
+};
+
+#ifndef MATCHING_ONE_PAIR_LIBRARY
+int main(int argc,char**argv){
+    try{
+        if(argc==2 && std::string(argv[1])=="control"){
+            const std::array<int,4> masks{{4403,4405,4371,4402}};
+            const std::array<int,4> expected{{2,3,4,6}};
+            for(int i=0;i<4;++i){
+                Geometry g(4,i>=2);ProjectedPairs counter(4,i>=2);std::vector<int> order;
+                for(int v=0;v<16;++v)if(masks[i]>>v&1){order.push_back(v);g.insert(v);}
+                int k=order.size();auto r=counter.count(order,k,g.direction());
+                std::unordered_set<EdgeKey> brute;
+                for(int v=0;v<16;++v)if(!(masks[i]>>v&1)){
+                    Geometry one=g;if(one.insert(v)==2)continue;
+                    for(int w=v+1;w<16;++w)if(!(masks[i]>>w&1)){
+                        Geometry alone=g;if(alone.insert(w)==2)continue;
+                        Geometry two=one;if(two.insert(w)==2)brute.insert(pair_key(v,w));
+                    }
+                }
+                if(r.nu!=0 || r.edges!=unsigned(expected[i]) || counter.edges!=brute)
+                    throw std::logic_error("physical pair witness mismatch");
+                std::cout<<"mask="<<masks[i]<<" nu="<<r.nu<<" e="<<r.edges<<" nonlocal_candidates="<<r.candidates<<'\n';
+            }
+            return 0;
+        }
+        if(argc!=6)throw std::invalid_argument("L prefixes existing_permutation_seed b output.csv");
+        int L=std::stoi(argv[1]),N=L*L,b=std::stoi(argv[4]);
+        auto samples=std::stoull(argv[2]),seed=std::stoull(argv[3]);
+        if(L<3 || b<1 || b>N-2 || !samples)throw std::invalid_argument("bounds");
+        Geometry g(L,false);ProjectedPairs counter(L,false);std::vector<int>order(N);
+        std::mt19937_64 rng(seed);std::ofstream out(argv[5]);
+        if(!out)throw std::runtime_error("cannot open output");
+        out<<"J1,rank_b,dx_b,dy_b,nu_b,synergy_edges,nonlocal_candidates\n";
+        auto start=std::chrono::steady_clock::now();
+        for(std::uint64_t i=0;i<samples;++i){
+            std::iota(order.begin(),order.end(),0);
+            for(int v=N-1;v>0;--v)std::swap(order[v],order[bounded(rng,v+1)]);
+            g.reset();int j1=0,rank=0;
+            for(int k=1;k<=b;++k){rank=g.insert(order[k-1]);if(rank && !j1)j1=k;if(rank==2)break;}
+            if(rank!=1){out<<j1<<','<<rank<<",0,0,-1,-1,-1\n";continue;}
+            auto d=g.direction();auto r=counter.count(order,b,d);
+            out<<j1<<",1,"<<d.first<<','<<d.second<<','<<r.nu<<','<<r.edges<<','<<r.candidates<<'\n';
+        }
+        out.close();if(!out)throw std::runtime_error("write failed");
+        std::cout<<"prefixes="<<samples<<" seconds="
+                 <<std::chrono::duration<double>(std::chrono::steady_clock::now()-start).count()<<'\n';
+        return 0;
+    }catch(const std::exception&e){std::cerr<<e.what()<<'\n';return 1;}
+}
+#endif
diff --git a/analysis/exact-completion-pairs-20260929/execution-receipt.json b/analysis/exact-completion-pairs-20260929/execution-receipt.json
new file mode 100644
index 000000000..302ee16cc
--- /dev/null
+++ b/analysis/exact-completion-pairs-20260929/execution-receipt.json
@@ -0,0 +1,28 @@
+{
+  "status": "completed_and_scored",
+  "new_random_prefixes": 0,
+  "source_prefixes": 140000,
+  "local_completed_batches": [0,1,2,3,4,5,6,7],
+  "cloud_completed_batches": [8,9,10,11,12,13],
+  "cloud_instance": "DevEnvC_TV2N0X",
+  "cloud_id": "4a8d1d443419434889e49148ed0a7ba6",
+  "remote_directory": "/workspace/Matching-One-TV2N0X/exact-pairs-20260930",
+  "cloud_workers": 6,
+  "cloud_cpu_quota_cores": 14.5,
+  "cloud_memory_limit_bytes": 26843545600,
+  "cloud_wall_seconds": 420.380964801996,
+  "cloud_source_commit": "0684c3b4eee0a647e643540ecfcdcd2918a70977",
+  "input_bundle_sha256": "eb6a6d62a4ba76d65bba78b19dfc6f484f191b959c3c937c77859346f0b488ac",
+  "return_bundle_sha256": "01827565b6cefee6dddf022d8501480f4e920dd0269c166cda17fac3f4961ba9",
+  "combined_manifest_sha256": "d6f388f7c0734c3ff6458ee48f5d4039b8f9d9449714fe1bec95d4d355bad28e",
+  "result_sha256": "08088c5fbc63f8098ac4cf7a023d2ba05e72c09ffe51e72bd6856581389c6e04",
+  "local_scorer_python": "/Users/lc/python-envs/research-py311/bin/python",
+  "scoring_passes": 1,
+  "old_result_identity": "all140000 first-five-field tuples equal; original primary and secondary point/SE reproduced",
+  "interruptions": "User pause ended the first local process; 4 completed batches retained. Resume completed another4. Cloud-first request stopped only owned incomplete local workers;6 remaining batches replayed remotely from original seeds. Partial batches never scored.",
+  "preserved_local_partial_directory": "/private/tmp/matching-exact-pairs-interrupted-w2_bqml_",
+  "remote_data_preserved": true,
+  "cloud_final_state": "Ready, verified after owned tunnel stopped and power-off",
+  "research_processes_remaining_before_shutdown": false,
+  "other_machines_operated": false
+}
diff --git a/analysis/exact-completion-pairs-20260929/finish_on_cloud.py b/analysis/exact-completion-pairs-20260929/finish_on_cloud.py
new file mode 100644
index 000000000..6961b96be
--- /dev/null
+++ b/analysis/exact-completion-pairs-20260929/finish_on_cloud.py
@@ -0,0 +1,94 @@
+#!/usr/bin/env python3
+"""Finish only missing source batches on TV2N0X after cloud-first rerouting."""
+import argparse
+from concurrent.futures import ThreadPoolExecutor, as_completed
+from datetime import datetime, timezone
+import gzip
+import json
+import os
+import platform
+import shlex
+import subprocess
+import tempfile
+import time
+from pathlib import Path
+
+from run import HERE, OLD, sha
+
+
+def main():
+    ap = argparse.ArgumentParser()
+    ap.add_argument("--source-commit", required=True)
+    args = ap.parse_args()
+    manifest = HERE / "data/run.json"
+    run = json.loads(manifest.read_text())
+    assert run["status"] != "completed"
+    for name, digest in run["source_sha256"].items():
+        assert sha(HERE / name) == digest
+    original = json.loads((OLD / "data/run.json").read_text())
+    assert sha(OLD / "data/run.json") == run["original_manifest_sha256"]
+    finished = {entry["batch"] for entry in run["batches"]}
+    for entry in run["batches"]:
+        assert sha(HERE / "data" / entry["file"]) == entry["sha256"]
+    remaining = [entry for entry in original["batches"] if entry["batch"] not in finished]
+    assert len(remaining) == 6 and finished == set(range(8))
+    receipt = dict(instance="DevEnvC_TV2N0X", instance_id="4a8d1d443419434889e49148ed0a7ba6",
+                   source_commit=args.source_commit, runner_sha256=sha(Path(__file__)),
+                   pid=os.getpid(), platform=platform.platform(), python=platform.python_version(),
+                   compiler=subprocess.check_output(["g++", "--version"], text=True).strip(),
+                   utc_started=datetime.now(timezone.utc).isoformat(), workers=6,
+                   retained_local_batches=sorted(finished), completed_cloud_batches=[],
+                   reason="Owner requests medium/large computation on Huawei first; local incomplete workers stopped",
+                   new_random_prefixes=0)
+    run["cloud_continuation"] = receipt
+    def save():
+        manifest.write_text(json.dumps(run, indent=2)+"\n")
+    start = time.monotonic()
+    try:
+        with tempfile.TemporaryDirectory(prefix="exact-pair-cloud-", dir=HERE) as temp:
+            temp = Path(temp)
+            binary = temp / "engine"
+            subprocess.run(["g++", "-O3", "-std=c++17", str(HERE / "engine.cpp"), "-o", str(binary)], check=True)
+            receipt["binary_sha256"] = sha(binary)
+            # Same old-prefix seed; cost check only, never extra evidence.
+            benchmark = [str(binary), "512", "16", remaining[0]["seed"], str(run["contract"]["b"]), str(temp / "benchmark.csv")]
+            receipt["benchmark_log"] = subprocess.check_output(benchmark, text=True).strip()
+            run["status"] = "running"; save()
+            def job(entry):
+                batch = entry["batch"]
+                raw = temp / f"exact-square-L512-b{batch:02d}.csv"
+                target = HERE / "data" / (raw.name+".gz")
+                assert not target.exists()
+                command = [str(binary), "512", str(entry["samples"]), entry["seed"], str(run["contract"]["b"]), str(raw)]
+                begin = time.monotonic()
+                log = subprocess.check_output(command, text=True).strip()
+                seconds = time.monotonic()-begin
+                payload = raw.read_bytes()
+                assert payload.count(b"\n") == entry["samples"]+1
+                with target.open("xb") as out:
+                    with gzip.GzipFile(fileobj=out, mode="wb", mtime=0) as gz:
+                        gz.write(payload)
+                print(f"batch={batch}: {log}", flush=True)
+                return dict(batch=batch, samples=entry["samples"], seed=entry["seed"],
+                            file=target.name, sha256=sha(target), original_file=entry["file"],
+                            dependency_group=entry["dependency_group"], runtime_seconds=seconds,
+                            engine_log=log, command=shlex.join(command), execution="TV2N0X")
+            with ThreadPoolExecutor(max_workers=6) as pool:
+                for future in as_completed([pool.submit(job, entry) for entry in remaining]):
+                    entry = future.result(); run["batches"].append(entry)
+                    run["batches"].sort(key=lambda e: e["batch"])
+                    receipt["completed_cloud_batches"].append(entry["batch"]); save()
+        run["status"] = "completed"
+    except BaseException as error:
+        run["status"] = "failed"; receipt["error"] = str(error)
+        raise
+    finally:
+        receipt["wall_seconds"] = time.monotonic()-start
+        run["utc_finished"] = datetime.now(timezone.utc).isoformat()
+        run["timing_note"] = "Mixed execution: local successful batches0..7, cloud8..13. Owner pause and abandoned partial work excluded from successful batch timings; cloud wall includes compile/benchmark."
+        save()
+    print(f"Cloud continuation completed in {receipt['wall_seconds']:.3f}s", flush=True)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/analysis/exact-completion-pairs-20260929/results/RESULT.md b/analysis/exact-completion-pairs-20260929/results/RESULT.md
new file mode 100644
index 000000000..94c3cd4bb
--- /dev/null
+++ b/analysis/exact-completion-pairs-20260929/results/RESULT.md
@@ -0,0 +1,40 @@
+# Same-prefix exact completion-pair refinement
+
+All 140000 rows aligned across 14 original batches; every first-five-field tuple matched.
+
+Early minus late; fixed a,b and original overlap weights. No new random prefixes.
+
+| Metric | Estimate | Batch jackknife SE |
+|---|---:|---:|
+| RB_deltaY | -0.00344938566765506 | 0.00077225276269743 |
+| original_probe_deltaY | 0.00668182660775089 | 0.0120870551771217 |
+| paired_RB_minus_probe | -0.0101312122754059 | 0.0123956612572906 |
+| RB_deltaE | -183.963984858095 | 41.2052909135037 |
+| RB_deltaZ2 | 3.22818100446661e-08 | 7.23066188814621e-09 |
+| RB_earlyY | 0.0955745614429761 | 0.000450216499320379 |
+| RB_lateY | 0.0990239471106312 | 0.000525721971141274 |
+| secondary_deltaNu | -0.0398651161631196 | 0.569877945474596 |
+
+Descriptive SE_RB/SE_probe: 0.0638908941326869. Variance ratio: 0.004082046353074206.
+
+| Support / cohort | Rank-one | Eligible | Supported |
+|---|---:|---:|---:|
+| primary / early | 24201 | 24201 | 24109 |
+| primary / late | 30827 | 30827 | 30354 |
+| primary / combined | 55028 | 55028 | 54463 |
+| secondary / early | 24201 | 24201 | 24201 |
+| secondary / late | 30827 | 30827 | 30821 |
+| secondary / combined | 55028 | 55028 | 55022 |
+
+Original committed primary and secondary point/SE reproduced (absolute tolerances 1e-14 / 1e-12).
+This is a source/estimand identity check, not scientific replication.
+
+JSON retains exact integer cell sums, mean_e, meanY, all 14 aligned deletion vectors/supports, full 8x8 covariance, and input/source hashes and commits. First seven metrics share exact (D,c) cells; the secondary uses all rank-one prefixes within D, including c=m.
+
+Y=2e/(m-c); original prefix probe mean=(Y0+Y1+Y2+Y3)/4. RB_deltaE averages cell mean-edge differences; RB_deltaZ2=-2*RB_deltaE/[m(m-1)]. c=m has no conditional safe mean. Singular covariance is expected; no inverse is used.
+
+- Estimator refinement after the original readout; same 140k prefixes and batch blocks, zero new samples.
+- No independent confirmation: reproducing old primary/secondary is source/estimand identity, not scientific replication.
+- Exact means conditional mean 2e/(m-c); finite-prefix uncertainty remains. Four actual old probes are averaged per prefix.
+- SE_RB/SE_probe is descriptive; variance reduction does not guarantee empirical delete-batch jackknife SE shrinkage.
+- No automatic Markov claim: one weighted moment can cancel across cells; finite L512 count clock only, no continuum claim.
diff --git a/analysis/exact-completion-pairs-20260929/results/result.json b/analysis/exact-completion-pairs-20260929/results/result.json
new file mode 100644
index 000000000..a175040fe
--- /dev/null
+++ b/analysis/exact-completion-pairs-20260929/results/result.json
@@ -0,0 +1,49272 @@
+{
+  "schema": "matching-one.exact-pair-score.v1",
+  "status": "scored",
+  "parameters": {
+    "N": 262144,
+    "a": 154646,
+    "b": 155385,
+    "probes": 4,
+    "batches": 14,
+    "samples_per_batch": 10000
+  },
+  "created_utc": "2026-09-30T01:54:01.580101+00:00",
+  "m": 106759,
+  "metric_order": [
+    "RB_deltaY",
+    "original_probe_deltaY",
+    "paired_RB_minus_probe",
+    "RB_deltaE",
+    "RB_deltaZ2",
+    "RB_earlyY",
+    "RB_lateY",
+    "secondary_deltaNu"
+  ],
+  "point_vector": [
+    -0.0034493856676550577,
+    0.006681826607750889,
+    -0.010131212275405947,
+    -183.96398485809462,
+    3.22818100446661e-08,
+    0.09557456144297612,
+    0.09902394711063117,
+    -0.03986511616311962
+  ],
+  "point_estimates": {
+    "RB_deltaY": -0.0034493856676550577,
+    "original_probe_deltaY": 0.006681826607750889,
+    "paired_RB_minus_probe": -0.010131212275405947,
+    "RB_deltaE": -183.96398485809462,
+    "RB_deltaZ2": 3.22818100446661e-08,
+    "RB_earlyY": 0.09557456144297612,
+    "RB_lateY": 0.09902394711063117,
+    "secondary_deltaNu": -0.03986511616311962
+  },
+  "standard_errors": {
+    "RB_deltaY": 0.0007722527626974296,
+    "original_probe_deltaY": 0.012087055177121728,
+    "paired_RB_minus_probe": 0.012395661257290627,
+    "RB_deltaE": 41.20529091350368,
+    "RB_deltaZ2": 7.230661888146215e-09,
+    "RB_earlyY": 0.00045021649932037885,
+    "RB_lateY": 0.0005257219711412741,
+    "secondary_deltaNu": 0.5698779454745956
+  },
+  "primary": {
+    "status": "scoreable",
+    "reason": null,
+    "stratification": "exact (D,nu_b), nu_b<m",
+    "cell_counts": {
+      "common": 1190,
+      "early_only": 70,
+      "late_only": 229
+    },
+    "overlap_weight_sum": 13086.989109440276,
+    "coverage": {
+      "early": {
+        "rank1_prefixes": 24201,
+        "eligible_prefixes": 24201,
+        "supported_prefixes": 24109,
+        "unsupported_eligible_prefixes": 92,
+        "supported_fraction_of_eligible": 0.9961985041940415,
+        "supported_fraction_of_rank1": 0.9961985041940415,
+        "dead_prefixes": 0
+      },
+      "late": {
+        "rank1_prefixes": 30827,
+        "eligible_prefixes": 30827,
+        "supported_prefixes": 30354,
+        "unsupported_eligible_prefixes": 473,
+        "supported_fraction_of_eligible": 0.9846563077821391,
+        "supported_fraction_of_rank1": 0.9846563077821391,
+        "dead_prefixes": 0
+      },
+      "combined": {
+        "rank1_prefixes": 55028,
+        "eligible_prefixes": 55028,
+        "supported_prefixes": 54463,
+        "unsupported_eligible_prefixes": 565,
+        "supported_fraction_of_eligible": 0.9897324998182744,
+        "supported_fraction_of_rank1": 0.9897324998182744
+      }
+    },
+    "weighted_early_mean": 0.09557456144297612,
+    "weighted_late_mean": 0.09902394711063117,
+    "delta": -0.0034493856676550577,
+    "deltaE": -183.96398485809462,
+    "deltaZ2": 3.22818100446661e-08,
+    "original_probe_deltaY": 0.006681826607750889,
+    "cells": [
+      {
+        "direction": [
+          0,
+          1
+        ],
+        "nu_b": 0,
+        "common_support": true,
+        "overlap_weight": 653.1930487344163,
+        "normalized_weight": 0.04991163691450134,
+        "safe_sites": 106759,
+        "early": {
+          "n": 1173,
+          "sum_e": 9054976,
+          "sum_old_probe_Y": 559,
+          "mean_e": 7719.502131287298,
+          "meanY": 0.14461548218487055,
+          "mean_old_prefixprobe_Y": 0.11913895993179881
+        },
+        "late": {
+          "n": 1474,
+          "sum_e": 11194952,
+          "sum_old_probe_Y": 359,
+          "mean_e": 7594.947082767978,
+          "meanY": 0.14228209486353335,
+          "mean_old_prefixprobe_Y": 0.0608887381275441
+        },
+        "RB_deltaY": 0.002333387321337199,
+        "original_probe_deltaY": 0.05825022180425471,
+        "paired_RB_minus_probe": -0.05591683448291751,
+        "RB_deltaE": 124.55504851931983,
+        "RB_deltaZ2": -2.1856791260019992e-08
+      },
+      {
+        "direction": [
+          0,
+          1
+        ],
+        "nu_b": 1,
+        "common_support": true,
+        "overlap_weight": 5.454545454545454,
+        "normalized_weight": 0.00041679147196743883,
+        "safe_sites": 106758,
+        "early": {
+          "n": 10,
+          "sum_e": 88893,
+          "sum_old_probe_Y": 0,
+          "mean_e": 8889.3,
+          "meanY": 0.1665317821615242,
+          "mean_old_prefixprobe_Y": 0.0
+        },
+        "late": {
+          "n": 12,
+          "sum_e": 76025,
+          "sum_old_probe_Y": 0,
+          "mean_e": 6335.416666666667,
+          "meanY": 0.1186874363825974,
+          "mean_old_prefixprobe_Y": 0.0
+        },
+        "RB_deltaY": 0.047844345778926795,
+        "original_probe_deltaY": 0.0,
+        "paired_RB_minus_probe": 0.047844345778926795,
+        "RB_deltaE": 2553.8833333333323,
+        "RB_deltaZ2": -4.4815280940180006e-07
+      },
+      {
+        "direction": [
+          0,
+          1
+        ],
+        "nu_b": 2,
+        "common_support": true,
+        "overlap_weight": 12.745098039215685,
+        "normalized_weight": 0.0009738754982245712,
+        "safe_sites": 106757,
+        "early": {
+          "n": 25,
+          "sum_e": 161543,
+          "sum_old_probe_Y": 0,
+          "mean_e": 6461.72,
+          "meanY": 0.12105473177402887,
+          "mean_old_prefixprobe_Y": 0.0
+        },
+        "late": {
+          "n": 26,
+          "sum_e": 162808,
+          "sum_old_probe_Y": 0,
+          "mean_e": 6261.846153846154,
+          "meanY": 0.11731026825119016,
+          "mean_old_prefixprobe_Y": 0.0
+        },
+        "RB_deltaY": 0.0037444635228387063,
+        "original_probe_deltaY": 0.0,
+        "paired_RB_minus_probe": 0.0037444635228387063,
+        "RB_deltaE": 199.873846153846,
+        "RB_deltaZ2": -3.507365607139035e-08
+      },
+      {
+        "direction": [
+          0,
+          1
+        ],
+        "nu_b": 3,
+        "common_support": true,
+        "overlap_weight": 13.508771929824562,
+        "normalized_weight": 0.0010322291718023997,
+        "safe_sites": 106756,
+        "early": {
+          "n": 22,
+          "sum_e": 149837,
+          "sum_old_probe_Y": 0,
+          "mean_e": 6810.772727272727,
+          "meanY": 0.12759512771690074,
+          "mean_old_prefixprobe_Y": 0.0
+        },
+        "late": {
+          "n": 35,
+          "sum_e": 267449,
+          "sum_old_probe_Y": 71,
+          "mean_e": 7641.4,
+          "meanY": 0.14315635655138823,
+          "mean_old_prefixprobe_Y": 0.5071428571428571
+        },
+        "RB_deltaY": -0.01556122883448749,
+        "original_probe_deltaY": -0.5071428571428571,
+        "paired_RB_minus_probe": 0.4915816283083696,
+        "RB_deltaE": -830.6272727272726,
+        "RB_deltaZ2": 1.457576158549983e-07
+      },
+      {
+        "direction": [
+          0,
+          1
+        ],
+        "nu_b": 4,
+        "common_support": true,
+        "overlap_weight": 11.304347826086957,
+        "normalized_weight": 0.0008637852245122284,
+        "safe_sites": 106755,
+        "early": {
+          "n": 20,
+          "sum_e": 120582,
+          "sum_old_probe_Y": 0,
+          "mean_e": 6029.1,
+          "meanY": 0.11295208655332303,
+          "mean_old_prefixprobe_Y": 0.0
+        },
+        "late": {
+          "n": 26,
+          "sum_e": 223170,
+          "sum_old_probe_Y": 102,
+          "mean_e": 8583.461538461539,
+          "meanY": 0.1608067357680959,
+          "mean_old_prefixprobe_Y": 0.9807692307692307
+        },
+        "RB_deltaY": -0.047854649214772865,
+        "original_probe_deltaY": -0.9807692307692307,
+        "paired_RB_minus_probe": 0.9329145815544578,
+        "RB_deltaE": -2554.3615384615387,
+        "RB_deltaZ2": 4.482367243437549e-07
+      },
+      {
+        "direction": [
+          0,
+          1
+        ],
+        "nu_b": 5,
+        "common_support": true,
+        "overlap_weight": 17.835616438356166,
+        "normalized_weight": 0.001362851018611338,
+        "safe_sites": 106754,
+        "early": {
+          "n": 31,
+          "sum_e": 241454,
+          "sum_old_probe_Y": 0,
+          "mean_e": 7788.8387096774195,
+          "meanY": 0.1459212527807374,
+          "mean_old_prefixprobe_Y": 0.0
+        },
+        "late": {
+          "n": 42,
+          "sum_e": 219504,
+          "sum_old_probe_Y": 13,
+          "mean_e": 5226.285714285715,
+          "meanY": 0.09791269112699691,
+          "mean_old_prefixprobe_Y": 0.07738095238095238
+        },
+        "RB_deltaY": 0.048008561653740484,
+        "original_probe_deltaY": -0.07738095238095238,
+        "paired_RB_minus_probe": 0.12538951403469287,
+        "RB_deltaE": 2562.552995391705,
+        "RB_deltaZ2": -4.49674152744823e-07
+      },
+      {
+        "direction": [
+          0,
+          1
+        ],
+        "nu_b": 6,
+        "common_support": true,
+        "overlap_weight": 13.245283018867925,
+        "normalized_weight": 0.0010120955177775357,
+        "safe_sites": 106753,
+        "early": {
+          "n": 27,
+          "sum_e": 180162,
+          "sum_old_probe_Y": 0,
+          "mean_e": 6672.666666666667,
+          "meanY": 0.12501131896371376,
+          "mean_old_prefixprobe_Y": 0.0
+        },
+        "late": {
+          "n": 26,
+          "sum_e": 168393,
+          "sum_old_probe_Y": 0,
+          "mean_e": 6476.653846153846,
+          "meanY": 0.12133905082112627,
+          "mean_old_prefixprobe_Y": 0.0
+        },
+        "RB_deltaY": 0.00367226814258749,
+        "original_probe_deltaY": 0.0,
+        "paired_RB_minus_probe": 0.00367226814258749,
+        "RB_deltaE": 196.01282051282124,
+        "RB_deltaZ2": -3.439612727999517e-08
+      },
+      {
+        "direction": [
+          0,
+          1
+        ],
+        "nu_b": 7,
+        "common_support": true,
+        "overlap_weight": 18.986842105263158,
+        "normalized_weight": 0.0014508182093287626,
+        "safe_sites": 106752,
+        "early": {
+          "n": 37,
+          "sum_e": 239356,
+          "sum_old_probe_Y": 0,
+          "mean_e": 6469.081081081081,
+          "meanY": 0.1211983116209735,
+          "mean_old_prefixprobe_Y": 0.0
+        },
+        "late": {
+          "n": 39,
+          "sum_e": 267084,
+          "sum_old_probe_Y": 51,
+          "mean_e": 6848.307692307692,
+          "meanY": 0.12830312672938574,
+          "mean_old_prefixprobe_Y": 0.3269230769230769
+        },
+        "RB_deltaY": -0.007104815108412238,
+        "original_probe_deltaY": -0.3269230769230769,
+        "paired_RB_minus_probe": 0.31981826181466466,
+        "RB_deltaE": -379.22661122661157,
+        "RB_deltaZ2": 6.654629403110175e-08
+      },
+      {
+        "direction": [
+          0,
+          1
+        ],
+        "nu_b": 8,
+        "common_support": true,
+        "overlap_weight": 18.16216216216216,
+        "normalized_weight": 0.0013878029553077965,
+        "safe_sites": 106751,
+        "early": {
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+        "dependency_group": "safe-insertion-independent-20260929/square/L512/batch3",
+        "sum_nonlocal_candidates": 20322566,
+        "original": {
+          "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/safe-insertion-production-20260929/data/production-square-L512-b03.csv.gz",
+          "bytes": 32670,
+          "sha256": "dd483db8df4b968e3244983c8462420484807ebca99b118923afb2d6d67b8f00"
+        },
+        "replay": {
+          "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/exact-completion-pairs-20260929/data/exact-square-L512-b03.csv.gz",
+          "bytes": 52692,
+          "sha256": "f3023270c7962a3c026114cf27b398396142c23432810ecdccabcaefc55f74b7"
+        }
+      },
+      {
+        "batch": 4,
+        "rows": 10000,
+        "all_first5_identical": true,
+        "seed": "13568940959106988201",
+        "dependency_group": "safe-insertion-independent-20260929/square/L512/batch4",
+        "sum_nonlocal_candidates": 20340820,
+        "original": {
+          "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/safe-insertion-production-20260929/data/production-square-L512-b04.csv.gz",
+          "bytes": 32769,
+          "sha256": "617da5123f5090265e2e07ec40762ffbb0edf4d9b8cf7dccecc64708a198ec69"
+        },
+        "replay": {
+          "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/exact-completion-pairs-20260929/data/exact-square-L512-b04.csv.gz",
+          "bytes": 52689,
+          "sha256": "9b8271e270d02130cd29da0983ddcc34c6f0441e0e6994822dcdd22c313062c8"
+        }
+      },
+      {
+        "batch": 5,
+        "rows": 10000,
+        "all_first5_identical": true,
+        "seed": "14221577344058561937",
+        "dependency_group": "safe-insertion-independent-20260929/square/L512/batch5",
+        "sum_nonlocal_candidates": 20652861,
+        "original": {
+          "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/safe-insertion-production-20260929/data/production-square-L512-b05.csv.gz",
+          "bytes": 32875,
+          "sha256": "3c94ed5c60759faaa225af35cd8e12d6996c07d7913ee2385fc78d60c4e8ddd3"
+        },
+        "replay": {
+          "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/exact-completion-pairs-20260929/data/exact-square-L512-b05.csv.gz",
+          "bytes": 53175,
+          "sha256": "5f275599206d6db5099bf138477821dfd4ea516f411fefe36c5de1f07477069a"
+        }
+      },
+      {
+        "batch": 6,
+        "rows": 10000,
+        "all_first5_identical": true,
+        "seed": "2117620933003635365",
+        "dependency_group": "safe-insertion-independent-20260929/square/L512/batch6",
+        "sum_nonlocal_candidates": 20156253,
+        "original": {
+          "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/safe-insertion-production-20260929/data/production-square-L512-b06.csv.gz",
+          "bytes": 32702,
+          "sha256": "de9eee185c039bfefbd29ebc1cd87c7962f475860ce1aa68bf67ef5a13cd18f4"
+        },
+        "replay": {
+          "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/exact-completion-pairs-20260929/data/exact-square-L512-b06.csv.gz",
+          "bytes": 52711,
+          "sha256": "8a9ceb8fb776d7a318565864145efa3a4f8131b95c984c4a5d6a4c05b3e2ab84"
+        }
+      },
+      {
+        "batch": 7,
+        "rows": 10000,
+        "all_first5_identical": true,
+        "seed": "10091810389689459151",
+        "dependency_group": "safe-insertion-independent-20260929/square/L512/batch7",
+        "sum_nonlocal_candidates": 20872496,
+        "original": {
+          "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/safe-insertion-production-20260929/data/production-square-L512-b07.csv.gz",
+          "bytes": 32615,
+          "sha256": "c95346db4ed6c55b20bfc6906b50ba8bbfbfee71488c1a5c0e0a23e6caa3b757"
+        },
+        "replay": {
+          "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/exact-completion-pairs-20260929/data/exact-square-L512-b07.csv.gz",
+          "bytes": 52773,
+          "sha256": "46f60c48a55d02559dc5a21f53d4a49eda3feb3f456837b322cc3a133d302b9e"
+        }
+      },
+      {
+        "batch": 8,
+        "rows": 10000,
+        "all_first5_identical": true,
+        "seed": "11008454278899210265",
+        "dependency_group": "safe-insertion-independent-20260929/square/L512/batch8",
+        "sum_nonlocal_candidates": 20071629,
+        "original": {
+          "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/safe-insertion-production-20260929/data/production-square-L512-b08.csv.gz",
+          "bytes": 32677,
+          "sha256": "30372e4f839caa13574317aae26dbdeb733e90d306d6c3518898b69bb9ed33e1"
+        },
+        "replay": {
+          "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/exact-completion-pairs-20260929/data/exact-square-L512-b08.csv.gz",
+          "bytes": 52613,
+          "sha256": "06a6924043ef47a1c664305490004c7cab7b137fa9a85e0735914a98921267e0"
+        }
+      },
+      {
+        "batch": 9,
+        "rows": 10000,
+        "all_first5_identical": true,
+        "seed": "17332517276911702520",
+        "dependency_group": "safe-insertion-independent-20260929/square/L512/batch9",
+        "sum_nonlocal_candidates": 20181560,
+        "original": {
+          "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/safe-insertion-production-20260929/data/production-square-L512-b09.csv.gz",
+          "bytes": 32987,
+          "sha256": "51eec39ed48afa403935abd2f6433156c5485ab133dcd8d03b4f8a47e814f27e"
+        },
+        "replay": {
+          "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/exact-completion-pairs-20260929/data/exact-square-L512-b09.csv.gz",
+          "bytes": 52987,
+          "sha256": "525d35bfc340547d0e0617ab1074b34818bcbd1762f438f6e9ffd4170305d17f"
+        }
+      },
+      {
+        "batch": 10,
+        "rows": 10000,
+        "all_first5_identical": true,
+        "seed": "17211728716206152366",
+        "dependency_group": "safe-insertion-independent-20260929/square/L512/batch10",
+        "sum_nonlocal_candidates": 19911083,
+        "original": {
+          "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/safe-insertion-production-20260929/data/production-square-L512-b10.csv.gz",
+          "bytes": 32713,
+          "sha256": "4e51e12a3efa950ececd20eb2f07b6c331d02f503d3fb9e6730341f0d56bd556"
+        },
+        "replay": {
+          "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/exact-completion-pairs-20260929/data/exact-square-L512-b10.csv.gz",
+          "bytes": 52440,
+          "sha256": "80d47a4fabc6a5419b7e5cfc7f1964fbe08c8080f015456012a9f775e2f8e4bf"
+        }
+      },
+      {
+        "batch": 11,
+        "rows": 10000,
+        "all_first5_identical": true,
+        "seed": "13292054708231777266",
+        "dependency_group": "safe-insertion-independent-20260929/square/L512/batch11",
+        "sum_nonlocal_candidates": 20334271,
+        "original": {
+          "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/safe-insertion-production-20260929/data/production-square-L512-b11.csv.gz",
+          "bytes": 32781,
+          "sha256": "e91819af15ede7b1b73e35339312f8214ef73d87447e07ad3b9f82ddef8a420a"
+        },
+        "replay": {
+          "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/exact-completion-pairs-20260929/data/exact-square-L512-b11.csv.gz",
+          "bytes": 52499,
+          "sha256": "5d0910c2cb54a063a0e685cacc121676adf0abecb9bbf559707cd95d60dc9d8f"
+        }
+      },
+      {
+        "batch": 12,
+        "rows": 10000,
+        "all_first5_identical": true,
+        "seed": "9255174391098419870",
+        "dependency_group": "safe-insertion-independent-20260929/square/L512/batch12",
+        "sum_nonlocal_candidates": 20156691,
+        "original": {
+          "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/safe-insertion-production-20260929/data/production-square-L512-b12.csv.gz",
+          "bytes": 32789,
+          "sha256": "c23466b9802fc28cc0f2044d6cd9c595e1d79f3e7d1ddf50b5d4b395d0d0ebd3"
+        },
+        "replay": {
+          "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/exact-completion-pairs-20260929/data/exact-square-L512-b12.csv.gz",
+          "bytes": 52648,
+          "sha256": "02fc628489c24d33b6ca8a2bb9fc76bac2f61e9fa9fb1f0630b3660d8cff59db"
+        }
+      },
+      {
+        "batch": 13,
+        "rows": 10000,
+        "all_first5_identical": true,
+        "seed": "4111027655712514342",
+        "dependency_group": "safe-insertion-independent-20260929/square/L512/batch13",
+        "sum_nonlocal_candidates": 20274965,
+        "original": {
+          "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/safe-insertion-production-20260929/data/production-square-L512-b13.csv.gz",
+          "bytes": 32776,
+          "sha256": "01e55fa4457010955e71cbc38ef8d43568ec85d0eb30157735ddc9be19254008"
+        },
+        "replay": {
+          "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/exact-completion-pairs-20260929/data/exact-square-L512-b13.csv.gz",
+          "bytes": 52666,
+          "sha256": "cc846063ddacbe3a147d076b1979b5aadd44fa337dc4beb1967618b7f27871b2"
+        }
+      }
+    ]
+  },
+  "source_manifest": {
+    "original_result_commit": "2a1677e1efe1137dffc07c3c4ae4ceebf15ecfd3",
+    "original_acquisition_commit": "55fa265629701e04b879b774ae7d3b97fefa845f",
+    "replay_source_commit": "8b5eee0d",
+    "scoring_head_commit": "a79a16c383631b1d6db209d1923ff009ac8d21dc",
+    "files": [
+      {
+        "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/exact-completion-pairs-20260929/analyze.py",
+        "bytes": 15574,
+        "sha256": "b6fcf38a79b1ae94789af89b27d77c8fda2a7635013c7420ea91b2f005544ad9"
+      },
+      {
+        "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/exact-completion-pairs-20260929/engine.cpp",
+        "bytes": 7915,
+        "sha256": "48def78ee470acef233fd4c4d2543f1d89c1869b552a4a434ab2aa220be95682"
+      },
+      {
+        "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/exact-completion-pairs-20260929/run.py",
+        "bytes": 5290,
+        "sha256": "edd9e6b8ffeab865f8d89de3dfab6d1f8353b41d258f4f3f211239af73e3a469"
+      },
+      {
+        "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/safe-insertion-production-20260929/analyze.py",
+        "bytes": 25821,
+        "sha256": "bba96add5df9c37ffb7a59ff80c9bdcace2911191dec6278daa1844ab0772e8f"
+      },
+      {
+        "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/safe-insertion-production-20260929/engine.cpp",
+        "bytes": 4124,
+        "sha256": "75644ef25ea539a63f713a14890807a480b15f5d7b40c6cdc19409ebd3de69d2"
+      },
+      {
+        "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/safe-insertion-production-20260929/run.py",
+        "bytes": 6149,
+        "sha256": "45f0f1bd25ab9437203ae26eba44cdbd6b68554f68cd90dd959edcb797a9a1ca"
+      },
+      {
+        "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/completion-hazard-production-20260929/engine.cpp",
+        "bytes": 7379,
+        "sha256": "42ac8a013cb731f4da4e3121ca429fd475555063e876638ad5d375d195448115"
+      },
+      {
+        "path": "/private/tmp/matching-frontier-20260929-Jk39nH/notes/completion-pair-synergy-20260929.md",
+        "bytes": 11868,
+        "sha256": "3d775b622bbaeb8611d6866a47ebafa3c9832a0f3e511c21de0f64ea8c060892"
+      }
+    ],
+    "note": "Commit references and working-file hashes are distinct; replay code may be uncommitted."
+  },
+  "warnings": [
+    "Estimator refinement after the original readout; same 140k prefixes and batch blocks, zero new samples.",
+    "No independent confirmation: reproducing old primary/secondary is source/estimand identity, not scientific replication.",
+    "Exact means conditional mean 2e/(m-c); finite-prefix uncertainty remains. Four actual old probes are averaged per prefix.",
+    "SE_RB/SE_probe is descriptive; variance reduction does not guarantee empirical delete-batch jackknife SE shrinkage.",
+    "No automatic Markov claim: one weighted moment can cancel across cells; finite L512 count clock only, no continuum claim."
+  ]
+}
diff --git a/analysis/exact-completion-pairs-20260929/resume.log b/analysis/exact-completion-pairs-20260929/resume.log
new file mode 100644
index 000000000..b6e85fccd
--- /dev/null
+++ b/analysis/exact-completion-pairs-20260929/resume.log
@@ -0,0 +1,4 @@
+batch=4: prefixes=10000 seconds=232.437
+batch=6: prefixes=10000 seconds=232.956
+batch=7: prefixes=10000 seconds=233.58
+batch=5: prefixes=10000 seconds=235.203
diff --git a/analysis/exact-completion-pairs-20260929/resume.py b/analysis/exact-completion-pairs-20260929/resume.py
new file mode 100644
index 000000000..e2be052c1
--- /dev/null
+++ b/analysis/exact-completion-pairs-20260929/resume.py
@@ -0,0 +1,92 @@
+#!/usr/bin/env python3
+"""Finish missing whole batches after the owner's pause; retain completed rows."""
+from concurrent.futures import ThreadPoolExecutor, as_completed
+from datetime import datetime, timezone
+import gzip
+import json
+import os
+from pathlib import Path
+import shlex
+import subprocess
+import tempfile
+import time
+
+from run import HERE, OLD, sha
+
+
+def main():
+    path = HERE / "data/run.json"
+    run = json.loads(path.read_text())
+    if run["status"] == "completed":
+        raise SystemExit("Replay already complete")
+    for pid in [run["pid"]] + [v["pid"] for v in run.get("resumptions", [])]:
+        try:
+            os.kill(pid, 0)
+        except ProcessLookupError:
+            continue
+        raise SystemExit(f"Recorded process {pid} is alive; inspect it rather than launching a duplicate")
+    for name, digest in run["source_sha256"].items():
+        assert sha(HERE / name) == digest, f"changed source: {name}"
+    source = json.loads((OLD / "data/run.json").read_text())
+    assert sha(OLD / "data/run.json") == run["original_manifest_sha256"]
+    finished = {entry["batch"] for entry in run["batches"]}
+    for entry in run["batches"]:
+        assert sha(HERE / "data" / entry["file"]) == entry["sha256"]
+    remaining = [entry for entry in source["batches"] if entry["batch"] not in finished]
+    begin = time.monotonic()
+    receipt = {"pid": os.getpid(), "utc_started": datetime.now(timezone.utc).isoformat(),
+               "reason": "Owner paused the task; stopped processes no longer existed on resume",
+               "retained_batches": sorted(finished), "replayed_batches": [e["batch"] for e in remaining],
+               "source_commit": subprocess.check_output(["git", "rev-parse", "HEAD"], text=True).strip(),
+               "source_sha256": sha(Path(__file__)), "workers": 4,
+               "new_random_prefixes": 0, "partial_files": "Original abandoned temporary CSVs retained; no partial batches scored"}
+    run.setdefault("resumptions", []).append(receipt)
+    def save():
+        path.write_text(json.dumps(run, indent=2)+"\n")
+    run["status"] = "running"; save()
+    try:
+        with tempfile.TemporaryDirectory(prefix="matching-exact-pairs-resume-") as temp:
+            temp = Path(temp)
+            binary = temp / "engine"
+            subprocess.run(["clang++", "-O3", "-std=c++17", str(HERE / "engine.cpp"), "-o", str(binary)], check=True)
+            receipt["binary_sha256"] = sha(binary)
+            def job(entry):
+                batch = entry["batch"]
+                raw = temp / f"exact-square-L512-b{batch:02d}.csv"
+                target = HERE / "data" / (raw.name+".gz")
+                if target.exists():
+                    raise RuntimeError(f"Unregistered output already exists: {target}")
+                command = [str(binary), str(run["contract"]["L"]), str(entry["samples"]),
+                           entry["seed"], str(run["contract"]["b"]), str(raw)]
+                start = time.monotonic()
+                log = subprocess.check_output(command, text=True).strip()
+                seconds = time.monotonic()-start
+                payload = raw.read_bytes()
+                assert payload.count(b"\n") == entry["samples"]+1
+                with target.open("xb") as out:
+                    with gzip.GzipFile(fileobj=out, mode="wb", mtime=0) as gz:
+                        gz.write(payload)
+                print(f"batch={batch}: {log}", flush=True)
+                return dict(batch=batch, samples=entry["samples"], seed=entry["seed"],
+                            file=target.name, sha256=sha(target), original_file=entry["file"],
+                            dependency_group=entry["dependency_group"], runtime_seconds=seconds,
+                            engine_log=log, command=shlex.join(command))
+            with ThreadPoolExecutor(max_workers=4) as pool:
+                for future in as_completed([pool.submit(job, entry) for entry in remaining]):
+                    run["batches"].append(future.result())
+                    run["batches"].sort(key=lambda e: e["batch"]); save()
+        run["status"] = "completed"
+    except BaseException as error:
+        run["status"] = "failed"; receipt["error"] = str(error)
+        raise
+    finally:
+        receipt["wall_seconds"] = time.monotonic()-begin
+        run["utc_finished"] = datetime.now(timezone.utc).isoformat()
+        run["elapsed_seconds_including_owner_pause"] = (datetime.now(timezone.utc)-datetime.fromisoformat(run["utc_started"])).total_seconds()
+        run["timing_note"] = "Per-batch times cover successful evaluations only; elapsed includes owner pause. Abandoned partial work is not a second sample."
+        save()
+    print(f"Completed remaining batches in {receipt['wall_seconds']:.3f}s", flush=True)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/analysis/exact-completion-pairs-20260929/run.log b/analysis/exact-completion-pairs-20260929/run.log
new file mode 100644
index 000000000..076b21b77
--- /dev/null
+++ b/analysis/exact-completion-pairs-20260929/run.log
@@ -0,0 +1,4 @@
+batch=1: prefixes=10000 seconds=221.25
+batch=3: prefixes=10000 seconds=221.769
+batch=0: prefixes=10000 seconds=222.415
+batch=2: prefixes=10000 seconds=222.874
diff --git a/analysis/exact-completion-pairs-20260929/run.py b/analysis/exact-completion-pairs-20260929/run.py
new file mode 100644
index 000000000..d355be838
--- /dev/null
+++ b/analysis/exact-completion-pairs-20260929/run.py
@@ -0,0 +1,108 @@
+#!/usr/bin/env python3
+"""Replay existing prefixes and integrate out safe-site probe randomness.
+
+This is not an independent acquisition. All seed/sample identities come
+from the completed safe-insertion block. No old files are overwritten.
+"""
+import argparse
+from concurrent.futures import ThreadPoolExecutor, as_completed
+from datetime import datetime, timezone
+import gzip
+import hashlib
+import json
+import os
+from pathlib import Path
+import platform
+import shlex
+import subprocess
+import tempfile
+import time
+
+HERE = Path(__file__).resolve().parent
+OLD = HERE.parent / "safe-insertion-production-20260929"
+
+
+def sha(path):
+    return hashlib.sha256(path.read_bytes()).hexdigest()
+
+
+def main():
+    ap = argparse.ArgumentParser()
+    ap.add_argument("--source-commit", required=True)
+    ap.add_argument("--workers", type=int, default=4)
+    ap.add_argument("--compiler", default="clang++")
+    args = ap.parse_args()
+    if not 1 <= args.workers <= 4:
+        raise SystemExit("This local replay uses at most four workers")
+    source = OLD / "data/run.json"
+    original = json.loads(source.read_text())
+    assert original["status"] == "completed"
+    contract = original["contract"]
+    directory = HERE / "data"
+    directory.mkdir(exist_ok=True)
+    manifest = directory / "run.json"
+    if manifest.exists():
+        raise SystemExit("Do not overwrite or silently resume an existing replay")
+    started = time.monotonic()
+    run = {"schema": "matching-one.exact-pair-replay.v1", "status": "building",
+           "source_commit": args.source_commit, "pid": os.getpid(),
+           "source_sha256": {name: sha(HERE / name) for name in ("engine.cpp", "run.py")},
+           "included_geometry_sha256": sha(HERE.parent / "completion-hazard-production-20260929/engine.cpp"),
+           "original_manifest": "analysis/safe-insertion-production-20260929/data/run.json",
+           "original_manifest_sha256": sha(source), "original_acquisition_commit": original["source_commit"],
+           "contract": contract, "new_random_prefixes": 0,
+           "interpretation": "Same source prefixes, exact conditional mean of safe-insertion probes. Post-readout estimator refinement, not new independent evidence.",
+           "workers": args.workers, "platform": platform.platform(), "python": platform.python_version(),
+           "command": shlex.join(["python3", str(Path(__file__).resolve()), *os.sys.argv[1:]]),
+           "compiler": subprocess.check_output([args.compiler, "--version"], text=True).strip(),
+           "utc_started": datetime.now(timezone.utc).isoformat(), "batches": []}
+    def save():
+        manifest.write_text(json.dumps(run, indent=2)+"\n")
+    save()
+    try:
+        with tempfile.TemporaryDirectory(prefix="exact-pair-build-", dir=HERE) as temp:
+            temp = Path(temp)
+            binary = temp / "engine"
+            build = [args.compiler, "-O3", "-std=c++17", str(HERE / "engine.cpp"), "-o", str(binary)]
+            subprocess.run(build, check=True)
+            run["build_command"] = shlex.join(build);run["binary_sha256"] = sha(binary)
+            # Physical witness control and first128 cost/source check were
+            # already done once locally before this full replay, not repeated.
+            def job(entry):
+                batch = entry["batch"]
+                raw = temp / f"exact-square-L512-b{batch:02d}.csv"
+                command = [str(binary), str(contract["L"]), str(entry["samples"]),
+                           entry["seed"], str(contract["b"]), str(raw)]
+                begin = time.monotonic()
+                log = subprocess.check_output(command, text=True).strip()
+                seconds = time.monotonic()-begin
+                data = raw.read_bytes()
+                if data.count(b"\n") != entry["samples"]+1:
+                    raise RuntimeError("incomplete replay")
+                target = directory / (raw.name+".gz")
+                with target.open("wb") as out:
+                    with gzip.GzipFile(fileobj=out, mode="wb", mtime=0) as gz:
+                        gz.write(data)
+                print(f"batch={batch}: {log}", flush=True)
+                return {"batch": batch, "samples": entry["samples"], "seed": entry["seed"],
+                        "file": target.name, "sha256": sha(target), "original_file": entry["file"],
+                        "dependency_group": entry["dependency_group"], "runtime_seconds": seconds,
+                        "engine_log": log, "command": shlex.join(command)}
+            run["status"] = "running";save()
+            with ThreadPoolExecutor(max_workers=args.workers) as pool:
+                futures = [pool.submit(job, entry) for entry in original["batches"]]
+                for future in as_completed(futures):
+                    run["batches"].append(future.result())
+                    run["batches"].sort(key=lambda x: x["batch"]);save()
+        run["status"] = "completed"
+    except BaseException as error:
+        run["status"] = "failed";run["error"] = str(error)
+        raise
+    finally:
+        run["wall_seconds"] = time.monotonic()-started
+        run["utc_finished"] = datetime.now(timezone.utc).isoformat();save()
+    print(f"Completed same-prefix replay in {run['wall_seconds']:.3f}s", flush=True)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/analysis/exact-pair-independent-archive-20260930/README.md b/analysis/exact-pair-independent-archive-20260930/README.md
new file mode 100644
index 000000000..353b3fd64
--- /dev/null
+++ b/analysis/exact-pair-independent-archive-20260930/README.md
@@ -0,0 +1,77 @@
+# 固定负向目标：独立既有档案复核
+
+唯一科学任务：将前块的固定预测 `deltaY < 0` 应用于旧 completion-hazard
+档案的14×10000个排列，补读精确协同边数e。该档案其它结果早已读过；
+本次是此前未测过e的固定目标读出，不是全新前瞻采样或原封未看的holdout。
+
+已完成：deltaY=-0.003199368397±0.000620090643（14批jackknife SE），
+负向预测在此独立既有档案中再现。前块为-0.003449385668±0.000772252763；
+本块减前块=+0.000250017271±0.000990397261，两块点差仅0.2524个比较SE，未合并。
+140000行全部对应；共同支持54120/54723=98.8981%，1202个共同格。
+具体数字、原始统计及结论边界见[结果](results/RESULT.md)和[完整JSON](results/result.json)。
+
+参数不变：square site、L=512、N=262144、a=154646、b=155385。
+只在rank1且有安全空站点的前缀内，以精确整数方向D和当前完成数c分格：
+早组J1≤a，晚组a<J1≤b，Ybar=2sum_e/[(N-b-c)n]，w=nE*nL/(nE+nL)。
+这里c是完成数，不是旧manifest中第三个时间cutoff 156120。
+
+保存deltaY、deltaE、deltaZ2、earlyY、lateY及5×5完整协方差。
+14次对齐整批删除均重新求共同支持和权重；点估计不作jackknife偏差修正。
+deltaZ2=-2deltaE/[(N-b)(N-b-1)]，是同一几何公式的派生量。
+不合并两独立块；仅用两块点差及sqrt(SE1²+SE2²)比较。
+
+## 来源与行对应
+
+- `inputs/archive/`：completion-hazard-production-20260929旧manifest及14个压缩生产批次。
+- `inputs/reference-result.json`：前块原结果的未改动副本；只读取，不重新评分。
+- `source/`：两个原engine.cpp的逐字节副本，保留include相对结构。
+- `source-record.json`：实际源hash、准备时HEAD、各源文件最近提交、批次来源。
+- `data/`：本次精确重放原始行、每批命令和退出状态、完整run.json。
+- `results/result.json`：完整评分、整数格充分统计、批次依赖、协方差和删除结果。
+- `results/RESULT.md`：简短数值报告。
+
+旧rank_b=0若J1>b、=2若J2≤b、否则=1；只把旧rank0的未来J1映射为0。
+其余J1和dx_b、dy_b、nu_b原样逐行核对。旧排列的Fisher–Yates消耗独立mt19937_64
+流，tau使用另一time_rng；沿用旧seed重放，不改变几何或随机数实现。
+无伪造四探针，无原scorer调用，无新seed或独立样本。
+
+## 实际运行命令
+
+本地准备与最终评分使用研究Python3.11.15；只需标准库：
+
+```sh
+/Users/lc/python-envs/research-py311/bin/python analysis/exact-pair-independent-archive-20260930/runner.py prepare
+/Users/lc/python-envs/research-py311/bin/python analysis/exact-pair-independent-archive-20260930/score.py
+```
+
+云端仅TV2N0X（4a8d1d443419434889e49148ed0a7ba6），新路径
+`/workspace/Matching-One-TV2N0X/exact-pair-independent-20260930`。
+启动前实时Ready且隧道已停；Running后仅一次ssh-key-reset，成功且密钥mode600。
+初始进程清单无其他研究作业。实测16 affinity CPU、14.5核配额、25GiB内存；使用14 workers。
+
+容器重启后系统g++未保留。首次benchmark命令在读取编译器版本时终止，尚未重放任何行。
+随后把编译工具安装到本次路径的独立installroot；未更改现有研究目录或系统包集合：
+
+```sh
+yum -y --installroot=/workspace/Matching-One-TV2N0X/exact-pair-independent-20260930/toolchain --releasever=2.0 --setopt=reposdir=/etc/yum.repos.d --setopt=logdir=/workspace/Matching-One-TV2N0X/exact-pair-independent-20260930/toolchain-logs --nodocs install gcc-c++ time
+```
+
+第一次安装命令因log目录尚未创建立即退出，创建该目录后安装成功。
+GCC10.3.1在该目录内chroot编译原源，Python3.9.9驱动重放；具体编译命令及二进制hash见benchmark.json。
+
+```sh
+cd /workspace/Matching-One-TV2N0X/exact-pair-independent-20260930
+OMP_NUM_THREADS=1 OPENBLAS_NUM_THREADS=1 MKL_NUM_THREADS=1 python3 runner.py benchmark
+nohup env OMP_NUM_THREADS=1 OPENBLAS_NUM_THREADS=1 MKL_NUM_THREADS=1 python3 runner.py replay > run.log 2>&1 < /dev/null &
+```
+
+一次32行成本/对应检查，使用旧batch0同seed，耗时1.020989秒，峰值75960KiB，全部映射对应。
+此重复前缀的科学权重为0。完整重放PID1015，14批全部成功，墙钟520.875642秒。
+原始行及每批命令均已取回。
+完整评分合并执行140000行对应检查，不额外重跑原枚举、旧评分或全套测试。
+
+此次仅写本目录，无commit/push、Issue操作、现有engine/文档/ledger修改或PR合并。
+PR838读取时为OPEN/Draft；资源收尾事实见execution.json。
+
+2026-09-30 02:25:19 UTC（北京时间10:25:19）已实时核实TV2N0X为Ready、己方隧道已停。
+关机前无其他研究进程，远端数据与任务内编译环境保留。本范围未完成事项：无。
diff --git a/analysis/exact-pair-independent-archive-20260930/benchmark.json b/analysis/exact-pair-independent-archive-20260930/benchmark.json
new file mode 100644
index 000000000..037168b04
--- /dev/null
+++ b/analysis/exact-pair-independent-archive-20260930/benchmark.json
@@ -0,0 +1,51 @@
+{
+  "environment": {
+    "utc": "2026-09-30T02:12:22.665118+00:00",
+    "platform": "Linux-5.10.0-182.0.0.95.r3582_286.hce2.aarch64-aarch64-with-glibc2.34",
+    "python": "3.9.9 (main, May 16 2026, 13:00:28) \n[GCC 10.3.1]",
+    "machine": "aarch64",
+    "hostname": "4a8d1d443419434889e49148ed0a7ba6",
+    "cpu_quota_cores": 14.5,
+    "memory_limit_bytes": 26843545600,
+    "affinity_cpus": 16,
+    "compiler": "g++ (GCC) 10.3.1\nCopyright (C) 2020 Free Software Foundation, Inc.\nThis is free software; see the source for copying conditions.  There is NO\nwarranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.\n\n",
+    "ps": "    PID    PPID COMMAND             ELAPSED %CPU %MEM\n      1       0 tini                  06:23  0.0  0.0\n      6       1 start_docker.sh       06:23  0.0  0.0\n     11       1 sudo                  06:23  0.0  0.0\n     16      11 devenvd               06:23  0.0  0.0\n     17       1 sshd                  06:23  0.0  0.0\n    969       6 sleep                 00:03  0.0  0.0\n    971      17 sshd                  00:00  0.0  0.0\n    974     971 sshd                  00:00  0.0  0.0\n    975     974 python3               00:00  0.0  0.0\n    985     975 ps                    00:00  0.0  0.0\n",
+    "thread_environment": {
+      "OMP_NUM_THREADS": "1",
+      "OPENBLAS_NUM_THREADS": "1",
+      "MKL_NUM_THREADS": "1"
+    }
+  },
+  "build_command": [
+    "chroot",
+    "/workspace/Matching-One-TV2N0X/exact-pair-independent-20260930/toolchain",
+    "/usr/bin/g++",
+    "-O3",
+    "-std=c++17",
+    "/project/source/exact-completion-pairs-20260929/engine.cpp",
+    "-o",
+    "/project/engine"
+  ],
+  "binary_sha256": "72ddbdb1c2e26352ed8f62b820bfcdc741750bc5b50cf459f4b0adefe019fab7",
+  "command": [
+    "/workspace/Matching-One-TV2N0X/exact-pair-independent-20260930/toolchain/usr/bin/time",
+    "-v",
+    "/workspace/Matching-One-TV2N0X/exact-pair-independent-20260930/build/engine",
+    "512",
+    "32",
+    "18419746559907930925",
+    "155385",
+    "/workspace/Matching-One-TV2N0X/exact-pair-independent-20260930/build/same-seed-32.csv"
+  ],
+  "stdout": "prefixes=32 seconds=1.00033\n",
+  "time_log": "\tCommand being timed: \"/workspace/Matching-One-TV2N0X/exact-pair-independent-20260930/build/engine 512 32 18419746559907930925 155385 /workspace/Matching-One-TV2N0X/exact-pair-independent-20260930/build/same-seed-32.csv\"\n\tUser time (seconds): 1.01\n\tSystem time (seconds): 0.00\n\tPercent of CPU this job got: 99%\n\tElapsed (wall clock) time (h:mm:ss or m:ss): 0:01.01\n\tAverage shared text size (kbytes): 0\n\tAverage unshared data size (kbytes): 0\n\tAverage stack size (kbytes): 0\n\tAverage total size (kbytes): 0\n\tMaximum resident set size (kbytes): 75960\n\tAverage resident set size (kbytes): 0\n\tMajor (requiring I/O) page faults: 0\n\tMinor (reclaiming a frame) page faults: 10253\n\tVoluntary context switches: 0\n\tInvoluntary context switches: 2\n\tSwaps: 0\n\tFile system inputs: 0\n\tFile system outputs: 8\n\tSocket messages sent: 0\n\tSocket messages received: 0\n\tSignals delivered: 0\n\tPage size (bytes): 4096\n\tExit status: 0\n",
+  "wall_seconds": 1.020989065989852,
+  "max_rss_kib": 75960,
+  "same_seed": "18419746559907930925",
+  "batch": 0,
+  "rows": 32,
+  "all_mapped_first5_match": true,
+  "scientific_weight": 0,
+  "recommended_workers": 14,
+  "approximate_replay_seconds": 319.05908312182873
+}
diff --git a/analysis/exact-pair-independent-archive-20260930/data/batch-00.json b/analysis/exact-pair-independent-archive-20260930/data/batch-00.json
new file mode 100644
index 000000000..9a6c75af6
--- /dev/null
+++ b/analysis/exact-pair-independent-archive-20260930/data/batch-00.json
@@ -0,0 +1,19 @@
+{
+  "batch": 0,
+  "seed": "18419746559907930925",
+  "samples": 10000,
+  "original_file": "production-square-L512-b00.csv.gz",
+  "dependency_group": "completion-independent-20260929/square/L512/batch0",
+  "command": [
+    "/workspace/Matching-One-TV2N0X/exact-pair-independent-20260930/build/engine",
+    "512",
+    "10000",
+    "18419746559907930925",
+    "155385",
+    "/workspace/Matching-One-TV2N0X/exact-pair-independent-20260930/data/exact-square-L512-b00.csv"
+  ],
+  "returncode": 0,
+  "stdout": "prefixes=10000 seconds=497.94\n",
+  "stderr": "",
+  "wall_seconds": 498.0191309801303
+}
diff --git a/analysis/exact-pair-independent-archive-20260930/data/batch-01.json b/analysis/exact-pair-independent-archive-20260930/data/batch-01.json
new file mode 100644
index 000000000..5c46118c8
--- /dev/null
+++ b/analysis/exact-pair-independent-archive-20260930/data/batch-01.json
@@ -0,0 +1,19 @@
+{
+  "batch": 1,
+  "seed": "13442365983088886094",
+  "samples": 10000,
+  "original_file": "production-square-L512-b01.csv.gz",
+  "dependency_group": "completion-independent-20260929/square/L512/batch1",
+  "command": [
+    "/workspace/Matching-One-TV2N0X/exact-pair-independent-20260930/build/engine",
+    "512",
+    "10000",
+    "13442365983088886094",
+    "155385",
+    "/workspace/Matching-One-TV2N0X/exact-pair-independent-20260930/data/exact-square-L512-b01.csv"
+  ],
+  "returncode": 0,
+  "stdout": "prefixes=10000 seconds=496.606\n",
+  "stderr": "",
+  "wall_seconds": 496.69426990509965
+}
diff --git a/analysis/exact-pair-independent-archive-20260930/data/batch-02.json b/analysis/exact-pair-independent-archive-20260930/data/batch-02.json
new file mode 100644
index 000000000..c8721ddb2
--- /dev/null
+++ b/analysis/exact-pair-independent-archive-20260930/data/batch-02.json
@@ -0,0 +1,19 @@
+{
+  "batch": 2,
+  "seed": "12192384056273429425",
+  "samples": 10000,
+  "original_file": "production-square-L512-b02.csv.gz",
+  "dependency_group": "completion-independent-20260929/square/L512/batch2",
+  "command": [
+    "/workspace/Matching-One-TV2N0X/exact-pair-independent-20260930/build/engine",
+    "512",
+    "10000",
+    "12192384056273429425",
+    "155385",
+    "/workspace/Matching-One-TV2N0X/exact-pair-independent-20260930/data/exact-square-L512-b02.csv"
+  ],
+  "returncode": 0,
+  "stdout": "prefixes=10000 seconds=518.555\n",
+  "stderr": "",
+  "wall_seconds": 518.6520514099393
+}
diff --git a/analysis/exact-pair-independent-archive-20260930/data/batch-03.json b/analysis/exact-pair-independent-archive-20260930/data/batch-03.json
new file mode 100644
index 000000000..c242560ec
--- /dev/null
+++ b/analysis/exact-pair-independent-archive-20260930/data/batch-03.json
@@ -0,0 +1,19 @@
+{
+  "batch": 3,
+  "seed": "16795263932832913938",
+  "samples": 10000,
+  "original_file": "production-square-L512-b03.csv.gz",
+  "dependency_group": "completion-independent-20260929/square/L512/batch3",
+  "command": [
+    "/workspace/Matching-One-TV2N0X/exact-pair-independent-20260930/build/engine",
+    "512",
+    "10000",
+    "16795263932832913938",
+    "155385",
+    "/workspace/Matching-One-TV2N0X/exact-pair-independent-20260930/data/exact-square-L512-b03.csv"
+  ],
+  "returncode": 0,
+  "stdout": "prefixes=10000 seconds=512.749\n",
+  "stderr": "",
+  "wall_seconds": 512.8563893151004
+}
diff --git a/analysis/exact-pair-independent-archive-20260930/data/batch-04.json b/analysis/exact-pair-independent-archive-20260930/data/batch-04.json
new file mode 100644
index 000000000..921bd5147
--- /dev/null
+++ b/analysis/exact-pair-independent-archive-20260930/data/batch-04.json
@@ -0,0 +1,19 @@
+{
+  "batch": 4,
+  "seed": "5606540329541986450",
+  "samples": 10000,
+  "original_file": "production-square-L512-b04.csv.gz",
+  "dependency_group": "completion-independent-20260929/square/L512/batch4",
+  "command": [
+    "/workspace/Matching-One-TV2N0X/exact-pair-independent-20260930/build/engine",
+    "512",
+    "10000",
+    "5606540329541986450",
+    "155385",
+    "/workspace/Matching-One-TV2N0X/exact-pair-independent-20260930/data/exact-square-L512-b04.csv"
+  ],
+  "returncode": 0,
+  "stdout": "prefixes=10000 seconds=520.684\n",
+  "stderr": "",
+  "wall_seconds": 520.7907340640668
+}
diff --git a/analysis/exact-pair-independent-archive-20260930/data/batch-05.json b/analysis/exact-pair-independent-archive-20260930/data/batch-05.json
new file mode 100644
index 000000000..4590177a4
--- /dev/null
+++ b/analysis/exact-pair-independent-archive-20260930/data/batch-05.json
@@ -0,0 +1,19 @@
+{
+  "batch": 5,
+  "seed": "176734671850227651",
+  "samples": 10000,
+  "original_file": "production-square-L512-b05.csv.gz",
+  "dependency_group": "completion-independent-20260929/square/L512/batch5",
+  "command": [
+    "/workspace/Matching-One-TV2N0X/exact-pair-independent-20260930/build/engine",
+    "512",
+    "10000",
+    "176734671850227651",
+    "155385",
+    "/workspace/Matching-One-TV2N0X/exact-pair-independent-20260930/data/exact-square-L512-b05.csv"
+  ],
+  "returncode": 0,
+  "stdout": "prefixes=10000 seconds=519.204\n",
+  "stderr": "",
+  "wall_seconds": 519.3030597630423
+}
diff --git a/analysis/exact-pair-independent-archive-20260930/data/batch-06.json b/analysis/exact-pair-independent-archive-20260930/data/batch-06.json
new file mode 100644
index 000000000..c3350fa7e
--- /dev/null
+++ b/analysis/exact-pair-independent-archive-20260930/data/batch-06.json
@@ -0,0 +1,19 @@
+{
+  "batch": 6,
+  "seed": "14922363405798405830",
+  "samples": 10000,
+  "original_file": "production-square-L512-b06.csv.gz",
+  "dependency_group": "completion-independent-20260929/square/L512/batch6",
+  "command": [
+    "/workspace/Matching-One-TV2N0X/exact-pair-independent-20260930/build/engine",
+    "512",
+    "10000",
+    "14922363405798405830",
+    "155385",
+    "/workspace/Matching-One-TV2N0X/exact-pair-independent-20260930/data/exact-square-L512-b06.csv"
+  ],
+  "returncode": 0,
+  "stdout": "prefixes=10000 seconds=520.065\n",
+  "stderr": "",
+  "wall_seconds": 520.1712592260446
+}
diff --git a/analysis/exact-pair-independent-archive-20260930/data/batch-07.json b/analysis/exact-pair-independent-archive-20260930/data/batch-07.json
new file mode 100644
index 000000000..68e145a5a
--- /dev/null
+++ b/analysis/exact-pair-independent-archive-20260930/data/batch-07.json
@@ -0,0 +1,19 @@
+{
+  "batch": 7,
+  "seed": "3540279280158270918",
+  "samples": 10000,
+  "original_file": "production-square-L512-b07.csv.gz",
+  "dependency_group": "completion-independent-20260929/square/L512/batch7",
+  "command": [
+    "/workspace/Matching-One-TV2N0X/exact-pair-independent-20260930/build/engine",
+    "512",
+    "10000",
+    "3540279280158270918",
+    "155385",
+    "/workspace/Matching-One-TV2N0X/exact-pair-independent-20260930/data/exact-square-L512-b07.csv"
+  ],
+  "returncode": 0,
+  "stdout": "prefixes=10000 seconds=508.834\n",
+  "stderr": "",
+  "wall_seconds": 508.9134591759648
+}
diff --git a/analysis/exact-pair-independent-archive-20260930/data/batch-08.json b/analysis/exact-pair-independent-archive-20260930/data/batch-08.json
new file mode 100644
index 000000000..ea1b1e14c
--- /dev/null
+++ b/analysis/exact-pair-independent-archive-20260930/data/batch-08.json
@@ -0,0 +1,19 @@
+{
+  "batch": 8,
+  "seed": "182843979474712324",
+  "samples": 10000,
+  "original_file": "production-square-L512-b08.csv.gz",
+  "dependency_group": "completion-independent-20260929/square/L512/batch8",
+  "command": [
+    "/workspace/Matching-One-TV2N0X/exact-pair-independent-20260930/build/engine",
+    "512",
+    "10000",
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+  "stdout": "prefixes=10000 seconds=514.132\n",
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+}
diff --git a/analysis/exact-pair-independent-archive-20260930/data/batch-09.json b/analysis/exact-pair-independent-archive-20260930/data/batch-09.json
new file mode 100644
index 000000000..cbff29c94
--- /dev/null
+++ b/analysis/exact-pair-independent-archive-20260930/data/batch-09.json
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+{
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+  "seed": "4164285986391024372",
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+  "original_file": "production-square-L512-b09.csv.gz",
+  "dependency_group": "completion-independent-20260929/square/L512/batch9",
+  "command": [
+    "/workspace/Matching-One-TV2N0X/exact-pair-independent-20260930/build/engine",
+    "512",
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+    "155385",
+    "/workspace/Matching-One-TV2N0X/exact-pair-independent-20260930/data/exact-square-L512-b09.csv"
+  ],
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+  "stdout": "prefixes=10000 seconds=520.72\n",
+  "stderr": "",
+  "wall_seconds": 520.8245567351114
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diff --git a/analysis/exact-pair-independent-archive-20260930/data/batch-10.json b/analysis/exact-pair-independent-archive-20260930/data/batch-10.json
new file mode 100644
index 000000000..60d16cf62
--- /dev/null
+++ b/analysis/exact-pair-independent-archive-20260930/data/batch-10.json
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+{
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+  "original_file": "production-square-L512-b10.csv.gz",
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+  "command": [
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diff --git a/analysis/exact-pair-independent-archive-20260930/data/batch-11.json b/analysis/exact-pair-independent-archive-20260930/data/batch-11.json
new file mode 100644
index 000000000..12b111a6f
--- /dev/null
+++ b/analysis/exact-pair-independent-archive-20260930/data/batch-11.json
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+{
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+  "original_file": "production-square-L512-b11.csv.gz",
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+    "/workspace/Matching-One-TV2N0X/exact-pair-independent-20260930/data/exact-square-L512-b11.csv"
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+  "stdout": "prefixes=10000 seconds=520.011\n",
+  "stderr": "",
+  "wall_seconds": 520.1165391458198
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diff --git a/analysis/exact-pair-independent-archive-20260930/data/batch-12.json b/analysis/exact-pair-independent-archive-20260930/data/batch-12.json
new file mode 100644
index 000000000..73056b506
--- /dev/null
+++ b/analysis/exact-pair-independent-archive-20260930/data/batch-12.json
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+{
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+  "original_file": "production-square-L512-b12.csv.gz",
+  "dependency_group": "completion-independent-20260929/square/L512/batch12",
+  "command": [
+    "/workspace/Matching-One-TV2N0X/exact-pair-independent-20260930/build/engine",
+    "512",
+    "10000",
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+    "155385",
+    "/workspace/Matching-One-TV2N0X/exact-pair-independent-20260930/data/exact-square-L512-b12.csv"
+  ],
+  "returncode": 0,
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+  "stderr": "",
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diff --git a/analysis/exact-pair-independent-archive-20260930/data/batch-13.json b/analysis/exact-pair-independent-archive-20260930/data/batch-13.json
new file mode 100644
index 000000000..02d2a8a56
--- /dev/null
+++ b/analysis/exact-pair-independent-archive-20260930/data/batch-13.json
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+{
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+  "original_file": "production-square-L512-b13.csv.gz",
+  "dependency_group": "completion-independent-20260929/square/L512/batch13",
+  "command": [
+    "/workspace/Matching-One-TV2N0X/exact-pair-independent-20260930/build/engine",
+    "512",
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+    "10329155733035115099",
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+    "/workspace/Matching-One-TV2N0X/exact-pair-independent-20260930/data/exact-square-L512-b13.csv"
+  ],
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+  "stdout": "prefixes=10000 seconds=515.136\n",
+  "stderr": "",
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diff --git a/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b00.csv b/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b00.csv
new file mode 100644
index 000000000..34c2a92cf
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diff --git a/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b01.csv b/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b01.csv
new file mode 100644
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diff --git a/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b02.csv b/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b02.csv
new file mode 100644
index 000000000..cfa8f63cc
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diff --git a/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b03.csv b/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b03.csv
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index 000000000..f3df67a2e
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new file mode 100644
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diff --git a/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b04.csv b/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b04.csv
new file mode 100644
index 000000000..b76b8438c
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diff --git a/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b05.csv b/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b05.csv
new file mode 100644
index 000000000..eabd2a8a9
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diff --git a/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b05.csv.gz b/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b05.csv.gz
new file mode 100644
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diff --git a/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b06.csv b/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b06.csv
new file mode 100644
index 000000000..b6a166fa4
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diff --git a/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b07.csv b/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b07.csv
new file mode 100644
index 000000000..9d75fec68
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diff --git a/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b08.csv b/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b08.csv
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diff --git a/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b09.csv b/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b09.csv
new file mode 100644
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diff --git a/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b10.csv b/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b10.csv
new file mode 100644
index 000000000..cc252b537
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diff --git a/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b11.csv b/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b11.csv
new file mode 100644
index 000000000..aea763735
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new file mode 100644
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diff --git a/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b12.csv b/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b12.csv
new file mode 100644
index 000000000..1ee499856
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diff --git a/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b12.csv.gz b/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b12.csv.gz
new file mode 100644
index 000000000..4ac8c932b
Binary files /dev/null and b/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b12.csv.gz differ
diff --git a/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b13.csv b/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b13.csv
new file mode 100644
index 000000000..3b512deee
--- /dev/null
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diff --git a/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b13.csv.gz b/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b13.csv.gz
new file mode 100644
index 000000000..0c6832311
Binary files /dev/null and b/analysis/exact-pair-independent-archive-20260930/data/exact-square-L512-b13.csv.gz differ
diff --git a/analysis/exact-pair-independent-archive-20260930/data/run.json b/analysis/exact-pair-independent-archive-20260930/data/run.json
new file mode 100644
index 000000000..e26132c6e
--- /dev/null
+++ b/analysis/exact-pair-independent-archive-20260930/data/run.json
@@ -0,0 +1,333 @@
+{
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+}
diff --git a/analysis/exact-pair-independent-archive-20260930/data/started.json b/analysis/exact-pair-independent-archive-20260930/data/started.json
new file mode 100644
index 000000000..d8f4a8299
--- /dev/null
+++ b/analysis/exact-pair-independent-archive-20260930/data/started.json
@@ -0,0 +1,36 @@
+{
+  "status": "running",
+  "started_utc": "2026-09-30T02:12:46.643643+00:00",
+  "pid": 1015,
+  "workers": 14,
+  "environment": {
+    "utc": "2026-09-30T02:12:46.639231+00:00",
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diff --git a/analysis/exact-pair-independent-archive-20260930/execution.json b/analysis/exact-pair-independent-archive-20260930/execution.json
new file mode 100644
index 000000000..dfed54527
--- /dev/null
+++ b/analysis/exact-pair-independent-archive-20260930/execution.json
@@ -0,0 +1,96 @@
+{
+  "status": "complete",
+  "completed_utc": "2026-09-30T02:25:19Z",
+  "write_scope": "analysis/exact-pair-independent-archive-20260930/",
+  "local_root": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/exact-pair-independent-archive-20260930",
+  "source_head_at_preparation": "2d315636cb18b7f3c4f7e44dd31b14f6d3a896cf",
+  "shared_head_at_final_read": "eb9884dcc0a550b5d87fb247084e059504e76b2a",
+  "shared_head_note": "Other agent advanced the mechanism mainline; this task did not commit or push.",
+  "engine_last_commit": "8b5eee0d47b6c220c2ba46a5bae08d2d19b392a4",
+  "geometry_and_original_acquisition_commit": "9520c9bdf702a62435f1054e64184b1df4fc0ab3",
+  "actual_remote_compiled_source_sha256": {
+    "exact-completion-pairs-20260929/engine.cpp": "48def78ee470acef233fd4c4d2543f1d89c1869b552a4a434ab2aa220be95682",
+    "completion-hazard-production-20260929/engine.cpp": "42ac8a013cb731f4da4e3121ca429fd475555063e876638ad5d375d195448115"
+  },
+  "compiled_source_note": "sha256sum read both uploaded sources and the copies inside toolchain/project/source; both pairs matched. Sources were copied byte for byte.",
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+  "binary_sha256": "72ddbdb1c2e26352ed8f62b820bfcdc741750bc5b50cf459f4b0adefe019fab7",
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+    "workers": 14,
+    "batches": 14,
+    "samples_per_batch": 10000,
+    "new_samples": 0,
+    "wall_seconds": 520.8756419259589,
+    "pid": 1015,
+    "exit_status": "all 14 engines exit 0",
+    "data_retrieved": true
+  },
+  "validation": {
+    "benchmark_old_seed_rows": 32,
+    "benchmark_scientific_weight": 0,
+    "benchmark_seconds": 1.020989065989852,
+    "benchmark_max_rss_KiB": 75960,
+    "full_mapped_rows": 140000,
+    "all_first5_match": true,
+    "full_score_runs": 1,
+    "old_census_runs": 0,
+    "full_test_suites": 0,
+    "old_scoring_runs": 0
+  },
+  "numerical_result": {
+    "deltaY": -0.003199368397006982,
+    "SE": 0.0006200906428905627,
+    "archive_minus_reference": 0.0002500172706480755,
+    "difference_SE": 0.0009903972611504154,
+    "pooled_blocks": false
+  },
+  "scope_note": "Previously unmeasured e from an existing independent random archive whose other outcomes had already been read. Not prospective new sampling or untouched holdout.",
+  "resources": {
+    "instance": "DevEnvC_TV2N0X",
+    "uuid": "4a8d1d443419434889e49148ed0a7ba6",
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+    "affinity_cpus": 16,
+    "memory_limit_bytes": 26843545600,
+    "system_package_changes": false,
+    "toolchain": "Task-local yum installroot; preserved remotely.",
+    "initial_state": "Ready",
+    "ssh_key_resets": 1,
+    "ssh_key_reset_result": "success; mode 600",
+    "managed_tunnel_pid": 96593,
+    "managed_tunnel_port": 10026,
+    "pre_shutdown_no_other_research_jobs": true,
+    "pre_shutdown_process_evidence": "Matching tunnel: DevEnvC_TV2N0X at 127.0.0.1:10026 (PID 96593).\n** WARNING: connection is not using a post-quantum key exchange algorithm.\r\n** This session may be vulnerable to \"store now, decrypt later\" attacks.\r\n** The server may need to be upgraded. See https://openssh.com/pq.html\r\n    PID    PPID COMMAND         STAT     ELAPSED %CPU %MEM\n      1       0 tini            Ss         16:41  0.0  0.0\n      6       1 start_docker.sh S          16:41  0.0  0.0\n     11       1 sudo            S          16:41  0.0  0.0\n     16      11 devenvd         Sl         16:41  0.0  0.1\n     17       1 sshd            S          16:41  0.0  0.0\n   1489       6 sleep           S          00:01  0.0  0.0\n   1490      17 sshd            Ss         00:00  0.0  0.0\n   1493    1490 sshd            S          00:00  0.0  0.0\n   1494    1493 bash            Ss         00:00  0.0  0.0\n   1502    1494 ps              R          00:00  0.0  0.0\nFilesystem      Size  Used Avail Use% Mounted on\n/dev/sda         40G   27G   13G  68% /workspace\n2026-09-30T02:22:40Z\n",
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+    "power_off_result": "Dev environment 4a8d1d443419434889e49148ed0a7ba6 is Stopping, please execute 'hdspace devenv list' to check the status\nCheck huawei list for Running/Ready; an accepted request can still be Starting/Stopping.\n",
+    "tunnel_stop_command": "/Users/lc/.codex/skills/connect-huawei-codebuddy/scripts/huawei TV2N0X stop",
+    "tunnel_stop_result": "Stopped only the managed tunnel for DevEnvC_TV2N0X.\n",
+    "final_state": "Ready",
+    "final_tunnel_state": "stopped",
+    "verified_utc": "2026-09-30T02:25:19Z",
+    "targeted_final_evidence": [
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+      "Selected key path: /Users/lc/.devenv/.ssh/IdentityFile/4a8d1d443419434889e49148ed0a7ba6 (content never displayed).",
+      "Selected tunnel is stopped: DevEnvC_TV2N0X.",
+      "2026-09-30T02:25:19Z"
+    ],
+    "remote_data_preserved": true,
+    "disk_reset": false,
+    "other_machine_operations": 0
+  },
+  "repository": {
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+    "state": "OPEN",
+    "isDraft": true,
+    "merged": false,
+    "commits": 0,
+    "pushes": 0,
+    "issue_operations": 0,
+    "existing_engine_document_ledger_edits": 0
+  },
+  "unfinished_items": []
+}
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diff --git a/analysis/exact-pair-independent-archive-20260930/inputs/archive/run.json b/analysis/exact-pair-independent-archive-20260930/inputs/archive/run.json
new file mode 100644
index 000000000..505b75208
--- /dev/null
+++ b/analysis/exact-pair-independent-archive-20260930/inputs/archive/run.json
@@ -0,0 +1,263 @@
+{
+  "schema": "matching-one.completion-hazard-run.v1",
+  "status": "completed",
+  "pid": 410,
+  "utc_started": "2026-09-29T14:43:58.518550+00:00",
+  "source_commit": "9520c9bdf702a62435f1054e64184b1df4fc0ab3",
+  "source_sha256": {
+    "engine.cpp": "42ac8a013cb731f4da4e3121ca429fd475555063e876638ad5d375d195448115",
+    "run.py": "9a9a06dec6d698e2827c0ac04034479d197c408e679b48deac9ee088d3c6ad70",
+    "contract.json": "6cde2b4b3adff1500ac02cf4b6b8e679259fd36a1e5676855fc314e486596964"
+  },
+  "contract": {
+    "schema": "matching-one.completion-hazard-contract.v1",
+    "namespace": "completion-independent-20260929",
+    "frozen_before_new_data": true,
+    "parent_commit": "e772037d09678a2dda299ca99f24dc6f6a765af9",
+    "lattice": "square",
+    "L": 512,
+    "N": 262144,
+    "a": 154646,
+    "b": 155385,
+    "c": 156120,
+    "cutoff_source": "analysis/birth-markov-kernel-20260929/result.json; old cloud square L512 pooled .25/.50/.75 birth-mixture quantiles, held fixed in this new block",
+    "batches": 14,
+    "samples_per_batch": 10000,
+    "workers": 14,
+    "sampling": "uniform Fisher-Yates permutation; independent mt19937_64 stream for uniform integer tau in [b,c-1]",
+    "early": "J1<=a and J2>b",
+    "late": "a<J1<=b and J2>b",
+    "primary": [
+      "early-minus-late survival P(J2>c)",
+      "overlap-weighted early-minus-late survival within exact direction",
+      "overlap-weighted early-minus-late survival within exact (direction,integer nu_b)"
+    ],
+    "mechanistic_readouts": [
+      "immediate mean nu_b contrast scaled by (c-b)/(N-b), not a finite-lag explanation",
+      "I=(c-b)*1{J2>tau}*nu_tau/(N-tau); hazard-predicted survival contrast mean(I_late)-mean(I_early)",
+      "actual survival contrast minus hazard-predicted contrast with same-batch covariance"
+    ],
+    "strata_weights": "w=e*l/(e+l), only strata with both groups; recompute on every batch deletion; not a mediated fraction of the pooled contrast",
+    "prospective_model_contrast": "rank-only, rank+direction, and rank+direction+instantaneous completion-count predictive sufficiency each imply their corresponding fixed finite-lag contrast is zero; a zero contrast does not establish full Markovness",
+    "analysis": "aligned delete-one batch estimates and full covariance; fixed sample count; no p-value stopping, cutoff scan, free exponent, regression feature search or pooling with old archives",
+    "benchmark": "32 independent samples, excluded from scientific block; runtime alone decides resource feasibility",
+    "scope": "one existing large size, one lattice and insertion clock; not an original-U identification or continuum/universality certificate"
+  },
+  "command": "python3 /workspace/Matching-One-TV2N0X/completion-hazard-20260929/run.py --source-commit 9520c9bdf702a62435f1054e64184b1df4fc0ab3",
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+  "python": "3.9.9",
+  "compiler": "g++ (GCC) 10.3.1\nCopyright (C) 2020 Free Software Foundation, Inc.\nThis is free software; see the source for copying conditions.  There is NO\nwarranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.",
+  "workers": 14,
+  "budget_seconds": 900,
+  "seed_rule": "first 64 SHA256 bits of matching-one/completion-independent-20260929/{role}/square/L512/{batch}/{stream}; stream=permutation or audit-time",
+  "batches": [
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+      "batch": 0,
+      "samples": 10000,
+      "seed": "18419746559907930925",
+      "time_seed": "12723381226631050454",
+      "role": "production",
+      "runtime_seconds": 308.89561684895307,
+      "engine_log": "samples=10000 seconds=308.887",
+      "command": "/workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/engine sample square 512 10000 18419746559907930925 12723381226631050454 155385 156120 /workspace/Matching-One-TV2N0X/completion-hazard-20260929/completion-build-paewxyus/production-square-L512-b00.csv",
+      "sha256": "32f16cc37728d67fdbe3446a48da1bb4d36c48fc4f051648ac8e36e2595d4026",
+      "dependency_group": "completion-independent-20260929/square/L512/batch0"
+    },
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+      "sha256": "fdc708228c1e0535787aa3426ea725b276ecef0b8ba193215b1bf93512518892",
+      "dependency_group": "completion-independent-20260929/square/L512/batch1"
+    },
+    {
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+      "batch": 2,
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+      "seed": "12192384056273429425",
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diff --git a/analysis/exact-pair-independent-archive-20260930/inputs/reference-result.json b/analysis/exact-pair-independent-archive-20260930/inputs/reference-result.json
new file mode 100644
index 000000000..a175040fe
--- /dev/null
+++ b/analysis/exact-pair-independent-archive-20260930/inputs/reference-result.json
@@ -0,0 +1,49272 @@
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+    "note": "Commit references and working-file hashes are distinct; replay code may be uncommitted."
+  },
+  "warnings": [
+    "Estimator refinement after the original readout; same 140k prefixes and batch blocks, zero new samples.",
+    "No independent confirmation: reproducing old primary/secondary is source/estimand identity, not scientific replication.",
+    "Exact means conditional mean 2e/(m-c); finite-prefix uncertainty remains. Four actual old probes are averaged per prefix.",
+    "SE_RB/SE_probe is descriptive; variance reduction does not guarantee empirical delete-batch jackknife SE shrinkage.",
+    "No automatic Markov claim: one weighted moment can cancel across cells; finite L512 count clock only, no continuum claim."
+  ]
+}
diff --git a/analysis/exact-pair-independent-archive-20260930/inputs/reference-run.json b/analysis/exact-pair-independent-archive-20260930/inputs/reference-run.json
new file mode 100644
index 000000000..1191abe1a
--- /dev/null
+++ b/analysis/exact-pair-independent-archive-20260930/inputs/reference-run.json
@@ -0,0 +1,297 @@
+{
+  "schema": "matching-one.exact-pair-replay.v1",
+  "status": "completed",
+  "source_commit": "8b5eee0d",
+  "pid": 90782,
+  "source_sha256": {
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+  "original_manifest_sha256": "8096d3d0a327604dd921e9c82d022daa685e2c7d6c4bf6e5b87c195b1d3dbf89",
+  "original_acquisition_commit": "55fa265629701e04b879b774ae7d3b97fefa845f",
+  "contract": {
+    "schema": "matching-one.safe-insertion-contract.v1",
+    "namespace": "safe-insertion-independent-20260929",
+    "parent_commit": "eb1e424a43f4bc9dee51c610549c4556af3874ea",
+    "frozen_before_new_data": true,
+    "lattice": "square",
+    "L": 512,
+    "N": 262144,
+    "a": 154646,
+    "b": 155385,
+    "cutoff_source": "unchanged old square-L512 count cutoffs, as in completion-hazard-production-20260929",
+    "probes": 4,
+    "batches": 14,
+    "samples_per_batch": 10000,
+    "workers": 14,
+    "primary": "overlap-weighted early-minus-late mean safe-insertion completion increment Y within exact (primitive direction,integer nu_b)",
+    "early": "rank_b=1 and J1<=a",
+    "late": "rank_b=1 and a<J1<=b",
+    "primary_eligibility": "nu_b<N-b; cells with nu_b=N-b have no safe conditional step and are separately counted",
+    "sampling": "independent uniform site permutations; four with-replacement uniform safe-site probes in separate copies of each rank-one prefix, selected by vacancy rejection using a distinct RNG stream",
+    "probe_unit": "one original prefix; average its four increments before inference, never treat them as independent configurations",
+    "weights": "w=n_early*n_late/(n_early+n_late) on common exact cells; normalize over cells and recompute all support and weights on aligned whole-batch deletion",
+    "mechanistic_map": "in each cell, Delta mean e=(N-b-nu_b)*DeltaY/2, Delta two-step survival=-(N-b-nu_b)*DeltaY/[(N-b)*(N-b-1)]",
+    "secondary": "independent replication of the previously retrospective exact-direction weighted mean nu_b contrast; all rank-one prefixes, including any with no safe step",
+    "model_prediction": "(rank,direction,nu) weak Markov closure requires the primary cohort contrast to be zero. Nonzero population contrast violates a necessary condition; an unresolved contrast does not establish full closure. No prospective sign assumed from L4 because square/triangular signs differ.",
+    "analysis": "one fixed primary mean; full aligned 14-batch delete-one covariance with derived edge/two-step contrasts and declared secondary; no old/new pooling, cutoff/feature/subgroup search, result-dependent sample extension, or exponent fitting",
+    "benchmark": "32 new excluded prefixes, runtime only, separate seed role; anticipated ideal-parallel production time must not exceed 900 seconds to start this block",
+    "scope": "one lattice, one existing size and count clock; no direct finite-window-survival replacement, original-U identification or continuum certificate"
+  },
+  "new_random_prefixes": 0,
+  "interpretation": "Same source prefixes, exact conditional mean of safe-insertion probes. Post-readout estimator refinement, not new independent evidence.",
+  "workers": 4,
+  "platform": "macOS-27.0-arm64-arm-64bit",
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+      "execution": "TV2N0X"
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+      "runtime_seconds": 381.7914042969933,
+      "engine_log": "prefixes=10000 seconds=381.735",
+      "command": "/workspace/Matching-One-TV2N0X/exact-pairs-20260930/analysis/exact-completion-pairs-20260929/exact-pair-cloud-q6j7j4co/engine 512 10000 17332517276911702520 155385 /workspace/Matching-One-TV2N0X/exact-pairs-20260930/analysis/exact-completion-pairs-20260929/exact-pair-cloud-q6j7j4co/exact-square-L512-b09.csv",
+      "execution": "TV2N0X"
+    },
+    {
+      "batch": 10,
+      "samples": 10000,
+      "seed": "17211728716206152366",
+      "file": "exact-square-L512-b10.csv.gz",
+      "sha256": "80d47a4fabc6a5419b7e5cfc7f1964fbe08c8080f015456012a9f775e2f8e4bf",
+      "original_file": "production-square-L512-b10.csv.gz",
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch10",
+      "runtime_seconds": 401.4404009709833,
+      "engine_log": "prefixes=10000 seconds=401.385",
+      "command": "/workspace/Matching-One-TV2N0X/exact-pairs-20260930/analysis/exact-completion-pairs-20260929/exact-pair-cloud-q6j7j4co/engine 512 10000 17211728716206152366 155385 /workspace/Matching-One-TV2N0X/exact-pairs-20260930/analysis/exact-completion-pairs-20260929/exact-pair-cloud-q6j7j4co/exact-square-L512-b10.csv",
+      "execution": "TV2N0X"
+    },
+    {
+      "batch": 11,
+      "samples": 10000,
+      "seed": "13292054708231777266",
+      "file": "exact-square-L512-b11.csv.gz",
+      "sha256": "5d0910c2cb54a063a0e685cacc121676adf0abecb9bbf559707cd95d60dc9d8f",
+      "original_file": "production-square-L512-b11.csv.gz",
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch11",
+      "runtime_seconds": 416.1216379549587,
+      "engine_log": "prefixes=10000 seconds=416.066",
+      "command": "/workspace/Matching-One-TV2N0X/exact-pairs-20260930/analysis/exact-completion-pairs-20260929/exact-pair-cloud-q6j7j4co/engine 512 10000 13292054708231777266 155385 /workspace/Matching-One-TV2N0X/exact-pairs-20260930/analysis/exact-completion-pairs-20260929/exact-pair-cloud-q6j7j4co/exact-square-L512-b11.csv",
+      "execution": "TV2N0X"
+    },
+    {
+      "batch": 12,
+      "samples": 10000,
+      "seed": "9255174391098419870",
+      "file": "exact-square-L512-b12.csv.gz",
+      "sha256": "02fc628489c24d33b6ca8a2bb9fc76bac2f61e9fa9fb1f0630b3660d8cff59db",
+      "original_file": "production-square-L512-b12.csv.gz",
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch12",
+      "runtime_seconds": 417.67217682197224,
+      "engine_log": "prefixes=10000 seconds=417.618",
+      "command": "/workspace/Matching-One-TV2N0X/exact-pairs-20260930/analysis/exact-completion-pairs-20260929/exact-pair-cloud-q6j7j4co/engine 512 10000 9255174391098419870 155385 /workspace/Matching-One-TV2N0X/exact-pairs-20260930/analysis/exact-completion-pairs-20260929/exact-pair-cloud-q6j7j4co/exact-square-L512-b12.csv",
+      "execution": "TV2N0X"
+    },
+    {
+      "batch": 13,
+      "samples": 10000,
+      "seed": "4111027655712514342",
+      "file": "exact-square-L512-b13.csv.gz",
+      "sha256": "cc846063ddacbe3a147d076b1979b5aadd44fa337dc4beb1967618b7f27871b2",
+      "original_file": "production-square-L512-b13.csv.gz",
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch13",
+      "runtime_seconds": 415.4595349589363,
+      "engine_log": "prefixes=10000 seconds=415.405",
+      "command": "/workspace/Matching-One-TV2N0X/exact-pairs-20260930/analysis/exact-completion-pairs-20260929/exact-pair-cloud-q6j7j4co/engine 512 10000 4111027655712514342 155385 /workspace/Matching-One-TV2N0X/exact-pairs-20260930/analysis/exact-completion-pairs-20260929/exact-pair-cloud-q6j7j4co/exact-square-L512-b13.csv",
+      "execution": "TV2N0X"
+    }
+  ],
+  "build_command": "clang++ -O3 -std=c++17 /private/tmp/matching-frontier-20260929-Jk39nH/analysis/exact-completion-pairs-20260929/engine.cpp -o /private/tmp/matching-frontier-20260929-Jk39nH/analysis/exact-completion-pairs-20260929/exact-pair-build-w2_bqml_/engine",
+  "binary_sha256": "2fa12e7a14075feda924271292c7a098be98a5848354322fda86043bc0ced36e",
+  "resumptions": [
+    {
+      "pid": 93063,
+      "utc_started": "2026-09-30T01:38:11.899918+00:00",
+      "reason": "Owner paused the task; stopped processes no longer existed on resume",
+      "retained_batches": [
+        0,
+        1,
+        2,
+        3
+      ],
+      "replayed_batches": [
+        4,
+        5,
+        6,
+        7,
+        8,
+        9,
+        10,
+        11,
+        12,
+        13
+      ],
+      "source_commit": "267f1ecfe50d4a3004185943bbc03034d51d6dc5",
+      "source_sha256": "544a0320c06aca1f0ceb22883bb0af53fc7c917c4f4a58c559710f8b2e3a7469",
+      "workers": 4,
+      "new_random_prefixes": 0,
+      "partial_files": "Original abandoned temporary CSVs retained; no partial batches scored",
+      "binary_sha256": "2fa12e7a14075feda924271292c7a098be98a5848354322fda86043bc0ced36e"
+    }
+  ],
+  "cloud_continuation": {
+    "instance": "DevEnvC_TV2N0X",
+    "instance_id": "4a8d1d443419434889e49148ed0a7ba6",
+    "source_commit": "0684c3b4eee0a647e643540ecfcdcd2918a70977",
+    "runner_sha256": "f119d53ea67c3c1b3ad5f40afa82e28ac676240064d40c91ebeba43b43471221",
+    "pid": 247,
+    "platform": "Linux-5.10.0-182.0.0.95.r3582_286.hce2.aarch64-aarch64-with-glibc2.34",
+    "python": "3.9.9",
+    "compiler": "g++ (GCC) 10.3.1\nCopyright (C) 2020 Free Software Foundation, Inc.\nThis is free software; see the source for copying conditions.  There is NO\nwarranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.",
+    "utc_started": "2026-09-30T01:46:19.133424+00:00",
+    "workers": 6,
+    "retained_local_batches": [
+      0,
+      1,
+      2,
+      3,
+      4,
+      5,
+      6,
+      7
+    ],
+    "completed_cloud_batches": [
+      9,
+      8,
+      10,
+      13,
+      11,
+      12
+    ],
+    "reason": "Owner requests medium/large computation on Huawei first; local incomplete workers stopped",
+    "new_random_prefixes": 0,
+    "binary_sha256": "72ddbdb1c2e26352ed8f62b820bfcdc741750bc5b50cf459f4b0adefe019fab7",
+    "benchmark_log": "prefixes=16 seconds=0.397452",
+    "wall_seconds": 420.380964801996
+  },
+  "utc_finished": "2026-09-30T01:53:19.514408+00:00",
+  "timing_note": "Mixed execution: local successful batches0..7, cloud8..13. Owner pause and abandoned partial work excluded from successful batch timings; cloud wall includes compile/benchmark."
+}
diff --git a/analysis/exact-pair-independent-archive-20260930/results/RESULT.md b/analysis/exact-pair-independent-archive-20260930/results/RESULT.md
new file mode 100644
index 000000000..2f44380ab
--- /dev/null
+++ b/analysis/exact-pair-independent-archive-20260930/results/RESULT.md
@@ -0,0 +1,23 @@
+# 独立既有档案的固定目标读出
+
+早组减晚组；±为14个对齐整批删除的标准误。
+
+| 指标 | 本档案 | SE |
+|---|---:|---:|
+| deltaY | -0.00319936839700698 | 0.000620090642890563 |
+| deltaE | -170.64900963119 | 33.0941230302755 |
+| deltaZ2 | 2.99453119450194e-08 | 5.80732252610344e-09 |
+| earlyY | 0.0951156042700857 | 0.000575740212820995 |
+| lateY | 0.0983149726670927 | 0.000284385318087397 |
+
+前块 deltaY = -0.00344938566765506 ± 0.00077225276269743；本块减前块 = 0.000250017270648076 ± 0.000990397261150415（sqrt(SE1²+SE2²)）。
+本块 deltaY/SE = -5.159517；两块未合并。
+
+140000行按约定映射后，前五字段全部逐行对应。共同支持：54120/54723（98.898087%）；共同格数1202。
+
+按exact(D,c)格直接累加整数n、sum_e；Ybar=2sum_e/[(N-b-c)n]；w=nE*nL/(nE+nL)。
+每次删除整批重新求共同支持和权重；保存5×5完整协方差、14次删除向量/支持/权重、整数格统计和来源依赖。
+
+这是另一独立随机块上此前未测过e的固定目标读出。该档案的其它结果早已读过，不能称全新前瞻采样或原封未看的holdout。
+没有新采样、旧census或全套测试；32行成本检查为旧batch0的重复，科学权重为0。
+deltaZ2为同几何公式的派生量；结果仅属于有限L512插入时钟，不作连续极限或完整Markov判决。
diff --git a/analysis/exact-pair-independent-archive-20260930/results/result.json b/analysis/exact-pair-independent-archive-20260930/results/result.json
new file mode 100644
index 000000000..2fe9d62a9
--- /dev/null
+++ b/analysis/exact-pair-independent-archive-20260930/results/result.json
@@ -0,0 +1,398385 @@
+{
+  "status": "scored",
+  "created_utc": "2026-09-30T02:22:03.376149+00:00",
+  "fixed": {
+    "L": 512,
+    "N": 262144,
+    "a": 154646,
+    "b": 155385,
+    "batches": 14,
+    "samples_per_batch": 10000
+  },
+  "m": 106759,
+  "metric_order": [
+    "deltaY",
+    "deltaE",
+    "deltaZ2",
+    "earlyY",
+    "lateY"
+  ],
+  "point_estimates": {
+    "deltaY": -0.003199368397006982,
+    "deltaE": -170.64900963119013,
+    "deltaZ2": 2.994531194501944e-08,
+    "earlyY": 0.09511560427008574,
+    "lateY": 0.09831497266709271
+  },
+  "standard_errors": {
+    "deltaY": 0.0006200906428905627,
+    "deltaE": 33.09412303027552,
+    "deltaZ2": 5.807322526103436e-09,
+    "earlyY": 0.0005757402128209948,
+    "lateY": 0.00028438531808739747
+  },
+  "primary": {
+    "cell_counts": {
+      "common": 1202,
+      "early_only": 88,
+      "late_only": 229
+    },
+    "overlap_weight_sum": 13003.570558065729,
+    "coverage": {
+      "early": {
+        "rank1_prefixes": 24218,
+        "eligible_prefixes": 24218,
+        "supported_prefixes": 24105,
+        "unsupported_eligible_prefixes": 113,
+        "no_safe_site_prefixes": 0,
+        "supported_fraction_of_eligible": 0.9953340490544224,
+        "supported_fraction_of_rank1": 0.9953340490544224
+      },
+      "late": {
+        "rank1_prefixes": 30505,
+        "eligible_prefixes": 30505,
+        "supported_prefixes": 30015,
+        "unsupported_eligible_prefixes": 490,
+        "no_safe_site_prefixes": 0,
+        "supported_fraction_of_eligible": 0.9839370594984429,
+        "supported_fraction_of_rank1": 0.9839370594984429
+      },
+      "combined": {
+        "rank1_prefixes": 54723,
+        "eligible_prefixes": 54723,
+        "supported_prefixes": 54120,
+        "unsupported_eligible_prefixes": 603,
+        "no_safe_site_prefixes": 0,
+        "supported_fraction_of_eligible": 0.9889808672770133,
+        "supported_fraction_of_rank1": 0.9889808672770133
+      }
+    },
+    "rank_counts": [
+      42574,
+      54723,
+      42703
+    ],
+    "cells": [
+      {
+        "direction": [
+          0,
+          1
+        ],
+        "c": 0,
+        "safe_sites": 106759,
+        "common_support": true,
+        "overlap_weight": 626.2414471097129,
+        "early": {
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+          "sum_e": 8225845,
+          "meanY": 0.13808351245415554
+        },
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+          "meanY": 0.15361475410304157
+        },
+        "deltaY": -0.015531241648886047,
+        "deltaE": -829.0499135967127,
+        "deltaZ2": 1.454808225040376e-07,
+        "normalized_weight": 0.048159191686107626
+      },
+      {
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+          1
+        ],
+        "c": 1,
+        "safe_sites": 106758,
+        "common_support": true,
+        "overlap_weight": 4.0,
+        "early": {
+          "n": 6,
+          "sum_e": 23509,
+          "meanY": 0.07340277387486964
+        },
+        "late": {
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+          "sum_e": 118542,
+          "meanY": 0.185063414451376
+        },
+        "deltaY": -0.11166064057650636,
+        "deltaE": -5960.333333333333,
+        "deltaZ2": 1.045913136845665e-06,
+        "normalized_weight": 0.0003076078206473005
+      },
+      {
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+          1
+        ],
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+        "safe_sites": 106757,
+        "common_support": true,
+        "overlap_weight": 13.796610169491526,
+        "early": {
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+          "sum_e": 136345,
+          "meanY": 0.1161047987485598
+        },
+        "late": {
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+          "meanY": 0.08910713874322818
+        },
+        "deltaY": 0.026997660005331634,
+        "deltaE": 1441.0945945945946,
+        "deltaZ2": -2.528817909384986e-07,
+        "normalized_weight": 0.001060986296639418
+      },
+      {
+        "direction": [
+          0,
+          1
+        ],
+        "c": 3,
+        "safe_sites": 106756,
+        "common_support": true,
+        "overlap_weight": 8.571428571428571,
+        "early": {
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+          "meanY": 0.141476825658511
+        },
+        "late": {
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+          "meanY": 0.10958697091186131
+        },
+        "deltaY": 0.03188985474664968,
+        "deltaE": 1702.2166666666667,
+        "deltaZ2": -2.9870322243011664e-07,
+        "normalized_weight": 0.0006591596156727868
+      },
+      {
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+          1
+        ],
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+        "common_support": true,
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+        "normalized_weight": 0.0014133332300011105
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+        "deltaZ2": 2.239001945714472e-08,
+        "normalized_weight": 0.0013195929697333473
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+      {
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+        "common_support": true,
+        "overlap_weight": 19.21518987341772,
+        "early": {
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+          "sha256": "73aaa04f05ee279532f7c781a97acda5191840cf79fc2cee72f2a6c9caaed7f1",
+          "last_file_commit": "87f6644d3b57c7dbd67c44c63bcceffb6b7693ac"
+        },
+        {
+          "original": "analysis/completion-hazard-production-20260929/data/production-square-L512-b12.csv.gz",
+          "snapshot": "inputs/archive/production-square-L512-b12.csv.gz",
+          "sha256": "75f556befdd4986e26f6594f7d2ec44fa6517b9b19021b9cbaa612116471875c",
+          "last_file_commit": "87f6644d3b57c7dbd67c44c63bcceffb6b7693ac"
+        },
+        {
+          "original": "analysis/completion-hazard-production-20260929/data/production-square-L512-b13.csv.gz",
+          "snapshot": "inputs/archive/production-square-L512-b13.csv.gz",
+          "sha256": "d5ffad750a152bd62325ae22375496b861da96f8687ded8ad7c0a8220d06a202",
+          "last_file_commit": "87f6644d3b57c7dbd67c44c63bcceffb6b7693ac"
+        }
+      ],
+      "reference_seeds_disjoint": true,
+      "prediction": "deltaY < 0",
+      "new_random_prefixes": 0,
+      "archive_status": "Other outcomes in this existing independent block were previously read; e had not been measured. Not prospective sampling or untouched holdout."
+    },
+    "scorer_sha256": "0114bb3ee588e1ff672690a5bc4462b16da7f1360e27688d3099da37ca9f7ea3",
+    "runner_sha256": "c27446cda4cb48e5eba3e298d1b972d04df60fe36f2a0f7e7aa572cc630d58c4",
+    "reference_result_sha256": "08088c5fbc63f8098ac4cf7a023d2ba05e72c09ffe51e72bd6856581389c6e04",
+    "replay_manifest_sha256": "8b536e8a9c542a86458d2ece4dc5cd0536000acc385cc88bf6c5f9e389930c8e",
+    "command": [
+      "/Users/lc/python-envs/research-py311/bin/python",
+      "analysis/exact-pair-independent-archive-20260930/score.py"
+    ],
+    "python": "3.11.15 (main, Jun 23 2026, 15:46:51) [Clang 22.1.3 ]",
+    "platform": "macOS-27.0-arm64-arm-64bit"
+  },
+  "scope": [
+    "Fixed target readout of previously unmeasured e in an existing independent random block.",
+    "Other results from this archive were previously inspected. Not new prospective sampling or untouched holdout.",
+    "No new samples, no block pooling, no cutoff/feature/model search, no artificial probe observations.",
+    "deltaZ2 = -2 deltaE/[m(m-1)] is derived from the same configurations, not independent evidence.",
+    "Finite L512 count-clock readout; no continuum, fixed-label-time or full Markov conclusion."
+  ]
+}
diff --git a/analysis/exact-pair-independent-archive-20260930/run.log b/analysis/exact-pair-independent-archive-20260930/run.log
new file mode 100644
index 000000000..bbda89e66
--- /dev/null
+++ b/analysis/exact-pair-independent-archive-20260930/run.log
@@ -0,0 +1,15 @@
+{"batch": 1, "seconds": 496.69426990509965, "status": "completed"}
+{"batch": 0, "seconds": 498.0191309801303, "status": "completed"}
+{"batch": 7, "seconds": 508.9134591759648, "status": "completed"}
+{"batch": 3, "seconds": 512.8563893151004, "status": "completed"}
+{"batch": 8, "seconds": 514.2208237138111, "status": "completed"}
+{"batch": 13, "seconds": 515.23845415283, "status": "completed"}
+{"batch": 10, "seconds": 515.5714685129933, "status": "completed"}
+{"batch": 2, "seconds": 518.6520514099393, "status": "completed"}
+{"batch": 5, "seconds": 519.3030597630423, "status": "completed"}
+{"batch": 11, "seconds": 520.1165391458198, "status": "completed"}
+{"batch": 6, "seconds": 520.1712592260446, "status": "completed"}
+{"batch": 12, "seconds": 520.217798746191, "status": "completed"}
+{"batch": 4, "seconds": 520.7907340640668, "status": "completed"}
+{"batch": 9, "seconds": 520.8245567351114, "status": "completed"}
+{"status": "completed", "wall_seconds": 520.8756419259589}
diff --git a/analysis/exact-pair-independent-archive-20260930/runner.py b/analysis/exact-pair-independent-archive-20260930/runner.py
new file mode 100644
index 000000000..cdd49cdf9
--- /dev/null
+++ b/analysis/exact-pair-independent-archive-20260930/runner.py
@@ -0,0 +1,209 @@
+#!/usr/bin/env python3
+"""One bounded replay of existing permutations; standard library only."""
+import argparse
+from concurrent.futures import ThreadPoolExecutor, as_completed
+import csv
+from datetime import datetime, timezone
+import gzip
+import hashlib
+import json
+import math
+import os
+from pathlib import Path
+import platform
+import shutil
+import subprocess
+import sys
+import time
+
+ROOT = Path(__file__).resolve().parent
+FIXED = dict(L=512, N=262144, a=154646, b=155385, batches=14, samples_per_batch=10000)
+OLD_HEADER = ['J1', 'J2', 'dx_b', 'dy_b', 'nu_b', 'tau', 'nu_tau']
+NEW_HEADER = ['J1', 'rank_b', 'dx_b', 'dy_b', 'nu_b', 'synergy_edges', 'nonlocal_candidates']
+
+
+def require(ok, message):
+    if not ok:
+        raise ValueError(message)
+
+
+def utc():
+    return datetime.now(timezone.utc).isoformat()
+
+
+def digest(path):
+    return hashlib.sha256(Path(path).read_bytes()).hexdigest()
+
+
+def save(path, obj):
+    path.parent.mkdir(parents=True, exist_ok=True)
+    with path.open('x') as f:
+        json.dump(obj, f, indent=2, allow_nan=False)
+        f.write('\n')
+
+
+def read(path):
+    return json.loads(path.read_text())
+
+
+def mapped(row):
+    j1, j2, dx, dy, c, _, _ = map(int, row)
+    rank = 0 if j1 > FIXED['b'] else (2 if j2 <= FIXED['b'] else 1)
+    return [0 if rank == 0 else j1, rank, dx, dy, c]
+
+
+def prepare():
+    repo = ROOT.parent.parent
+    old = ROOT.parent / 'completion-hazard-production-20260929'
+    reference = ROOT.parent / 'exact-completion-pairs-20260929'
+    manifest = read(old / 'data/run.json')
+    require(manifest['status'] == 'completed', 'source incomplete')
+    require(all(manifest['contract'][k] == v for k, v in FIXED.items()), 'fixed constants differ')
+    batches = sorted(manifest['batches'], key=lambda b: b['batch'])
+    require([b['batch'] for b in batches] == list(range(14)), 'source batch IDs')
+    other = read(reference / 'data/run.json')
+    require(not ({b['seed'] for b in batches} & {b['seed'] for b in other['batches']}), 'seed overlap')
+    paths = [old / 'engine.cpp', reference / 'engine.cpp', old / 'data/run.json',
+             reference / 'data/run.json', reference / 'results/result.json']
+    paths += [old / 'data' / b['file'] for b in batches]
+    records = []
+    for src in paths:
+        if src.suffix == '.cpp':
+            dst = ROOT / 'source' / src.parent.name / src.name
+        elif src.parent.name == 'data' and src.parent.parent == old:
+            dst = ROOT / 'inputs/archive' / src.name
+        elif src.parent.name == 'data':
+            dst = ROOT / 'inputs/reference-run.json'
+        else:
+            dst = ROOT / 'inputs/reference-result.json'
+        require(not dst.exists(), 'refuse overwrite: ' + str(dst))
+        dst.parent.mkdir(parents=True, exist_ok=True)
+        shutil.copyfile(src, dst)
+        relative = str(src.relative_to(repo))
+        records.append(dict(original=relative, snapshot=str(dst.relative_to(ROOT)), sha256=digest(dst),
+                            last_file_commit=subprocess.check_output(['git', '-C', str(repo), 'log', '-1', '--format=%H', '--', relative], text=True).strip()))
+    for batch in batches:
+        require(digest(ROOT / 'inputs/archive' / batch['file']) == batch['sha256'], 'source hash mismatch')
+    save(ROOT / 'source-record.json', dict(prepared_utc=utc(), fixed=FIXED,
+         head_commit=subprocess.check_output(['git', '-C', str(repo), 'rev-parse', 'HEAD'], text=True).strip(),
+         branch=subprocess.check_output(['git', '-C', str(repo), 'branch', '--show-current'], text=True).strip(),
+         old_acquisition_commit=manifest['source_commit'], files=records,
+         reference_seeds_disjoint=True, prediction='deltaY < 0', new_random_prefixes=0,
+         archive_status='Other outcomes in this existing independent block were previously read; e had not been measured. Not prospective sampling or untouched holdout.'))
+    print('Prepared unchanged sources and all 14 archived input batches.', flush=True)
+
+
+def environment():
+    cg = Path('/sys/fs/cgroup')
+    def value(*names):
+        for name in names:
+            path = cg / name
+            if path.exists():
+                return path.read_text().strip()
+        return None
+    quota = value('cpu/cpu.cfs_quota_us')
+    period = value('cpu/cpu.cfs_period_us')
+    maximum = value('cpu.max')
+    cpu = int(quota) / int(period) if quota and int(quota) > 0 else None
+    if cpu is None and maximum and not maximum.startswith('max'):
+        q, p = maximum.split()
+        cpu = int(q) / int(p)
+    mem = value('memory/memory.limit_in_bytes', 'memory.max')
+    require(cpu is not None and mem not in (None, 'max'), 'unmeasured resource limits')
+    return dict(utc=utc(), platform=platform.platform(), python=sys.version,
+                machine=platform.machine(), hostname=platform.node(), cpu_quota_cores=cpu,
+                memory_limit_bytes=int(mem), affinity_cpus=len(os.sched_getaffinity(0)),
+                compiler=subprocess.check_output(['chroot', str(ROOT / 'toolchain'), '/usr/bin/g++', '--version'], text=True),
+                ps=subprocess.check_output(['ps', '-eo', 'pid,ppid,comm,etime,pcpu,pmem'], text=True),
+                thread_environment={k: os.environ.get(k) for k in ('OMP_NUM_THREADS', 'OPENBLAS_NUM_THREADS', 'MKL_NUM_THREADS')})
+
+
+def benchmark():
+    env = environment()
+    build = ROOT / 'build'
+    build.mkdir(exist_ok=False)
+    # The container root loses packages at restart. Compile within this task's
+    # persistent installroot, leaving every other remote workspace untouched.
+    project = ROOT / 'toolchain/project'
+    project.mkdir(exist_ok=False)
+    shutil.copytree(ROOT / 'source', project / 'source')
+    cmd = ['chroot', str(ROOT / 'toolchain'), '/usr/bin/g++', '-O3', '-std=c++17',
+           '/project/source/exact-completion-pairs-20260929/engine.cpp', '-o', '/project/engine']
+    subprocess.run(cmd, check=True)
+    binary = build / 'engine'
+    shutil.copyfile(project / 'engine', binary)
+    binary.chmod(0o755)
+    b = read(ROOT / 'inputs/archive/run.json')['batches'][0]
+    out = build / 'same-seed-32.csv'
+    run = [str(ROOT / 'toolchain/usr/bin/time'), '-v', str(binary), '512', '32', b['seed'], '155385', str(out)]
+    start = time.monotonic()
+    proc = subprocess.run(run, text=True, capture_output=True, check=True)
+    seconds = time.monotonic() - start
+    with gzip.open(ROOT / 'inputs/archive' / b['file'], 'rt') as f, out.open() as g:
+        a, z = csv.reader(f), csv.reader(g)
+        require(next(a) == OLD_HEADER and next(z) == NEW_HEADER, 'header mismatch')
+        rows = list(z)
+        require(len(rows) == 32, 'benchmark rows')
+        for i, row in enumerate(rows):
+            require(mapped(next(a)) == list(map(int, row[:5])), 'benchmark alignment row ' + str(i))
+    rss = int(next(line.split(':', 1)[1] for line in proc.stderr.splitlines() if 'Maximum resident set size (kbytes)' in line))
+    workers = min(14, math.floor(env['cpu_quota_cores']), env['affinity_cpus'], int(env['memory_limit_bytes'] * .8 // (rss * 1024)))
+    require(workers >= 1, 'no feasible workers')
+    save(ROOT / 'benchmark.json', dict(environment=env, build_command=cmd, binary_sha256=digest(binary),
+         command=run, stdout=proc.stdout, time_log=proc.stderr, wall_seconds=seconds, max_rss_kib=rss,
+         same_seed=b['seed'], batch=b['batch'], rows=32, all_mapped_first5_match=True,
+         scientific_weight=0, recommended_workers=workers,
+         approximate_replay_seconds=seconds / 32 * 10000 * math.ceil(14 / workers)))
+    print(json.dumps(read(ROOT / 'benchmark.json')), flush=True)
+
+
+def replay():
+    benchmark_record = read(ROOT / 'benchmark.json')
+    env = environment()
+    workers = min(benchmark_record['recommended_workers'], math.floor(env['cpu_quota_cores']), env['affinity_cpus'], 14)
+    source = read(ROOT / 'inputs/archive/run.json')
+    data = ROOT / 'data'
+    data.mkdir(exist_ok=False)
+    record = dict(status='running', started_utc=utc(), pid=os.getpid(), workers=workers, environment=env,
+                  fixed=FIXED, new_random_prefixes=0, source_manifest_sha256=digest(ROOT / 'inputs/archive/run.json'),
+                  binary_sha256=digest(ROOT / 'build/engine'), runner_sha256=digest(Path(__file__)), batches=[])
+    save(data / 'started.json', record)
+    start = time.monotonic()
+    def one(batch):
+        file = 'exact-square-L512-b%02d.csv' % batch['batch']
+        raw = data / file
+        command = [str(ROOT / 'build/engine'), '512', '10000', batch['seed'], '155385', str(raw)]
+        t0 = time.monotonic()
+        proc = subprocess.run(command, capture_output=True, text=True)
+        row = dict(batch=batch['batch'], seed=batch['seed'], samples=10000, original_file=batch['file'],
+                   dependency_group=batch['dependency_group'], command=command,
+                   returncode=proc.returncode, stdout=proc.stdout, stderr=proc.stderr, wall_seconds=time.monotonic()-t0)
+        save(data / ('batch-%02d.json' % batch['batch']), row)
+        require(proc.returncode == 0, 'engine failed: ' + file)
+        compressed = data / (file + '.gz')
+        with raw.open('rb') as f, compressed.open('xb') as output:
+            with gzip.GzipFile(filename='', mode='wb', fileobj=output, mtime=0) as g:
+                shutil.copyfileobj(f, g)
+        row.update(file=compressed.name, sha256=digest(compressed))
+        print(json.dumps(dict(batch=row['batch'], seconds=row['wall_seconds'], status='completed')), flush=True)
+        return row
+    try:
+        with ThreadPoolExecutor(max_workers=workers) as pool:
+            futures = [pool.submit(one, b) for b in source['batches']]
+            for future in as_completed(futures):
+                record['batches'].append(future.result())
+        record['batches'].sort(key=lambda b: b['batch'])
+        record.update(status='completed', wall_seconds=time.monotonic()-start, finished_utc=utc())
+        save(data / 'run.json', record)
+        print(json.dumps(dict(status='completed', wall_seconds=record['wall_seconds'])), flush=True)
+    except BaseException as exc:
+        record.update(status='failed', error=repr(exc), finished_utc=utc())
+        save(data / 'failed.json', record)
+        raise
+
+
+if __name__ == '__main__':
+    ap = argparse.ArgumentParser(description=__doc__)
+    ap.add_argument('mode', choices=['prepare', 'benchmark', 'replay'])
+    args = ap.parse_args()
+    globals()[args.mode]()
diff --git a/analysis/exact-pair-independent-archive-20260930/score.py b/analysis/exact-pair-independent-archive-20260930/score.py
new file mode 100644
index 000000000..74d062247
--- /dev/null
+++ b/analysis/exact-pair-independent-archive-20260930/score.py
@@ -0,0 +1,198 @@
+#!/usr/bin/env python3
+"""Single score of the fixed negative target on the existing independent archive."""
+import sys
+sys.dont_write_bytecode = True
+import csv
+import gzip
+from itertools import zip_longest
+import math
+from pathlib import Path
+import platform
+from runner import ROOT, FIXED, OLD_HEADER, NEW_HEADER, digest, mapped, read, require, save, utc
+
+METRICS = ('deltaY', 'deltaE', 'deltaZ2', 'earlyY', 'lateY')
+M = FIXED['N'] - FIXED['b']
+
+
+def load_blocks():
+    archive = read(ROOT / 'inputs/archive/run.json')
+    replay = read(ROOT / 'data/run.json')
+    require(replay['status'] == 'completed' and replay['new_random_prefixes'] == 0, 'incomplete replay')
+    require(replay['source_manifest_sha256'] == digest(ROOT / 'inputs/archive/run.json'), 'source manifest differs')
+    aa = sorted(archive['batches'], key=lambda r: r['batch'])
+    rr = sorted(replay['batches'], key=lambda r: r['batch'])
+    require([r['batch'] for r in aa] == [r['batch'] for r in rr] == list(range(14)), 'batch IDs differ')
+    blocks, alignment = [], []
+    for old, new in zip(aa, rr):
+        batch = old['batch']
+        require(old['seed'] == new['seed'] and old['dependency_group'] == new['dependency_group'], 'batch dependency mismatch')
+        require(new['original_file'] == old['file'], 'source file mismatch')
+        op, np = ROOT / 'inputs/archive' / old['file'], ROOT / 'data' / new['file']
+        require(digest(op) == old['sha256'] and digest(np) == new['sha256'], 'input hash differs')
+        cells, rank_counts, rank1, eligible = {}, [0, 0, 0], [0, 0], [0, 0]
+        rows, candidates, rank0_mapped = 0, 0, 0
+        with gzip.open(op, 'rt') as f, gzip.open(np, 'rt') as g:
+            a, z = csv.reader(f), csv.reader(g)
+            require(next(a) == OLD_HEADER and next(z) == NEW_HEADER, 'column mismatch')
+            for line, (o, n) in enumerate(zip_longest(a, z), 2):
+                label = 'batch %d line %d' % (batch, line)
+                require(o is not None and n is not None and len(o) == 7 and len(n) == 7, label + ': row mismatch')
+                v = list(map(int, n))
+                require(mapped(o) == v[:5], label + ': first5 mismatch')
+                j1, rank, dx, dy, c, e, candidate = v
+                rows += 1
+                rank_counts[rank] += 1
+                if rank != 1:
+                    require(v[2:] == [0, 0, -1, -1, -1], label + ': sentinel mismatch')
+                    rank0_mapped += int(rank == 0)
+                    continue
+                require(1 <= j1 <= FIXED['b'] and 0 <= c <= M, label + ': rank1 bounds')
+                require(math.gcd(dx, dy) == 1 and (dx > 0 or (dx == 0 and dy > 0)), label + ': direction convention')
+                s = M - c
+                require(0 <= e <= s * (s - 1) // 2 and candidate >= 0, label + ': pair count')
+                group = int(j1 > FIXED['a'])
+                rank1[group] += 1
+                candidates += candidate
+                if s == 0:
+                    continue
+                eligible[group] += 1
+                cell = cells.setdefault((dx, dy, c), [[0, 0], [0, 0]])
+                cell[group][0] += 1
+                cell[group][1] += e
+        require(rows == old['samples'] == new['samples'] == 10000, 'batch size differs')
+        blocks.append(dict(batch=batch, cells=cells, rank_counts=rank_counts, rank1=rank1, eligible=eligible))
+        alignment.append(dict(batch=batch, rows=rows, all_mapped_first5_match=True, rank0_J1_mapped_to_zero=rank0_mapped,
+                              seed=old['seed'], dependency_group=old['dependency_group'], sum_nonlocal_candidates=candidates,
+                              original_file=old['file'], original_sha256=old['sha256'],
+                              replay_file=new['file'], replay_sha256=new['sha256']))
+    return blocks, alignment
+
+
+def estimate(blocks, omit=None):
+    cells, risk, eligible, ranks = {}, [0, 0], [0, 0], [0, 0, 0]
+    for block in blocks:
+        if block['batch'] == omit:
+            continue
+        for j in range(2):
+            risk[j] += block['rank1'][j]
+            eligible[j] += block['eligible'][j]
+        for j in range(3):
+            ranks[j] += block['rank_counts'][j]
+        for key, groups in block['cells'].items():
+            cell = cells.setdefault(key, [[0, 0], [0, 0]])
+            for j in range(2):
+                for k in range(2):
+                    cell[j][k] += groups[j][k]
+    terms, weights, rows = [[] for _ in METRICS], [], []
+    supported = [0, 0]
+    counts = dict(common=0, early_only=0, late_only=0)
+    for (dx, dy, c), groups in sorted(cells.items()):
+        (ne, se), (nl, sl) = groups
+        s = M - c
+        common = bool(ne and nl)
+        w = ne * nl / (ne + nl) if common else 0.0
+        counts['common' if common else ('early_only' if ne else 'late_only')] += 1
+        row = dict(direction=[dx, dy], c=c, safe_sites=s, common_support=common, overlap_weight=w,
+                   early=dict(n=ne, sum_e=se, meanY=2 * se / (s * ne) if ne else None),
+                   late=dict(n=nl, sum_e=sl, meanY=2 * sl / (s * nl) if nl else None))
+        if common:
+            # Integer numerator before division; no fabricated probe observations.
+            de = (se * nl - sl * ne) / (ne * nl)
+            dy_value = 2 * de / s
+            vector = [dy_value, de, -2 * de / (M * (M - 1)), row['early']['meanY'], row['late']['meanY']]
+            weights.append(w)
+            for term, value in zip(terms, vector):
+                term.append(w * value)
+            row.update(dict(zip(METRICS[:3], vector[:3])))
+            supported[0] += ne
+            supported[1] += nl
+        rows.append(row)
+    total_w = math.fsum(weights)
+    require(total_w > 0, 'no common support for deletion ' + str(omit))
+    point = [math.fsum(term) / total_w for term in terms]
+    point[2] = -2 * point[1] / (M * (M - 1))
+    for row in rows:
+        row['normalized_weight'] = row['overlap_weight'] / total_w
+    coverage = {}
+    for group, j in [('early', 0), ('late', 1), ('combined', None)]:
+        r, e, s = (sum(values) if j is None else values[j] for values in (risk, eligible, supported))
+        coverage[group] = dict(rank1_prefixes=r, eligible_prefixes=e, supported_prefixes=s,
+                               unsupported_eligible_prefixes=e-s, no_safe_site_prefixes=r-e,
+                               supported_fraction_of_eligible=s/e if e else None,
+                               supported_fraction_of_rank1=s/r if r else None)
+    support = dict(cell_counts=counts, overlap_weight_sum=total_w, coverage=coverage, rank_counts=ranks)
+    return point, support, rows
+
+
+def main():
+    require(not (ROOT / 'results/result.json').exists(), 'score already exists; no repeat scoring')
+    blocks, alignment = load_blocks()
+    point, support, rows = estimate(blocks)
+    deletions = []
+    for batch in range(14):
+        vector, info, cells = estimate(blocks, omit=batch)
+        deletions.append(dict(omitted_batch=batch, point_vector=vector, support=info,
+             common_cells=[dict(direction=r['direction'], c=r['c'], nE=r['early']['n'], nL=r['late']['n'],
+                                weight=r['overlap_weight'], normalized_weight=r['normalized_weight'])
+                           for r in cells if r['common_support']]))
+    means = [math.fsum(d['point_vector'][i] for d in deletions) / 14 for i in range(5)]
+    covariance = [[13 / 14 * math.fsum((d['point_vector'][i]-means[i]) * (d['point_vector'][j]-means[j])
+                                      for d in deletions) for j in range(5)] for i in range(5)]
+    se = [math.sqrt(covariance[i][i]) for i in range(5)]
+    reference = read(ROOT / 'inputs/reference-result.json')
+    compare = {}
+    for i, name in enumerate(METRICS):
+        ref_point, ref_se = reference['point_estimates']['RB_' + name], reference['standard_errors']['RB_' + name]
+        uncertainty = math.hypot(ref_se, se[i])
+        compare[name] = dict(reference=ref_point, reference_SE=ref_se, archive=point[i], archive_SE=se[i],
+                             archive_minus_reference=point[i]-ref_point, independent_difference_SE=uncertainty,
+                             difference_over_SE=(point[i]-ref_point)/uncertainty)
+    result = dict(status='scored', created_utc=utc(), fixed=FIXED, m=M, metric_order=METRICS,
+                  point_estimates=dict(zip(METRICS, point)), standard_errors=dict(zip(METRICS, se)),
+                  primary=dict(**support, cells=rows),
+                  joint_uncertainty=dict(method='aligned whole-batch delete-one; exact (D,c) support and overlap weights recomputed',
+                       covariance_formula='(B-1)/B sum (theta_minus_batch - mean)(theta_minus_batch - mean)^T',
+                       point_estimator='pooled plug-in; no jackknife bias correction', covariance=covariance,
+                       metric_order=METRICS, delete_one_mean=means, delete_one=deletions),
+                  aligned_batches=alignment,
+                  batch_sufficient_statistics=[dict(batch=b['batch'], rank_counts=b['rank_counts'], rank1=b['rank1'], eligible=b['eligible'],
+                      cells=[dict(direction=list(k[:2]), c=k[2], early=dict(n=v[0][0], sum_e=v[0][1]),
+                                  late=dict(n=v[1][0], sum_e=v[1][1])) for k, v in sorted(b['cells'].items())]) for b in blocks],
+                  independent_block_comparison=compare,
+                  prediction=dict(fixed_sign='negative', observed_sign='negative' if point[0] < 0 else 'nonnegative',
+                                  primary_delta_over_SE=point[0]/se[0]),
+                  provenance=dict(source_record=read(ROOT / 'source-record.json'),
+                      scorer_sha256=digest(Path(__file__)), runner_sha256=digest(ROOT / 'runner.py'),
+                      reference_result_sha256=digest(ROOT / 'inputs/reference-result.json'),
+                      replay_manifest_sha256=digest(ROOT / 'data/run.json'), command=[sys.executable, *sys.argv],
+                      python=sys.version, platform=platform.platform()),
+                  scope=['Fixed target readout of previously unmeasured e in an existing independent random block.',
+                         'Other results from this archive were previously inspected. Not new prospective sampling or untouched holdout.',
+                         'No new samples, no block pooling, no cutoff/feature/model search, no artificial probe observations.',
+                         'deltaZ2 = -2 deltaE/[m(m-1)] is derived from the same configurations, not independent evidence.',
+                         'Finite L512 count-clock readout; no continuum, fixed-label-time or full Markov conclusion.'])
+    save(ROOT / 'results/result.json', result)
+    lines = ['# 独立既有档案的固定目标读出', '', '早组减晚组；±为14个对齐整批删除的标准误。', '',
+             '| 指标 | 本档案 | SE |', '|---|---:|---:|']
+    for name, value, error in zip(METRICS, point, se):
+        lines.append('| %s | %.15g | %.15g |' % (name, value, error))
+    cmp = compare['deltaY']
+    lines += ['', '前块 deltaY = %.15g ± %.15g；本块减前块 = %.15g ± %.15g（sqrt(SE1²+SE2²)）。' %
+              (cmp['reference'], cmp['reference_SE'], cmp['archive_minus_reference'], cmp['independent_difference_SE']),
+              '本块 deltaY/SE = %.6f；两块未合并。' % (point[0]/se[0]), '',
+              '140000行按约定映射后，前五字段全部逐行对应。共同支持：%d/%d（%.6f%%）；共同格数%d。' %
+              (support['coverage']['combined']['supported_prefixes'], support['rank_counts'][1],
+               100*support['coverage']['combined']['supported_fraction_of_rank1'], support['cell_counts']['common']), '',
+              '按exact(D,c)格直接累加整数n、sum_e；Ybar=2sum_e/[(N-b-c)n]；w=nE*nL/(nE+nL)。',
+              '每次删除整批重新求共同支持和权重；保存5×5完整协方差、14次删除向量/支持/权重、整数格统计和来源依赖。', '',
+              '这是另一独立随机块上此前未测过e的固定目标读出。该档案的其它结果早已读过，不能称全新前瞻采样或原封未看的holdout。',
+              '没有新采样、旧census或全套测试；32行成本检查为旧batch0的重复，科学权重为0。',
+              'deltaZ2为同几何公式的派生量；结果仅属于有限L512插入时钟，不作连续极限或完整Markov判决。', '']
+    with (ROOT / 'results/RESULT.md').open('x') as f:
+        f.write('\n'.join(lines))
+    print({k: result[k] for k in ('point_estimates', 'standard_errors', 'prediction')})
+    print(compare['deltaY'])
+
+
+if __name__ == '__main__':
+    main()
diff --git a/analysis/exact-pair-independent-archive-20260930/source-record.json b/analysis/exact-pair-independent-archive-20260930/source-record.json
new file mode 100644
index 000000000..5d0eadb5a
--- /dev/null
+++ b/analysis/exact-pair-independent-archive-20260930/source-record.json
@@ -0,0 +1,134 @@
+{
+  "prepared_utc": "2026-09-30T02:07:44.078122+00:00",
+  "fixed": {
+    "L": 512,
+    "N": 262144,
+    "a": 154646,
+    "b": 155385,
+    "batches": 14,
+    "samples_per_batch": 10000
+  },
+  "head_commit": "2d315636cb18b7f3c4f7e44dd31b14f6d3a896cf",
+  "branch": "research/frontier-and-birth-gap-20260929",
+  "old_acquisition_commit": "9520c9bdf702a62435f1054e64184b1df4fc0ab3",
+  "files": [
+    {
+      "original": "analysis/completion-hazard-production-20260929/engine.cpp",
+      "snapshot": "source/completion-hazard-production-20260929/engine.cpp",
+      "sha256": "42ac8a013cb731f4da4e3121ca429fd475555063e876638ad5d375d195448115",
+      "last_file_commit": "9520c9bdf702a62435f1054e64184b1df4fc0ab3"
+    },
+    {
+      "original": "analysis/exact-completion-pairs-20260929/engine.cpp",
+      "snapshot": "source/exact-completion-pairs-20260929/engine.cpp",
+      "sha256": "48def78ee470acef233fd4c4d2543f1d89c1869b552a4a434ab2aa220be95682",
+      "last_file_commit": "8b5eee0d47b6c220c2ba46a5bae08d2d19b392a4"
+    },
+    {
+      "original": "analysis/completion-hazard-production-20260929/data/run.json",
+      "snapshot": "inputs/archive/run.json",
+      "sha256": "7c63af9d9c7dcf5eff4b0f14644fe83560d73190532d8528f3e0cf0c5f0af2df",
+      "last_file_commit": "87f6644d3b57c7dbd67c44c63bcceffb6b7693ac"
+    },
+    {
+      "original": "analysis/exact-completion-pairs-20260929/data/run.json",
+      "snapshot": "inputs/reference-run.json",
+      "sha256": "d6f388f7c0734c3ff6458ee48f5d4039b8f9d9449714fe1bec95d4d355bad28e",
+      "last_file_commit": "7f71af8599ce6e7c0f426cb6f7a069dbc0cf6100"
+    },
+    {
+      "original": "analysis/exact-completion-pairs-20260929/results/result.json",
+      "snapshot": "inputs/reference-result.json",
+      "sha256": "08088c5fbc63f8098ac4cf7a023d2ba05e72c09ffe51e72bd6856581389c6e04",
+      "last_file_commit": "7f71af8599ce6e7c0f426cb6f7a069dbc0cf6100"
+    },
+    {
+      "original": "analysis/completion-hazard-production-20260929/data/production-square-L512-b00.csv.gz",
+      "snapshot": "inputs/archive/production-square-L512-b00.csv.gz",
+      "sha256": "32f16cc37728d67fdbe3446a48da1bb4d36c48fc4f051648ac8e36e2595d4026",
+      "last_file_commit": "87f6644d3b57c7dbd67c44c63bcceffb6b7693ac"
+    },
+    {
+      "original": "analysis/completion-hazard-production-20260929/data/production-square-L512-b01.csv.gz",
+      "snapshot": "inputs/archive/production-square-L512-b01.csv.gz",
+      "sha256": "fdc708228c1e0535787aa3426ea725b276ecef0b8ba193215b1bf93512518892",
+      "last_file_commit": "87f6644d3b57c7dbd67c44c63bcceffb6b7693ac"
+    },
+    {
+      "original": "analysis/completion-hazard-production-20260929/data/production-square-L512-b02.csv.gz",
+      "snapshot": "inputs/archive/production-square-L512-b02.csv.gz",
+      "sha256": "57b5f9ef9bcc6f21a5a14f5e7b6c24fccb94388f349943738916cc1700ab4897",
+      "last_file_commit": "87f6644d3b57c7dbd67c44c63bcceffb6b7693ac"
+    },
+    {
+      "original": "analysis/completion-hazard-production-20260929/data/production-square-L512-b03.csv.gz",
+      "snapshot": "inputs/archive/production-square-L512-b03.csv.gz",
+      "sha256": "65ec5cee09f76a03c69b702a21bb6f27dd70930b48e3b7f047de33103032eaa4",
+      "last_file_commit": "87f6644d3b57c7dbd67c44c63bcceffb6b7693ac"
+    },
+    {
+      "original": "analysis/completion-hazard-production-20260929/data/production-square-L512-b04.csv.gz",
+      "snapshot": "inputs/archive/production-square-L512-b04.csv.gz",
+      "sha256": "033189fab591caa248c66c9710d63104036dc3d26b88e495d965615a84ecc4fb",
+      "last_file_commit": "87f6644d3b57c7dbd67c44c63bcceffb6b7693ac"
+    },
+    {
+      "original": "analysis/completion-hazard-production-20260929/data/production-square-L512-b05.csv.gz",
+      "snapshot": "inputs/archive/production-square-L512-b05.csv.gz",
+      "sha256": "b38b95839aab1c6f6575a9b94702c41a032a2cc00430c6eb904840472488fbd4",
+      "last_file_commit": "87f6644d3b57c7dbd67c44c63bcceffb6b7693ac"
+    },
+    {
+      "original": "analysis/completion-hazard-production-20260929/data/production-square-L512-b06.csv.gz",
+      "snapshot": "inputs/archive/production-square-L512-b06.csv.gz",
+      "sha256": "330829d82ec4dad029f53dc07fe8bda5252bb35118aa9b507012c73df88e5a35",
+      "last_file_commit": "87f6644d3b57c7dbd67c44c63bcceffb6b7693ac"
+    },
+    {
+      "original": "analysis/completion-hazard-production-20260929/data/production-square-L512-b07.csv.gz",
+      "snapshot": "inputs/archive/production-square-L512-b07.csv.gz",
+      "sha256": "a292a7a8be0fb5f79f2449745709c51942fc4261fd50e542569d33d04005ee30",
+      "last_file_commit": "87f6644d3b57c7dbd67c44c63bcceffb6b7693ac"
+    },
+    {
+      "original": "analysis/completion-hazard-production-20260929/data/production-square-L512-b08.csv.gz",
+      "snapshot": "inputs/archive/production-square-L512-b08.csv.gz",
+      "sha256": "113325911eb95469467bfe73496d5b57448413125dcea383b55b0ff7875db4ac",
+      "last_file_commit": "87f6644d3b57c7dbd67c44c63bcceffb6b7693ac"
+    },
+    {
+      "original": "analysis/completion-hazard-production-20260929/data/production-square-L512-b09.csv.gz",
+      "snapshot": "inputs/archive/production-square-L512-b09.csv.gz",
+      "sha256": "b5712154670f8179ddf32487f35a794ea16b5ef4f0cdad2025c3f8e447f321da",
+      "last_file_commit": "87f6644d3b57c7dbd67c44c63bcceffb6b7693ac"
+    },
+    {
+      "original": "analysis/completion-hazard-production-20260929/data/production-square-L512-b10.csv.gz",
+      "snapshot": "inputs/archive/production-square-L512-b10.csv.gz",
+      "sha256": "98bb914abc5188216cd5b9b4584ba46539bd4c16c1b4d745b6a8033748243668",
+      "last_file_commit": "87f6644d3b57c7dbd67c44c63bcceffb6b7693ac"
+    },
+    {
+      "original": "analysis/completion-hazard-production-20260929/data/production-square-L512-b11.csv.gz",
+      "snapshot": "inputs/archive/production-square-L512-b11.csv.gz",
+      "sha256": "73aaa04f05ee279532f7c781a97acda5191840cf79fc2cee72f2a6c9caaed7f1",
+      "last_file_commit": "87f6644d3b57c7dbd67c44c63bcceffb6b7693ac"
+    },
+    {
+      "original": "analysis/completion-hazard-production-20260929/data/production-square-L512-b12.csv.gz",
+      "snapshot": "inputs/archive/production-square-L512-b12.csv.gz",
+      "sha256": "75f556befdd4986e26f6594f7d2ec44fa6517b9b19021b9cbaa612116471875c",
+      "last_file_commit": "87f6644d3b57c7dbd67c44c63bcceffb6b7693ac"
+    },
+    {
+      "original": "analysis/completion-hazard-production-20260929/data/production-square-L512-b13.csv.gz",
+      "snapshot": "inputs/archive/production-square-L512-b13.csv.gz",
+      "sha256": "d5ffad750a152bd62325ae22375496b861da96f8687ded8ad7c0a8220d06a202",
+      "last_file_commit": "87f6644d3b57c7dbd67c44c63bcceffb6b7693ac"
+    }
+  ],
+  "reference_seeds_disjoint": true,
+  "prediction": "deltaY < 0",
+  "new_random_prefixes": 0,
+  "archive_status": "Other outcomes in this existing independent block were previously read; e had not been measured. Not prospective sampling or untouched holdout."
+}
diff --git a/analysis/exact-pair-independent-archive-20260930/source/completion-hazard-production-20260929/engine.cpp b/analysis/exact-pair-independent-archive-20260930/source/completion-hazard-production-20260929/engine.cpp
new file mode 100644
index 000000000..f690e4d83
--- /dev/null
+++ b/analysis/exact-pair-independent-archive-20260930/source/completion-hazard-production-20260929/engine.cpp
@@ -0,0 +1,164 @@
+// Completion geometry on independent uniform-permutation filtrations.
+// Lifted union-find convention retained from ../birth-gap-20260929/engine.cpp.
+// The old engine and archives are unchanged. C++17, standard library only.
+#include <algorithm>
+#include <array>
+#include <chrono>
+#include <cstdint>
+#include <fstream>
+#include <iostream>
+#include <numeric>
+#include <random>
+#include <stdexcept>
+#include <string>
+#include <vector>
+
+struct Neighbor { int vertex, dx, dy; };
+struct Observation { int j1=0, j2=0, dx_b=0, dy_b=0, nu_b=-1, tau=0, nu_tau=-1; };
+
+class Geometry {
+    int L, N, degree;
+    std::vector<std::array<Neighbor,6>> neighbor;
+    std::vector<int> parent, size;
+    std::vector<std::int64_t> dx, dy;
+    std::vector<unsigned char> active;
+    int rank=0;
+    std::int64_t bx=0, by=0;
+    struct Root { int vertex; std::int64_t x,y; };
+    Root find(int v) {
+        if (parent[v]==v) return {v,0,0};
+        Root r=find(parent[v]);
+        dx[v]+=r.x; dy[v]+=r.y; parent[v]=r.vertex;
+        return {r.vertex,dx[v],dy[v]};
+    }
+    void edge(int v, Neighbor e) {
+        Root a=find(v), b=find(e.vertex);
+        std::int64_t x=a.x+e.dx-b.x, y=a.y+e.dy-b.y;
+        if (a.vertex==b.vertex) {
+            if (x%L || y%L) throw std::logic_error("nonperiodic cycle");
+            x/=L; y/=L;
+            if (!(x||y) || rank==2) return;
+            if (!rank) {bx=x;by=y;rank=1;}
+            else if (bx*y!=by*x) rank=2;
+        } else {
+            if (size[a.vertex]<size[b.vertex]) {std::swap(a,b);x=-x;y=-y;}
+            parent[b.vertex]=a.vertex;dx[b.vertex]=x;dy[b.vertex]=y;
+            size[a.vertex]+=size[b.vertex];
+        }
+    }
+public:
+    Geometry(int length,bool triangular):L(length),N(L*L),degree(triangular?6:4),
+        neighbor(N),parent(N),size(N),dx(N),dy(N),active(N) {
+        if (L<3 || L>1024) throw std::invalid_argument("L outside [3,1024]");
+        const int sx[]={1,-1,0,0,1,-1},sy[]={0,0,1,-1,1,-1};
+        for(int y=0;y<L;++y)for(int x=0;x<L;++x)for(int j=0;j<degree;++j)
+            neighbor[x+L*y][j]={((x+sx[j]+L)%L)+L*((y+sy[j]+L)%L),sx[j],sy[j]};
+    }
+    void reset(){std::fill(active.begin(),active.end(),0);rank=0;bx=by=0;}
+    int insert(int v) {
+        if(active[v])throw std::logic_error("double insertion");
+        active[v]=1;parent[v]=v;size[v]=1;dx[v]=dy[v]=0;
+        for(int j=0;j<degree;++j)if(active[neighbor[v][j].vertex])edge(v,neighbor[v][j]);
+        return rank;
+    }
+    std::pair<int,int> direction()const{
+        if(rank!=1)return {0,0};
+        auto g=std::gcd(bx,by);auto x=bx/g,y=by/g;
+        if(x<0 || (x==0 && y<0)){x=-x;y=-y;}
+        return {static_cast<int>(x),static_cast<int>(y)};
+    }
+    int completion_count() {
+        if(rank!=1)return -1;
+        int result=0;
+        // A new cycle through v arises precisely from two neighbours in
+        // the same old component. Cycles between different old components
+        // cannot arise from a single star. Existing windings are all along b.
+        for(int v=0;v<N;++v)if(!active[v]){
+            std::array<Root,6> roots; int used=0; bool completes=false;
+            for(int j=0;j<degree && !completes;++j){
+                Neighbor e=neighbor[v][j];if(!active[e.vertex])continue;
+                Root r=find(e.vertex);r.x=e.dx-r.x;r.y=e.dy-r.y;
+                for(int h=0;h<used;++h)if(roots[h].vertex==r.vertex){
+                    auto x=r.x-roots[h].x,y=r.y-roots[h].y;
+                    if(x%L || y%L)throw std::logic_error("hypothetical nonperiodic cycle");
+                    if(bx*y!=by*x){completes=true;break;}
+                }
+                roots[used++]=r;
+            }
+            result+=completes;
+        }
+        return result;
+    }
+    Observation observe(const std::vector<int>&order,int b,int tau){
+        reset();Observation o;o.tau=tau;
+        for(int k=1;k<=N;++k){
+            insert(order[k-1]);
+            if(rank>=1 && !o.j1)o.j1=k;
+            if(rank==2){o.j2=k;return o;}
+            if(k==b){
+                auto d=direction();o.dx_b=d.first;o.dy_b=d.second;
+                o.nu_b=completion_count();
+            }
+            if(k==tau)o.nu_tau=(tau==b?o.nu_b:completion_count());
+        }
+        throw std::logic_error("full torus not rank two");
+    }
+    int control(std::ostream&out){
+        if(L!=3)throw std::logic_error("control L3 only");
+        out<<"mask,rank,dx,dy,nu2\n";int controls=0;
+        for(int mask=0;mask<(1<<N);++mask){
+            reset();for(int v=0;v<N;++v)if(mask>>v&1)insert(v);
+            auto d=direction();int nu=completion_count();
+            if(rank==1){
+                int brute=0;
+                for(int v=0;v<N;++v)if(!(mask>>v&1)){
+                    Geometry copy=*this;brute+=(copy.insert(v)==2);++controls;
+                }
+                if(nu!=brute)throw std::logic_error("completion probe mismatch");
+            }
+            out<<mask<<','<<rank<<','<<d.first<<','<<d.second<<','<<nu<<'\n';
+        }
+        return controls;
+    }
+};
+
+std::uint64_t bounded(std::mt19937_64&rng,std::uint64_t bound){
+    std::uint64_t threshold=(-bound)%bound;
+    while(true){auto r=rng();if(r>=threshold)return r%bound;}
+}
+
+int main(int argc,char**argv){
+    try{
+        if(argc<3)throw std::invalid_argument("mode lattice ...");
+        std::string mode=argv[1],lattice=argv[2];
+        if(lattice!="square" && lattice!="triangular")throw std::invalid_argument("lattice");
+        if(mode=="control"){
+            if(argc!=4)throw std::invalid_argument("control square|triangular output.csv");
+            Geometry g(3,lattice=="triangular");std::ofstream out(argv[3]);
+            if(!out)throw std::runtime_error("cannot open control output");
+            int checks=g.control(out);out.close();
+            if(!out)throw std::runtime_error("control write failed");
+            std::cout<<"512 subsets; "<<checks<<" single-site completion probes agree\n";return 0;
+        }
+        if(mode!="sample" || argc!=10)
+            throw std::invalid_argument("sample square|triangular L samples permutation_seed time_seed b c output.csv");
+        int L=std::stoi(argv[3]),N=L*L,b=std::stoi(argv[7]),c=std::stoi(argv[8]);
+        auto samples=std::stoull(argv[4]),seed=std::stoull(argv[5]),time_seed=std::stoull(argv[6]);
+        if(!samples || b<1 || b>=c || c>N)throw std::invalid_argument("sample/time bounds");
+        Geometry g(L,lattice=="triangular");std::vector<int>order(N);
+        std::mt19937_64 rng(seed),time_rng(time_seed);
+        std::ofstream out(argv[9]);if(!out)throw std::runtime_error("cannot open output");
+        out<<"J1,J2,dx_b,dy_b,nu_b,tau,nu_tau\n";
+        auto start=std::chrono::steady_clock::now();
+        for(std::uint64_t i=0;i<samples;++i){
+            std::iota(order.begin(),order.end(),0);
+            for(int v=N-1;v>0;--v)std::swap(order[v],order[bounded(rng,v+1)]);
+            int tau=b+bounded(time_rng,c-b);
+            auto o=g.observe(order,b,tau);
+            out<<o.j1<<','<<o.j2<<','<<o.dx_b<<','<<o.dy_b<<','<<o.nu_b<<','<<o.tau<<','<<o.nu_tau<<'\n';
+        }
+        out.close();if(!out)throw std::runtime_error("output write failed");
+        double sec=std::chrono::duration<double>(std::chrono::steady_clock::now()-start).count();
+        std::cout<<"samples="<<samples<<" seconds="<<sec<<'\n';return 0;
+    }catch(const std::exception&e){std::cerr<<e.what()<<'\n';return 1;}
+}
diff --git a/analysis/exact-pair-independent-archive-20260930/source/exact-completion-pairs-20260929/engine.cpp b/analysis/exact-pair-independent-archive-20260930/source/exact-completion-pairs-20260929/engine.cpp
new file mode 100644
index 000000000..b02a4ac86
--- /dev/null
+++ b/analysis/exact-pair-independent-archive-20260930/source/exact-completion-pairs-20260929/engine.cpp
@@ -0,0 +1,159 @@
+// Conditional mean of the existing safe-insertion target, with no safe probes.
+// Source permutations and lifted union-find are reused without modification.
+#define main original_completion_engine_main
+#include "../completion-hazard-production-20260929/engine.cpp"
+#undef main
+#include <unordered_map>
+#include <unordered_set>
+
+using EdgeKey = std::uint64_t;
+EdgeKey pair_key(int a,int b){if(a>b)std::swap(a,b);return (EdgeKey(a)<<32)|unsigned(b);}
+
+struct Contact {
+    std::array<int,6> component{};
+    std::array<std::int64_t,6> height{};
+    int n=0;
+};
+
+struct PairResult {
+    int nu=0;
+    std::uint64_t edges=0, candidates=0;
+};
+
+class ProjectedPairs {
+    int L,N,z;
+    std::vector<std::array<Neighbor,6>> neighbor;
+    std::vector<unsigned char> occupied,safe;
+    std::vector<int> component,stack;
+    std::vector<std::int64_t> potential;
+    std::vector<Contact> contact;
+public:
+    std::unordered_set<EdgeKey> edges;
+    ProjectedPairs(int length,bool triangular):L(length),N(L*L),z(triangular?6:4),
+        neighbor(N),occupied(N),safe(N),component(N),potential(N),contact(N){
+        const int sx[]={1,-1,0,0,1,-1},sy[]={0,0,1,-1,1,-1};
+        for(int y=0;y<L;++y)for(int x=0;x<L;++x)for(int j=0;j<z;++j)
+            neighbor[x+L*y][j]={((x+sx[j]+L)%L)+L*((y+sy[j]+L)%L),sx[j],sy[j]};
+        stack.reserve(N);
+    }
+    PairResult count(const std::vector<int>& order,int k,std::pair<int,int> d){
+        auto ell=[d](int x,int y){return std::int64_t(d.first)*y-std::int64_t(d.second)*x;};
+        std::fill(occupied.begin(),occupied.end(),0);
+        std::fill(safe.begin(),safe.end(),0);
+        std::fill(component.begin(),component.end(),-1);
+        for(int i=0;i<k;++i)occupied[order[i]]=1;
+        // Rank one guarantees a consistent transverse potential even when
+        // more than one occupied component carries the persistent direction.
+        for(int root=0;root<N;++root)if(occupied[root] && component[root]<0){
+            component[root]=root;potential[root]=0;stack.clear();stack.push_back(root);
+            while(!stack.empty()){
+                int u=stack.back();stack.pop_back();
+                for(int j=0;j<z;++j){auto e=neighbor[u][j];if(!occupied[e.vertex])continue;
+                    auto h=potential[u]+ell(e.dx,e.dy);
+                    if(component[e.vertex]<0){component[e.vertex]=root;potential[e.vertex]=h;stack.push_back(e.vertex);}
+                    else if(component[e.vertex]!=root || potential[e.vertex]!=h)
+                        throw std::logic_error("inconsistent rank-one transverse potential");
+                }
+            }
+        }
+        PairResult out;
+        for(int v=0;v<N;++v)if(!occupied[v]){
+            auto& row=contact[v];row.n=0;bool completes=false;
+            for(int j=0;j<z && !completes;++j){
+                auto e=neighbor[v][j];if(!occupied[e.vertex])continue;
+                int c=component[e.vertex];auto a=potential[e.vertex]-ell(e.dx,e.dy);
+                int h=0;while(h<row.n && row.component[h]!=c)++h;
+                if(h<row.n){if(row.height[h]!=a)completes=true;}
+                else {row.component[row.n]=c;row.height[row.n++]=a;}
+            }
+            if(completes)++out.nu;
+            else safe[v]=1;
+        }
+        // Each root pair has groups of vertices with equal relative height.
+        // Only cross-group pairs are enumerated; every one is a true edge.
+        using HeightGroups=std::unordered_map<std::int64_t,std::vector<int>>;
+        std::unordered_map<EdgeKey,HeightGroups> buckets;
+        for(int v=0;v<N;++v)if(safe[v]){
+            const auto& row=contact[v];
+            for(int i=0;i<row.n;++i)for(int j=i+1;j<row.n;++j){
+                int c=row.component[i],cc=row.component[j];
+                auto h=row.height[i]-row.height[j];
+                if(c>cc){std::swap(c,cc);h=-h;}
+                buckets[pair_key(c,cc)][h].push_back(v);
+            }
+        }
+        edges.clear();
+        for(const auto& bucket:buckets){
+            const auto& groups=bucket.second;
+            for(auto a=groups.begin();a!=groups.end();++a){
+                auto b=a;++b;
+                for(;b!=groups.end();++b)
+                    for(int v:a->second)for(int w:b->second){
+                        edges.insert(pair_key(v,w));++out.candidates;
+                    }
+            }
+        }
+        // A direct lattice edge provides one more relative-height constraint.
+        // One shared old component is enough for a conflict with that edge.
+        for(int v=0;v<N;++v)if(safe[v]){
+            const auto& a=contact[v];
+            for(int j=0;j<z;++j){
+                auto e=neighbor[v][j];int w=e.vertex;
+                if(w<=v || !safe[w])continue;
+                const auto& b=contact[w];bool mismatch=false;
+                for(int p=0;p<a.n && !mismatch;++p)for(int q=0;q<b.n;++q)
+                    if(a.component[p]==b.component[q] && a.height[p]-b.height[q]+ell(e.dx,e.dy)!=0){mismatch=true;break;}
+                if(mismatch)edges.insert(pair_key(v,w));
+            }
+        }
+        out.edges=edges.size();return out;
+    }
+};
+
+int main(int argc,char**argv){
+    try{
+        if(argc==2 && std::string(argv[1])=="control"){
+            const std::array<int,4> masks{{4403,4405,4371,4402}};
+            const std::array<int,4> expected{{2,3,4,6}};
+            for(int i=0;i<4;++i){
+                Geometry g(4,i>=2);ProjectedPairs counter(4,i>=2);std::vector<int> order;
+                for(int v=0;v<16;++v)if(masks[i]>>v&1){order.push_back(v);g.insert(v);}
+                int k=order.size();auto r=counter.count(order,k,g.direction());
+                std::unordered_set<EdgeKey> brute;
+                for(int v=0;v<16;++v)if(!(masks[i]>>v&1)){
+                    Geometry one=g;if(one.insert(v)==2)continue;
+                    for(int w=v+1;w<16;++w)if(!(masks[i]>>w&1)){
+                        Geometry alone=g;if(alone.insert(w)==2)continue;
+                        Geometry two=one;if(two.insert(w)==2)brute.insert(pair_key(v,w));
+                    }
+                }
+                if(r.nu!=0 || r.edges!=unsigned(expected[i]) || counter.edges!=brute)
+                    throw std::logic_error("physical pair witness mismatch");
+                std::cout<<"mask="<<masks[i]<<" nu="<<r.nu<<" e="<<r.edges<<" nonlocal_candidates="<<r.candidates<<'\n';
+            }
+            return 0;
+        }
+        if(argc!=6)throw std::invalid_argument("L prefixes existing_permutation_seed b output.csv");
+        int L=std::stoi(argv[1]),N=L*L,b=std::stoi(argv[4]);
+        auto samples=std::stoull(argv[2]),seed=std::stoull(argv[3]);
+        if(L<3 || b<1 || b>N-2 || !samples)throw std::invalid_argument("bounds");
+        Geometry g(L,false);ProjectedPairs counter(L,false);std::vector<int>order(N);
+        std::mt19937_64 rng(seed);std::ofstream out(argv[5]);
+        if(!out)throw std::runtime_error("cannot open output");
+        out<<"J1,rank_b,dx_b,dy_b,nu_b,synergy_edges,nonlocal_candidates\n";
+        auto start=std::chrono::steady_clock::now();
+        for(std::uint64_t i=0;i<samples;++i){
+            std::iota(order.begin(),order.end(),0);
+            for(int v=N-1;v>0;--v)std::swap(order[v],order[bounded(rng,v+1)]);
+            g.reset();int j1=0,rank=0;
+            for(int k=1;k<=b;++k){rank=g.insert(order[k-1]);if(rank && !j1)j1=k;if(rank==2)break;}
+            if(rank!=1){out<<j1<<','<<rank<<",0,0,-1,-1,-1\n";continue;}
+            auto d=g.direction();auto r=counter.count(order,b,d);
+            out<<j1<<",1,"<<d.first<<','<<d.second<<','<<r.nu<<','<<r.edges<<','<<r.candidates<<'\n';
+        }
+        out.close();if(!out)throw std::runtime_error("write failed");
+        std::cout<<"prefixes="<<samples<<" seconds="
+                 <<std::chrono::duration<double>(std::chrono::steady_clock::now()-start).count()<<'\n';
+        return 0;
+    }catch(const std::exception&e){std::cerr<<e.what()<<'\n';return 1;}
+}
diff --git a/analysis/geometric-source-window-20260930/benchmark.json b/analysis/geometric-source-window-20260930/benchmark.json
new file mode 100644
index 000000000..ac32537f2
--- /dev/null
+++ b/analysis/geometric-source-window-20260930/benchmark.json
@@ -0,0 +1,40 @@
+{
+  "environment": {
+    "platform": "Linux-5.10.0-182.0.0.95.r3582_286.hce2.aarch64-aarch64-with-glibc2.34",
+    "machine": "aarch64",
+    "python": "3.9.9",
+    "cpu_quota": 14.5,
+    "memory_bytes": 26843545600,
+    "hostname": "83750ac4eae34bbbbc64b894b854dd8c"
+  },
+  "command": [
+    "/workspace/Matching-One-TgFr7R/geometric-source-window-20260930/analysis/geometric-source-window-20260930/build/engine",
+    "512",
+    "32",
+    "9722445355900932119",
+    "155385",
+    "735",
+    "/workspace/Matching-One-TgFr7R/geometric-source-window-20260930/analysis/geometric-source-window-20260930/build/old-seed-32.csv"
+  ],
+  "build_command": [
+    "chroot",
+    "/workspace/Matching-One-TgFr7R/geometric-source-window-20260930/toolchain",
+    "/usr/bin/g++",
+    "-O3",
+    "-std=c++17",
+    "/project/analysis/geometric-source-window-20260930/engine.cpp",
+    "-o",
+    "/project/engine"
+  ],
+  "compiler": "g++ (GCC) 10.3.1\nCopyright (C) 2020 Free Software Foundation, Inc.\nThis is free software; see the source for copying conditions.  There is NO\nwarranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.\n\n",
+  "stdout": "prefixes=32 lag=735 seconds=1.04858\n",
+  "wall_seconds": 1.0649478120030835,
+  "children_peak_rss_kib": 185640,
+  "workers": 14,
+  "predicted_ideal_seconds": 332.7961912509636,
+  "rows": 32,
+  "aligned_first5": true,
+  "scientific_weight": 0,
+  "source_commit": "d9824b5bf478fc8acad238be2b9f31ce99787966",
+  "binary_sha256": "db3b6ea76b9e5e9dc3bb9010c4256f65e577124677a44b92fa255975b434a24b"
+}
diff --git a/analysis/geometric-source-window-20260930/contract.json b/analysis/geometric-source-window-20260930/contract.json
new file mode 100644
index 000000000..d396c5e6c
--- /dev/null
+++ b/analysis/geometric-source-window-20260930/contract.json
@@ -0,0 +1,28 @@
+{
+  "schema": "matching-one.geometric-source-window-contract.v1",
+  "date": "2026-09-30",
+  "parent_commit": "eb9884dcc0a550b5d87fb247084e059504e76b2a",
+  "scientific_question": "Does a clock-preserving single geometric kick transmit to the original finite-window rank survival at L512?",
+  "lattice": "square occupied-site NN torus",
+  "L": 512,
+  "N": 262144,
+  "b": 155385,
+  "lag": 735,
+  "end_count": 156120,
+  "lag_origin": "original completion-hazard contract c-b, unchanged; no lag search",
+  "input_manifest": "analysis/safe-insertion-production-20260929/data/run.json",
+  "dependency_group": "safe_insertion_independent_L512",
+  "reuse_status": "previously analysed prefixes; new mechanism response on their existing full permutations, not new independent production or untouched holdout",
+  "source": "One kick at b: uniform completing probabilities 1/m; safe probabilities (s/m)*exp(theta*d)/sum_safe exp(theta*d). Original uniform source thereafter.",
+  "primary": "Mean derivative at theta=0 of P(rank remains one at b+735), conditional on rank_b=1 under the original prefix law",
+  "per_prefix_estimator": "1{rank_{b+h}=1} * [sum_{v in next h vacancies} d_v/h - 2e/s]; exactly zero if s=0",
+  "derivation": "Condition on the unordered next-h-site set; its first site is uniform among those h. Average the normalized source likelihood score over these h possible first sites. Endpoint survival depends on the set, not the order.",
+  "accompanying_readouts": [
+    "unperturbed rank-one survival at exactly the same endpoint",
+    "exact two-step derivative -(s*sum_d2-4e^2)/(s*m*(m-1)), zero for s=0"
+  ],
+  "inference": "14 original aligned whole-batch deletions; conditional risk denominators recomputed on each deletion; full 3x3 covariance. No early/late subgroup scan, no theta grid, no new cutoff, no expansion based on result.",
+  "interpretation": "Nonzero 735-step response establishes finite-size delayed rank transmission by a source that preserves the occupation clock and immediate hazard. Local negativity alone does not fix this sign. Does not identify a continuum limit, original-U mechanism or explain a fraction of passive history memory.",
+  "compute": "140000 existing permutations in 14 batches; Huawei TgFr7R only in a new task directory; one short old-seed cost probe of zero additional scientific weight; max14workers based on live cgroup limits",
+  "support_boundary": "Independent negative-history replication remains owned by the separate TV2N0X worker and is not changed or pooled with this mechanism computation."
+}
diff --git a/analysis/geometric-source-window-20260930/data/batch-00.json b/analysis/geometric-source-window-20260930/data/batch-00.json
new file mode 100644
index 000000000..581d23a5e
--- /dev/null
+++ b/analysis/geometric-source-window-20260930/data/batch-00.json
@@ -0,0 +1,20 @@
+{
+  "batch": 0,
+  "seed": "9722445355900932119",
+  "samples": 10000,
+  "original_file": "production-square-L512-b00.csv.gz",
+  "dependency_group": "safe-insertion-independent-20260929/square/L512/batch0",
+  "command": [
+    "/workspace/Matching-One-TgFr7R/geometric-source-window-20260930/analysis/geometric-source-window-20260930/build/engine",
+    "512",
+    "10000",
+    "9722445355900932119",
+    "155385",
+    "735",
+    "/workspace/Matching-One-TgFr7R/geometric-source-window-20260930/analysis/geometric-source-window-20260930/data/window-b00.csv"
+  ],
+  "returncode": 0,
+  "stdout": "prefixes=10000 lag=735 seconds=495.679\n",
+  "stderr": "",
+  "wall_seconds": 495.76140430604573
+}
diff --git a/analysis/geometric-source-window-20260930/data/batch-01.json b/analysis/geometric-source-window-20260930/data/batch-01.json
new file mode 100644
index 000000000..93e9cacb6
--- /dev/null
+++ b/analysis/geometric-source-window-20260930/data/batch-01.json
@@ -0,0 +1,20 @@
+{
+  "batch": 1,
+  "seed": "5026903940319098550",
+  "samples": 10000,
+  "original_file": "production-square-L512-b01.csv.gz",
+  "dependency_group": "safe-insertion-independent-20260929/square/L512/batch1",
+  "command": [
+    "/workspace/Matching-One-TgFr7R/geometric-source-window-20260930/analysis/geometric-source-window-20260930/build/engine",
+    "512",
+    "10000",
+    "5026903940319098550",
+    "155385",
+    "735",
+    "/workspace/Matching-One-TgFr7R/geometric-source-window-20260930/analysis/geometric-source-window-20260930/data/window-b01.csv"
+  ],
+  "returncode": 0,
+  "stdout": "prefixes=10000 lag=735 seconds=503.279\n",
+  "stderr": "",
+  "wall_seconds": 503.37715200701496
+}
diff --git a/analysis/geometric-source-window-20260930/data/batch-02.json b/analysis/geometric-source-window-20260930/data/batch-02.json
new file mode 100644
index 000000000..7035af9bc
--- /dev/null
+++ b/analysis/geometric-source-window-20260930/data/batch-02.json
@@ -0,0 +1,20 @@
+{
+  "batch": 2,
+  "seed": "7334852068188275347",
+  "samples": 10000,
+  "original_file": "production-square-L512-b02.csv.gz",
+  "dependency_group": "safe-insertion-independent-20260929/square/L512/batch2",
+  "command": [
+    "/workspace/Matching-One-TgFr7R/geometric-source-window-20260930/analysis/geometric-source-window-20260930/build/engine",
+    "512",
+    "10000",
+    "7334852068188275347",
+    "155385",
+    "735",
+    "/workspace/Matching-One-TgFr7R/geometric-source-window-20260930/analysis/geometric-source-window-20260930/data/window-b02.csv"
+  ],
+  "returncode": 0,
+  "stdout": "prefixes=10000 lag=735 seconds=513.945\n",
+  "stderr": "",
+  "wall_seconds": 514.0260947649949
+}
diff --git a/analysis/geometric-source-window-20260930/data/batch-03.json b/analysis/geometric-source-window-20260930/data/batch-03.json
new file mode 100644
index 000000000..f1dbc78cb
--- /dev/null
+++ b/analysis/geometric-source-window-20260930/data/batch-03.json
@@ -0,0 +1,20 @@
+{
+  "batch": 3,
+  "seed": "7761250788365968736",
+  "samples": 10000,
+  "original_file": "production-square-L512-b03.csv.gz",
+  "dependency_group": "safe-insertion-independent-20260929/square/L512/batch3",
+  "command": [
+    "/workspace/Matching-One-TgFr7R/geometric-source-window-20260930/analysis/geometric-source-window-20260930/build/engine",
+    "512",
+    "10000",
+    "7761250788365968736",
+    "155385",
+    "735",
+    "/workspace/Matching-One-TgFr7R/geometric-source-window-20260930/analysis/geometric-source-window-20260930/data/window-b03.csv"
+  ],
+  "returncode": 0,
+  "stdout": "prefixes=10000 lag=735 seconds=509.862\n",
+  "stderr": "",
+  "wall_seconds": 509.95311440201476
+}
diff --git a/analysis/geometric-source-window-20260930/data/batch-04.json b/analysis/geometric-source-window-20260930/data/batch-04.json
new file mode 100644
index 000000000..363a197c7
--- /dev/null
+++ b/analysis/geometric-source-window-20260930/data/batch-04.json
@@ -0,0 +1,20 @@
+{
+  "batch": 4,
+  "seed": "13568940959106988201",
+  "samples": 10000,
+  "original_file": "production-square-L512-b04.csv.gz",
+  "dependency_group": "safe-insertion-independent-20260929/square/L512/batch4",
+  "command": [
+    "/workspace/Matching-One-TgFr7R/geometric-source-window-20260930/analysis/geometric-source-window-20260930/build/engine",
+    "512",
+    "10000",
+    "13568940959106988201",
+    "155385",
+    "735",
+    "/workspace/Matching-One-TgFr7R/geometric-source-window-20260930/analysis/geometric-source-window-20260930/data/window-b04.csv"
+  ],
+  "returncode": 0,
+  "stdout": "prefixes=10000 lag=735 seconds=509.858\n",
+  "stderr": "",
+  "wall_seconds": 509.94920543499757
+}
diff --git a/analysis/geometric-source-window-20260930/data/batch-05.json b/analysis/geometric-source-window-20260930/data/batch-05.json
new file mode 100644
index 000000000..eb66595e4
--- /dev/null
+++ b/analysis/geometric-source-window-20260930/data/batch-05.json
@@ -0,0 +1,20 @@
+{
+  "batch": 5,
+  "seed": "14221577344058561937",
+  "samples": 10000,
+  "original_file": "production-square-L512-b05.csv.gz",
+  "dependency_group": "safe-insertion-independent-20260929/square/L512/batch5",
+  "command": [
+    "/workspace/Matching-One-TgFr7R/geometric-source-window-20260930/analysis/geometric-source-window-20260930/build/engine",
+    "512",
+    "10000",
+    "14221577344058561937",
+    "155385",
+    "735",
+    "/workspace/Matching-One-TgFr7R/geometric-source-window-20260930/analysis/geometric-source-window-20260930/data/window-b05.csv"
+  ],
+  "returncode": 0,
+  "stdout": "prefixes=10000 lag=735 seconds=511.782\n",
+  "stderr": "",
+  "wall_seconds": 511.87585210602265
+}
diff --git a/analysis/geometric-source-window-20260930/data/batch-06.json b/analysis/geometric-source-window-20260930/data/batch-06.json
new file mode 100644
index 000000000..675da46ba
--- /dev/null
+++ b/analysis/geometric-source-window-20260930/data/batch-06.json
@@ -0,0 +1,20 @@
+{
+  "batch": 6,
+  "seed": "2117620933003635365",
+  "samples": 10000,
+  "original_file": "production-square-L512-b06.csv.gz",
+  "dependency_group": "safe-insertion-independent-20260929/square/L512/batch6",
+  "command": [
+    "/workspace/Matching-One-TgFr7R/geometric-source-window-20260930/analysis/geometric-source-window-20260930/build/engine",
+    "512",
+    "10000",
+    "2117620933003635365",
+    "155385",
+    "735",
+    "/workspace/Matching-One-TgFr7R/geometric-source-window-20260930/analysis/geometric-source-window-20260930/data/window-b06.csv"
+  ],
+  "returncode": 0,
+  "stdout": "prefixes=10000 lag=735 seconds=507.565\n",
+  "stderr": "",
+  "wall_seconds": 507.6606401609606
+}
diff --git a/analysis/geometric-source-window-20260930/data/batch-07.json b/analysis/geometric-source-window-20260930/data/batch-07.json
new file mode 100644
index 000000000..4eed655d9
--- /dev/null
+++ b/analysis/geometric-source-window-20260930/data/batch-07.json
@@ -0,0 +1,20 @@
+{
+  "batch": 7,
+  "seed": "10091810389689459151",
+  "samples": 10000,
+  "original_file": "production-square-L512-b07.csv.gz",
+  "dependency_group": "safe-insertion-independent-20260929/square/L512/batch7",
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+        "10000",
+        "9255174391098419870",
+        "155385",
+        "735",
+        "/workspace/Matching-One-TgFr7R/geometric-source-window-20260930/analysis/geometric-source-window-20260930/data/window-b12.csv"
+      ],
+      "returncode": 0,
+      "stdout": "prefixes=10000 lag=735 seconds=511.293\n",
+      "stderr": "",
+      "wall_seconds": 511.39416568900924,
+      "file": "window-b12.csv.gz",
+      "sha256": "309731d31f024cfb1201c05f8ab5605f56d8bef1932ee0e5ef1c37bfd1429e64"
+    },
+    {
+      "batch": 13,
+      "seed": "4111027655712514342",
+      "samples": 10000,
+      "original_file": "production-square-L512-b13.csv.gz",
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch13",
+      "command": [
+        "/workspace/Matching-One-TgFr7R/geometric-source-window-20260930/analysis/geometric-source-window-20260930/build/engine",
+        "512",
+        "10000",
+        "4111027655712514342",
+        "155385",
+        "735",
+        "/workspace/Matching-One-TgFr7R/geometric-source-window-20260930/analysis/geometric-source-window-20260930/data/window-b13.csv"
+      ],
+      "returncode": 0,
+      "stdout": "prefixes=10000 lag=735 seconds=511.583\n",
+      "stderr": "",
+      "wall_seconds": 511.679738339968,
+      "file": "window-b13.csv.gz",
+      "sha256": "f08bcb50a7c5b56a89cf9b9347dd7f38fd597832d67e76015856484ad39f6cf3"
+    }
+  ],
+  "wall_seconds": 519.1446849519853,
+  "finished_utc": "2026-09-30T02:42:02.887470+00:00"
+}
diff --git a/analysis/geometric-source-window-20260930/data/started.json b/analysis/geometric-source-window-20260930/data/started.json
new file mode 100644
index 000000000..f7237fa57
--- /dev/null
+++ b/analysis/geometric-source-window-20260930/data/started.json
@@ -0,0 +1,27 @@
+{
+  "schema": "matching-one.geometric-source-window-run.v1",
+  "status": "running",
+  "pid": 995,
+  "started_utc": "2026-09-30T02:33:23.742283+00:00",
+  "environment": {
+    "platform": "Linux-5.10.0-182.0.0.95.r3582_286.hce2.aarch64-aarch64-with-glibc2.34",
+    "machine": "aarch64",
+    "python": "3.9.9",
+    "cpu_quota": 14.5,
+    "memory_bytes": 26843545600,
+    "hostname": "83750ac4eae34bbbbc64b894b854dd8c"
+  },
+  "source_commit": "d9824b5bf478fc8acad238be2b9f31ce99787966",
+  "binary_sha256": "db3b6ea76b9e5e9dc3bb9010c4256f65e577124677a44b92fa255975b434a24b",
+  "workers": 14,
+  "source_sha256": {
+    "analysis/geometric-source-window-20260930/engine.cpp": "1269a3fda3d7a9dcd302fa872ad4437488aeb4af5d5ffd254f2b3f31e754d0b1",
+    "analysis/geometric-source-window-20260930/run.py": "314d4929d8ff45c6e74464cad5a65061510fb331f4fb6521b3000e9d94f7d0f0",
+    "analysis/geometric-source-window-20260930/contract.json": "8f044bc068e9edc09b8a1b56acb0d0cd399431b42d14c4d3c60cc77eab5892eb",
+    "analysis/exact-completion-pairs-20260929/engine.cpp": "58610c299052b8ceded1606e4a7618363b90eef60a6ad9efa2b8256aee72f6e3",
+    "analysis/completion-hazard-production-20260929/engine.cpp": "42ac8a013cb731f4da4e3121ca429fd475555063e876638ad5d375d195448115"
+  },
+  "original_manifest_sha256": "8096d3d0a327604dd921e9c82d022daa685e2c7d6c4bf6e5b87c195b1d3dbf89",
+  "new_random_prefixes": 0,
+  "batches": []
+}
diff --git a/analysis/geometric-source-window-20260930/data/window-b00.csv.gz b/analysis/geometric-source-window-20260930/data/window-b00.csv.gz
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diff --git a/analysis/geometric-source-window-20260930/data/window-b12.csv.gz b/analysis/geometric-source-window-20260930/data/window-b12.csv.gz
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diff --git a/analysis/geometric-source-window-20260930/data/window-b13.csv.gz b/analysis/geometric-source-window-20260930/data/window-b13.csv.gz
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diff --git a/analysis/geometric-source-window-20260930/engine.cpp b/analysis/geometric-source-window-20260930/engine.cpp
new file mode 100644
index 000000000..eeae47a82
--- /dev/null
+++ b/analysis/geometric-source-window-20260930/engine.cpp
@@ -0,0 +1,40 @@
+// One source kick, original uniform continuation, existing permutation seeds.
+// Reuses the exact projected-pair and lifted topology implementations.
+#define MATCHING_ONE_PAIR_LIBRARY
+#include "../exact-completion-pairs-20260929/engine.cpp"
+
+int main(int argc,char**argv){
+    try {
+        if(argc!=7)throw std::invalid_argument("L prefixes existing_seed b lag output.csv");
+        const int L=std::stoi(argv[1]),N=L*L,b=std::stoi(argv[4]),h=std::stoi(argv[5]);
+        const auto samples=std::stoull(argv[2]),seed=std::stoull(argv[3]);
+        if(!samples || b<1 || h<2 || b+h>N)throw std::invalid_argument("bounds");
+        Geometry g(L,false);ProjectedPairs counter(L,false);
+        std::vector<int>order(N),degree(N);std::mt19937_64 rng(seed);
+        std::ofstream out(argv[6]);if(!out)throw std::runtime_error("output");
+        out<<"J1,rank_b,dx_b,dy_b,nu_b,synergy_edges,sum_degree_squared,survival_h,window_degree_sum\n";
+        const auto start=std::chrono::steady_clock::now();
+        for(std::uint64_t row=0;row<samples;++row){
+            std::iota(order.begin(),order.end(),0);
+            for(int v=N-1;v>0;--v)std::swap(order[v],order[bounded(rng,v+1)]);
+            g.reset();int j1=0,rank=0;
+            for(int k=1;k<=b;++k){rank=g.insert(order[k-1]);if(rank && !j1)j1=k;if(rank==2)break;}
+            if(rank!=1){out<<j1<<','<<rank<<",0,0,-1,-1,-1,-1,-1\n";continue;}
+            const auto d=g.direction();const auto r=counter.count(order,b,d);
+            std::fill(degree.begin(),degree.end(),0);
+            for(auto edge:counter.edges){++degree[int(edge>>32)];++degree[int(edge&0xffffffffULL)];}
+            std::uint64_t squared=0,window_sum=0;
+            for(int value:degree)squared+=std::uint64_t(value)*value;
+            for(int k=b;k<b+h;++k){
+                const int v=order[k];window_sum+=degree[v];
+                if(rank==1)rank=g.insert(v);
+            }
+            out<<j1<<",1,"<<d.first<<','<<d.second<<','<<r.nu<<','<<r.edges<<','
+               <<squared<<','<<(rank==1)<<','<<window_sum<<'\n';
+        }
+        out.close();if(!out)throw std::runtime_error("write failed");
+        std::cout<<"prefixes="<<samples<<" lag="<<h<<" seconds="
+                 <<std::chrono::duration<double>(std::chrono::steady_clock::now()-start).count()<<'\n';
+        return 0;
+    }catch(const std::exception&e){std::cerr<<e.what()<<'\n';return 1;}
+}
diff --git a/analysis/geometric-source-window-20260930/execution.json b/analysis/geometric-source-window-20260930/execution.json
new file mode 100644
index 000000000..ad16b8a8a
--- /dev/null
+++ b/analysis/geometric-source-window-20260930/execution.json
@@ -0,0 +1,72 @@
+{
+  "status": "completed_and_resources_released",
+  "source_commit": "d9824b5bf478fc8acad238be2b9f31ce99787966",
+  "contract_commit": "18ca9e132f08b89071291d74cbd10133d99b2e5d",
+  "runner_and_scorer_commit": "5929403a002c6e956cb74f1b6bb8d827bcc84109",
+  "remote_root": "/workspace/Matching-One-TgFr7R/geometric-source-window-20260930",
+  "instance": "DevEnvC_TgFr7R",
+  "instance_uuid": "83750ac4eae34bbbbc64b894b854dd8c",
+  "local_ssh_port": 10027,
+  "initial_state": "Ready",
+  "startup": "Running verified after power-on; one successful ssh-key-reset, local key mode600; only this named instance's tunnel started",
+  "tunnel_pid": 98390,
+  "runner_pid": 995,
+  "resources": {
+    "architecture": "aarch64",
+    "python": "3.9.9",
+    "compiler": "GCC10.3.1 in task-local installroot",
+    "system_packages_changed": false,
+    "cpu_quota_cores": 14.5,
+    "memory_limit_bytes": 26843545600,
+    "workers": 14,
+    "worker_peak_observed_rss_kib_approx": 80000,
+    "thread_environment": "OMP_NUM_THREADS=OPENBLAS_NUM_THREADS=MKL_NUM_THREADS=1"
+  },
+  "benchmark": {
+    "old_prefixes": 32,
+    "scientific_weight": 0,
+    "wall_seconds": 1.0649478120030835,
+    "all_first5_match": true,
+    "note": "children_peak_rss in benchmark includes the compiler and is a conservative concurrency input, not the engine-only peak"
+  },
+  "replay": {
+    "started_utc": "2026-09-30T02:33:23.742283+00:00",
+    "finished_utc": "2026-09-30T02:42:02.887470+00:00",
+    "wall_seconds": 519.1446849519853,
+    "batches": 14,
+    "prefixes": 140000,
+    "new_independent_prefixes": 0,
+    "exit_status": "all14engines0; data/run.json completed",
+    "command": "python3 analysis/geometric-source-window-20260930/run.py run --source-commit d9824b5bf478fc8acad238be2b9f31ce99787966",
+    "binary_sha256": "db3b6ea76b9e5e9dc3bb9010c4256f65e577124677a44b92fa255975b434a24b"
+  },
+  "transfer": {
+    "input_bundle_sha256": "74a2d7955f8de882b607d81d227145b2b69b06ca2e5491633f57bbc2c0de42fb",
+    "return_bundle_sha256": "ac8bbdd755fed0d1e26dcb3c6189d860e92d90429df59b4b2baf0cf2734e180f",
+    "local_and_remote_sha_match": true,
+    "run_manifest_sha256": "c1285f6d260a1a2ec9eafc3b166829a132bfdf6491be40495caa10ca66da6e68",
+    "result_sha256": "f4942305786b10bab6080e4f35141096dbc9b3a450db04ffeaa9fe6e74af88b6",
+    "raw_remote_files_preserved": true
+  },
+  "analysis": {
+    "local_python": "research-py311 Python3.11",
+    "full_score_runs": 1,
+    "all140000_first5_match": true,
+    "risk_prefixes": 55028,
+    "lag": 735,
+    "primary": -0.0110420319107,
+    "batch_se": 0.000157666446747,
+    "full_covariance": "results/result.json",
+    "support_review": "verification.md; one exact specified L4,h3 control and source review; no production rerun"
+  },
+  "cleanup": {
+    "pre_shutdown_no_other_research_jobs_verified_utc": "2026-09-30T02:43:40Z",
+    "own_tunnel_stopped": true,
+    "power_off_requested": true,
+    "final_cloud_state": "Ready",
+    "final_cloud_state_verified_utc": "2026-09-30T02:47:22Z",
+    "other_machine_operations_by_main": 0,
+    "note": "The separate support worker operated TV2N0X and recorded its own completed/Ready receipt; main did not operate it."
+  },
+  "scope": "New finite-window modified-source readout on the existing safe-insertion dependency group, not a new Monte Carlo production block, independent history replication, original-U identification or continuum proof"
+}
diff --git a/analysis/geometric-source-window-20260930/results/RESULT.md b/analysis/geometric-source-window-20260930/results/RESULT.md
new file mode 100644
index 000000000..018e08b58
--- /dev/null
+++ b/analysis/geometric-source-window-20260930/results/RESULT.md
@@ -0,0 +1,13 @@
+# Finite-window geometric-source response
+
+Same existing prefixes; new mechanism readout, not an independent block.
+
+| Metric | Estimate | Whole-batch SE |
+|---|---:|---:|
+| finite_window_susceptibility | -0.0110420319107 | 0.000157666446747 |
+| baseline_survival | 0.444755397252 | 0.00214095866922 |
+| two_step_susceptibility | -5.90719899852e-05 | 4.69939324914e-07 |
+
+55028 rank-one prefixes / 140000 total, 14 batches, lag=735.
+Full3x3 covariance, aligned deletions and unchanged prefix correspondence are in result.json.
+An infinitesimal explicitly modified-source response; not a finite-theta effect, continuum limit, original-U claim or explained fraction of birth memory.
diff --git a/analysis/geometric-source-window-20260930/results/result.json b/analysis/geometric-source-window-20260930/results/result.json
new file mode 100644
index 000000000..c61f1292f
--- /dev/null
+++ b/analysis/geometric-source-window-20260930/results/result.json
@@ -0,0 +1,285 @@
+{
+  "schema": "matching-one.geometric-source-window.v1",
+  "names": [
+    "finite_window_susceptibility",
+    "baseline_survival",
+    "two_step_susceptibility"
+  ],
+  "estimate": [
+    -0.011042031910713413,
+    0.4447553972523079,
+    -5.907198998515381e-05
+  ],
+  "batch_delete_one_se": [
+    0.00015766644674739815,
+    0.0021409586692189685,
+    4.699393249136158e-07
+  ],
+  "covariance": [
+    [
+      2.4858708429950136e-08,
+      -8.96987757789178e-08,
+      5.7710721136104395e-12
+    ],
+    [
+      -8.96987757789178e-08,
+      4.5837040233038575e-06,
+      2.36648941089081e-10
+    ],
+    [
+      5.7710721136104395e-12,
+      2.36648941089081e-10,
+      2.2084296910026496e-13
+    ]
+  ],
+  "deletions": [
+    [
+      -0.011034036332805083,
+      0.4450467728678226,
+      -5.8992063942352397e-05
+    ],
+    [
+      -0.010931526597622887,
+      0.44460088388282687,
+      -5.913656783387466e-05
+    ],
+    [
+      -0.0110162960819242,
+      0.44398332126147644,
+      -5.8925538565685974e-05
+    ],
+    [
+      -0.01104680292650645,
+      0.4448727984344423,
+      -5.915551811319931e-05
+    ],
+    [
+      -0.011045699435666433,
+      0.44442270058708416,
+      -5.914213152704974e-05
+    ],
+    [
+      -0.011059576882568546,
+      0.44561169691505625,
+      -5.911495590873768e-05
+    ],
+    [
+      -0.011102191687864417,
+      0.44538851582842265,
+      -5.911122538946851e-05
+    ],
+    [
+      -0.011034890307642377,
+      0.44469464079418847,
+      -5.8821334487592915e-05
+    ],
+    [
+      -0.010988889405652283,
+      0.4438247323822384,
+      -5.928005836894249e-05
+    ],
+    [
+      -0.01105720405045589,
+      0.4440484815257191,
+      -5.9131152802413355e-05
+    ],
+    [
+      -0.011112551555232743,
+      0.44437492668048334,
+      -5.925287983185336e-05
+    ],
+    [
+      -0.011022094919176714,
+      0.44559413202933984,
+      -5.889424828134892e-05
+    ],
+    [
+      -0.011066569160725822,
+      0.4455920035991628,
+      -5.910307184729292e-05
+    ],
+    [
+      -0.011070176000574805,
+      0.4445205345438181,
+      -5.89469222840797e-05
+    ]
+  ],
+  "batch_sufficient_statistics": [
+    {
+      "batch": 0,
+      "prefix_rows": 10000,
+      "risk": 3930,
+      "sums": [
+        -43.803743449063646,
+        1733.0,
+        -0.23623698157672066
+      ]
+    },
+    {
+      "batch": 1,
+      "prefix_rows": 10000,
+      "risk": 3890,
+      "sums": [
+        -48.60452483349849,
+        1738.0,
+        -0.22648765901436152
+      ]
+    },
+    {
+      "batch": 2,
+      "prefix_rows": 10000,
+      "risk": 3945,
+      "sums": [
+        -44.87547922980377,
+        1794.0,
+        -0.24052017835210712
+      ]
+    },
+    {
+      "batch": 3,
+      "prefix_rows": 10000,
+      "risk": 3928,
+      "sums": [
+        -43.129302438258115,
+        1741.0,
+        -0.22776648931855917
+      ]
+    },
+    {
+      "batch": 4,
+      "prefix_rows": 10000,
+      "risk": 3928,
+      "sums": [
+        -43.185690820182984,
+        1764.0,
+        -0.22845054387080163
+      ]
+    },
+    {
+      "batch": 5,
+      "prefix_rows": 10000,
+      "risk": 4006,
+      "sums": [
+        -43.339200280325414,
+        1738.0,
+        -0.2344501845274296
+      ]
+    },
+    {
+      "batch": 6,
+      "prefix_rows": 10000,
+      "risk": 3949,
+      "sums": [
+        -40.53208275831119,
+        1724.0,
+        -0.23127118323438137
+      ]
+    },
+    {
+      "batch": 7,
+      "prefix_rows": 10000,
+      "risk": 3957,
+      "sums": [
+        -44.05804908113389,
+        1763.0,
+        -0.2465490912871859
+      ]
+    },
+    {
+      "batch": 8,
+      "prefix_rows": 10000,
+      "risk": 3929,
+      "sums": [
+        -46.099672243311716,
+        1795.0,
+        -0.22146176230845188
+      ]
+    },
+    {
+      "batch": 9,
+      "prefix_rows": 10000,
+      "risk": 3957,
+      "sums": [
+        -42.91846392190498,
+        1796.0,
+        -0.23072636013099093
+      ]
+    },
+    {
+      "batch": 10,
+      "prefix_rows": 10000,
+      "risk": 3882,
+      "sums": [
+        -39.25837013880375,
+        1746.0,
+        -0.2200656730230718
+      ]
+    },
+    {
+      "batch": 11,
+      "prefix_rows": 10000,
+      "risk": 3903,
+      "sums": [
+        -44.116329239828204,
+        1693.0,
+        -0.23964502151908018
+      ]
+    },
+    {
+      "batch": 12,
+      "prefix_rows": 10000,
+      "risk": 3905,
+      "sums": [
+        -41.86471677895151,
+        1694.0,
+        -0.2290871228538877
+      ]
+    },
+    {
+      "batch": 13,
+      "prefix_rows": 10000,
+      "risk": 3919,
+      "sums": [
+        -41.835306769360045,
+        1755.0,
+        -0.23789521388601417
+      ]
+    }
+  ],
+  "prefix_rows": 140000,
+  "risk_prefixes": 55028,
+  "all_first5_match_old": true,
+  "lag": 735,
+  "clock": "insertion count",
+  "source": "one normalized safe-degree kick",
+  "dependency_group": "safe_insertion_independent_L512",
+  "new_independent_prefixes": 0,
+  "contract": {
+    "schema": "matching-one.geometric-source-window-contract.v1",
+    "date": "2026-09-30",
+    "parent_commit": "eb9884dcc0a550b5d87fb247084e059504e76b2a",
+    "scientific_question": "Does a clock-preserving single geometric kick transmit to the original finite-window rank survival at L512?",
+    "lattice": "square occupied-site NN torus",
+    "L": 512,
+    "N": 262144,
+    "b": 155385,
+    "lag": 735,
+    "end_count": 156120,
+    "lag_origin": "original completion-hazard contract c-b, unchanged; no lag search",
+    "input_manifest": "analysis/safe-insertion-production-20260929/data/run.json",
+    "dependency_group": "safe_insertion_independent_L512",
+    "reuse_status": "previously analysed prefixes; new mechanism response on their existing full permutations, not new independent production or untouched holdout",
+    "source": "One kick at b: uniform completing probabilities 1/m; safe probabilities (s/m)*exp(theta*d)/sum_safe exp(theta*d). Original uniform source thereafter.",
+    "primary": "Mean derivative at theta=0 of P(rank remains one at b+735), conditional on rank_b=1 under the original prefix law",
+    "per_prefix_estimator": "1{rank_{b+h}=1} * [sum_{v in next h vacancies} d_v/h - 2e/s]; exactly zero if s=0",
+    "derivation": "Condition on the unordered next-h-site set; its first site is uniform among those h. Average the normalized source likelihood score over these h possible first sites. Endpoint survival depends on the set, not the order.",
+    "accompanying_readouts": [
+      "unperturbed rank-one survival at exactly the same endpoint",
+      "exact two-step derivative -(s*sum_d2-4e^2)/(s*m*(m-1)), zero for s=0"
+    ],
+    "inference": "14 original aligned whole-batch deletions; conditional risk denominators recomputed on each deletion; full 3x3 covariance. No early/late subgroup scan, no theta grid, no new cutoff, no expansion based on result.",
+    "interpretation": "Nonzero 735-step response establishes finite-size delayed rank transmission by a source that preserves the occupation clock and immediate hazard. Local negativity alone does not fix this sign. Does not identify a continuum limit, original-U mechanism or explain a fraction of passive history memory.",
+    "compute": "140000 existing permutations in 14 batches; Huawei TgFr7R only in a new task directory; one short old-seed cost probe of zero additional scientific weight; max14workers based on live cgroup limits",
+    "support_boundary": "Independent negative-history replication remains owned by the separate TV2N0X worker and is not changed or pooled with this mechanism computation."
+  }
+}
diff --git a/analysis/geometric-source-window-20260930/run.py b/analysis/geometric-source-window-20260930/run.py
new file mode 100644
index 000000000..f12daeda5
--- /dev/null
+++ b/analysis/geometric-source-window-20260930/run.py
@@ -0,0 +1,159 @@
+#!/usr/bin/env python3
+"""Bounded Huawei replay of one fixed mechanism readout, standard library."""
+import argparse
+from concurrent.futures import ThreadPoolExecutor, as_completed
+import csv
+from datetime import datetime, timezone
+import gzip
+import hashlib
+import json
+import math
+import os
+from pathlib import Path
+import platform
+import resource
+import shutil
+import subprocess
+import time
+
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parent.parent
+OLD = HERE.parent / "safe-insertion-production-20260929/data"
+SOURCES = [HERE / "engine.cpp", HERE / "run.py", HERE / "contract.json",
+           HERE.parent / "exact-completion-pairs-20260929/engine.cpp",
+           HERE.parent / "completion-hazard-production-20260929/engine.cpp"]
+
+
+def sha(path):
+    return hashlib.sha256(path.read_bytes()).hexdigest()
+
+
+def save(path, value):
+    with path.open("x") as f:
+        json.dump(value, f, indent=2, allow_nan=False)
+        f.write("\n")
+
+
+def env():
+    cg = Path("/sys/fs/cgroup")
+    if (cg / "cpu/cpu.cfs_quota_us").exists():
+        cores = int((cg / "cpu/cpu.cfs_quota_us").read_text())/int((cg / "cpu/cpu.cfs_period_us").read_text())
+        memory = int((cg / "memory/memory.limit_in_bytes").read_text())
+    else:
+        quota, period = (cg / "cpu.max").read_text().split()
+        cores, memory = int(quota)/int(period), int((cg / "memory.max").read_text())
+    if cores <= 0 or memory <= 0:
+        raise ValueError("unmeasured limits")
+    return dict(platform=platform.platform(), machine=platform.machine(), python=platform.python_version(),
+                cpu_quota=cores, memory_bytes=memory, hostname=platform.node())
+
+
+def command(batch, samples, output):
+    return [str(HERE / "build/engine"), "512", str(samples), str(batch["seed"]), "155385", "735", str(output)]
+
+
+def main():
+    parser = argparse.ArgumentParser()
+    parser.add_argument("mode", choices=["benchmark", "run"])
+    parser.add_argument("--source-commit", required=True)
+    args = parser.parse_args()
+    manifest = json.loads((OLD / "run.json").read_text())
+    batches = sorted(manifest["batches"], key=lambda x: x["batch"])
+    assert manifest["status"] == "completed" and len(batches) == 14
+    assert all(b["samples"] == 10000 for b in batches)
+    environment = env()
+    if args.mode == "benchmark":
+        build = HERE / "build"
+        build.mkdir(exist_ok=False)
+        compiler = shutil.which("g++")
+        if compiler:
+            compile_cmd = [compiler, "-O3", "-std=c++17", str(HERE / "engine.cpp"), "-o", str(build / "engine")]
+            version_cmd = [compiler, "--version"]
+        else:
+            toolchain = ROOT / "toolchain"
+            project = toolchain / "project"
+            project.mkdir(exist_ok=False)
+            for src in SOURCES:
+                if src.suffix != ".cpp":
+                    continue
+                dst = project / src.relative_to(ROOT)
+                dst.parent.mkdir(parents=True, exist_ok=True)
+                shutil.copyfile(src, dst)
+            compile_cmd = ["chroot", str(toolchain), "/usr/bin/g++", "-O3", "-std=c++17",
+                           "/project/analysis/geometric-source-window-20260930/engine.cpp", "-o", "/project/engine"]
+            version_cmd = ["chroot", str(toolchain), "/usr/bin/g++", "--version"]
+        subprocess.run(compile_cmd, check=True)
+        if not compiler:
+            shutil.copyfile(project / "engine", build / "engine")
+            (build / "engine").chmod(0o755)
+        output = build / "old-seed-32.csv"
+        cmd = command(batches[0], 32, output)
+        start = time.monotonic()
+        proc = subprocess.run(cmd, check=True, capture_output=True, text=True)
+        seconds = time.monotonic()-start
+        with output.open() as f, gzip.open(OLD / batches[0]["file"], "rt") as g:
+            new, old = csv.DictReader(f), csv.DictReader(g)
+            keys = ["J1", "rank_b", "dx_b", "dy_b", "nu_b"]
+            rows = list(new)
+            assert len(rows) == 32
+            for row in rows:
+                previous = next(old)
+                assert all(row[k] == previous[k] for k in keys)
+        rss = resource.getrusage(resource.RUSAGE_CHILDREN).ru_maxrss
+        workers = min(14, math.floor(environment["cpu_quota"]), int(environment["memory_bytes"]*.8//(rss*1024)))
+        assert workers >= 1
+        result = dict(environment=environment, command=cmd, build_command=compile_cmd,
+                      compiler=subprocess.check_output(version_cmd, text=True), stdout=proc.stdout,
+                      wall_seconds=seconds, children_peak_rss_kib=rss, workers=workers,
+                      predicted_ideal_seconds=seconds/32*10000*math.ceil(14/workers),
+                      rows=32, aligned_first5=True, scientific_weight=0, source_commit=args.source_commit,
+                      binary_sha256=sha(build / "engine"))
+        save(HERE / "benchmark.json", result)
+        print(json.dumps(result), flush=True)
+        return
+    bench = json.loads((HERE / "benchmark.json").read_text())
+    assert sha(HERE / "build/engine") == bench["binary_sha256"]
+    workers = min(bench["workers"], math.floor(environment["cpu_quota"]), 14)
+    data = HERE / "data"
+    data.mkdir(exist_ok=False)
+    record = dict(schema="matching-one.geometric-source-window-run.v1", status="running", pid=os.getpid(),
+                  started_utc=datetime.now(timezone.utc).isoformat(), environment=environment,
+                  source_commit=args.source_commit, binary_sha256=bench["binary_sha256"], workers=workers,
+                  source_sha256={str(p.relative_to(ROOT)): sha(p) for p in SOURCES},
+                  original_manifest_sha256=sha(OLD / "run.json"), new_random_prefixes=0, batches=[])
+    save(data / "started.json", record)
+    def one(batch):
+        output = data / ("window-b%02d.csv" % batch["batch"])
+        cmd = command(batch, 10000, output)
+        begin = time.monotonic()
+        process = subprocess.run(cmd, capture_output=True, text=True)
+        row = dict(batch=batch["batch"], seed=batch["seed"], samples=10000, original_file=batch["file"],
+                   dependency_group=batch["dependency_group"], command=cmd, returncode=process.returncode,
+                   stdout=process.stdout, stderr=process.stderr, wall_seconds=time.monotonic()-begin)
+        save(data / ("batch-%02d.json" % batch["batch"]), row)
+        if process.returncode:
+            raise RuntimeError(row)
+        compressed = output.with_suffix(".csv.gz")
+        with output.open("rb") as f, compressed.open("xb") as g:
+            with gzip.GzipFile(filename="", mode="wb", fileobj=g, mtime=0) as z:
+                shutil.copyfileobj(f, z)
+        row.update(file=compressed.name, sha256=sha(compressed))
+        print(json.dumps(dict(batch=row["batch"], status="completed", seconds=row["wall_seconds"])), flush=True)
+        return row
+    begin = time.monotonic()
+    try:
+        with ThreadPoolExecutor(max_workers=workers) as pool:
+            for task in as_completed([pool.submit(one, b) for b in batches]):
+                record["batches"].append(task.result())
+        record["batches"].sort(key=lambda b: b["batch"])
+        record["status"] = "completed"
+    except Exception as exc:
+        record.update(status="failed", error=str(exc))
+        raise
+    finally:
+        record.update(wall_seconds=time.monotonic()-begin, finished_utc=datetime.now(timezone.utc).isoformat())
+        save(data / "run.json", record)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/analysis/geometric-source-window-20260930/score.py b/analysis/geometric-source-window-20260930/score.py
new file mode 100644
index 000000000..3fd3a6138
--- /dev/null
+++ b/analysis/geometric-source-window-20260930/score.py
@@ -0,0 +1,79 @@
+#!/usr/bin/env python3
+"""One fixed finite-window mechanism readout with aligned whole-batch covariance."""
+import csv
+import gzip
+import hashlib
+from itertools import zip_longest
+import json
+import math
+from pathlib import Path
+
+HERE = Path(__file__).resolve().parent
+OLD = HERE.parent / "safe-insertion-production-20260929/data"
+M, H = 262144-155385, 735
+NAMES = ["finite_window_susceptibility", "baseline_survival", "two_step_susceptibility"]
+
+
+def main():
+    manifest = json.loads((HERE / "data/run.json").read_text())
+    assert manifest["status"] == "completed"
+    batches = []
+    for batch in manifest["batches"]:
+        path = HERE / "data" / batch["file"]
+        assert hashlib.sha256(path.read_bytes()).hexdigest() == batch["sha256"]
+        rows, values = 0, []
+        with gzip.open(path, "rt") as f, gzip.open(OLD / batch["original_file"], "rt") as g:
+            for new, old in zip_longest(csv.DictReader(f), csv.DictReader(g)):
+                assert new is not None and old is not None
+                assert all(new[k] == old[k] for k in ["J1", "rank_b", "dx_b", "dy_b", "nu_b"])
+                rows += 1
+                if int(new["rank_b"]) != 1:
+                    continue
+                c, e, d2, survives, total = (int(new[k]) for k in ["nu_b", "synergy_edges", "sum_degree_squared", "survival_h", "window_degree_sum"])
+                s = M-c
+                assert 0 <= c <= M and e >= 0 and survives in (0, 1)
+                if not s:
+                    assert survives == 0 and e == 0 and d2 == 0
+                    values.append([0., 0., 0.])
+                else:
+                    variance_numerator = s*d2-4*e*e
+                    assert variance_numerator >= 0
+                    values.append([survives*(total/H-2*e/s), float(survives),
+                                   -variance_numerator/(s*M*(M-1))])
+        assert rows == batch["samples"] == 10000
+        batches.append(dict(batch=batch["batch"], prefix_rows=rows, risk=len(values),
+                            sums=[math.fsum(v[j] for v in values) for j in range(3)]))
+    assert sorted(b["batch"] for b in batches) == list(range(14))
+    def evaluate(omit=None):
+        selected = [b for b in batches if b["batch"] != omit]
+        n = sum(b["risk"] for b in selected)
+        return [math.fsum(b["sums"][j] for b in selected)/n for j in range(3)]
+    estimate = evaluate()
+    deletions = [evaluate(i) for i in range(14)]
+    center = [math.fsum(v[j] for v in deletions)/14 for j in range(3)]
+    covariance = [[13/14*math.fsum((v[i]-center[i])*(v[j]-center[j]) for v in deletions)
+                   for j in range(3)] for i in range(3)]
+    se = [math.sqrt(covariance[i][i]) for i in range(3)]
+    result = dict(schema="matching-one.geometric-source-window.v1", names=NAMES,
+                  estimate=estimate, batch_delete_one_se=se, covariance=covariance,
+                  deletions=deletions, batch_sufficient_statistics=batches,
+                  prefix_rows=sum(b["prefix_rows"] for b in batches), risk_prefixes=sum(b["risk"] for b in batches),
+                  all_first5_match_old=True, lag=H, clock="insertion count", source="one normalized safe-degree kick",
+                  dependency_group="safe_insertion_independent_L512", new_independent_prefixes=0,
+                  contract=json.loads((HERE / "contract.json").read_text()))
+    target = HERE / "results"
+    target.mkdir(exist_ok=False)
+    (target / "result.json").write_text(json.dumps(result, indent=2, allow_nan=False)+"\n")
+    lines = ["# Finite-window geometric-source response", "",
+             "Same existing prefixes; new mechanism readout, not an independent block.", "",
+             "| Metric | Estimate | Whole-batch SE |", "|---|---:|---:|"]
+    lines += [f"| {name} | {value:.12g} | {error:.12g} |" for name, value, error in zip(NAMES, estimate, se)]
+    lines += ["", f"{result['risk_prefixes']} rank-one prefixes / {result['prefix_rows']} total, 14 batches, lag={H}.",
+              "Full3x3 covariance, aligned deletions and unchanged prefix correspondence are in result.json.",
+              "An infinitesimal explicitly modified-source response; not a finite-theta effect, continuum limit, original-U claim or explained fraction of birth memory.", ""]
+    (target / "RESULT.md").write_text("\n".join(lines))
+    print("\n".join(lines))
+
+
+if __name__ == "__main__":
+    main()
diff --git a/analysis/geometric-source-window-20260930/verification.md b/analysis/geometric-source-window-20260930/verification.md
new file mode 100644
index 000000000..93a83462f
--- /dev/null
+++ b/analysis/geometric-source-window-20260930/verification.md
@@ -0,0 +1,70 @@
+# 固定窗口几何源：旁路核验
+
+2026-09-30，独立支持worker。结论：**所审版本未发现实质公式或实现错位**。
+源归一化、unordered next-h集合条件平均、原排列延续、三个条件risk均值及整批协方差实现同一固定合同。
+这是源码审查加一个指定构型的精确分数对照，不是生产数据、云端构建或执行结果的认证。
+
+审查时HEAD为`d9824b5bf478fc8acad238be2b9f31ce99787966`；下述五文件相对
+`5929403a002c6e956cb74f1b6bb8d827bcc84109`无变化。只写本文件，无commit/push。
+
+## 发现与对应
+
+| 核对项 | 结论与依据 |
+|---|---|
+| 固定合同 | contract为square NN occupied-site torus，L=512、N=262144、b=155385、h=735、终点156120。run.py:51–52的实参及score.py:13的M=106759、H=735一致。输入均为safe-insertion-production旧14批，未混入独立completion-hazard复核块。 |
+| kick归一化 | 完成点总质量c/m；安全点总质量(s/m)×Z/Z=s/m，合计1。θ=0每空点概率1/m；安全点P′₀(v)=(d_v−2e/s)/m，完成点导数0。因此likelihood score为d_v−2e/s，已包含正确归一化。最终均匀源期望不能再额外乘s/m。 |
+| unordered集合平均 | note:15–36正确。均匀h-subset B的概率为1/C(m,h)，kick后为Σ[v∈B]Pθ(v)/C(m−1,h−1)，比值为(m/h)ΣPθ(v)。存活B全部由初始安全点构成，导数score恰为Σd_v/h−2e/s；不存活B乘survival后为0。度数均取初始A，既不沿途重算，也不把两步曲率外推至735步。 |
+| e、degree及平方和 | engine.cpp:23–27从去重的无序协同边给两端各加1；故Σ_safe d=2e。degree每个前缀清零，sum_degree_squared为Σd²；乘法先转uint64_t。score.py:39先用Python整数计算sΣd²−4e²，随后才除法，没有先以浮点做这两个大数的相减。 |
+| 原排列及真实延续 | engine.cpp:18–21与旧safe-insertion engine.cpp:68–76使用同一完整Fisher–Yates、mt19937_64 seed和bounded算法。旧probe_rng独立；旧probe只读取const order并修改Geometry副本，不消耗排列rng、不改order。新counter也不消耗rng、不修改g；故order[b:b+h]恢复同一完整排列的原后缀。 |
+| 终点存活及score | engine.cpp:28–33累加恰好h个后缀站点的**初始**degree，并对原g继续逐点插入。达到rank2后跳过后续Geometry更新不会改变终点survival，因为rank单调；window_sum仍累加全部h项，随后由survival=0消去。score.py:41实现指定条件平均，不是非零θ有限差分。 |
+| 三个条件risk均值 | score.py:30–50只排除rank_b≠1，所有rank1前缀均进入同一risk分母；未按终点存活或s>0再筛选。s=0时三项均为0，但该前缀仍计入risk，符合条件于原rank_b=1的目标。总体点值为各批分子和/各批risk和，不是等权平均各批条件均值。 |
+| 两步伴随量 | −(sΣd²−4e²)/(s m(m−1))=−s Var_safe(d)/(m(m−1))。它与有限窗口响应及baseline_survival按同一批、同一rank1风险人口联合保存；不是额外独立证据。 |
+| batch covariance | score.py:47–59每次删除整批后重新加总分子和risk分母，得到14个三维比率向量。以删除向量均值为中心、系数13/14构造完整3×3协方差，SE取对角平方根；没有丢掉跨指标协方差，也没有误把14批当140000个独立删除单位。 |
+
+实现边界：通用note允许h=1；当前engine要求h≥2，这不影响固定h=735合同。
+run.py和score.py的参数是匹配当前合同的硬编码，本核验不自动适用于未来修改后的参数。
+run.py的children_peak_rss还可能包含编译子进程峰值，因而可能保守限制worker；不改变估计量。
+这些均不构成当前科学读出的阻碍，不要求增加测试或重跑生产。
+
+## 唯一实际计算对照
+
+使用本地研究解释器`/Users/lc/python-envs/research-py311/bin/python -B -`，代码从stdin运行；
+只用标准库`fractions.Fraction`、`itertools.combinations`、`math.comb`，不写临时文件。
+通过独立的格点提升坐标/回路绕数求rank，没有调用旧derive.py、旧census或生产scorer。
+
+固定square L4，v=x+4y，唯一输入为既定构型
+`A={0,1,4,5,8,12}`，**只核对h=3**：
+
+- m=10、c=0、s=10；协同边为(2,3)、(6,7)，e=2。
+- 四个degree=1、其余六个degree=0；Σd²=4、mean_d=2/5；安全score总和精确为0。
+- 只计算120=C(10,3)个终点集合，存活概率精确为5/6。
+- 直接kick导数：对10个首点及各自36个后续二元素集合，按
+  `P′₀(v)=(d_v−2/5)/10`与后缀概率`1/C(9,2)`累加360项，得到**−7/180**。
+- 条件平均读出：对相同120个终点集合累加
+  `survival×(Σd_v/3−2/5)/C(10,3)`，得到**−7/180**。
+- 两结果以Fraction精确相等，且与已写出的A、h=3数值相符；没有数值差分或容差判定。
+
+这个单构型对照核实归一化单kick与集合平均公式的连接；它没有编译/执行本次C++，
+也不声称已验证L512生产行、735步结果或云运行状态。当前源码对应性由上表的逐段审查提供。
+
+## 实际操作范围
+
+已读指定五文件；为确认源与依赖语义，另读源kick定义、旧safe-insertion engine及被include的pair/geometry接口。
+做了上述唯一精确小对照，并只读检查版本差异及下面的文件hash。
+无SSH、无云端操作、无旧档案数据重放/评分、无旧L4 census、无三构型全滞后重跑、无扩样或cutoff搜索。
+本核验未发现需立即中断主线的实质问题；生产执行及结果核对仍由主线程负责。
+
+审查文件SHA256（区分主线程后续修改）：
+
+```text
+notes/geometric-source-window-identity-20260930.md
+6b0c6f7e0b2b69aa106bc1256a697fc805907868a8fb3349a7cc4606bddf21be
+analysis/geometric-source-window-20260930/engine.cpp
+1269a3fda3d7a9dcd302fa872ad4437488aeb4af5d5ffd254f2b3f31e754d0b1
+analysis/geometric-source-window-20260930/score.py
+2c8fb9f5a6c31b8af74d497a04d74e78ea2e5dabe7fc7e2789cd3f7279d1769e
+analysis/geometric-source-window-20260930/run.py
+314d4929d8ff45c6e74464cad5a65061510fb331f4fb6521b3000e9d94f7d0f0
+analysis/geometric-source-window-20260930/contract.json
+8f044bc068e9edc09b8a1b56acb0d0cd399431b42d14c4d3c60cc77eab5892eb
+```
diff --git a/analysis/one-sided-birth-source-20260930/RESULT.md b/analysis/one-sided-birth-source-20260930/RESULT.md
new file mode 100644
index 000000000..f80e804dd
--- /dev/null
+++ b/analysis/one-sided-birth-source-20260930/RESULT.md
@@ -0,0 +1,27 @@
+# One-sided birth source: moving-root E_top response
+
+Square L4, one count6 kick on 64 symmetry-related geometries; activation probability 8/1001.
+First birth unchanged; deltaP0=0 and deltaM=deltaEtop=deltaP2=g(p)>0 at every interior p.
+
+Exact g(p), in positive p^k(1-p)^(16-k) terms:
+
+- k=8: coefficient 96/35
+- k=9: coefficient 32/9
+- k=10: coefficient 1088/525
+- k=11: coefficient 256/385
+- k=12: coefficient 16/165
+
+Illustrative floating-point evaluation at the baseline balance root:
+
+- p_balance: 0.5906721123
+- baseline_Etop: 0.6454881965
+- M_prime: 4.979143122
+- P0_prime: -2.459746053
+- delta_M: 0.0001705764924
+- delta_Etop: 0.0001705764924
+- root_derivative: -3.425820231e-05
+- moving_root_Etop_derivative: 0.0001685329558
+- normal_factor: 0.9880198227
+
+Sign proof uses exact coefficients and strict monotonicity, not decimal root precision.
+No old census was rerun. This is a constructed dynamic source, not the original norm4/U identification experiment.
diff --git a/analysis/one-sided-birth-source-20260930/derive.py b/analysis/one-sided-birth-source-20260930/derive.py
new file mode 100644
index 000000000..ec97b9e0b
--- /dev/null
+++ b/analysis/one-sided-birth-source-20260930/derive.py
@@ -0,0 +1,94 @@
+#!/usr/bin/env python3
+"""Transport existing exact responses into an actual moving-root E_top map.
+
+Reads existing finite tables; no topology census or new sampling. Exact
+positive coefficients prove the sign. Root decimals are illustrations only.
+"""
+from fractions import Fraction as F
+import hashlib
+import json
+from math import comb
+from pathlib import Path
+
+HERE = Path(__file__).resolve().parent
+BASE = HERE.parent / "completion-pair-synergy-20260929/result.json"
+KICK = HERE.parent / "birth-selection-mechanism-20260930/result.json"
+
+
+def main():
+    model = next(x for x in json.loads(BASE.read_text())["models"] if x["lattice"] == "square")
+    start = next(x for x in json.loads(KICK.read_text())["rows"] if x["name"] == "A")
+    points = [(v % 4, v//4) for v in start["occupied"]]
+    orbit = set()
+    for reflected in [False, True]:
+        for rotation in range(4):
+            transformed = []
+            for x, y in points:
+                if reflected:
+                    x = -x
+                for _ in range(rotation):
+                    x, y = -y, x
+                transformed.append((x, y))
+            for tx in range(4):
+                for ty in range(4):
+                    orbit.add(tuple(sorted((x+tx) % 4+4*((y+ty) % 4) for x, y in transformed)))
+    activation = F(len(orbit), comb(16, 6))
+    h_to_derivative = {x["lag_insertions"]: F(x["first_kick_derivative_at_zero"])
+                       for x in start["all_lags_single_kick"]}
+    bernstein = [F(0) if k <= 6 else -activation*h_to_derivative[k-6] for k in range(17)]
+    powers = [comb(16, k)*v for k, v in enumerate(bernstein)]
+    assert all(v >= 0 for v in bernstein) and any(v > 0 for v in bernstein)
+    counts = model["rank_counts"]
+    def probabilities(p):
+        prob = [0., 0., 0.]
+        derivative = [0., 0., 0.]
+        for k, rank, number in counts:
+            prob[rank] += number*p**k*(1-p)**(16-k)
+            derivative[rank] += number*((k*p**(k-1)*(1-p)**(16-k) if k else 0)
+                                       - ((16-k)*p**k*(1-p)**(15-k) if k < 16 else 0))
+        return prob, derivative
+    lo, hi = .01, .99
+    for _ in range(40):
+        p = (lo+hi)/2
+        prob, _ = probabilities(p)
+        if prob[2]-prob[0] > 0:
+            hi = p
+        else:
+            lo = p
+    p = (lo+hi)/2
+    prob, derivative = probabilities(p)
+    g = sum(float(v)*p**k*(1-p)**(16-k) for k, v in enumerate(powers))
+    mp = derivative[2]-derivative[0]
+    normal_factor = -2*derivative[0]/mp
+    output = {
+        "schema": "matching-one.one-sided-birth-source.v1", "date": "2026-09-30",
+        "source": "At count6 only, kick when occupied set is in the translation/dihedral orbit of A={0,1,4,5,8,12}; otherwise uniform. Uniform continuation after the kick.",
+        "source_model": "square NN occupied-site L4; theta0 is the original uniform-permutation/iid-label model",
+        "orbit_size": len(orbit), "prefix_activation_probability": str(activation),
+        "delta_P2_Bernstein_degree16_coefficients": [str(v) for v in bernstein],
+        "delta_P2_coefficients_of_p_k_times_1_minus_p_16_minus_k": [str(v) for v in powers],
+        "exact_sign": "All coefficients nonnegative and some positive: delta_P2(p)>0 for every0<p<1. Delta_P0=0 exactly.",
+        "root_illustration": {"p_balance": p, "baseline_Etop": prob[0]+prob[2],
+                              "M_prime": mp, "P0_prime": derivative[0], "delta_M": g, "delta_Etop": g,
+                              "root_derivative": -g/mp, "moving_root_Etop_derivative": g*normal_factor,
+                              "normal_factor": normal_factor},
+        "source_artifact_sha256": {str(f.relative_to(HERE.parent.parent)): hashlib.sha256(f.read_bytes()).hexdigest()
+                                   for f in [BASE, KICK]},
+        "scope": "Exact finite, symmetry-invariant constructed dynamic source. Not natural-archive evidence, original norm4 source, U six-coordinate normalization, large-size scaling or CFT identification."
+    }
+    (HERE / "result.json").write_text(json.dumps(output, indent=2)+"\n")
+    lines = ["# One-sided birth source: moving-root E_top response", "",
+             f"Square L4, one count6 kick on {len(orbit)} symmetry-related geometries; activation probability {activation}.",
+             "First birth unchanged; deltaP0=0 and deltaM=deltaEtop=deltaP2=g(p)>0 at every interior p.", "",
+             "Exact g(p), in positive p^k(1-p)^(16-k) terms:", ""]
+    lines += [f"- k={k}: coefficient {v}" for k, v in enumerate(powers) if v]
+    lines += ["", "Illustrative floating-point evaluation at the baseline balance root:", ""]
+    lines += [f"- {key}: {value:.10g}" for key, value in output["root_illustration"].items()]
+    lines += ["", "Sign proof uses exact coefficients and strict monotonicity, not decimal root precision.",
+              "No old census was rerun. This is a constructed dynamic source, not the original norm4/U identification experiment.", ""]
+    (HERE / "RESULT.md").write_text("\n".join(lines))
+    print("\n".join(lines))
+
+
+if __name__ == "__main__":
+    main()
diff --git a/analysis/one-sided-birth-source-20260930/result.json b/analysis/one-sided-birth-source-20260930/result.json
new file mode 100644
index 000000000..02e1508a2
--- /dev/null
+++ b/analysis/one-sided-birth-source-20260930/result.json
@@ -0,0 +1,63 @@
+{
+  "schema": "matching-one.one-sided-birth-source.v1",
+  "date": "2026-09-30",
+  "source": "At count6 only, kick when occupied set is in the translation/dihedral orbit of A={0,1,4,5,8,12}; otherwise uniform. Uniform continuation after the kick.",
+  "source_model": "square NN occupied-site L4; theta0 is the original uniform-permutation/iid-label model",
+  "orbit_size": 64,
+  "prefix_activation_probability": "8/1001",
+  "delta_P2_Bernstein_degree16_coefficients": [
+    "0",
+    "0",
+    "0",
+    "0",
+    "0",
+    "0",
+    "0",
+    "0",
+    "16/75075",
+    "2/6435",
+    "136/525525",
+    "16/105105",
+    "4/75075",
+    "0",
+    "0",
+    "0",
+    "0"
+  ],
+  "delta_P2_coefficients_of_p_k_times_1_minus_p_16_minus_k": [
+    "0",
+    "0",
+    "0",
+    "0",
+    "0",
+    "0",
+    "0",
+    "0",
+    "96/35",
+    "32/9",
+    "1088/525",
+    "256/385",
+    "16/165",
+    "0",
+    "0",
+    "0",
+    "0"
+  ],
+  "exact_sign": "All coefficients nonnegative and some positive: delta_P2(p)>0 for every0<p<1. Delta_P0=0 exactly.",
+  "root_illustration": {
+    "p_balance": 0.5906721123312584,
+    "baseline_Etop": 0.6454881965464061,
+    "M_prime": 4.979143122349653,
+    "P0_prime": -2.4597460525088057,
+    "delta_M": 0.00017057649239640227,
+    "delta_Etop": 0.00017057649239640227,
+    "root_derivative": -3.425820230608422e-05,
+    "moving_root_Etop_derivative": 0.00016853295577687747,
+    "normal_factor": 0.9880198227152202
+  },
+  "source_artifact_sha256": {
+    "analysis/completion-pair-synergy-20260929/result.json": "0aabdbfcc79f906bc3161d25fddbd859d63faf0e07f4c6377adfa275070ba800",
+    "analysis/birth-selection-mechanism-20260930/result.json": "b18565c1f5a9aff5ad241069d64e57699d6e84f472259630aa3ba54ac86832bb"
+  },
+  "scope": "Exact finite, symmetry-invariant constructed dynamic source. Not natural-archive evidence, original norm4 source, U six-coordinate normalization, large-size scaling or CFT identification."
+}
diff --git a/analysis/research_frontier.json b/analysis/research_frontier.json
new file mode 100644
index 000000000..fcd043fda
--- /dev/null
+++ b/analysis/research_frontier.json
@@ -0,0 +1,842 @@
+{
+  "schema": "matching-one.research-frontier.v1",
+  "as_of": "2026-09-30",
+  "current_pr": 838,
+  "main_base": "d31fa5fe9584b77595e4a78b9560aa2ff77cf0b3",
+  "scope": "current attention and research map; not a replacement of historical result contracts",
+  "priority_semantics": "attention only; no locks, research permissions or automatic rejection",
+  "allocation_update": "2026-09-30 owner direction: one support worker handles precision and replication; main investigator leads mechanism derivation and discriminating interventions, without waiting for precision results. Medium/large compute goes to Huawei first.",
+  "entry_points": {
+    "overview": "README.md",
+    "claims": "docs/STATUS.md",
+    "frontier": "docs/RESEARCH-FRONTIER.md",
+    "next": "docs/ROADMAP.md",
+    "historical_evidence": "analysis/research_ledger.yaml"
+  },
+  "branch_sources": [
+    {
+      "id": "P838_safe_transport_selection",
+      "pr": 838,
+      "commit": "e56c0b8361628a066abea4ba4d3655c74e3a62bf",
+      "branch": "research/frontier-and-birth-gap-20260929",
+      "path": "notes/safe-transport-selection-readout-20260930.md",
+      "integration": "open_pr",
+      "url": "https://github.com/LightChainr/Matching-One/blob/e56c0b8361628a066abea4ba4d3655c74e3a62bf/notes/safe-transport-selection-readout-20260930.md"
+    },
+    {
+      "id": "P838_window_source_readout",
+      "pr": 838,
+      "commit": "bbfe8c89a4c2e63f4b29d7961721a2d2f5311ade",
+      "branch": "research/frontier-and-birth-gap-20260929",
+      "path": "notes/geometric-source-window-readout-20260930.md",
+      "integration": "open_pr",
+      "url": "https://github.com/LightChainr/Matching-One/blob/bbfe8c89a4c2e63f4b29d7961721a2d2f5311ade/notes/geometric-source-window-readout-20260930.md"
+    },
+    {
+      "id": "P838_one_sided_normal_source",
+      "pr": 838,
+      "commit": "467b18255f0dbf04916db1cd6682a01f70c09f9e",
+      "branch": "research/frontier-and-birth-gap-20260929",
+      "path": "notes/one-sided-birth-source-normal-response-20260930.md",
+      "integration": "open_pr",
+      "url": "https://github.com/LightChainr/Matching-One/blob/467b18255f0dbf04916db1cd6682a01f70c09f9e/notes/one-sided-birth-source-normal-response-20260930.md"
+    },
+    {
+      "id": "P838_independent_pair_replication",
+      "pr": 838,
+      "commit": "d9824b5bf478fc8acad238be2b9f31ce99787966",
+      "branch": "research/frontier-and-birth-gap-20260929",
+      "path": "analysis/exact-pair-independent-archive-20260930/results/RESULT.md",
+      "integration": "open_pr",
+      "url": "https://github.com/LightChainr/Matching-One/blob/d9824b5bf478fc8acad238be2b9f31ce99787966/analysis/exact-pair-independent-archive-20260930/results/RESULT.md"
+    },
+    {
+      "id": "P838_window_source_identity",
+      "pr": 838,
+      "commit": "18ca9e132f08b89071291d74cbd10133d99b2e5d",
+      "branch": "research/frontier-and-birth-gap-20260929",
+      "path": "notes/geometric-source-window-identity-20260930.md",
+      "integration": "open_pr",
+      "url": "https://github.com/LightChainr/Matching-One/blob/18ca9e132f08b89071291d74cbd10133d99b2e5d/notes/geometric-source-window-identity-20260930.md"
+    },
+    {
+      "id": "P838_geometric_source",
+      "pr": 838,
+      "commit": "4e4b6a03fd5e31ee01384b18ecede12a071d3608",
+      "branch": "research/frontier-and-birth-gap-20260929",
+      "path": "notes/birth-selection-intervention-20260930.md",
+      "integration": "open_pr",
+      "url": "https://github.com/LightChainr/Matching-One/blob/4e4b6a03fd5e31ee01384b18ecede12a071d3608/notes/birth-selection-intervention-20260930.md"
+    },
+    {
+      "id": "P838_exact_pair_readout",
+      "pr": 838,
+      "commit": "7f71af8599ce6e7c0f426cb6f7a069dbc0cf6100",
+      "branch": "research/frontier-and-birth-gap-20260929",
+      "path": "notes/exact-completion-pairs-readout-20260930.md",
+      "integration": "open_pr",
+      "url": "https://github.com/LightChainr/Matching-One/blob/7f71af8599ce6e7c0f426cb6f7a069dbc0cf6100/notes/exact-completion-pairs-readout-20260930.md"
+    },
+    {
+      "id": "P838_exact_pairs",
+      "pr": 838,
+      "commit": "8b5eee0d47b6c220c2ba46a5bae08d2d19b392a4",
+      "branch": "research/frontier-and-birth-gap-20260929",
+      "path": "notes/projected-component-pair-criterion-20260929.md",
+      "integration": "open_pr",
+      "url": "https://github.com/LightChainr/Matching-One/blob/8b5eee0d47b6c220c2ba46a5bae08d2d19b392a4/notes/projected-component-pair-criterion-20260929.md"
+    },
+    {
+      "id": "P838_label_curvature",
+      "pr": 838,
+      "commit": "ed9f3e281909a9f292818c3492d6219a4471282a",
+      "branch": "research/frontier-and-birth-gap-20260929",
+      "path": "notes/completion-pair-label-curvature-20260930.md",
+      "integration": "open_pr",
+      "url": "https://github.com/LightChainr/Matching-One/blob/ed9f3e281909a9f292818c3492d6219a4471282a/notes/completion-pair-label-curvature-20260930.md"
+    },
+    {
+      "id": "P739",
+      "pr": 739,
+      "commit": "907a9d94ee0c3f20b00188647872a067abffd275",
+      "branch": "analysis/geometric-balance-manuscript-20260913",
+      "path": "docs/manuscripts/geometric-balance/manuscript.md",
+      "integration": "open_pr",
+      "url": "https://github.com/LightChainr/Matching-One/blob/907a9d94ee0c3f20b00188647872a067abffd275/docs/manuscripts/geometric-balance/manuscript.md"
+    },
+    {
+      "id": "P771",
+      "pr": 771,
+      "commit": "29f92a93658ad163aeb0fad227e8bd6d3c2ea5e3",
+      "branch": "analysis/geometric-balance-round2-integrated-20260914",
+      "path": "docs/manuscripts/geometric-balance/round2-claim-ledger-20260914.md",
+      "integration": "open_pr",
+      "url": "https://github.com/LightChainr/Matching-One/blob/29f92a93658ad163aeb0fad227e8bd6d3c2ea5e3/docs/manuscripts/geometric-balance/round2-claim-ledger-20260914.md"
+    },
+    {
+      "id": "P773",
+      "pr": 773,
+      "commit": "9ef22d1cc531c09dedf9b1195ab6aad8ef79668f",
+      "branch": "analysis/topological-scaling-program-20260914",
+      "path": "notes/rank-birth-flux-conservation-20260914.md",
+      "integration": "open_pr",
+      "url": "https://github.com/LightChainr/Matching-One/blob/9ef22d1cc531c09dedf9b1195ab6aad8ef79668f/notes/rank-birth-flux-conservation-20260914.md"
+    },
+    {
+      "id": "P822",
+      "pr": 822,
+      "commit": "b23bd0c44a7ae6d6969ee95d3402d7aa1a25a962",
+      "branch": "analysis/delta1-mechanism-20260919",
+      "path": "analysis/queue-20260919/814/NOTE.md",
+      "integration": "open_pr",
+      "url": "https://github.com/LightChainr/Matching-One/blob/b23bd0c44a7ae6d6969ee95d3402d7aa1a25a962/analysis/queue-20260919/814/NOTE.md"
+    },
+    {
+      "id": "P823",
+      "pr": 823,
+      "commit": "3d0176a9f394f7d7228766709731493fd15095c5",
+      "branch": "proof/white-matching-separator-20260919",
+      "path": "analysis/queue-20260919/815/supplement/NOTE.md",
+      "integration": "open_pr",
+      "url": "https://github.com/LightChainr/Matching-One/blob/3d0176a9f394f7d7228766709731493fd15095c5/analysis/queue-20260919/815/supplement/NOTE.md"
+    },
+    {
+      "id": "P825",
+      "pr": 825,
+      "commit": "3bf94a6900fe2866626eaf6c3e7f2a00180bb775",
+      "branch": "analysis/rank-L7-gate-20260919",
+      "path": "analysis/queue-20260919/816/NOTE.md",
+      "integration": "open_pr",
+      "url": "https://github.com/LightChainr/Matching-One/blob/3bf94a6900fe2866626eaf6c3e7f2a00180bb775/analysis/queue-20260919/816/NOTE.md"
+    },
+    {
+      "id": "P828",
+      "pr": 828,
+      "commit": "9e59eef0007382c07f879fb9eadeb5e5be141f5b",
+      "branch": "research/control-rev2-20260919",
+      "path": "docs/research-status/control-20260919-rev2.md",
+      "integration": "open_pr",
+      "url": "https://github.com/LightChainr/Matching-One/blob/9e59eef0007382c07f879fb9eadeb5e5be141f5b/docs/research-status/control-20260919-rev2.md"
+    },
+    {
+      "id": "P829",
+      "pr": 829,
+      "commit": "2fd4186de14d8566ef828c11f937a5bab06b41da",
+      "branch": "analysis/p809-endpoint-reversal-20260923",
+      "path": "analysis/p809-endpoint-reversal-20260923/NOTE.md",
+      "integration": "open_pr",
+      "url": "https://github.com/LightChainr/Matching-One/blob/2fd4186de14d8566ef828c11f937a5bab06b41da/analysis/p809-endpoint-reversal-20260923/NOTE.md"
+    },
+    {
+      "id": "P830",
+      "pr": 830,
+      "commit": "48ec90124865525d5823ccd7b431b4adcc9be4c7",
+      "branch": "analysis/p809-passive-path-20260923",
+      "path": "analysis/p809-passive-path-20260923/NOTE.md",
+      "integration": "open_pr",
+      "url": "https://github.com/LightChainr/Matching-One/blob/48ec90124865525d5823ccd7b431b4adcc9be4c7/analysis/p809-passive-path-20260923/NOTE.md"
+    },
+    {
+      "id": "P831",
+      "pr": 831,
+      "commit": "e099524bdc7e9c8844c56d88e5d7b411225f85c9",
+      "branch": "analysis/p809-parametric-observability-20260923",
+      "path": "analysis/p809-parametric-observability-20260923/NOTE.md",
+      "integration": "open_pr",
+      "url": "https://github.com/LightChainr/Matching-One/blob/e099524bdc7e9c8844c56d88e5d7b411225f85c9/analysis/p809-parametric-observability-20260923/NOTE.md"
+    },
+    {
+      "id": "P832",
+      "pr": 832,
+      "commit": "fc17b825447adee725a04f081eed1178f554ee46",
+      "branch": "analysis/p809-positive-order-gap-20260923",
+      "path": "analysis/p809-positive-order-gap-20260923/NOTE.md",
+      "integration": "open_pr",
+      "url": "https://github.com/LightChainr/Matching-One/blob/fc17b825447adee725a04f081eed1178f554ee46/analysis/p809-positive-order-gap-20260923/NOTE.md"
+    },
+    {
+      "id": "P834",
+      "pr": 834,
+      "commit": "8f40247491af737c03aa3b1156db5918a7024093",
+      "branch": "analysis/p833-width5-positive-cylinder-20260923",
+      "path": "analysis/p833-width5-positive-cylinder-20260923/NOTE.md",
+      "integration": "open_pr",
+      "url": "https://github.com/LightChainr/Matching-One/blob/8f40247491af737c03aa3b1156db5918a7024093/analysis/p833-width5-positive-cylinder-20260923/NOTE.md"
+    },
+    {
+      "id": "P835",
+      "pr": 835,
+      "commit": "5d063e3a0053035a648db99969c4c4972e7d8584",
+      "branch": "analysis/p833-orphan-source-contract-20260924",
+      "path": "analysis/p833-orphan-source-contract-20260924/NOTE.md",
+      "integration": "open_pr",
+      "url": "https://github.com/LightChainr/Matching-One/blob/5d063e3a0053035a648db99969c4c4972e7d8584/analysis/p833-orphan-source-contract-20260924/NOTE.md"
+    },
+    {
+      "id": "P836",
+      "pr": 836,
+      "commit": "19b6771437276ce0522d21dd1a5cbf1b7c7c5c1c",
+      "branch": "analysis/p833-noise-depth-literature-20260924",
+      "path": "analysis/p833-noise-depth-literature-20260924/NOTE.md",
+      "integration": "open_pr",
+      "url": "https://github.com/LightChainr/Matching-One/blob/19b6771437276ce0522d21dd1a5cbf1b7c7c5c1c/analysis/p833-noise-depth-literature-20260924/NOTE.md"
+    },
+    {
+      "id": "P837",
+      "pr": 837,
+      "commit": "0ed7e4f7a6e68ba628ce26dd1e9cf1e65d64c9f3",
+      "branch": "analysis/p833-fixed-noise-wedge-20260924",
+      "path": "analysis/p833-fixed-noise-wedge-20260924/INTERFACE-REFINEMENT.md",
+      "integration": "open_pr",
+      "url": "https://github.com/LightChainr/Matching-One/blob/0ed7e4f7a6e68ba628ce26dd1e9cf1e65d64c9f3/analysis/p833-fixed-noise-wedge-20260924/INTERFACE-REFINEMENT.md"
+    }
+  ],
+  "observer_sectors": [
+    {
+      "id": "rank_filtration",
+      "state": "ambient image rank 0/1/2",
+      "source": "iid Bernoulli monotone coupling / uniform site permutation",
+      "geometry": "fixed honest torus",
+      "readout": [
+        "J1",
+        "J2",
+        "T1",
+        "T2",
+        "two-time rank-one overlap"
+      ],
+      "not_equivalent_to": [
+        "temporary prefix closure",
+        "original U"
+      ]
+    },
+    {
+      "id": "marked_rank_filtration",
+      "state": "rank0, persistent primitive unoriented rank1 direction, absorbing rank2",
+      "source": "same uniform permutation or iid Uniform site labels; clocks kept distinct",
+      "geometry": "fixed honest torus; exact L3 geometry and independent square-L512 conditional readout",
+      "readout": [
+        "direction-marked H_d(s,t)",
+        "completion-pivotal count nu2",
+        "rank-one completion phase"
+      ],
+      "not_equivalent_to": [
+        "unmarked rank filtration",
+        "future direction revealed before birth",
+        "original U",
+        "label process with occupied count also observed"
+      ]
+    },
+    {
+      "id": "prefix_prediction",
+      "state": "lifted connectivity with temporary closure",
+      "source": "explicit uniform or addressed site source language",
+      "geometry": "growing periodic strip",
+      "readout": [
+        "endpoint rank",
+        "rank trajectory"
+      ],
+      "not_equivalent_to": [
+        "all-source deterministic state",
+        "fixed-torus insertion filtration"
+      ]
+    },
+    {
+      "id": "tensor_noise",
+      "state": "artificial three-row preparations",
+      "source": "epsilon-bounded independent future sites",
+      "geometry": "cylinder with temporary closure",
+      "readout": [
+        "rank-only bit query"
+      ],
+      "not_equivalent_to": [
+        "complete boundary observation",
+        "typical critical Bernoulli preparation",
+        "full recursive realization order"
+      ]
+    },
+    {
+      "id": "original_u",
+      "source": "original named source B and six-coordinate thermal jet",
+      "geometry": "original norm-4/norm-5 contracts",
+      "readout": "physical normalizer, rank-one denominator and pooled moving-root counterterm",
+      "not_equivalent_to": [
+        "root H4",
+        "aggregate balanced rank",
+        "primitive C3 channel"
+      ]
+    }
+  ],
+  "scientific_nodes": [
+    {
+      "id": "joint_birth_process",
+      "attention": "P0",
+      "issue": 778,
+      "integration": "open_pr",
+      "sources": [
+        "P773",
+        "P823",
+        "P828",
+        "P838_exact_pairs",
+        "P838_label_curvature",
+        "P838_exact_pair_readout",
+        "P838_geometric_source",
+        "P838_independent_pair_replication",
+        "P838_window_source_identity",
+        "P838_window_source_readout",
+        "P838_one_sided_normal_source",
+        "P838_safe_transport_selection"
+      ],
+      "known": "Author noncoalescence/clock proofs and exact L4 weak-closure failures. Exact same-prefix conditional averaging now resolves the L512 completion-creation target at -0.00344939 SE0.00077225, supporting finite enlarged-observer history dependence. This is post-readout refinement, not independent confirmation; direction-only secondary still does not reproduce the earlier selected signal.",
+      "boundary": "No copula uniqueness, exact square exponent or cross-lattice universality theorem. L3 completion-phase closure is count-clock, from the uniform empty start, not a continuum or hidden-count label realization. Population width comparability is not finite-sample calibration.",
+      "new_branch_paths": [
+        "notes/birth-gap-noncoalescence-20260929.md",
+        "notes/birth-gap-pilot-20260929.md",
+        "notes/birth-gap-cloud-20260929.md",
+        "notes/two-time-pivotal-density-bound-20260929.md",
+        "notes/square-birth-transfer-and-iqr-clock-20260929.md",
+        "notes/birth-kernel-markov-characterization-20260929.md",
+        "notes/birth-history-memory-readout-20260929.md",
+        "analysis/birth-markov-kernel-20260929/result.json",
+        "analysis/birth-markov-kernel-20260929/exact-clock.json",
+        "notes/completion-pivotal-hazard-kernel-20260929.md",
+        "notes/birth-completion-geometry-20260929.md",
+        "analysis/birth-completion-geometry-20260929/result.json",
+        "analysis/birth-completion-geometry-20260929/marked-clock.json",
+        "notes/completion-hazard-independent-block-20260929.md",
+        "analysis/completion-hazard-production-20260929/contract.json",
+        "analysis/completion-hazard-production-20260929/data/run.json",
+        "analysis/completion-hazard-production-20260929/results/result.json",
+        "notes/directional-markov-completion-20260929.md",
+        "notes/directional-hazard-contrast-20260929.md",
+        "analysis/directional-hazard-contrast-20260929/result.json",
+        "notes/completion-pair-synergy-20260929.md",
+        "notes/completion-pair-synergy-L4-20260929.md",
+        "analysis/completion-pair-synergy-20260929/result.json",
+        "notes/safe-insertion-independent-block-20260929.md",
+        "analysis/safe-insertion-production-20260929/contract.json",
+        "analysis/safe-insertion-production-20260929/data/run.json",
+        "analysis/safe-insertion-production-20260929/results/result.json",
+        "notes/projected-component-pair-criterion-20260929.md",
+        "notes/completion-pair-label-curvature-20260930.md",
+        "notes/exact-completion-pairs-readout-20260930.md",
+        "analysis/exact-completion-pairs-20260929/data/run.json",
+        "analysis/exact-completion-pairs-20260929/results/result.json"
+      ],
+      "observer": "rank_filtration",
+      "acquisition": [
+        "exact construction",
+        "128000-filtration local pilot",
+        "560000-filtration independent CPU block through L512",
+        "two-time arm-density proof and primary-literature probability inputs",
+        "square matching-arm transfer and population-clock proof; no new samples",
+        "existing cloud-block three-time analysis and exact L3 archive polynomial transform; no enumeration rerun",
+        "new L3 512-subset geometry per lattice, prefix DP, exact phase realization and marked-label polynomial transform; no Monte Carlo or 9! rerun",
+        "new independent 140000-filtration square-L512 block on TV2N0X, precommitted source/contract 9520c9bdf702a62435f1054e64184b1df4fc0ab3; completion geometry at b and independent uniform audit time tau",
+        "one retrospective necessary-condition contrast on the same saved nu_b, cohorts, cutoffs and exact-D weights; no new acquisition, feature or threshold scan",
+        "one exact 65536-subset L4 census per lattice and integer prefix DP, 1.23 seconds total; no Monte Carlo, permutation enumeration or L512 reanalysis",
+        "new independent 140000-prefix square-L512 safe-insertion block, four probes per eligible prefix; acquisition commit55fa2656 before samples, scoring commitf340aa55 before readout; 519.843 seconds on TV2N0X",
+        "same140000 prefixes replayed with exact pair enumeration; 8 local completed batches retained,6 missing batches on TV2N0X in420.381s. No new independent prefixes; all140k original first-five-field tuples matched."
+      ],
+      "tension": "Same static laws and all adjacent transition tables can coexist with different limit diagonal atoms.",
+      "finite_result": "Soft gap mass approximately proportional to resolution down to delta=.00625, L<=512; no resolved plateau, not proof of noncoalescence.",
+      "theory_result": "Different-site birth pairs have locally uniformly bounded time density; direct diagonal mass is bounded by alpha6/alpha4. Tightness excludes limit coalescence for triangular and square NN/matching models on their own pivotal clocks. Population insertion-IQR is comparable to N*a_eta. Author proofs, not independent certification.",
+      "memory_result": "L512 early-minus-late rank-one survival: square +0.0142234 SE0.0051069, triangular +0.0070228 SE0.0051459, one (.25,.50,.75) mixture-quantile triple. Exact L3 label kernels are separated for square and nonseparated for triangular. The evidential levels remain distinct.",
+      "probability_input_sources": [
+        "https://arxiv.org/abs/0711.4948",
+        "https://arxiv.org/abs/1305.5526",
+        "https://doi.org/10.1007/BF01205674",
+        "https://arxiv.org/abs/2008.01606"
+      ],
+      "completion_geometry_result": "Triangular L3 early/late fast-type fractions 1/4 versus 3/7 explain survival difference 5/112; nu2 phase rows close strongly within rank one and yield a four-state weak count-clock realization. Triangular H_d=H/3. Square H_axis=3p^3(1-q)^3[(1+q)^3-p^3], H_diag=3p^6(1-q)^3: pooled label Markovness does not survive axial marking.",
+      "independent_geometry_result": "Raw early-late survival +0.0068571 SE0.0044443; exact-D overlap -0.0041500 SE0.0045119; exact-(D,nu2) overlap +0.0012161 SE0.0047636. Random-time integrated prediction +0.0155022 SE0.0035050; paired closure residual -0.0086451 SE0.0063892. No old/new pooling; different conditional weights are not an explained fraction.",
+      "directional_hazard_result": "Old post-hoc within-D mean nu_b=+1.629168 SE0.390976, not reproduced by the new independent predeclared contrast -0.039865 SE0.569878. Preserve both; no pooling or accepted direction-only exclusion. Within-(D,nu) immediate equality is tautological.",
+      "pair_synergy_result": "At matched (k,D,nu2), exact uniform-start early-minus-late two-step survival is -4/4131 square L4 and 2/121 triangular L4, while immediate survival equals 1. Safe-pair edge count e gives survival2=[(m-nu)(m-nu-1)-2e]/[m(m-1)] and safe-step mean completion increment 2e/(m-nu).",
+      "safe_insertion_result": "Primary same-(D,nu) increment deltaY=0.00668183 SE0.01208706; derived deltaZ2=-6.25403e-8 SE1.13192e-7. 55028 risk prefixes, 220112 probes, only1233 nonzero; 98.973% common-support coverage. A null weighted mean does not prove full closure.",
+      "label_curvature_identity": "Fixed-entry H_tt=E[I(c(c-1)-2e)]/(1-t)^2. Risk-set mean count derivative=(2 mean e-Var(c))/(1-t); hazard derivative=(mean c+2 mean e-Var(c))/(1-t)^2. Exact finite identities, not a count-to-label substitution or continuum limit.",
+      "exact_pair_refinement": "Same target deltaY=-0.003449385668 SE0.000772252763, about4.47 batchSE; SE15.65x smaller. Derived deltaZ2=3.228181004e-8 SE7.230661888e-9. Original direction-only secondary unchanged. Same data, not extra confirmation or a continuum conclusion.",
+      "geometric_source_result": "Safe-site exponential degree tilt preserves the whole occupation clock and current rank exit hazard, but has delayed survival response -s Var_safe(degree)/(m(m-1)). All-lag single-kick responses computed on three specified square-L4 preparations. Birth contrast equals a backward-retention/forward-future covariance; safe-path weighting separates geometry transport and survival selection.",
+      "independent_pair_replication": "Other independent140k archive reproduces fixed negative target: deltaY=-0.0031993684 SE0.0006200906. Archive-minus-first=0.0002500173 SE0.0009903973; no pooling. All140k prefixes match. Other archive readouts had been seen; first e readout, not new prospective production. Support worker completed520.876s onTV2N0X and verifiedReady.",
+      "window_source_result": "Completed735-step derivative=-0.01104203191 SE0.00015766645; baseline survival0.4447553973 SE0.0021409587. Same55028risk/140000prefixes,14batches,3x3covariance. TgFr7R14workers519.145s, retrieved and Ready verified. New mechanism readout, not independent block or finite-theta effect.",
+      "one_sided_source_result": "First birth unchanged, deltaM=deltaEtop=g. Moving-root normal factor=-2P0'/M'>0. Explicit symmetric squareL4 source has g(p)>0 at every interior p, rank2 physical response Jacobian and normalEtop derivative0.00016853296 at balance. Constructed dynamic source, not original norm4/U mapping.",
+      "safe_transport_selection_result": "Completed: same1198cells give natural=-0.002205065286 SE0.000787091700, safe-reference=-0.002537058794 SE0.000636492674, selection=0.000331993508 SE0.000287375294. Safe-reference memory persists; selection-only disfavoured for this estimand, selection term itself unresolved.97666new conditional futures of old entrances, not independent preparations.",
+      "source_identification_bounds": "At fixed(k,D,c), Y and survival2 are affine. At one p, two first-birth-preserving source columns in(M,Etop) are g_r*(1,1); moving-root columns remain proportional. Thermal p plus ONE geometric-source rank2 is not identification of TWO geometric sources.",
+      "next": "Distinguish realizable entrance-law and safe-growth sources through nonredundant same-contract response columns, including endpoint normalization. Do not swap early/late labels at fixed birth count, reuse proportional readouts or paste the old735-step column into a changed source contract. Replication,735-step transmission and safe/selection decomposition are completed; no automatic precision or size ladder.",
+      "information_gain": "high: identifies process structure beyond the completed noncoalescence argument",
+      "cost": "small exact algebra; bounded pilot; substantial two-time probability theory"
+    },
+    {
+      "id": "source_noise_state",
+      "attention": "P1",
+      "integration": "open_pr",
+      "sources": [
+        "P829",
+        "P830",
+        "P831",
+        "P832",
+        "P834",
+        "P835",
+        "P836",
+        "P837"
+      ],
+      "observer": "tensor_noise",
+      "known": "All-k deterministic coding obstruction and physical product queries; distributed stochastic parity code has tensor rank three, product-family positive initial memory three, and exact single-readout distinction bounds.",
+      "boundary": "Original five-column artificial preparation. Tensor rank three is not a full physical-language upper bound. Sharp monotone TV concerns one Boolean nonadaptive output, not transcripts or feedback; abstract threshold attainment is not physical realization.",
+      "tension": "Stochastic sharing evades the deterministic rank-four obstruction, but product TV decays exponentially and general monotone single-output TV decays as k^-1/2; the product family loses the old positive/linear memory gap too.",
+      "discussion": "https://github.com/LightChainr/Matching-One/pull/837#issuecomment-5814562555",
+      "new_branch_paths": [
+        "notes/deterministic-block-encoding-obstruction-20260929.md",
+        "analysis/deterministic-block-encoding-20260929/check.py",
+        "analysis/deterministic-block-encoding-20260929/result.json",
+        "notes/stochastic-parity-block-encoding-20260929.md",
+        "analysis/stochastic-parity-block-encoding-20260929/check.py",
+        "analysis/stochastic-parity-block-encoding-20260929/result.json"
+      ],
+      "next": "Error-versus-memory under a concrete realizable query family; if using transcripts or feedback state that changed contract. Stochastic rank-three existence alone is now answered; report A/L/error/width/height and positive memory.",
+      "acquisition": [
+        "reuse exact finite engines",
+        "all-k algebraic proof and all-width physical construction",
+        "one exact census of 65536 two-block encoders",
+        "stochastic parity and sharp monotone proofs; one k=2,3,4 exact control, 468 tensor moments and 194 monotone functions"
+      ],
+      "information_gain": "high: tests whether reliability preserves the representation advantage",
+      "cost": "bounded one-block algebra/topology first"
+    },
+    {
+      "id": "geometric_threshold_laws",
+      "attention": "P1 bounded writing",
+      "integration": "open_pr",
+      "sources": [
+        "P739",
+        "P771"
+      ],
+      "observer": "rank_filtration",
+      "known": "Author arbitrary-period root and full-law iff proofs; fixed-p homological free energy and birth clocks.",
+      "boundary": "Author proof supplied is not independent all-proof certification; fixed-p does not grant nearcritical uniformity.",
+      "next": "Integrate supplied statements and actual remaining probability inputs, not repeat oblique necessity as missing.",
+      "acquisition": [
+        "existing proof synthesis"
+      ],
+      "information_gain": "consolidates a mature result; lower need for new production",
+      "cost": "bounded writing/theory"
+    },
+    {
+      "id": "original_u_identification",
+      "attention": "P1 theory",
+      "integration": "hypothesis",
+      "sources": [
+        "P771",
+        "P773",
+        "P822",
+        "P828"
+      ],
+      "issue": 275,
+      "observer": "original_u",
+      "known": "Historical production exists; root thermal tangent may be removed by moving-root normalization.",
+      "boundary": "Candidate operator identity and original-U normal forward maps remain unresolved.",
+      "tension": "Strong root H4 need not survive original-U quotient; equivalent black/white pair sources are not independent.",
+      "next": "Two represented same-source normal columns or actual projected Ward/mixed-response coefficient.",
+      "acquisition": [
+        "existing data",
+        "small exact/source calculation",
+        "theory"
+      ],
+      "information_gain": "high if columns differ, zero precision benefit if images agree",
+      "cost": "theory/small matrix rather than broad sampling"
+    }
+  ],
+  "decision_experiments": [
+    {
+      "id": "birth_soft_diagonal",
+      "node": "joint_birth_process",
+      "status": "implemented_in_this_branch",
+      "statistic": "Z_L(delta)=E[(1-G_L/delta)_+]",
+      "source_paths": [
+        "notes/birth-gap-noncoalescence-20260929.md",
+        "analysis/birth-gap-20260929/analyze.py"
+      ],
+      "mechanism_contrast": "vanishing uniform small-ball mass versus aggregation; direct atom alone does not decide",
+      "interpretation": "local pilot and independent finer cloud block are exploratory, not proof of double limit"
+    },
+    {
+      "id": "two_time_birth_density",
+      "node": "joint_birth_process",
+      "status": "author_proof_with_explicit_probability_inputs",
+      "source_paths": [
+        "notes/two-time-pivotal-density-bound-20260929.md",
+        "notes/square-birth-transfer-and-iqr-clock-20260929.md"
+      ],
+      "bound": "P(gap<=delta,both births in compact window)<=C_window*delta+C_window*alpha6/alpha4",
+      "interpretation": "Triangular and square NN/matching corollaries include tightness. Does not infer a global delta bound by dropping the window tail, or a unique copula."
+    },
+    {
+      "id": "population_iqr_clock",
+      "node": "joint_birth_process",
+      "status": "author_proof_using_standard_arm_bounds",
+      "source_paths": [
+        "notes/square-birth-transfer-and-iqr-clock-20260929.md"
+      ],
+      "bound": "W_population asymp (N+1)*a_eta; rescaled insertion and label times differ by o_P(1)",
+      "interpretation": "Population normalization cannot manufacture or remove a limiting zero-gap atom; no ratio limit, finite-sample guarantee or cross-modulus universality claim."
+    },
+    {
+      "id": "coded_response_rank",
+      "node": "source_noise_state",
+      "status": "all_k_classification_and_physical_product_query_proof",
+      "statistic": "(A,L,eta,w,h) under one fixed preparation/source/readout contract",
+      "mechanism_contrast": "Only a single informative block preserves rank three for deterministic four-state encoders; distributed encoding yields rank four",
+      "source_paths": [
+        "notes/deterministic-block-encoding-obstruction-20260929.md",
+        "analysis/deterministic-block-encoding-20260929/result.json"
+      ],
+      "finite_control": "One exact k=2 census, 65536 encoders; 43680 injective: 192 rank3, 43488 rank4. Physical query construction is proved, not numerically re-enumerated."
+    },
+    {
+      "id": "stochastic_parity_readout",
+      "node": "source_noise_state",
+      "status": "all_k_author_proofs_and_one_small_k_exact_control",
+      "source_paths": [
+        "notes/stochastic-parity-block-encoding-20260929.md",
+        "analysis/stochastic-parity-block-encoding-20260929/result.json"
+      ],
+      "mechanism_contrast": "Private randomization can preserve rank three with distributed shares, but exact rank does not preserve one-shot reliability or a positive/linear gap.",
+      "rank_and_memory": "Explicit product-query family: tensor rank=minimum positive initial memory=3; larger physical language only bounded between 3 and 4.",
+      "bound": "TV_product=2^(1-k); TV_single_monotone<=binom(k-1,floor((k-1)/2))/2^(k-1)",
+      "finite_control": "One k=2,3,4 run: 468 exact tensor moments and 194 monotone Boolean functions; no old-census rerun or cloud job.",
+      "interpretation": "Single nonadaptive final rank only. Full transcript, feedback and physical threshold implementation are separate questions; observed-share BSC is not a derived instruction-noise channel."
+    },
+    {
+      "id": "rank_one_history_memory",
+      "node": "joint_birth_process",
+      "status": "author_characterization_exact_finite_controls_and_retrospective_archive_readout",
+      "source_paths": [
+        "notes/birth-kernel-markov-characterization-20260929.md",
+        "notes/birth-history-memory-readout-20260929.md",
+        "analysis/birth-markov-kernel-20260929/analyze.py",
+        "analysis/birth-markov-kernel-20260929/result.json",
+        "analysis/birth-markov-kernel-20260929/exact_clock.py",
+        "analysis/birth-markov-kernel-20260929/exact-clock.json"
+      ],
+      "null": "For current-rank-only time-inhomogeneous Markov models, H(a,c)H(b,b)=H(a,b)H(b,c) for every triple; all triples also suffice in natural rank history.",
+      "statistic": "Early-minus-late P(J2>c) within J1<=b<J2; one mixture-quantile triple, full aligned-batch covariance and cutoff re-estimation.",
+      "exact_result": "Stored L3 complete tables: square all count triples satisfy the identity and label H=6p^3(1-q^2)^3; triangular has two count violations and a nonzero label polynomial determinant.",
+      "interpretation": "Exact finite Markov and non-Markov controls do not determine the limit. Large-size association is small and retrospective, not independent certification. Label/count clocks and current rank/full microscopic state are different contracts."
+    },
+    {
+      "id": "completion_geometry_transmission",
+      "node": "joint_birth_process",
+      "observer": "marked_rank_filtration",
+      "status": "author_hazard_identity_and_exact_L3_mechanism",
+      "source_paths": [
+        "notes/completion-pivotal-hazard-kernel-20260929.md",
+        "notes/birth-completion-geometry-20260929.md",
+        "analysis/birth-completion-geometry-20260929/analyze.py",
+        "analysis/birth-completion-geometry-20260929/result.json",
+        "analysis/birth-completion-geometry-20260929/exact_marked_clock.py",
+        "analysis/birth-completion-geometry-20260929/marked-clock.json"
+      ],
+      "mechanism_contrast": "Direction alone versus completion-geometry mixing as the carrier of birth history; unmarked versus direction-marked weak Markovness under a fixed clock.",
+      "identity": "Exit probability nu2/(N-k); iid-label intensity nu2/(1-t); logarithmic H defect is minus the integral of birth-by-a minus birth-by-b cohort mean nu2 over 1-t.",
+      "finite_result": "Triangular direction does not remove memory; completion-type weights explain 5/112 and a binary rank-one phase closes count dynamics. Square marked axial label kernels are non-Markov despite pooled Markovness; all marked square count kernels remain Markov.",
+      "next_independent_readout": "Completed as completion_hazard_independent_L512 below; not still a missing acquisition.",
+      "interpretation": "512 subsets per lattice and exact symbolic thinning, not production validation. Four-state count realization is not minimality, arbitrary rank0 strong lumpability, label closure with hidden count or continuum sufficiency."
+    },
+    {
+      "id": "completion_hazard_independent_L512",
+      "node": "joint_birth_process",
+      "observer": "marked_rank_filtration",
+      "status": "completed_fixed_new_block",
+      "acquisition_commit": "9520c9bdf702a62435f1054e64184b1df4fc0ab3",
+      "source_paths": [
+        "notes/completion-hazard-independent-block-20260929.md",
+        "analysis/completion-hazard-production-20260929/contract.json",
+        "analysis/completion-hazard-production-20260929/ESTIMANDS.md",
+        "analysis/completion-hazard-production-20260929/data/run.json",
+        "analysis/completion-hazard-production-20260929/results/result.json"
+      ],
+      "design": "square L512, 14 x 10000, fixed insertion cutoffs 154646/155385/156120; exact D and nu2 at b, independent uniform integer tau in [b,c-1]; 32 benchmark samples excluded",
+      "result": "Pooled survival contrast smaller than old archive; exact-D and exact-(D,nu2) overlap-weighted finite-lag contrasts unresolved. Completion-count and integrated-hazard contrasts measurable; actual/predicted difference about 1.35 batch SE.",
+      "null_predictions": "Rank-only, rank+D and rank+D+nu2 sufficiency imply the corresponding conditional survival contrast is zero. One zero weighted contrast does not imply cellwise or recursive closure.",
+      "interpretation": "Prospective new-block readout, not continuum certification. Direction composition is a candidate, not proven mediation. Raw/conditional/hazard outputs share data and full covariance. Fixed run ended without result-dependent extension.",
+      "next": "The directional next-step necessity has since been checked retrospectively below; no redispatch of the completed acquisition or silent upgrade of its conditional zero to closure."
+    },
+    {
+      "id": "directional_next_step_necessity",
+      "node": "joint_birth_process",
+      "observer": "marked_rank_filtration",
+      "status": "retrospective_signal_not_reproduced_in_later_independent_block",
+      "source_paths": [
+        "notes/directional-markov-completion-20260929.md",
+        "notes/directional-hazard-contrast-20260929.md",
+        "analysis/directional-hazard-contrast-20260929/analyze.py",
+        "analysis/directional-hazard-contrast-20260929/result.json"
+      ],
+      "input_source_commit": "87f6644d3b57c7dbd67c44c63bcceffb6b7693ac",
+      "input_source_paths": [
+        "analysis/completion-hazard-production-20260929/data/run.json",
+        "analysis/completion-hazard-production-20260929/analyze.py"
+      ],
+      "integration": "open_pr",
+      "null_prediction": "At every count, within every persistent direction, mean nu2 must be independent of entry cutoff; all such conditions are necessary and sufficient for marked-rank Markovness under the uniform permutation source.",
+      "result": "Same-D overlap-weighted mean nu_b difference +1.629168 SE0.390976 sites; the unchanged finite-lag survival contrast remains -0.00414999 SE0.00451194.",
+      "interpretation": "The original same-block signal is preserved, but its independent predeclared replication is -0.039865 SE0.569878 and does not reproduce it. Direction-only L512 closure is not empirically excluded or proved. Long-window zero remains weaker than an all-count criterion.",
+      "next": "Independent replication and successor sampling are now delivered below; current large-size conclusions remain unresolved."
+    },
+    {
+      "id": "completion_pair_synergy",
+      "node": "joint_birth_process",
+      "observer": "rank_direction_completion_count_history",
+      "status": "author_identity_and_exact_L4_weak_markov_counterexamples",
+      "integration": "open_pr",
+      "source_paths": [
+        "notes/completion-pair-synergy-20260929.md",
+        "notes/completion-pair-synergy-L4-20260929.md",
+        "analysis/completion-pair-synergy-20260929/analyze.py",
+        "analysis/completion-pair-synergy-20260929/result.json"
+      ],
+      "reused_topology_commit": "8b37c20e5f2b1946cefbed98b09280a0cad25ed2",
+      "reused_topology_path": "analysis/birth-completion-geometry-20260929/analyze.py",
+      "identity": "A safe v creates degree(v) completion sites in the safe-pair graph. At fixed k,D,nu, two-step survival contrast is -2 Delta(mean e)/[m(m-1)]. Full degree distribution gives the one-step successor law, not automatic recursive closure.",
+      "exact_result": "Square k6,a4,D(0,1),nu0: survival2 difference -4/4131. Triangular k5,a4,D(0,1),nu0: 2/121. Both cohorts survive the next rank step with probability1. All fractions under the original uniform empty preparation.",
+      "computation": "65536 occupied subsets per lattice, integer prefix DP; all 43434 rank-one rows have the pair identity checked within the new calculation; one square physical witness has a hand-geometric check. No 16! enumeration, old-census rerun, new samples or cloud.",
+      "boundary": "Refutes universal weak (rank,D,nu) closure by finite examples. Does not extrapolate to L512, a continuum limit or an original-U mechanism. L3 closure and original prospective L512 readouts remain unchanged.",
+      "next": "The L512 acquisition and its exact conditional-mean refinement are both completed below. Finite L4 proofs, refined L512 evidence and continuum predictions are distinct."
+    },
+    {
+      "id": "safe_insertion_independent_L512",
+      "node": "joint_birth_process",
+      "observer": "rank_direction_completion_count_history",
+      "status": "completed_fixed_new_block_primary_unresolved_secondary_not_reproduced",
+      "integration": "open_pr",
+      "acquisition_commit": "55fa265629701e04b879b774ae7d3b97fefa845f",
+      "scorer_commit_before_readout": "f340aa55134213be71d22941fae2ccb7fd068cb7",
+      "source_paths": [
+        "notes/safe-insertion-independent-block-20260929.md",
+        "analysis/safe-insertion-production-20260929/contract.json",
+        "analysis/safe-insertion-production-20260929/ESTIMANDS.md",
+        "analysis/safe-insertion-production-20260929/data/run.json",
+        "analysis/safe-insertion-production-20260929/results/result.json",
+        "analysis/safe-insertion-production-20260929/execution-receipt.json"
+      ],
+      "design": "square L512, fixed a154646,b155385;14x10000 independent prefixes,4 uniform-safe probes per rank1 prefix, averaged within prefix; no old/new pooling",
+      "primary": "exact-(D,nu) overlap-weighted early-minus-late Y=nu_after_safe-nu_before:0.006681826608 SE0.01208705518",
+      "derived": "same probes imply mean-edge difference356.3978 SE645.0460 and two-step-survival difference-6.254031427e-8 SE1.131920089e-7; not independent binary outcomes",
+      "secondary": "predeclared independent same-D current nu contrast -0.03986511616 SE0.5698779455, not reproducing the old post-hoc +1.629168 SE0.390976",
+      "information": "1190 common exact cells retain54463/55028 risk prefixes;1233/220112 probes nonzero; full6x6 aligned covariance and14deletions retained",
+      "boundary": "No full-Markov, limiting, sign-reversal or engine-error inference. L4 exact failures and all original prior results remain preserved. Continuation outputs are sparse despite broad common-support coverage; no variance fraction is inferred from sparsity alone.",
+      "next": "Exact conditional averaging is now delivered below and resolves this same target. Preserve this original four-probe report rather than replacing it; the secondary remains unchanged."
+    },
+    {
+      "id": "exact_pair_conditional_mean_L512",
+      "node": "joint_birth_process",
+      "observer": "rank_direction_completion_count_history",
+      "status": "completed_same_target_refinement_resolved_finite_history_contrast",
+      "integration": "open_pr",
+      "result_commit": "7f71af8599ce6e7c0f426cb6f7a069dbc0cf6100",
+      "dependency_group": "safe_insertion_independent_L512",
+      "source_paths": [
+        "notes/exact-completion-pairs-readout-20260930.md",
+        "notes/projected-component-pair-criterion-20260929.md",
+        "analysis/exact-completion-pairs-20260929/data/run.json",
+        "analysis/exact-completion-pairs-20260929/results/result.json"
+      ],
+      "design": "Exact2e/(m-c) replaces four uniform-safe probes on the same140k prefixes; all original rows/cutoffs/exact cells/weights retained. Post-readout estimator refinement, zero new independent samples.",
+      "primary": "deltaY=-0.003449385668 SE0.000772252763; approximately4.47 batchSE from zero; early0.09557456,late0.09902395",
+      "derived": "deltaE=-183.963985 SE41.205291; deltaZ2=+3.228181004e-8 SE7.230661888e-9; exact transfer, not independent endpoints",
+      "precision": "SE_RB/SE_probe0.0638908941; empirical variance ratio0.00408204635. Paired estimator difference -0.01013121 SE0.01239566. Full8x8 covariance and14 aligned deletions retained.",
+      "interpretation": "Supports failure of finite enlarged-observer closure at L512, not a proof of unmarked rank non-Markovness, an independent replication, or a continuum limit. Direction-only secondary remains unreplicated.",
+      "execution": "8 completed local batches plus6 HuaweiTV2N0X batches,420.381s cloud,6workers. Medium/large computation now cloud-first; prior pause and interrupted partial evaluations do not add samples.",
+      "next": "Independent archival replication is complete and recorded separately. Main attention is clock-preserving finite-window geometric intervention, not additional precision cycles."
+    },
+    {
+      "id": "label_pair_selection_curvature",
+      "node": "joint_birth_process",
+      "observer": "fixed_entry_label_time_rank_one_cohort",
+      "status": "author_finite_identity_supplied_continuum_prediction_open",
+      "integration": "open_pr",
+      "source_commit": "ed9f3e281909a9f292818c3492d6219a4471282a",
+      "source_paths": ["notes/completion-pair-label-curvature-20260930.md"],
+      "mechanism_contrast": "pair creation increases future completion opportunities; survival selection of high-c geometries contributes a negative variance term",
+      "statistic": "nearcritical hazard derivative a_L^2*(mean c+2 mean e-Var(c))/(1-t)^2",
+      "boundary": "Requires the actual label-time cohort; original fixed-count replay cannot be relabelled by t=b/N. No uniform remainder, continuum moment limit or finite-window extrapolation supplied.",
+      "next": "Derive a numerical/sign/scaling distinction between candidate continuous kernels for this specified combination under one observer/source contract."
+    },
+    {
+      "id": "clock_preserving_geometric_intervention",
+      "node": "joint_birth_process",
+      "attention": "P0",
+      "status": "finite_source_map_and_large_L512_fixed_window_propagation_delivered",
+      "integration": "open_pr",
+      "source_commit": "4e4b6a03fd5e31ee01384b18ecede12a071d3608",
+      "source_paths": [
+        "notes/birth-selection-intervention-20260930.md",
+        "analysis/birth-selection-mechanism-20260930/result.json"
+      ],
+      "source": "Safe choices weighted exp(theta*degree) with total safe mass fixed; completing choices unchanged. At nonzero theta this is an explicitly modified source, not iid Bernoulli labels.",
+      "fixed_quantities": ["entire occupied-count arrival law", "instantaneous rank exit hazard at any fixed geometry"],
+      "prediction": "Immediate rank response zero; two-step survival derivative -s Var_safe(degree)/(m(m-1)); label survival curvature derivative -s Var_safe(degree)/(1-t)^2. No universal long-window sign asserted.",
+      "completed_readout": "For three specified square-L4 starts, actual lagged derivative peaks in magnitude at lag3 and is zero after lag6. Same birth law can give different futures; same c,e and local response can give different birth laws and lag3 responses.",
+      "next": "Both fixed735-step transfer and safe/selection decomposition are completed in separate entries. Next distinguish entrance versus safe-growth sources with nonredundant same-contract readouts; continuum scaled gain remains open, not an automatic size ladder.",
+      "boundary": "Exact finite modified-source mechanism, not natural L512 population evidence, continuum identification, original-U mapping or a new field identification. No new large production launched by this derivation."
+    },
+    {
+      "id": "fixed_sign_independent_pair_archive",
+      "node": "joint_birth_process",
+      "attention": "support_completed",
+      "status": "negative_prediction_reproduced",
+      "integration": "open_pr",
+      "source_commit": "d9824b5bf478fc8acad238be2b9f31ce99787966",
+      "dependency_group": "completion_hazard_independent_L512",
+      "source_paths": ["analysis/exact-pair-independent-archive-20260930/results/result.json", "analysis/exact-pair-independent-archive-20260930/execution.json"],
+      "primary": "deltaY=-0.003199368397 SE0.000620090643",
+      "comparison": "other archive minus first=0.000250017271 SE0.000990397261; no pooling",
+      "boundary": "Independent random block, other readouts already seen, fixed first e readout; not untouched holdout, new sampling, continuum proof or direction-only replication."
+    },
+    {
+      "id": "fixed735_geometric_source_transmission",
+      "node": "joint_birth_process",
+      "attention": "P0",
+      "status": "completed_resolved_delayed_geometric_transmission",
+      "integration": "open_pr",
+      "source_commit": "18ca9e132f08b89071291d74cbd10133d99b2e5d",
+      "result_commit": "bbfe8c89a4c2e63f4b29d7961721a2d2f5311ade",
+      "dependency_group": "safe_insertion_independent_L512",
+      "source_paths": ["notes/geometric-source-window-identity-20260930.md", "analysis/geometric-source-window-20260930/contract.json", "analysis/geometric-source-window-20260930/results/result.json", "analysis/geometric-source-window-20260930/execution.json"],
+      "primary": "Mean derivative of survival through735insertions under one safe-degree source kick, conditional onrank_b=1; b155385, endpoint156120",
+      "estimator": "Original uniform h-subset continuation gives survival*(sum_initial_degrees/h-2e/s), by exact conditional averaging over the first-arrival position.",
+      "result": "derivative=-0.0110420319107 SE0.000157666446747; baseline survival0.444755397252 SE0.00214095866922; two-step derivative-5.90719899852e-5 SE4.69939324914e-7. Full3x3covariance,14aligned deletions,55028riskprefixes, all140000first5matched.",
+      "boundary": "New mechanism readout on existing permutations; no independent data, new lag, theta grid, subgroup search, original-U or continuum claim."
+    },
+    {
+      "id": "safe_transport_selection_coupling",
+      "node": "joint_birth_process",
+      "attention": "P0",
+      "status": "completed_reference_memory_persists_selection_unresolved",
+      "integration": "open_pr",
+      "source_commit": "5054c02383242473834045fcbddc7379d83acfb9",
+      "result_commit": "e56c0b8361628a066abea4ba4d3655c74e3a62bf",
+      "dependency_group": "safe_insertion_independent_L512",
+      "source_paths": ["notes/safe-path-selection-coupling-20260930.md", "notes/safe-transport-selection-readout-20260930.md", "analysis/safe-transport-selection-20260930/contract.json", "analysis/safe-transport-selection-20260930/results/result.json", "analysis/safe-transport-selection-20260930/execution.json"],
+      "result": "Natural=-.002205065286 SE.000787091700; safe-reference=-.002537058794 SE.000636492674; selection=+.000331993508 SE.000287375294. Same1198cells/weights, full3x3covariance and14deletions;140000births match,97666reference endpoints,54850natural survivors. TgFr7R14workers664.094s, returned and Ready verified.",
+      "mechanism_contrast": "Selection-only for the specified contrast predicts zero unweighted safe-transport contrast; nonzero unweighted contrast requires entrance/transport contribution. Selection may reinforce or oppose it.",
+      "sampler": "Uniform random-order safe rejection keeps a reference path; first proposal completing kills a separate natural-survival flag. Accepted site is uniform safe and conditional expected flag equals the exact path weight. Only local hypothetical completion queries needed per proposal, not a full completion census per step.",
+      "boundary": "New conditional continuations of old first-birth prefixes share their dependency group; not independent preparations or unique birth-versus-evolution causal attribution."
+    },
+    {
+      "id": "entrance_versus_safe_growth_sources",
+      "node": "joint_birth_process",
+      "attention": "P0",
+      "status": "response_placement_formulas_and_rank_bounds_derived_realizable_source_pair_next",
+      "integration": "open_pr",
+      "source_commit": "2f639f69fa4730d2225308f21133948d6c9c1b62",
+      "source_paths": ["notes/safe-transport-selection-readout-20260930.md"],
+      "mechanism_contrast": "Entrance-law perturbation preserving each birth-count mass versus row-sum-zero safe-transition perturbation. Compare genuinely distinct response columns under one specified preparation, endpoint and normalization, not age-label swaps.",
+      "structural_bound": "Y/survival2 at fixed(k,D,c) are redundant. Two first-birth-preserving sources are also collinear in single-p static M/Etop and moving-root readouts. More precision cannot fix these ranks.",
+      "next": "Specify a realizable pair and a nonredundant joint-time or long-window readout before computation; include conditional-normalizer and any data-dependent external-weight derivatives.",
+      "boundary": "No claim that the proposed source columns have already been computed or separated; existing735-step susceptibility belongs to its own source contract."
+    },
+    {
+      "id": "one_sided_birth_moving_root_control",
+      "node": "joint_birth_process",
+      "attention": "mechanism_positive_control_completed",
+      "status": "explicit_finite_nonzero_normal_response_delivered",
+      "integration": "open_pr",
+      "source_commit": "467b18255f0dbf04916db1cd6682a01f70c09f9e",
+      "source_paths": ["notes/one-sided-birth-source-normal-response-20260930.md", "analysis/one-sided-birth-source-20260930/result.json"],
+      "result": "SquareL4 count6 kick on64 translation/dihedral-related preparations; alpha8/1001. Exact positive polynomialg; deltaM=deltaEtop=g and moving-root normal=g*(-2P0'/M'). Normal response0.00016853296, genuine rank2(p,theta)->(M,Etop)Jacobian.",
+      "boundary": "Constructed dynamic source at finiteL4, not the original norm4/U source/normalizer, continuum field identification or natural H4 attribution. L512fixed-count735-step result cannot be substituted for label-timeg."
+    },
+    {
+      "id": "normal_candidate_columns",
+      "node": "original_u_identification",
+      "status": "missing_theoretical_input",
+      "statistic": "same-source nuisance-profiled normal prediction images",
+      "mechanism_contrast": "distinct images identifiable; equal images remain unidentifiable regardless of precision"
+    }
+  ],
+  "dependency_groups": [
+    {
+      "id": "existing_paired_archives",
+      "rule": "marginals, roots, slopes, direct atoms, gap CDF and soft mass from one paired block are correlated views, not extra independent votes"
+    },
+    {
+      "id": "birth_gap_pilot",
+      "path": "analysis/birth-gap-20260929/data/run.json",
+      "rule": "separate streams by lattice/size/batch; within-batch derived statistics share full delete-one covariance; controls/benchmarks excluded"
+    },
+    {
+      "id": "birth_gap_cloud",
+      "path": "analysis/birth-gap-20260929/cloud-data-5k/run.json",
+      "rule": "cloud-independent-20260929 namespace; 14 batches per cell; no pooling with local pilot; derived CDF/Z/moments and later rank-history contrasts reuse the SAME block and are not independent evidence; excluded benchmarks are not evidence"
+    },
+    {
+      "id": "birth_L3_exact_geometry",
+      "path": "analysis/birth-completion-geometry-20260929/result.json",
+      "rule": "Deterministic occupied-set census and prefix DP share the same finite model as the earlier complete-permutation tables; source collapse is an exact identity, not independent stochastic evidence. Marked-label kernels derive from this same census."
+    },
+    {
+      "id": "completion_hazard_independent_L512",
+      "path": "analysis/completion-hazard-production-20260929/data/run.json",
+      "rule": "New completion-independent-20260929 namespace, 14 independent equal batches; all survival, directional and hazard readouts, including the later retrospective within-D nu_b necessity check, reuse this one block. Old birth-gap streams and the excluded benchmark are not pooled."
+    },
+    {
+      "id": "completion_pair_L4_exact_geometry",
+      "path": "analysis/completion-pair-synergy-20260929/result.json",
+      "rule": "Deterministic finite census and exact prefix-history weights, not a random block or independent replication of L512. Strong witnesses, weak-history contrasts and graph identities derive from the same L4 model."
+    },
+    {
+      "id": "safe_insertion_independent_L512",
+      "path": "analysis/safe-insertion-production-20260929/data/run.json",
+      "refinement_path": "analysis/exact-completion-pairs-20260929/data/run.json",
+      "conditional_continuation_path": "analysis/safe-transport-selection-20260930/data/run.json",
+      "rule": "One safe-insertion-independent-20260929 block,14 independent batches. Four probes, later exact-pair conditional means, derived survival, secondary nu and735-step source response reuse these prefixes. Safe/reference selection uses new conditional futures of those same first-birth prefixes and retains the same groups, not independent preparations. Each analysis retains aligned covariance; local/cloud execution does not change sample identity."
+    },
+    {
+      "id": "source_response_families",
+      "rule": "equivalent black/white pair source after normalizer/logit change is one normal direction"
+    },
+    {
+      "id": "research_pr_lineage",
+      "rule": "stacked proofs, copied tables and rerun controls are not independent external reviews"
+    }
+  ]
+}
diff --git a/analysis/research_ledger.yaml b/analysis/research_ledger.yaml
index 5395288aa..482ea8e52 100644
--- a/analysis/research_ledger.yaml
+++ b/analysis/research_ledger.yaml
@@ -1,5 +1,8 @@
 version: 5
 updated: 2026-08-29
+scope: historical_evidence_index_not_current_dispatch
+current_frontier: analysis/research_frontier.json
+current_frontier_as_of: 2026-09-29
 purpose: >
   Compact map of durable evidence and next information axes. This ranks work;
   it does not grant permission.
@@ -8,7 +11,7 @@ canonical_views:
   overview: README.md
   claim_ledger: docs/STATUS.md
   research_map: docs/RESEARCH-MAP.md
-  execution_synthesis: notes/SYNTHESIS-20260828.md
+  execution_synthesis: docs/ROADMAP.md
   roadmap: docs/ROADMAP.md
   artifact_registry: analysis/artifact_registry.yaml
   reproducibility: REPRODUCIBILITY.md
diff --git a/analysis/safe-insertion-production-20260929/ESTIMANDS.md b/analysis/safe-insertion-production-20260929/ESTIMANDS.md
new file mode 100644
index 000000000..c65a2477c
--- /dev/null
+++ b/analysis/safe-insertion-production-20260929/ESTIMANDS.md
@@ -0,0 +1,119 @@
+# Fixed prospective safe-insertion estimands
+
+One new square-site L512 insertion-clock block only: N=262144,
+a=154646, b=155385, m=N-b=106759, q=4 probes per safe prefix,
+B=14 independent equal-size batches of 10000 naturally sampled prefixes.
+The scorer reads a completed `data/run.json` and its declared compressed
+batches. This document and executable alone are not a production result.
+
+## Prefixes, cohorts, and probes
+
+Risk is `rank_b=1`. Early means `J1<=a`; late means `a<J1<=b`.
+Let `c=nu_b`, `s=m-c`. For each safe prefix (`s>0`), each probe chooses
+a uniformly random safe site in a fresh copy of that same prefix and records
+`Y=nu(A+v)-c`. Average the four probes within prefix:
+
+    Ybar_i = (Y_i0 + Y_i1 + Y_i2 + Y_i3)/4.
+
+The probes are continuations of one prefix, not four independent prefixes.
+Sampling must preserve the uniform-safe-site marginal even if probes repeat
+a site. The input does not reveal probe locations or prove uniformity.
+For `c=m`, no safe insertion exists; all four Y values must be -1 and Ybar
+is undefined. These dead prefixes are excluded from primary and included
+in secondary. Outside rank one, direction is (0,0), nu and all Y are -1;
+rank zero has J1=0, rank two has 1<=J1<=b.
+
+Rank-one D is the exact primitive unoriented integer direction: sign alone
+is canonicalized so its first nonzero coordinate is positive. Nonprimitive
+directions fail validation. Safe probes satisfy `0<=Y<=m-1-c`.
+
+## Primary and its exact consumers
+
+For each exact `(D,c)` cell h with s>0 and both cohorts present, use
+prefix counts n_Eh,n_Lh and cohort means of prefix Ybar:
+
+    w_h = n_Eh*n_Lh/(n_Eh+n_Lh),  W = sum_common w_h,
+    dY_h = mean_Eh(Ybar) - mean_Lh(Ybar),
+    deltaY = sum_common w_h*dY_h / W,
+    weighted_earlyY = sum_common w_h*mean_Eh(Ybar) / W,
+    weighted_lateY  = sum_common w_h*mean_Lh(Ybar) / W.
+
+For the completion graph on the s safe vacancies, with e unordered edges,
+the exact conditional mean is `E[Y|A]=2e/s`. Therefore within a matched cell:
+
+    dE_h  = s_h*dY_h/2,
+    dZ2_h = -s_h*dY_h/[m*(m-1)],
+    deltaE  = sum_common w_h*dE_h/W,
+    deltaZ2 = sum_common w_h*dZ2_h/W.
+
+The last quantity estimates the early-minus-late **unconditional two-next-
+insertion rank-one survival** contrast within matched cells, using
+`P(J2>b+2|A)=[s(s-1)-2e]/[m(m-1)]`. It is not a directly observed binary
+endpoint or survival conditional on the first step. Apply the s-dependent
+factors inside cells before weighting. Both consumers use precisely the
+primary support and weights. DeltaE and deltaZ2 are derived predictions
+from the same probes, not additional independent measurements.
+
+## Prespecified secondary replication
+
+Use every rank-one prefix, including c=m. Within exact D only, form
+`v_d=n_Ed*n_Ld/(n_Ed+n_Ld)` on shared cells and compute
+
+    secondary_deltaNu = sum_common v_d*(mean_Ed(c)-mean_Ld(c))/sum_common v_d.
+
+This repeats the old fixed direction-only completion-count contrast on a
+new independent block. Its support and weights are separately recomputed;
+it shares this new block with the primary. No old/new pooling occurs and
+differences between these differently weighted targets are not mediated
+fractions or explained percentages.
+
+## Joint uncertainty, support, and outputs
+
+The ordered vector is `(deltaY,deltaE,deltaZ2,secondary_deltaNu,
+weighted_earlyY,weighted_lateY)`. Delete an entire aligned batch j from all
+six coordinates, then recompute cell counts, support, means and weights:
+
+    theta_bar = (1/B) sum_j theta_(-j),
+    Cov_JK = (B-1)/B sum_j (theta_(-j)-theta_bar)(theta_(-j)-theta_bar)^T.
+
+Report pooled plug-in point estimates, all 14 deletion vectors, full 6x6
+covariance and its diagonal square-root SEs. The exact relations
+`deltaY=weighted_earlyY-weighted_lateY` and
+`deltaZ2=-2*deltaE/[m(m-1)]` make this covariance singular. No independence,
+matrix inverse, or separate-probe standard error is assumed.
+
+If either comparison has no common support in the full block or any
+deletion, write `not_scoreable` diagnostics with all deletion vectors,
+null the entire joint covariance/SEs, and exit 2. Undefined quantities
+remain null; no deletion or cohort is silently dropped. Shared support may
+otherwise change across deletions. Coverage uses both eligible prefixes
+and all rank-one prefixes as denominators, separately by cohort and combined.
+The overlap-weight sum is not asserted to be an effective sample size.
+
+JSON retains cell counts, exact integer probe sums and squared sums,
+prefix sums of squares, means, one-sided cells, support coverage, nonzero
+probe/prefix counts, per-batch totals, input hashes/manifests, and scoring
+sources. Production status, contract values, unique batch identities/files,
+hashes, row counts, directions, and all sentinel/range rules are checked.
+Structural errors exit 1 without scientific results. Input checks do not
+validate the geometry engine or establish independence of generation.
+
+Run after the parent has reviewed the scorer and acquired real data:
+
+    python analyze.py
+    python analyze.py --data-dir data --output-dir results-new
+    python analyze.py --output results-new/result.json --md results-new/RESULT.md
+
+Default inputs and outputs are relative to the script directory; an explicit
+data directory defaults to `contract.json` in its parent. Explicit relative
+CLI paths are relative to the invocation directory. `--contract` can name
+the input contract. Default outputs are `results/result.json` and
+`results/RESULT.md`; `--output-json`/`--output-md` are aliases. Existing
+outputs are refused; select fresh paths for another scoring pass.
+
+This primary examines one moment of one safe-step successor law. A
+population discrepancy challenges recursive `(rank,D,nu)` sufficiency at
+this fixed count; a weighted null may hide cell cancellation and does not
+establish even full successor-law equality, much less recursive closure.
+There is no subgroup search, cutoff search, fitted exponent, enlargement of
+the acquired block, label-time conclusion, or continuum claim.
diff --git a/analysis/safe-insertion-production-20260929/analyze.py b/analysis/safe-insertion-production-20260929/analyze.py
new file mode 100644
index 000000000..d90074015
--- /dev/null
+++ b/analysis/safe-insertion-production-20260929/analyze.py
@@ -0,0 +1,525 @@
+#!/usr/bin/env python3
+"""Score only the fixed prospective square-L512 safe-insertion block (stdlib).
+
+No acquisition, historical pooling, cutoff selection, or fitted models. Each
+prefix contributes its four-probe mean; all uncertainty uses aligned batches.
+"""
+
+import argparse
+import csv
+from dataclasses import dataclass, field, fields
+from datetime import datetime, timezone
+import gzip
+import hashlib
+import io
+import json
+import math
+from pathlib import Path
+import re
+import sys
+
+
+ROOT = Path(__file__).resolve().parent
+FIXED = dict(N=262144, a=154646, b=155385, probes=4,
+             batches=14, samples_per_batch=10000)
+COLUMNS = ("J1", "rank_b", "dx_b", "dy_b", "nu_b", "Y0", "Y1", "Y2", "Y3")
+METRICS = ("deltaY", "deltaE", "deltaZ2", "secondary_deltaNu",
+           "weighted_earlyY", "weighted_lateY")
+GROUPS = ("early", "late")
+
+
+def integer(value, label):
+    if type(value) is int:
+        return value
+    if isinstance(value, str) and re.fullmatch(r"-?[0-9]+", value):
+        return int(value)
+    raise ValueError(f"{label}: expected an integer, got {value!r}")
+
+
+def sha256(raw):
+    return hashlib.sha256(raw).hexdigest()
+
+
+def read_json(path):
+    raw = path.read_bytes()
+    doc = json.loads(raw)
+    if not isinstance(doc, dict):
+        raise ValueError(f"{path}: expected a JSON object")
+    return doc, {"path": str(path), "bytes": len(raw), "sha256": sha256(raw)}
+
+
+def check_contract(contract, label="contract.json"):
+    for key, expected in FIXED.items():
+        if type(contract.get(key)) is not int or contract[key] != expected:
+            raise ValueError(f"{label}.{key} must be the fixed integer {expected}")
+    for key, expected in (("lattice", "square"), ("L", 512)):
+        if key in contract and contract[key] != expected:
+            raise ValueError(f"{label}.{key} must be {expected!r}")
+
+
+@dataclass
+class Moments:
+    n: int = 0
+    sum_nu: int = 0
+    sum_nu_squared: int = 0
+    safe_prefixes: int = 0
+    sum_probe_Y: int = 0
+    sum_probe_Y_squared: int = 0
+    sum_squared_prefix_probe_sum: int = 0
+    nonzero_probes: int = 0
+    nonzero_prefixes: int = 0
+
+    def observe(self, nu, ys):
+        self.n += 1
+        self.sum_nu += nu
+        self.sum_nu_squared += nu * nu
+        if ys is not None:
+            total = sum(ys)
+            self.safe_prefixes += 1
+            self.sum_probe_Y += total
+            self.sum_probe_Y_squared += sum(y * y for y in ys)
+            self.sum_squared_prefix_probe_sum += total * total
+            self.nonzero_probes += sum(y > 0 for y in ys)
+            self.nonzero_prefixes += int(total > 0)
+
+    def merge(self, other):
+        for item in fields(self):
+            setattr(self, item.name, getattr(self, item.name) + getattr(other, item.name))
+
+    def report(self, q):
+        return {
+            "prefixes": self.n, "safe_prefixes": self.safe_prefixes,
+            "dead_prefixes": self.n - self.safe_prefixes,
+            "sum_nu_b": self.sum_nu, "sum_nu_b_squared": self.sum_nu_squared,
+            "mean_nu_b": self.sum_nu / self.n if self.n else None,
+            "valid_probes": q * self.safe_prefixes,
+            "nonzero_probes": self.nonzero_probes,
+            "nonzero_prefixes": self.nonzero_prefixes,
+            "sum_probe_Y": self.sum_probe_Y,
+            "sum_probe_Y_squared": self.sum_probe_Y_squared,
+            "sum_squared_prefix_probe_sum": self.sum_squared_prefix_probe_sum,
+            "sum_Ybar": self.sum_probe_Y / q,
+            "sum_Ybar_squared": self.sum_squared_prefix_probe_sum / (q * q),
+            "mean_Ybar": (self.sum_probe_Y / (q * self.safe_prefixes)
+                          if self.safe_prefixes else None),
+        }
+
+
+def pair():
+    return [Moments(), Moments()]
+
+
+@dataclass
+class BatchStats:
+    samples: int = 0
+    rank_counts: list = field(default_factory=lambda: [0, 0, 0])
+    sign_canonicalizations: int = 0
+    groups: list = field(default_factory=pair)
+    primary_cells: dict = field(default_factory=dict)
+    secondary_cells: dict = field(default_factory=dict)
+
+    def observe(self, values, parameters):
+        j1, rank, dx, dy, nu, *ys = values
+        a, b, m = parameters["a"], parameters["b"], parameters["N"] - parameters["b"]
+        if rank not in (0, 1, 2):
+            raise ValueError("rank_b must be 0, 1, or 2")
+        if rank == 0 and j1 != 0:
+            raise ValueError("rank zero requires J1=0")
+        if rank in (1, 2) and not 1 <= j1 <= b:
+            raise ValueError("rank one/two requires 1<=J1<=b")
+        if rank != 1:
+            if (dx, dy) != (0, 0) or nu != -1 or ys != [-1] * parameters["probes"]:
+                raise ValueError("outside rank one require direction=(0,0), nu_b=Y0=...=Y3=-1")
+        else:
+            if math.gcd(abs(dx), abs(dy)) != 1:
+                raise ValueError("rank-one direction must be nonzero and primitive")
+            if not 0 <= nu <= m:
+                raise ValueError("rank one requires 0<=nu_b<=m")
+            if nu == m:
+                if ys != [-1] * parameters["probes"]:
+                    raise ValueError("nu_b=m requires all Y=-1; safe continuation is undefined")
+            elif any(y < 0 or y > m - 1 - nu for y in ys):
+                raise ValueError("safe probes require 0<=Y<=m-1-nu_b")
+        self.samples += 1
+        self.rank_counts[rank] += 1
+        if rank != 1:
+            return
+        if dx < 0 or (dx == 0 and dy < 0):
+            dx, dy = -dx, -dy
+            self.sign_canonicalizations += 1
+        group = int(j1 > a)
+        probes = ys if nu < m else None
+        self.groups[group].observe(nu, probes)
+        self.secondary_cells.setdefault((dx, dy), pair())[group].observe(nu, probes)
+        if probes is not None:
+            self.primary_cells.setdefault((dx, dy, nu), pair())[group].observe(nu, probes)
+
+    def merge(self, other):
+        self.samples += other.samples
+        self.rank_counts = [x + y for x, y in zip(self.rank_counts, other.rank_counts)]
+        self.sign_canonicalizations += other.sign_canonicalizations
+        for target, source in zip(self.groups, other.groups):
+            target.merge(source)
+        for name in ("primary_cells", "secondary_cells"):
+            target = getattr(self, name)
+            for key, source in getattr(other, name).items():
+                for mine, theirs in zip(target.setdefault(key, pair()), source):
+                    mine.merge(theirs)
+
+    def totals(self, q):
+        risk = Moments()
+        for group in self.groups:
+            risk.merge(group)
+        return {
+            "rows": self.samples,
+            "rank_counts": dict(zip(("rank0", "rank1", "rank2"), self.rank_counts)),
+            "outside_rank1": self.rank_counts[0] + self.rank_counts[2],
+            "sign_canonicalizations": self.sign_canonicalizations,
+            "rank1": risk.report(q),
+            "cohorts": {key: group.report(q) for key, group in zip(GROUPS, self.groups)},
+        }
+
+
+def aggregate(blocks, omit=None):
+    total = BatchStats()
+    for index, block in enumerate(blocks):
+        if index != omit:
+            total.merge(block)
+    return total
+
+
+def batch_entries(run, data_dir, parameters):
+    if run.get("status") != "completed":
+        raise ValueError("run.status must be completed; partial production is not scoreable")
+    entries = run.get("batches")
+    if not isinstance(entries, list) or len(entries) != parameters["batches"]:
+        raise ValueError("run must contain all 14 contracted batches")
+    seen_ids, seen_paths, seen_hashes = set(), set(), set()
+    normalized = []
+    for entry in entries:
+        if not isinstance(entry, dict) or not {"file", "samples", "batch", "sha256"} <= entry.keys():
+            raise ValueError("each batch needs file, samples, batch, sha256")
+        identity = integer(entry["batch"], "batch ID")
+        if identity < 0 or identity in seen_ids:
+            raise ValueError(f"invalid or duplicate batch ID {identity}")
+        seen_ids.add(identity)
+        if integer(entry["samples"], "batch.samples") != parameters["samples_per_batch"]:
+            raise ValueError(f"batch {identity} must declare 10000 samples")
+        if not isinstance(entry["file"], str) or not entry["file"]:
+            raise ValueError(f"batch {identity}: file must be a nonempty relative path")
+        relative = Path(entry["file"])
+        path = (data_dir / relative).resolve()
+        if relative.is_absolute() or data_dir not in path.parents or path.suffix != ".gz":
+            raise ValueError(f"batch {identity}: expected a .gz file inside data-dir")
+        digest = entry["sha256"]
+        if not isinstance(digest, str) or not re.fullmatch(r"[0-9a-fA-F]{64}", digest):
+            raise ValueError(f"batch {identity}: invalid SHA256")
+        digest = digest.lower()
+        if path in seen_paths or digest in seen_hashes:
+            raise ValueError(f"batch {identity}: duplicate batch file or compressed content")
+        seen_paths.add(path)
+        seen_hashes.add(digest)
+        normalized.append((identity, path, digest, entry))
+    return sorted(normalized, key=lambda item: item[0])
+
+
+def read_batch(item, parameters):
+    identity, path, expected_hash, entry = item
+    raw = path.read_bytes()
+    if sha256(raw) != expected_hash:
+        raise ValueError(f"batch {identity}: SHA256 mismatch for {path}")
+    payload = gzip.decompress(raw).decode("utf-8-sig")
+    reader = csv.reader(io.StringIO(payload, newline=""), strict=True)
+    if next(reader, None) != list(COLUMNS):
+        raise ValueError(f"batch {identity}: require exact CSV header {','.join(COLUMNS)}")
+    stats = BatchStats()
+    for row in reader:
+        try:
+            if len(row) != len(COLUMNS):
+                raise ValueError(f"expected {len(COLUMNS)} fields, found {len(row)}")
+            stats.observe([integer(v, key) for key, v in zip(COLUMNS, row)], parameters)
+        except ValueError as error:
+            raise ValueError(f"batch {identity}, CSV line {reader.line_num}: {error}") from error
+    if stats.samples != parameters["samples_per_batch"]:
+        raise ValueError(f"batch {identity}: row count {stats.samples}, expected 10000")
+    manifest = {"batch": identity, "path": str(path), "bytes": len(raw),
+                "sha256": expected_hash, "sha256_verified": True,
+                "rows_verified": stats.samples, "declared": entry}
+    return stats, manifest
+
+
+def overlap(cells, groups, parameters, primary, detailed=False):
+    """Every call reselects common support and recomputes overlap weights."""
+    q, m = parameters["probes"], parameters["N"] - parameters["b"]
+    eligible = [g.safe_prefixes if primary else g.n for g in groups]
+    supported = [0, 0]
+    cell_counts = {"common": 0, "early_only": 0, "late_only": 0}
+    weights, weighted_early, weighted_late = [], [], []
+    weighted_delta, weighted_edges, weighted_survival = [], [], []
+    rows = []
+    for key, (early, late) in sorted(cells.items()):
+        common = bool(early.n and late.n)
+        weight = early.n * late.n / (early.n + late.n) if common else 0.0
+        mean_e = ((early.sum_probe_Y / q if primary else early.sum_nu) / early.n
+                  if early.n else None)
+        mean_l = ((late.sum_probe_Y / q if primary else late.sum_nu) / late.n
+                  if late.n else None)
+        delta = mean_e - mean_l if common else None
+        if common:
+            cell_counts["common"] += 1
+            supported[0] += early.n
+            supported[1] += late.n
+            weights.append(weight)
+            weighted_early.append(weight * mean_e)
+            weighted_late.append(weight * mean_l)
+            weighted_delta.append(weight * delta)
+            if primary:
+                s = m - key[2]
+                weighted_edges.append(weight * (s / 2) * delta)
+                weighted_survival.append(weight * (-s / (m * (m - 1))) * delta)
+        else:
+            cell_counts["early_only" if early.n else "late_only"] += 1
+        if detailed:
+            record = {"direction": list(key[:2]), "common_support": common,
+                      "overlap_weight": weight, "early": early.report(q),
+                      "late": late.report(q)}
+            if primary:
+                s = m - key[2]
+                record.update(nu_b=key[2], safe_sites=s, deltaY=delta,
+                              deltaE=s / 2 * delta if common else None,
+                              deltaZ2=-s / (m * (m - 1)) * delta if common else None)
+            else:
+                record["deltaNu"] = delta
+            rows.append(record)
+    weight_sum = math.fsum(weights)
+    available = weight_sum > 0
+    coverage = {}
+    for i, group in enumerate(GROUPS):
+        coverage[group] = {
+            "rank1_prefixes": groups[i].n, "eligible_prefixes": eligible[i],
+            "supported_prefixes": supported[i],
+            "unsupported_eligible_prefixes": eligible[i] - supported[i],
+            "supported_fraction_of_eligible": supported[i] / eligible[i] if eligible[i] else None,
+            "supported_fraction_of_rank1": supported[i] / groups[i].n if groups[i].n else None,
+            "dead_prefixes": groups[i].n - groups[i].safe_prefixes,
+        }
+    coverage["combined"] = {
+        "rank1_prefixes": sum(g.n for g in groups), "eligible_prefixes": sum(eligible),
+        "supported_prefixes": sum(supported),
+        "unsupported_eligible_prefixes": sum(eligible) - sum(supported),
+        "supported_fraction_of_eligible": sum(supported) / sum(eligible) if sum(eligible) else None,
+        "supported_fraction_of_rank1": (sum(supported) / sum(g.n for g in groups)
+                                        if sum(g.n for g in groups) else None),
+    }
+    result = {
+        "status": "scoreable" if available else "not_scoreable",
+        "reason": None if available else "no common early/late support in eligible cells",
+        "stratification": "exact (D,nu_b), nu_b<m" if primary else "exact D, all rank-one prefixes",
+        "cell_counts": cell_counts, "overlap_weight_sum": weight_sum,
+        "coverage": coverage,
+        "weighted_early_mean": math.fsum(weighted_early) / weight_sum if available else None,
+        "weighted_late_mean": math.fsum(weighted_late) / weight_sum if available else None,
+        "delta": math.fsum(weighted_delta) / weight_sum if available else None,
+    }
+    if primary:
+        result["deltaE"] = math.fsum(weighted_edges) / weight_sum if available else None
+        result["deltaZ2"] = math.fsum(weighted_survival) / weight_sum if available else None
+    if detailed:
+        for row in rows:
+            row["normalized_weight"] = row["overlap_weight"] / weight_sum if available else None
+        result["cells"] = rows
+    return result
+
+
+def estimate(stats, parameters, detailed=False):
+    primary = overlap(stats.primary_cells, stats.groups, parameters, True, detailed)
+    secondary = overlap(stats.secondary_cells, stats.groups, parameters, False, detailed)
+    vector = [primary["delta"], primary["deltaE"], primary["deltaZ2"],
+              secondary["delta"], primary["weighted_early_mean"], primary["weighted_late_mean"]]
+    return vector, primary, secondary
+
+
+def score(blocks, batch_ids, parameters):
+    total = aggregate(blocks)
+    point, primary, secondary = estimate(total, parameters, detailed=True)
+    stops = []
+    if any(v is None for v in point):
+        stops.append("full block: absent common support; see primary/secondary diagnostics")
+    replicates = []
+    for i, identity in enumerate(batch_ids):
+        vector, p, s = estimate(aggregate(blocks, omit=i), parameters)
+        if any(v is None for v in vector):
+            stops.append(f"delete batch {identity}: absent common support")
+        replicates.append({"deleted_batch": identity, "vector": vector,
+                           "primary_support": p, "secondary_support": s})
+    B, dimension = len(blocks), len(METRICS)
+    if B != parameters["batches"] or len(batch_ids) != B or len(set(batch_ids)) != B:
+        raise ValueError("scoring requires all distinct contracted batch IDs")
+    if any(block.samples != parameters["samples_per_batch"] for block in blocks):
+        raise ValueError("scoring requires equal contracted batch sizes")
+    covariance = [[None] * dimension for _ in METRICS]
+    jackknife_mean = [None] * dimension
+    standard_errors = [None] * dimension
+    if not stops:
+        vectors = [row["vector"] for row in replicates]
+        jackknife_mean = [math.fsum(v[k] for v in vectors) / B for k in range(dimension)]
+        covariance = [[(B - 1) / B * math.fsum(
+            (v[j] - jackknife_mean[j]) * (v[k] - jackknife_mean[k]) for v in vectors)
+            for k in range(dimension)] for j in range(dimension)]
+        standard_errors = [math.sqrt(covariance[k][k]) for k in range(dimension)]
+    return {
+        "schema": "matching-one.safe-insertion-score.v1",
+        "status": "not_scoreable" if stops else "scored",
+        "stop_reasons": stops, "parameters": dict(parameters),
+        "m": parameters["N"] - parameters["b"], "metric_order": list(METRICS),
+        "point_vector": point, "point_estimates": dict(zip(METRICS, point)),
+        "standard_errors": dict(zip(METRICS, standard_errors)),
+        "totals": total.totals(parameters["probes"]),
+        "batch_totals": [{"batch": identity, **block.totals(parameters["probes"])}
+                         for identity, block in zip(batch_ids, blocks)],
+        "primary": primary, "secondary": secondary,
+        "joint_uncertainty": {
+            "method": "aligned whole-batch delete-one jackknife; all weights/support recomputed",
+            "covariance_formula": "(B-1)/B * sum_j (theta_(-j)-mean)(theta_(-j)-mean)^T",
+            "metric_order": list(METRICS), "batch_count": B,
+            "status": "stopped_absent_support" if stops else "available",
+            "delete_one_mean_vector": jackknife_mean,
+            "covariance": covariance, "delete_one": replicates,
+            "point_estimator": "pooled plug-in; no jackknife bias correction",
+            "dependence": "four probes share one prefix; all six metrics share one new block",
+            "singularity": "deltaZ2=-2*deltaE/[m*(m-1)] and deltaY=weighted_earlyY-weighted_lateY",
+        },
+        "interpretation": [
+            "The primary tests one safe-step completion-creation mean at fixed exact (D,nu_b).",
+            "Derived edge and two-step-survival contrasts use cell-specific s before aggregation.",
+            "Secondary exact-D nu replication is independent of the old block, not of this primary.",
+            "A null weighted moment can hide cell cancellation and does not establish recursive closure.",
+            "Finite square L512 insertion clock only; no limiting, label-clock, or universality claim.",
+            "No historical data pooling, subgroup search, fitted exponent, or acquisition is performed.",
+        ],
+    }
+
+
+def source_manifest(run, contract_info):
+    local = []
+    for name in ("analyze.py", "ESTIMANDS.md", "engine.cpp", "run.py"):
+        path = ROOT / name
+        if path.is_file():
+            raw = path.read_bytes()
+            local.append({"path": str(path), "bytes": len(raw), "sha256": sha256(raw)})
+    declared = run.get("source_sha256", {})
+    if not isinstance(declared, dict):
+        raise ValueError("run.source_sha256 must be an object if provided")
+    if "contract.json" in declared and declared["contract.json"] != contract_info["sha256"]:
+        raise ValueError("run source contract.json hash disagrees with input contract")
+    return {
+        "local_files_at_scoring": local,
+        "acquisition_declared_source_commit": run.get("source_commit"),
+        "acquisition_declared_source_sha256": declared,
+        "note": "Local scoring hashes and acquisition-declared hashes are distinct provenance; "
+                "hash/range checks do not prove topology, uniform safe sampling, or seed independence.",
+    }
+
+
+def markdown(result):
+    def number(value):
+        return "undefined" if value is None else f"{value:.10g}"
+
+    lines = ["# Prospective square-L512 safe-insertion score", "",
+             f"Status: **{result['status']}**. Fixed 14 batches x 10000 prefixes; q=4.", "",
+             "All differences are early minus late. The point estimates are pooled plug-in estimates.", "",
+             "| Metric | Estimate | Aligned batch SE |", "|---|---:|---:|"]
+    for key in METRICS:
+        lines.append(f"| {key} | {number(result['point_estimates'][key])} | "
+                     f"{number(result['standard_errors'][key])} |")
+    if result["stop_reasons"]:
+        lines += ["", "Joint inference STOPPED; no deletions were omitted or replaced.", ""]
+        lines += [f"- {reason}" for reason in result["stop_reasons"]]
+    totals = result["totals"]
+    risk = totals["rank1"]
+    lines += ["", f"Rows: {totals['rows']}; rank counts: {totals['rank_counts']}.",
+              f"Rank-one prefixes: {risk['prefixes']}; safe: {risk['safe_prefixes']}; "
+              f"dead (nu=m): {risk['dead_prefixes']}.",
+              f"Valid probes: {risk['valid_probes']}; nonzero probes: {risk['nonzero_probes']}; "
+              f"prefixes with any nonzero probe: {risk['nonzero_prefixes']}.", "",
+              "| Comparison / cohort | Eligible | Supported | Coverage of eligible |",
+              "|---|---:|---:|---:|"]
+    for name in ("primary", "secondary"):
+        for group in GROUPS:
+            coverage = result[name]["coverage"][group]
+            lines.append(f"| {name} / {group} | {coverage['eligible_prefixes']} | "
+                         f"{coverage['supported_prefixes']} | "
+                         f"{number(coverage['supported_fraction_of_eligible'])} |")
+    lines += ["", "Primary excludes dead prefixes because a safe-step mean is undefined there; "
+              "secondary includes every rank-one prefix, including dead prefixes.", "",
+              "Within each shared exact (D,nu) cell: deltaE=(m-nu)*deltaY/2 and "
+              "deltaZ2=-(m-nu)*deltaY/[m(m-1)]. The same normalized overlap weights "
+              "combine all primary quantities. The exact-D secondary recomputes its own weights.", "",
+              "The JSON records exact integer sufficient statistics, all cells, support coverage, "
+              "all 14 aligned deletion vectors, the full 6x6 covariance and source/input manifests. "
+              "Probes are averaged within prefix; metrics are not independent and covariance is "
+              "necessarily singular. No covariance inverse or independent-metric test is used.", ""]
+    lines += [f"- {item}" for item in result["interpretation"]]
+    return "\n".join(lines) + "\n"
+
+
+def main(argv=None):
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument("--data-dir", type=Path, default=ROOT / "data")
+    parser.add_argument("--contract", type=Path,
+                        help="default: contract.json beside data-dir")
+    parser.add_argument("--output-dir", type=Path, default=ROOT / "results")
+    parser.add_argument("--output", "--output-json", dest="output", type=Path,
+                        help="JSON path (default: output-dir/result.json)")
+    parser.add_argument("--md", "--output-md", dest="md", type=Path,
+                        help="Markdown path (default: output-dir/RESULT.md)")
+    args = parser.parse_args(argv)
+    data_dir = args.data_dir.resolve()
+    output = (args.output or args.output_dir / "result.json").resolve()
+    report = (args.md or args.output_dir / "RESULT.md").resolve()
+    try:
+        if output == report:
+            raise ValueError("JSON and Markdown output paths must differ")
+        for path in (output, report):
+            if path.exists():
+                raise ValueError(f"refusing to overwrite {path}; choose new output paths")
+        contract_path = (args.contract or data_dir.parent / "contract.json").resolve()
+        contract, contract_info = read_json(contract_path)
+        check_contract(contract)
+        run, run_info = read_json(data_dir / "run.json")
+        if "contract" in run:
+            if not isinstance(run["contract"], dict):
+                raise ValueError("run.contract must be an object")
+            check_contract(run["contract"], "run.contract")
+        sources = source_manifest(run, contract_info)
+        entries = batch_entries(run, data_dir, FIXED)
+        blocks, manifests = [], []
+        for item in entries:
+            block, manifest = read_batch(item, FIXED)
+            blocks.append(block)
+            manifests.append(manifest)
+        result = score(blocks, [item[0] for item in entries], FIXED)
+        result["created_utc"] = datetime.now(timezone.utc).isoformat()
+        result["input_manifest"] = {"contract": contract_info, "run": run_info,
+                                    "contract_document": contract, "run_document": run,
+                                    "batches": manifests}
+        result["source_manifest"] = sources
+        result["outputs"] = {"json": str(output), "markdown": str(report)}
+        # Serialize before creating either output. Exclusive creation prevents overwrites.
+        encoded = json.dumps(result, ensure_ascii=False, indent=2, allow_nan=False) + "\n"
+        prose = markdown(result)
+        for path in (output, report):
+            path.parent.mkdir(parents=True, exist_ok=True)
+        with output.open("x", encoding="utf-8") as handle:
+            handle.write(encoded)
+        with report.open("x", encoding="utf-8") as handle:
+            handle.write(prose)
+        print(json.dumps({"status": result["status"], "json": str(output), "md": str(report)}))
+        return 2 if result["stop_reasons"] else 0
+    except (OSError, EOFError, ValueError, csv.Error) as error:
+        print(f"safe-insertion scoring error: {error}", file=sys.stderr)
+        return 1
+
+
+if __name__ == "__main__":
+    sys.exit(main())
diff --git a/analysis/safe-insertion-production-20260929/contract.json b/analysis/safe-insertion-production-20260929/contract.json
new file mode 100644
index 000000000..efd9d9cdd
--- /dev/null
+++ b/analysis/safe-insertion-production-20260929/contract.json
@@ -0,0 +1,29 @@
+{
+  "schema": "matching-one.safe-insertion-contract.v1",
+  "namespace": "safe-insertion-independent-20260929",
+  "parent_commit": "eb1e424a43f4bc9dee51c610549c4556af3874ea",
+  "frozen_before_new_data": true,
+  "lattice": "square",
+  "L": 512,
+  "N": 262144,
+  "a": 154646,
+  "b": 155385,
+  "cutoff_source": "unchanged old square-L512 count cutoffs, as in completion-hazard-production-20260929",
+  "probes": 4,
+  "batches": 14,
+  "samples_per_batch": 10000,
+  "workers": 14,
+  "primary": "overlap-weighted early-minus-late mean safe-insertion completion increment Y within exact (primitive direction,integer nu_b)",
+  "early": "rank_b=1 and J1<=a",
+  "late": "rank_b=1 and a<J1<=b",
+  "primary_eligibility": "nu_b<N-b; cells with nu_b=N-b have no safe conditional step and are separately counted",
+  "sampling": "independent uniform site permutations; four with-replacement uniform safe-site probes in separate copies of each rank-one prefix, selected by vacancy rejection using a distinct RNG stream",
+  "probe_unit": "one original prefix; average its four increments before inference, never treat them as independent configurations",
+  "weights": "w=n_early*n_late/(n_early+n_late) on common exact cells; normalize over cells and recompute all support and weights on aligned whole-batch deletion",
+  "mechanistic_map": "in each cell, Delta mean e=(N-b-nu_b)*DeltaY/2, Delta two-step survival=-(N-b-nu_b)*DeltaY/[(N-b)*(N-b-1)]",
+  "secondary": "independent replication of the previously retrospective exact-direction weighted mean nu_b contrast; all rank-one prefixes, including any with no safe step",
+  "model_prediction": "(rank,direction,nu) weak Markov closure requires the primary cohort contrast to be zero. Nonzero population contrast violates a necessary condition; an unresolved contrast does not establish full closure. No prospective sign assumed from L4 because square/triangular signs differ.",
+  "analysis": "one fixed primary mean; full aligned 14-batch delete-one covariance with derived edge/two-step contrasts and declared secondary; no old/new pooling, cutoff/feature/subgroup search, result-dependent sample extension, or exponent fitting",
+  "benchmark": "32 new excluded prefixes, runtime only, separate seed role; anticipated ideal-parallel production time must not exceed 900 seconds to start this block",
+  "scope": "one lattice, one existing size and count clock; no direct finite-window-survival replacement, original-U identification or continuum certificate"
+}
diff --git a/analysis/safe-insertion-production-20260929/data/benchmark-square-L512-b00.csv.gz b/analysis/safe-insertion-production-20260929/data/benchmark-square-L512-b00.csv.gz
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diff --git a/analysis/safe-insertion-production-20260929/data/run.json b/analysis/safe-insertion-production-20260929/data/run.json
new file mode 100644
index 000000000..64f3901d7
--- /dev/null
+++ b/analysis/safe-insertion-production-20260929/data/run.json
@@ -0,0 +1,259 @@
+{
+  "schema": "matching-one.safe-insertion-run.v1",
+  "status": "completed",
+  "pid": 316,
+  "utc_started": "2026-09-29T15:33:24.942039+00:00",
+  "source_commit": "55fa265629701e04b879b774ae7d3b97fefa845f",
+  "source_sha256": {
+    "engine.cpp": "75644ef25ea539a63f713a14890807a480b15f5d7b40c6cdc19409ebd3de69d2",
+    "run.py": "45f0f1bd25ab9437203ae26eba44cdbd6b68554f68cd90dd959edcb797a9a1ca",
+    "contract.json": "5b0eef57c673baebdcaaf7526f8529117687720a5c89b9b12ae794690b534ef6"
+  },
+  "included_geometry_sha256": "42ac8a013cb731f4da4e3121ca429fd475555063e876638ad5d375d195448115",
+  "contract": {
+    "schema": "matching-one.safe-insertion-contract.v1",
+    "namespace": "safe-insertion-independent-20260929",
+    "parent_commit": "eb1e424a43f4bc9dee51c610549c4556af3874ea",
+    "frozen_before_new_data": true,
+    "lattice": "square",
+    "L": 512,
+    "N": 262144,
+    "a": 154646,
+    "b": 155385,
+    "cutoff_source": "unchanged old square-L512 count cutoffs, as in completion-hazard-production-20260929",
+    "probes": 4,
+    "batches": 14,
+    "samples_per_batch": 10000,
+    "workers": 14,
+    "primary": "overlap-weighted early-minus-late mean safe-insertion completion increment Y within exact (primitive direction,integer nu_b)",
+    "early": "rank_b=1 and J1<=a",
+    "late": "rank_b=1 and a<J1<=b",
+    "primary_eligibility": "nu_b<N-b; cells with nu_b=N-b have no safe conditional step and are separately counted",
+    "sampling": "independent uniform site permutations; four with-replacement uniform safe-site probes in separate copies of each rank-one prefix, selected by vacancy rejection using a distinct RNG stream",
+    "probe_unit": "one original prefix; average its four increments before inference, never treat them as independent configurations",
+    "weights": "w=n_early*n_late/(n_early+n_late) on common exact cells; normalize over cells and recompute all support and weights on aligned whole-batch deletion",
+    "mechanistic_map": "in each cell, Delta mean e=(N-b-nu_b)*DeltaY/2, Delta two-step survival=-(N-b-nu_b)*DeltaY/[(N-b)*(N-b-1)]",
+    "secondary": "independent replication of the previously retrospective exact-direction weighted mean nu_b contrast; all rank-one prefixes, including any with no safe step",
+    "model_prediction": "(rank,direction,nu) weak Markov closure requires the primary cohort contrast to be zero. Nonzero population contrast violates a necessary condition; an unresolved contrast does not establish full closure. No prospective sign assumed from L4 because square/triangular signs differ.",
+    "analysis": "one fixed primary mean; full aligned 14-batch delete-one covariance with derived edge/two-step contrasts and declared secondary; no old/new pooling, cutoff/feature/subgroup search, result-dependent sample extension, or exponent fitting",
+    "benchmark": "32 new excluded prefixes, runtime only, separate seed role; anticipated ideal-parallel production time must not exceed 900 seconds to start this block",
+    "scope": "one lattice, one existing size and count clock; no direct finite-window-survival replacement, original-U identification or continuum certificate"
+  },
+  "command": "python3 /workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/run.py --source-commit 55fa265629701e04b879b774ae7d3b97fefa845f",
+  "platform": "Linux-5.10.0-182.0.0.95.r3582_286.hce2.aarch64-aarch64-with-glibc2.34",
+  "machine": "aarch64",
+  "python": "3.9.9",
+  "compiler": "g++ (GCC) 10.3.1\nCopyright (C) 2020 Free Software Foundation, Inc.\nThis is free software; see the source for copying conditions.  There is NO\nwarranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.",
+  "workers": 14,
+  "seed_rule": "first 64 SHA256 bits of matching-one/safe-insertion-independent-20260929/{role}/square/L512/{batch}/{stream}; stream=permutation or safe-probe",
+  "batches": [
+    {
+      "file": "production-square-L512-b00.csv.gz",
+      "batch": 0,
+      "samples": 10000,
+      "seed": "9722445355900932119",
+      "probe_seed": "17701568506019665282",
+      "role": "production",
+      "runtime_seconds": 504.8021701290272,
+      "engine_log": "samples=10000 seconds=504.791",
+      "command": "/workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/engine 512 10000 9722445355900932119 17701568506019665282 155385 /workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/production-square-L512-b00.csv",
+      "sha256": "dfea3bdaa73f93c05078c87b00bfcc1f65f262a353ea9623c58f0c339b904f5f",
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch0"
+    },
+    {
+      "file": "production-square-L512-b01.csv.gz",
+      "batch": 1,
+      "samples": 10000,
+      "seed": "5026903940319098550",
+      "probe_seed": "15814027654412906247",
+      "role": "production",
+      "runtime_seconds": 506.061842552037,
+      "engine_log": "samples=10000 seconds=506.049",
+      "command": "/workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/engine 512 10000 5026903940319098550 15814027654412906247 155385 /workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/production-square-L512-b01.csv",
+      "sha256": "59bd8a19407fcde243ace326f24c1c3ce6af6b7d426e7f2bbea9d94ba5f69bc7",
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch1"
+    },
+    {
+      "file": "production-square-L512-b02.csv.gz",
+      "batch": 2,
+      "samples": 10000,
+      "seed": "7334852068188275347",
+      "probe_seed": "6915056171437877797",
+      "role": "production",
+      "runtime_seconds": 513.5505838650279,
+      "engine_log": "samples=10000 seconds=513.522",
+      "command": "/workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/engine 512 10000 7334852068188275347 6915056171437877797 155385 /workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/production-square-L512-b02.csv",
+      "sha256": "73c50bc11bca07799f728af60e01535b4cc96561d0d7989979c4ec8ac451ad72",
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch2"
+    },
+    {
+      "file": "production-square-L512-b03.csv.gz",
+      "batch": 3,
+      "samples": 10000,
+      "seed": "7761250788365968736",
+      "probe_seed": "10799697386746431051",
+      "role": "production",
+      "runtime_seconds": 512.3049444580683,
+      "engine_log": "samples=10000 seconds=512.276",
+      "command": "/workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/engine 512 10000 7761250788365968736 10799697386746431051 155385 /workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/production-square-L512-b03.csv",
+      "sha256": "dd483db8df4b968e3244983c8462420484807ebca99b118923afb2d6d67b8f00",
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch3"
+    },
+    {
+      "file": "production-square-L512-b04.csv.gz",
+      "batch": 4,
+      "samples": 10000,
+      "seed": "13568940959106988201",
+      "probe_seed": "14598850575054591749",
+      "role": "production",
+      "runtime_seconds": 511.33745101303793,
+      "engine_log": "samples=10000 seconds=511.297",
+      "command": "/workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/engine 512 10000 13568940959106988201 14598850575054591749 155385 /workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/production-square-L512-b04.csv",
+      "sha256": "617da5123f5090265e2e07ec40762ffbb0edf4d9b8cf7dccecc64708a198ec69",
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch4"
+    },
+    {
+      "file": "production-square-L512-b05.csv.gz",
+      "batch": 5,
+      "samples": 10000,
+      "seed": "14221577344058561937",
+      "probe_seed": "14247619095644429396",
+      "role": "production",
+      "runtime_seconds": 516.7602939550998,
+      "engine_log": "samples=10000 seconds=516.724",
+      "command": "/workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/engine 512 10000 14221577344058561937 14247619095644429396 155385 /workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/production-square-L512-b05.csv",
+      "sha256": "3c94ed5c60759faaa225af35cd8e12d6996c07d7913ee2385fc78d60c4e8ddd3",
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch5"
+    },
+    {
+      "file": "production-square-L512-b06.csv.gz",
+      "batch": 6,
+      "samples": 10000,
+      "seed": "2117620933003635365",
+      "probe_seed": "6735888352822314438",
+      "role": "production",
+      "runtime_seconds": 512.6706185770454,
+      "engine_log": "samples=10000 seconds=512.63",
+      "command": "/workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/engine 512 10000 2117620933003635365 6735888352822314438 155385 /workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/production-square-L512-b06.csv",
+      "sha256": "de9eee185c039bfefbd29ebc1cd87c7962f475860ce1aa68bf67ef5a13cd18f4",
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch6"
+    },
+    {
+      "file": "production-square-L512-b07.csv.gz",
+      "batch": 7,
+      "samples": 10000,
+      "seed": "10091810389689459151",
+      "probe_seed": "14046470669360462118",
+      "role": "production",
+      "runtime_seconds": 508.336309403996,
+      "engine_log": "samples=10000 seconds=508.296",
+      "command": "/workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/engine 512 10000 10091810389689459151 14046470669360462118 155385 /workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/production-square-L512-b07.csv",
+      "sha256": "c95346db4ed6c55b20bfc6906b50ba8bbfbfee71488c1a5c0e0a23e6caa3b757",
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch7"
+    },
+    {
+      "file": "production-square-L512-b08.csv.gz",
+      "batch": 8,
+      "samples": 10000,
+      "seed": "11008454278899210265",
+      "probe_seed": "2129070058936452793",
+      "role": "production",
+      "runtime_seconds": 515.3228207649663,
+      "engine_log": "samples=10000 seconds=515.283",
+      "command": "/workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/engine 512 10000 11008454278899210265 2129070058936452793 155385 /workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/production-square-L512-b08.csv",
+      "sha256": "30372e4f839caa13574317aae26dbdeb733e90d306d6c3518898b69bb9ed33e1",
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch8"
+    },
+    {
+      "file": "production-square-L512-b09.csv.gz",
+      "batch": 9,
+      "samples": 10000,
+      "seed": "17332517276911702520",
+      "probe_seed": "10997333967657406408",
+      "role": "production",
+      "runtime_seconds": 516.0273232499603,
+      "engine_log": "samples=10000 seconds=515.988",
+      "command": "/workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/engine 512 10000 17332517276911702520 10997333967657406408 155385 /workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/production-square-L512-b09.csv",
+      "sha256": "51eec39ed48afa403935abd2f6433156c5485ab133dcd8d03b4f8a47e814f27e",
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch9"
+    },
+    {
+      "file": "production-square-L512-b10.csv.gz",
+      "batch": 10,
+      "samples": 10000,
+      "seed": "17211728716206152366",
+      "probe_seed": "18382265747265082668",
+      "role": "production",
+      "runtime_seconds": 515.5240167109296,
+      "engine_log": "samples=10000 seconds=515.482",
+      "command": "/workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/engine 512 10000 17211728716206152366 18382265747265082668 155385 /workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/production-square-L512-b10.csv",
+      "sha256": "4e51e12a3efa950ececd20eb2f07b6c331d02f503d3fb9e6730341f0d56bd556",
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch10"
+    },
+    {
+      "file": "production-square-L512-b11.csv.gz",
+      "batch": 11,
+      "samples": 10000,
+      "seed": "13292054708231777266",
+      "probe_seed": "13917358584408871786",
+      "role": "production",
+      "runtime_seconds": 515.0872192179086,
+      "engine_log": "samples=10000 seconds=515.05",
+      "command": "/workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/engine 512 10000 13292054708231777266 13917358584408871786 155385 /workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/production-square-L512-b11.csv",
+      "sha256": "e91819af15ede7b1b73e35339312f8214ef73d87447e07ad3b9f82ddef8a420a",
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch11"
+    },
+    {
+      "file": "production-square-L512-b12.csv.gz",
+      "batch": 12,
+      "samples": 10000,
+      "seed": "9255174391098419870",
+      "probe_seed": "16078787438699247451",
+      "role": "production",
+      "runtime_seconds": 514.9042300319998,
+      "engine_log": "samples=10000 seconds=514.866",
+      "command": "/workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/engine 512 10000 9255174391098419870 16078787438699247451 155385 /workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/production-square-L512-b12.csv",
+      "sha256": "c23466b9802fc28cc0f2044d6cd9c595e1d79f3e7d1ddf50b5d4b395d0d0ebd3",
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch12"
+    },
+    {
+      "file": "production-square-L512-b13.csv.gz",
+      "batch": 13,
+      "samples": 10000,
+      "seed": "4111027655712514342",
+      "probe_seed": "8225829675307131803",
+      "role": "production",
+      "runtime_seconds": 513.0653394469991,
+      "engine_log": "samples=10000 seconds=513.039",
+      "command": "/workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/engine 512 10000 4111027655712514342 8225829675307131803 155385 /workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/production-square-L512-b13.csv",
+      "sha256": "01e55fa4457010955e71cbc38ef8d43568ec85d0eb30157735ddc9be19254008",
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch13"
+    }
+  ],
+  "benchmarks": [
+    {
+      "file": "benchmark-square-L512-b00.csv.gz",
+      "batch": 0,
+      "samples": 32,
+      "seed": "10772643973641963268",
+      "probe_seed": "15816637589218709745",
+      "role": "benchmark",
+      "runtime_seconds": 1.1327592680463567,
+      "engine_log": "samples=32 seconds=1.12472",
+      "command": "/workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/engine 512 32 10772643973641963268 15816637589218709745 155385 /workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/benchmark-square-L512-b00.csv",
+      "sha256": "6b021d15cb39aecd8aec661d6062a066c8d6fdedf008a4a92e64090378c55427",
+      "dependency_group": "excluded-benchmark"
+    }
+  ],
+  "cgroup": {
+    "cpu/cpu.cfs_quota_us": "1450000",
+    "cpu/cpu.cfs_period_us": "100000",
+    "memory/memory.limit_in_bytes": "26843545600"
+  },
+  "build_command": "g++ -O3 -std=c++17 /workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/engine.cpp -o /workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/engine",
+  "binary_sha256": "50e609ac4da32bdd255d88b0dd13077f95ba1af35dda18389907d7595676377c",
+  "physical_witness_control": "mask=4403 next_nu_counts=6,4,0\nmask=4405 next_nu_counts=6,2,2",
+  "estimated_production_wall_seconds": 353.9872712644865,
+  "wall_seconds": 519.8431637079921,
+  "utc_finished": "2026-09-29T15:42:04.785202+00:00"
+}
diff --git a/analysis/safe-insertion-production-20260929/engine.cpp b/analysis/safe-insertion-production-20260929/engine.cpp
new file mode 100644
index 000000000..c03d58a8f
--- /dev/null
+++ b/analysis/safe-insertion-production-20260929/engine.cpp
@@ -0,0 +1,85 @@
+// Independent-prefix safe-insertion probes. Reuse unchanged tested geometry.
+#define main original_completion_engine_main
+#include "../completion-hazard-production-20260929/engine.cpp"
+#undef main
+
+struct SafeObservation {
+    int j1=0, rank_b=0, dx=0, dy=0, nu=-1;
+    std::array<int,4> y{{-1,-1,-1,-1}};
+};
+
+SafeObservation probe(Geometry& g,const std::vector<int>& order,int b,
+                      std::mt19937_64& probe_rng) {
+    g.reset(); SafeObservation o;
+    int N=static_cast<int>(order.size());
+    for(int k=1;k<=b;++k){
+        o.rank_b=g.insert(order[k-1]);
+        if(o.rank_b>=1 && !o.j1)o.j1=k;
+        if(o.rank_b==2)return o; // rank cannot subsequently decrease
+    }
+    if(o.rank_b!=1)return o;
+    auto d=g.direction();o.dx=d.first;o.dy=d.second;
+    o.nu=g.completion_count();
+    if(o.nu==N-b)return o; // no safe insertion; conditional probe undefined
+    for(int j=0;j<4;++j){
+        // An independent probe stream chooses uniformly among vacancies;
+        // reject completions, yielding uniform S(A). Copies share no edits.
+        while(true){
+            int v=order[b+bounded(probe_rng,N-b)];
+            Geometry copy=g;
+            if(copy.insert(v)==2)continue;
+            if(copy.direction()!=d)throw std::logic_error("safe direction changed");
+            o.y[j]=copy.completion_count()-o.nu;
+            if(o.y[j]<0)throw std::logic_error("completion count decreased");
+            break;
+        }
+    }
+    return o;
+}
+
+int main(int argc,char**argv){
+    try{
+        if(argc==2 && std::string(argv[1])=="control"){
+            // One known physical pair, not a repeat of the L4 census.
+            const std::array<int,2> masks{{4403,4405}};
+            const std::array<std::array<int,3>,2> expected{{{{6,4,0}},{{6,2,2}}}};
+            for(int i=0;i<2;++i){
+                Geometry g(4,false);int rank=0;
+                for(int v=0;v<16;++v)if(masks[i]>>v&1)rank=g.insert(v);
+                if(rank!=1 || g.direction()!=std::make_pair(0,1) || g.completion_count()!=0)
+                    throw std::logic_error("physical witness prefix mismatch");
+                std::array<int,3> counts{{0,0,0}};
+                for(int v=0;v<16;++v)if(!(masks[i]>>v&1)){
+                    Geometry copy=g;
+                    if(copy.insert(v)!=1)throw std::logic_error("witness unsafe");
+                    int nu=copy.completion_count();
+                    if(nu<0 || nu>2)throw std::logic_error("witness degree bounds");
+                    ++counts[nu];
+                }
+                if(counts!=expected[i])throw std::logic_error("witness successor counts");
+                std::cout<<"mask="<<masks[i]<<" next_nu_counts="<<counts[0]<<','<<counts[1]<<','<<counts[2]<<'\n';
+            }
+            return 0;
+        }
+        if(argc!=7)throw std::invalid_argument("L samples permutation_seed probe_seed b output.csv");
+        int L=std::stoi(argv[1]),N=L*L,b=std::stoi(argv[5]);
+        auto samples=std::stoull(argv[2]),seed=std::stoull(argv[3]),probe_seed=std::stoull(argv[4]);
+        if(!samples || b<1 || b>N-2)throw std::invalid_argument("sample/count bounds");
+        Geometry g(L,false);std::vector<int>order(N);
+        std::mt19937_64 rng(seed),probe_rng(probe_seed);
+        std::ofstream out(argv[6]);if(!out)throw std::runtime_error("cannot open output");
+        out<<"J1,rank_b,dx_b,dy_b,nu_b,Y0,Y1,Y2,Y3\n";
+        auto start=std::chrono::steady_clock::now();
+        for(std::uint64_t i=0;i<samples;++i){
+            std::iota(order.begin(),order.end(),0);
+            for(int v=N-1;v>0;--v)std::swap(order[v],order[bounded(rng,v+1)]);
+            auto o=probe(g,order,b,probe_rng);
+            out<<o.j1<<','<<o.rank_b<<','<<o.dx<<','<<o.dy<<','<<o.nu;
+            for(auto y:o.y)out<<','<<y;
+            out<<'\n';
+        }
+        out.close();if(!out)throw std::runtime_error("write failed");
+        double seconds=std::chrono::duration<double>(std::chrono::steady_clock::now()-start).count();
+        std::cout<<"samples="<<samples<<" seconds="<<seconds<<'\n';return 0;
+    }catch(const std::exception&e){std::cerr<<e.what()<<'\n';return 1;}
+}
diff --git a/analysis/safe-insertion-production-20260929/execution-receipt.json b/analysis/safe-insertion-production-20260929/execution-receipt.json
new file mode 100644
index 000000000..094a9a039
--- /dev/null
+++ b/analysis/safe-insertion-production-20260929/execution-receipt.json
@@ -0,0 +1,49 @@
+{
+  "schema": "matching-one.safe-insertion-execution.v1",
+  "status": "acquired_scored_downloaded",
+  "date": "2026-09-29",
+  "acquisition_commit": "55fa265629701e04b879b774ae7d3b97fefa845f",
+  "scorer_commit_before_target_readout": "f340aa55134213be71d22941fae2ccb7fd068cb7",
+  "host": "DevEnvC_TV2N0X",
+  "instance_id": "4a8d1d443419434889e49148ed0a7ba6",
+  "remote_root": "/workspace/Matching-One-TV2N0X/safe-insertion-20260929",
+  "remote_directory": "/workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929",
+  "acquisition_command": "python3 run.py --source-commit 55fa265629701e04b879b774ae7d3b97fefa845f",
+  "architecture": "aarch64",
+  "remote_python": "3.9.9",
+  "compiler": "GCC 10.3.1, installed gcc-c++ dependencies after restart",
+  "nproc": 16,
+  "cgroup_cpus": 14.5,
+  "cgroup_memory_gib": 25,
+  "workers": 14,
+  "observed_worker_rss_kib_range": [53420, 53752],
+  "rss_is_snapshot_not_peak": true,
+  "production_prefixes": 140000,
+  "rank_one_prefixes": 55028,
+  "valid_safe_probes": 220112,
+  "probes_per_eligible_prefix": 4,
+  "excluded_benchmark_prefixes": 32,
+  "acquisition_wall_seconds": 519.8431637079921,
+  "benchmark_estimated_ideal_parallel_seconds": 353.99,
+  "scoring_command": "/Users/lc/python-envs/research-py311/bin/python analysis/safe-insertion-production-20260929/analyze.py",
+  "scoring_location": "local existing Python 3.11 research environment",
+  "scoring_passes_on_complete_production": 1,
+  "checks": [
+    "Two known physical L4 witness rows, 20 vacant-site probes, once in remote runner",
+    "Scorer small in-memory arithmetic checks by implementation subagent before production scoring",
+    "Main-agent read of complete scorer and estimands before target readout",
+    "Single scoring pass verifies all declared rows, semantics, hashes and aligned batch structure",
+    "JSON parse and git whitespace check; no full suite or old census rerun"
+  ],
+  "run_manifest_sha256": "8096d3d0a327604dd921e9c82d022daa685e2c7d6c4bf6e5b87c195b1d3dbf89",
+  "result_json_sha256": "48134aa5d1a403b72cccfb7c32ac46f6cae58c4ca8c1dfb70a0c7badd53ae178",
+  "binary_sha256": "50e609ac4da32bdd255d88b0dd13077f95ba1af35dda18389907d7595676377c",
+  "included_geometry_sha256": "42ac8a013cb731f4da4e3121ca429fd475555063e876638ad5d375d195448115",
+  "source_freeze_scope": "contract and acquisition source before samples; scoring implementation committed while production ran, before any response readout",
+  "post_run": "No research processes remained in fresh name-only process listing; owned tunnel stopped, instance powered off and Ready verified. Remote raw data preserved.",
+  "ssh_key_reset_count": 1,
+  "other_instances_operated": false,
+  "old_archives_modified": false,
+  "old_new_pooling": false,
+  "result_dependent_extension": false
+}
diff --git a/analysis/safe-insertion-production-20260929/results/RESULT.md b/analysis/safe-insertion-production-20260929/results/RESULT.md
new file mode 100644
index 000000000..788f0cf8a
--- /dev/null
+++ b/analysis/safe-insertion-production-20260929/results/RESULT.md
@@ -0,0 +1,38 @@
+# Prospective square-L512 safe-insertion score
+
+Status: **scored**. Fixed 14 batches x 10000 prefixes; q=4.
+
+All differences are early minus late. The point estimates are pooled plug-in estimates.
+
+| Metric | Estimate | Aligned batch SE |
+|---|---:|---:|
+| deltaY | 0.006681826608 | 0.01208705518 |
+| deltaE | 356.3977798 | 645.0460174 |
+| deltaZ2 | -6.254031427e-08 | 1.131920089e-07 |
+| secondary_deltaNu | -0.03986511616 | 0.5698779455 |
+| weighted_earlyY | 0.09563676962 | 0.007712358034 |
+| weighted_lateY | 0.08895494301 | 0.01060802794 |
+
+Rows: 140000; rank counts: {'rank0': 42329, 'rank1': 55028, 'rank2': 42643}.
+Rank-one prefixes: 55028; safe: 55028; dead (nu=m): 0.
+Valid probes: 220112; nonzero probes: 1233; prefixes with any nonzero probe: 1215.
+
+| Comparison / cohort | Eligible | Supported | Coverage of eligible |
+|---|---:|---:|---:|
+| primary / early | 24201 | 24109 | 0.9961985042 |
+| primary / late | 30827 | 30354 | 0.9846563078 |
+| secondary / early | 24201 | 24201 | 1 |
+| secondary / late | 30827 | 30821 | 0.9998053654 |
+
+Primary excludes dead prefixes because a safe-step mean is undefined there; secondary includes every rank-one prefix, including dead prefixes.
+
+Within each shared exact (D,nu) cell: deltaE=(m-nu)*deltaY/2 and deltaZ2=-(m-nu)*deltaY/[m(m-1)]. The same normalized overlap weights combine all primary quantities. The exact-D secondary recomputes its own weights.
+
+The JSON records exact integer sufficient statistics, all cells, support coverage, all 14 aligned deletion vectors, the full 6x6 covariance and source/input manifests. Probes are averaged within prefix; metrics are not independent and covariance is necessarily singular. No covariance inverse or independent-metric test is used.
+
+- The primary tests one safe-step completion-creation mean at fixed exact (D,nu_b).
+- Derived edge and two-step-survival contrasts use cell-specific s before aggregation.
+- Secondary exact-D nu replication is independent of the old block, not of this primary.
+- A null weighted moment can hide cell cancellation and does not establish recursive closure.
+- Finite square L512 insertion clock only; no limiting, label-clock, or universality claim.
+- No historical data pooling, subgroup search, fitted exponent, or acquisition is performed.
diff --git a/analysis/safe-insertion-production-20260929/results/result.json b/analysis/safe-insertion-production-20260929/results/result.json
new file mode 100644
index 000000000..853f9823a
--- /dev/null
+++ b/analysis/safe-insertion-production-20260929/results/result.json
@@ -0,0 +1,74996 @@
+{
+  "schema": "matching-one.safe-insertion-score.v1",
+  "status": "scored",
+  "stop_reasons": [],
+  "parameters": {
+    "N": 262144,
+    "a": 154646,
+    "b": 155385,
+    "probes": 4,
+    "batches": 14,
+    "samples_per_batch": 10000
+  },
+  "m": 106759,
+  "metric_order": [
+    "deltaY",
+    "deltaE",
+    "deltaZ2",
+    "secondary_deltaNu",
+    "weighted_earlyY",
+    "weighted_lateY"
+  ],
+  "point_vector": [
+    0.006681826607750889,
+    356.39777976703607,
+    -6.254031426670283e-08,
+    -0.03986511616311962,
+    0.0956367696176017,
+    0.08895494300985081
+  ],
+  "point_estimates": {
+    "deltaY": 0.006681826607750889,
+    "deltaE": 356.39777976703607,
+    "deltaZ2": -6.254031426670283e-08,
+    "secondary_deltaNu": -0.03986511616311962,
+    "weighted_earlyY": 0.0956367696176017,
+    "weighted_lateY": 0.08895494300985081
+  },
+  "standard_errors": {
+    "deltaY": 0.012087055177121728,
+    "deltaE": 645.0460173744311,
+    "deltaZ2": 1.1319200885440885e-07,
+    "secondary_deltaNu": 0.5698779454745956,
+    "weighted_earlyY": 0.0077123580341316835,
+    "weighted_lateY": 0.010608027939709696
+  },
+  "totals": {
+    "rows": 140000,
+    "rank_counts": {
+      "rank0": 42329,
+      "rank1": 55028,
+      "rank2": 42643
+    },
+    "outside_rank1": 84972,
+    "sign_canonicalizations": 0,
+    "rank1": {
+      "prefixes": 55028,
+      "safe_prefixes": 55028,
+      "dead_prefixes": 0,
+      "sum_nu_b": 5496855,
+      "sum_nu_b_squared": 835742557,
+      "mean_nu_b": 99.89196409100822,
+      "valid_probes": 220112,
+      "nonzero_probes": 1233,
+      "nonzero_prefixes": 1215,
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+    "point_estimator": "pooled plug-in; no jackknife bias correction",
+    "dependence": "four probes share one prefix; all six metrics share one new block",
+    "singularity": "deltaZ2=-2*deltaE/[m*(m-1)] and deltaY=weighted_earlyY-weighted_lateY"
+  },
+  "interpretation": [
+    "The primary tests one safe-step completion-creation mean at fixed exact (D,nu_b).",
+    "Derived edge and two-step-survival contrasts use cell-specific s before aggregation.",
+    "Secondary exact-D nu replication is independent of the old block, not of this primary.",
+    "A null weighted moment can hide cell cancellation and does not establish recursive closure.",
+    "Finite square L512 insertion clock only; no limiting, label-clock, or universality claim.",
+    "No historical data pooling, subgroup search, fitted exponent, or acquisition is performed."
+  ],
+  "created_utc": "2026-09-29T15:42:50.251222+00:00",
+  "input_manifest": {
+    "contract": {
+      "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/safe-insertion-production-20260929/contract.json",
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+      "namespace": "safe-insertion-independent-20260929",
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+      "lattice": "square",
+      "L": 512,
+      "N": 262144,
+      "a": 154646,
+      "b": 155385,
+      "cutoff_source": "unchanged old square-L512 count cutoffs, as in completion-hazard-production-20260929",
+      "probes": 4,
+      "batches": 14,
+      "samples_per_batch": 10000,
+      "workers": 14,
+      "primary": "overlap-weighted early-minus-late mean safe-insertion completion increment Y within exact (primitive direction,integer nu_b)",
+      "early": "rank_b=1 and J1<=a",
+      "late": "rank_b=1 and a<J1<=b",
+      "primary_eligibility": "nu_b<N-b; cells with nu_b=N-b have no safe conditional step and are separately counted",
+      "sampling": "independent uniform site permutations; four with-replacement uniform safe-site probes in separate copies of each rank-one prefix, selected by vacancy rejection using a distinct RNG stream",
+      "probe_unit": "one original prefix; average its four increments before inference, never treat them as independent configurations",
+      "weights": "w=n_early*n_late/(n_early+n_late) on common exact cells; normalize over cells and recompute all support and weights on aligned whole-batch deletion",
+      "mechanistic_map": "in each cell, Delta mean e=(N-b-nu_b)*DeltaY/2, Delta two-step survival=-(N-b-nu_b)*DeltaY/[(N-b)*(N-b-1)]",
+      "secondary": "independent replication of the previously retrospective exact-direction weighted mean nu_b contrast; all rank-one prefixes, including any with no safe step",
+      "model_prediction": "(rank,direction,nu) weak Markov closure requires the primary cohort contrast to be zero. Nonzero population contrast violates a necessary condition; an unresolved contrast does not establish full closure. No prospective sign assumed from L4 because square/triangular signs differ.",
+      "analysis": "one fixed primary mean; full aligned 14-batch delete-one covariance with derived edge/two-step contrasts and declared secondary; no old/new pooling, cutoff/feature/subgroup search, result-dependent sample extension, or exponent fitting",
+      "benchmark": "32 new excluded prefixes, runtime only, separate seed role; anticipated ideal-parallel production time must not exceed 900 seconds to start this block",
+      "scope": "one lattice, one existing size and count clock; no direct finite-window-survival replacement, original-U identification or continuum certificate"
+    },
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+        "cutoff_source": "unchanged old square-L512 count cutoffs, as in completion-hazard-production-20260929",
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+        "workers": 14,
+        "primary": "overlap-weighted early-minus-late mean safe-insertion completion increment Y within exact (primitive direction,integer nu_b)",
+        "early": "rank_b=1 and J1<=a",
+        "late": "rank_b=1 and a<J1<=b",
+        "primary_eligibility": "nu_b<N-b; cells with nu_b=N-b have no safe conditional step and are separately counted",
+        "sampling": "independent uniform site permutations; four with-replacement uniform safe-site probes in separate copies of each rank-one prefix, selected by vacancy rejection using a distinct RNG stream",
+        "probe_unit": "one original prefix; average its four increments before inference, never treat them as independent configurations",
+        "weights": "w=n_early*n_late/(n_early+n_late) on common exact cells; normalize over cells and recompute all support and weights on aligned whole-batch deletion",
+        "mechanistic_map": "in each cell, Delta mean e=(N-b-nu_b)*DeltaY/2, Delta two-step survival=-(N-b-nu_b)*DeltaY/[(N-b)*(N-b-1)]",
+        "secondary": "independent replication of the previously retrospective exact-direction weighted mean nu_b contrast; all rank-one prefixes, including any with no safe step",
+        "model_prediction": "(rank,direction,nu) weak Markov closure requires the primary cohort contrast to be zero. Nonzero population contrast violates a necessary condition; an unresolved contrast does not establish full closure. No prospective sign assumed from L4 because square/triangular signs differ.",
+        "analysis": "one fixed primary mean; full aligned 14-batch delete-one covariance with derived edge/two-step contrasts and declared secondary; no old/new pooling, cutoff/feature/subgroup search, result-dependent sample extension, or exponent fitting",
+        "benchmark": "32 new excluded prefixes, runtime only, separate seed role; anticipated ideal-parallel production time must not exceed 900 seconds to start this block",
+        "scope": "one lattice, one existing size and count clock; no direct finite-window-survival replacement, original-U identification or continuum certificate"
+      },
+      "command": "python3 /workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/run.py --source-commit 55fa265629701e04b879b774ae7d3b97fefa845f",
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+      "compiler": "g++ (GCC) 10.3.1\nCopyright (C) 2020 Free Software Foundation, Inc.\nThis is free software; see the source for copying conditions.  There is NO\nwarranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.",
+      "workers": 14,
+      "seed_rule": "first 64 SHA256 bits of matching-one/safe-insertion-independent-20260929/{role}/square/L512/{batch}/{stream}; stream=permutation or safe-probe",
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+        "sha256": "51eec39ed48afa403935abd2f6433156c5485ab133dcd8d03b4f8a47e814f27e",
+        "sha256_verified": true,
+        "rows_verified": 10000,
+        "declared": {
+          "file": "production-square-L512-b09.csv.gz",
+          "batch": 9,
+          "samples": 10000,
+          "seed": "17332517276911702520",
+          "probe_seed": "10997333967657406408",
+          "role": "production",
+          "runtime_seconds": 516.0273232499603,
+          "engine_log": "samples=10000 seconds=515.988",
+          "command": "/workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/engine 512 10000 17332517276911702520 10997333967657406408 155385 /workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/production-square-L512-b09.csv",
+          "sha256": "51eec39ed48afa403935abd2f6433156c5485ab133dcd8d03b4f8a47e814f27e",
+          "dependency_group": "safe-insertion-independent-20260929/square/L512/batch9"
+        }
+      },
+      {
+        "batch": 10,
+        "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/safe-insertion-production-20260929/data/production-square-L512-b10.csv.gz",
+        "bytes": 32713,
+        "sha256": "4e51e12a3efa950ececd20eb2f07b6c331d02f503d3fb9e6730341f0d56bd556",
+        "sha256_verified": true,
+        "rows_verified": 10000,
+        "declared": {
+          "file": "production-square-L512-b10.csv.gz",
+          "batch": 10,
+          "samples": 10000,
+          "seed": "17211728716206152366",
+          "probe_seed": "18382265747265082668",
+          "role": "production",
+          "runtime_seconds": 515.5240167109296,
+          "engine_log": "samples=10000 seconds=515.482",
+          "command": "/workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/engine 512 10000 17211728716206152366 18382265747265082668 155385 /workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/production-square-L512-b10.csv",
+          "sha256": "4e51e12a3efa950ececd20eb2f07b6c331d02f503d3fb9e6730341f0d56bd556",
+          "dependency_group": "safe-insertion-independent-20260929/square/L512/batch10"
+        }
+      },
+      {
+        "batch": 11,
+        "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/safe-insertion-production-20260929/data/production-square-L512-b11.csv.gz",
+        "bytes": 32781,
+        "sha256": "e91819af15ede7b1b73e35339312f8214ef73d87447e07ad3b9f82ddef8a420a",
+        "sha256_verified": true,
+        "rows_verified": 10000,
+        "declared": {
+          "file": "production-square-L512-b11.csv.gz",
+          "batch": 11,
+          "samples": 10000,
+          "seed": "13292054708231777266",
+          "probe_seed": "13917358584408871786",
+          "role": "production",
+          "runtime_seconds": 515.0872192179086,
+          "engine_log": "samples=10000 seconds=515.05",
+          "command": "/workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/engine 512 10000 13292054708231777266 13917358584408871786 155385 /workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/production-square-L512-b11.csv",
+          "sha256": "e91819af15ede7b1b73e35339312f8214ef73d87447e07ad3b9f82ddef8a420a",
+          "dependency_group": "safe-insertion-independent-20260929/square/L512/batch11"
+        }
+      },
+      {
+        "batch": 12,
+        "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/safe-insertion-production-20260929/data/production-square-L512-b12.csv.gz",
+        "bytes": 32789,
+        "sha256": "c23466b9802fc28cc0f2044d6cd9c595e1d79f3e7d1ddf50b5d4b395d0d0ebd3",
+        "sha256_verified": true,
+        "rows_verified": 10000,
+        "declared": {
+          "file": "production-square-L512-b12.csv.gz",
+          "batch": 12,
+          "samples": 10000,
+          "seed": "9255174391098419870",
+          "probe_seed": "16078787438699247451",
+          "role": "production",
+          "runtime_seconds": 514.9042300319998,
+          "engine_log": "samples=10000 seconds=514.866",
+          "command": "/workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/engine 512 10000 9255174391098419870 16078787438699247451 155385 /workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/production-square-L512-b12.csv",
+          "sha256": "c23466b9802fc28cc0f2044d6cd9c595e1d79f3e7d1ddf50b5d4b395d0d0ebd3",
+          "dependency_group": "safe-insertion-independent-20260929/square/L512/batch12"
+        }
+      },
+      {
+        "batch": 13,
+        "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/safe-insertion-production-20260929/data/production-square-L512-b13.csv.gz",
+        "bytes": 32776,
+        "sha256": "01e55fa4457010955e71cbc38ef8d43568ec85d0eb30157735ddc9be19254008",
+        "sha256_verified": true,
+        "rows_verified": 10000,
+        "declared": {
+          "file": "production-square-L512-b13.csv.gz",
+          "batch": 13,
+          "samples": 10000,
+          "seed": "4111027655712514342",
+          "probe_seed": "8225829675307131803",
+          "role": "production",
+          "runtime_seconds": 513.0653394469991,
+          "engine_log": "samples=10000 seconds=513.039",
+          "command": "/workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/engine 512 10000 4111027655712514342 8225829675307131803 155385 /workspace/Matching-One-TV2N0X/safe-insertion-20260929/analysis/safe-insertion-production-20260929/safe-insertion-build-3vxfi8uh/production-square-L512-b13.csv",
+          "sha256": "01e55fa4457010955e71cbc38ef8d43568ec85d0eb30157735ddc9be19254008",
+          "dependency_group": "safe-insertion-independent-20260929/square/L512/batch13"
+        }
+      }
+    ]
+  },
+  "source_manifest": {
+    "local_files_at_scoring": [
+      {
+        "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/safe-insertion-production-20260929/analyze.py",
+        "bytes": 25821,
+        "sha256": "bba96add5df9c37ffb7a59ff80c9bdcace2911191dec6278daa1844ab0772e8f"
+      },
+      {
+        "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/safe-insertion-production-20260929/ESTIMANDS.md",
+        "bytes": 6016,
+        "sha256": "f7fb019c530599375bcb16cdafb75cf567c4cab4fecd6832cb651dee70aa4f22"
+      },
+      {
+        "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/safe-insertion-production-20260929/engine.cpp",
+        "bytes": 4124,
+        "sha256": "75644ef25ea539a63f713a14890807a480b15f5d7b40c6cdc19409ebd3de69d2"
+      },
+      {
+        "path": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/safe-insertion-production-20260929/run.py",
+        "bytes": 6149,
+        "sha256": "45f0f1bd25ab9437203ae26eba44cdbd6b68554f68cd90dd959edcb797a9a1ca"
+      }
+    ],
+    "acquisition_declared_source_commit": "55fa265629701e04b879b774ae7d3b97fefa845f",
+    "acquisition_declared_source_sha256": {
+      "engine.cpp": "75644ef25ea539a63f713a14890807a480b15f5d7b40c6cdc19409ebd3de69d2",
+      "run.py": "45f0f1bd25ab9437203ae26eba44cdbd6b68554f68cd90dd959edcb797a9a1ca",
+      "contract.json": "5b0eef57c673baebdcaaf7526f8529117687720a5c89b9b12ae794690b534ef6"
+    },
+    "note": "Local scoring hashes and acquisition-declared hashes are distinct provenance; hash/range checks do not prove topology, uniform safe sampling, or seed independence."
+  },
+  "outputs": {
+    "json": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/safe-insertion-production-20260929/results/result.json",
+    "markdown": "/private/tmp/matching-frontier-20260929-Jk39nH/analysis/safe-insertion-production-20260929/results/RESULT.md"
+  }
+}
diff --git a/analysis/safe-insertion-production-20260929/run.log b/analysis/safe-insertion-production-20260929/run.log
new file mode 100644
index 000000000..1a1c5e8f8
--- /dev/null
+++ b/analysis/safe-insertion-production-20260929/run.log
@@ -0,0 +1,19 @@
+mask=4403 next_nu_counts=6,4,0
+mask=4405 next_nu_counts=6,2,2
+benchmark batch=0: samples=32 seconds=1.12472
+Estimated ideal-parallel production wall: 353.99s
+production batch=0: samples=10000 seconds=504.791
+production batch=1: samples=10000 seconds=506.049
+production batch=7: samples=10000 seconds=508.296
+production batch=4: samples=10000 seconds=511.297
+production batch=3: samples=10000 seconds=512.276
+production batch=6: samples=10000 seconds=512.63
+production batch=13: samples=10000 seconds=513.039
+production batch=2: samples=10000 seconds=513.522
+production batch=12: samples=10000 seconds=514.866
+production batch=11: samples=10000 seconds=515.05
+production batch=8: samples=10000 seconds=515.283
+production batch=10: samples=10000 seconds=515.482
+production batch=9: samples=10000 seconds=515.988
+production batch=5: samples=10000 seconds=516.724
+Completed 14 batches in 519.84s
diff --git a/analysis/safe-insertion-production-20260929/run.py b/analysis/safe-insertion-production-20260929/run.py
new file mode 100644
index 000000000..3efa37cfe
--- /dev/null
+++ b/analysis/safe-insertion-production-20260929/run.py
@@ -0,0 +1,124 @@
+#!/usr/bin/env python3
+"""One fixed independent safe-insertion block, Python stdlib and C++17."""
+import argparse
+from concurrent.futures import ThreadPoolExecutor, as_completed
+from datetime import datetime, timezone
+import gzip
+import hashlib
+import json
+import os
+from pathlib import Path
+import platform
+import shlex
+import subprocess
+import tempfile
+import time
+
+HERE = Path(__file__).resolve().parent
+PARENT = HERE.parent / "completion-hazard-production-20260929/engine.cpp"
+
+
+def sha(path):
+    return hashlib.sha256(path.read_bytes()).hexdigest()
+
+
+def seed(role, batch, stream):
+    name = f"matching-one/safe-insertion-independent-20260929/{role}/square/L512/{batch}/{stream}"
+    return int.from_bytes(hashlib.sha256(name.encode()).digest()[:8], "big")
+
+
+def main():
+    ap = argparse.ArgumentParser()
+    ap.add_argument("--source-commit", required=True)
+    ap.add_argument("--compiler", default="g++")
+    ap.add_argument("--output", type=Path, default=HERE / "data")
+    args = ap.parse_args()
+    contract = json.loads((HERE / "contract.json").read_text())
+    args.output.mkdir(parents=True, exist_ok=True)
+    manifest = args.output / "run.json"
+    if manifest.exists():
+        raise SystemExit("Existing run is not overwritten or silently resumed")
+    started = time.monotonic()
+    workers = contract["workers"]
+    run = {
+        "schema": "matching-one.safe-insertion-run.v1", "status": "building", "pid": os.getpid(),
+        "utc_started": datetime.now(timezone.utc).isoformat(), "source_commit": args.source_commit,
+        "source_sha256": {name: sha(HERE / name) for name in ("engine.cpp", "run.py", "contract.json")},
+        "included_geometry_sha256": sha(PARENT), "contract": contract,
+        "command": shlex.join(["python3", str(Path(__file__).resolve()), *os.sys.argv[1:]]),
+        "platform": platform.platform(), "machine": platform.machine(), "python": platform.python_version(),
+        "compiler": subprocess.check_output([args.compiler, "--version"], text=True).strip(),
+        "workers": workers,
+        "seed_rule": "first 64 SHA256 bits of matching-one/safe-insertion-independent-20260929/{role}/square/L512/{batch}/{stream}; stream=permutation or safe-probe",
+        "batches": [], "benchmarks": [],
+    }
+    def save():
+        manifest.write_text(json.dumps(run, indent=2)+"\n")
+    for name in ("cpu/cpu.cfs_quota_us", "cpu/cpu.cfs_period_us", "memory/memory.limit_in_bytes", "cpu.max", "memory.max"):
+        path = Path("/sys/fs/cgroup") / name
+        if path.exists():
+            run.setdefault("cgroup", {})[name] = path.read_text().strip()
+    save()
+    try:
+        with tempfile.TemporaryDirectory(prefix="safe-insertion-build-", dir=HERE) as tmp:
+            tmp = Path(tmp)
+            binary = tmp / "engine"
+            build = [args.compiler, "-O3", "-std=c++17", str(HERE / "engine.cpp"), "-o", str(binary)]
+            subprocess.run(build, check=True)
+            run["build_command"] = shlex.join(build)
+            run["binary_sha256"] = sha(binary)
+            # Only the two physical L4 witness rows: 20 probes, not a census.
+            run["physical_witness_control"] = subprocess.check_output([str(binary), "control"], text=True).strip()
+            print(run["physical_witness_control"], flush=True)
+
+            def job(role, batch, samples):
+                stem = f"{role}-square-L512-b{batch:02d}"
+                raw = tmp / (stem+".csv")
+                ps, qs = seed(role, batch, "permutation"), seed(role, batch, "safe-probe")
+                command = [str(binary), str(contract["L"]), str(samples), str(ps), str(qs), str(contract["b"]), str(raw)]
+                begin = time.monotonic()
+                log = subprocess.check_output(command, text=True).strip()
+                elapsed = time.monotonic()-begin
+                target = args.output / (stem+".csv.gz")
+                content = raw.read_bytes()
+                if content.count(b"\n") != samples+1:
+                    raise RuntimeError("incomplete sample output")
+                with target.open("wb") as output:
+                    with gzip.GzipFile(fileobj=output, mode="wb", mtime=0) as gz:
+                        gz.write(content)
+                print(f"{role} batch={batch}: {log}", flush=True)
+                return {"file": target.name, "batch": batch, "samples": samples, "seed": str(ps), "probe_seed": str(qs),
+                        "role": role, "runtime_seconds": elapsed, "engine_log": log,
+                        "command": shlex.join(command), "sha256": sha(target),
+                        "dependency_group": f"safe-insertion-independent-20260929/square/L512/batch{batch}" if role=="production" else "excluded-benchmark"}
+
+            run["status"] = "benchmarking"; save()
+            benchmark = job("benchmark", 0, 32)
+            run["benchmarks"].append(benchmark)
+            estimate = benchmark["runtime_seconds"]/32 * contract["samples_per_batch"]*contract["batches"]/workers
+            run["estimated_production_wall_seconds"] = estimate
+            print(f"Estimated ideal-parallel production wall: {estimate:.2f}s", flush=True)
+            if estimate > 900:
+                run["status"] = "benchmark-budget-stop"; save()
+                return
+            run["status"] = "running"; save()
+            with ThreadPoolExecutor(max_workers=workers) as pool:
+                futures = [pool.submit(job, "production", b, contract["samples_per_batch"])
+                           for b in range(contract["batches"])]
+                for future in as_completed(futures):
+                    run["batches"].append(future.result())
+                    run["batches"].sort(key=lambda x: x["batch"])
+                    save()
+        run["status"] = "completed"
+    except BaseException as error:
+        run["status"] = "failed"; run["error"] = str(error)
+        raise
+    finally:
+        run["wall_seconds"] = time.monotonic()-started
+        run["utc_finished"] = datetime.now(timezone.utc).isoformat()
+        save()
+    print(f"Completed {len(run['batches'])} batches in {run['wall_seconds']:.2f}s", flush=True)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/analysis/safe-transport-selection-20260930/benchmark.json b/analysis/safe-transport-selection-20260930/benchmark.json
new file mode 100644
index 000000000..c95da0835
--- /dev/null
+++ b/analysis/safe-transport-selection-20260930/benchmark.json
@@ -0,0 +1,41 @@
+{
+  "environment": {
+    "platform": "Linux-5.10.0-182.0.0.95.r3582_286.hce2.aarch64-aarch64-with-glibc2.34",
+    "machine": "aarch64",
+    "python": "3.9.9",
+    "cpu_quota": 14.5,
+    "memory_bytes": 26843545600,
+    "hostname": "83750ac4eae34bbbbc64b894b854dd8c"
+  },
+  "command": [
+    "/workspace/Matching-One-TgFr7R/safe-transport-selection-20260930/analysis/safe-transport-selection-20260930/build/engine",
+    "512",
+    "32",
+    "9722445355900932119",
+    "7229524146119961523",
+    "155385",
+    "/workspace/Matching-One-TgFr7R/safe-transport-selection-20260930/analysis/safe-transport-selection-20260930/build/old-seed-32.csv"
+  ],
+  "build_command": [
+    "chroot",
+    "/workspace/Matching-One-TgFr7R/geometric-source-window-20260930/toolchain",
+    "/usr/bin/g++",
+    "-O3",
+    "-std=c++17",
+    "/safe-transport-selection-project/analysis/safe-transport-selection-20260930/engine.cpp",
+    "-o",
+    "/safe-transport-selection-project/engine"
+  ],
+  "compiler": "g++ (GCC) 10.3.1\nCopyright (C) 2020 Free Software Foundation, Inc.\nThis is free software; see the source for copying conditions.  There is NO\nwarranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.\n\n",
+  "stdout": "prefixes=32 seconds=1.35637\n",
+  "wall_seconds": 1.3772434319835156,
+  "children_peak_rss_kib": 190236,
+  "workers": 14,
+  "predicted_ideal_seconds": 430.38857249484863,
+  "within_cost_budget": true,
+  "rows": 32,
+  "original_births_match": true,
+  "scientific_weight": 0,
+  "source_commit": "5054c02383242473834045fcbddc7379d83acfb9",
+  "binary_sha256": "9cd09ffeafc7bda7f4e2468e39b30266fa0f56830dcf61b5c14d9cad97bd60ef"
+}
diff --git a/analysis/safe-transport-selection-20260930/contract.json b/analysis/safe-transport-selection-20260930/contract.json
new file mode 100644
index 000000000..a50366079
--- /dev/null
+++ b/analysis/safe-transport-selection-20260930/contract.json
@@ -0,0 +1,24 @@
+{
+  "schema": "matching-one.safe-transport-selection-contract.v1",
+  "question": "Does the early-minus-late completion-creation contrast persist in the uniform-safe reference chain without natural survival selection?",
+  "L": 512,
+  "N": 262144,
+  "a": 154646,
+  "b": 155385,
+  "batches": 14,
+  "samples_per_batch": 10000,
+  "source": "Replay every archived safe-insertion permutation up to first homology birth or b. For rank-one entrances J1<=b, use one fresh uniform-safe continuation to b, retaining a coupled natural survival flag. Do not restrict to original rank_b=1.",
+  "seed_rule": "first 64 SHA256 bits of matching-one/safe-transport-selection-20260930/continuation/square/L512/{batch}",
+  "coupling": "At each count, sample vacancies in uniform random order without replacement until first safe site. Kill the natural flag permanently if the first proposal completes. Insert the first safe site. All-completing states go to a separately counted cemetery. No retry cap.",
+  "endpoint": "Y=2*synergy_edges/(N-b-nu_b), within C=(primitive direction,integer nu_b)",
+  "groups": "early J1<=a; late a<J1<=b; rank-one entrance required",
+  "eligibility": "reference reaches b at rank one and nu_b<N-b. Direct rank-zero-to-two entrances, no entrance, reference cemetery and no safe endpoint are separately counted.",
+  "weights": "Common cells with both natural-alive cohorts: w=n_natural_early*n_natural_late/(n_natural_early+n_natural_late). Same normalized weights and support for all three contrasts.",
+  "primary_vector": ["natural_contrast", "safe_reference_contrast", "selection_contrast=natural-safe"],
+  "prediction": "A selection-only explanation for this reference decomposition requires safe_reference_contrast=0. Persistence requires entrance/within-safe transport information as well; zero does not establish all-history closure.",
+  "inference": "Recompute cells, means, eligibility, common support and weights after each aligned whole-batch deletion. Save full 3x3 covariance and all14 deletions. No adaptive subgroup, feature, cutoff, sample extension or size ladder.",
+  "dependency": "Same original first-birth prefixes and batch groups as safe-insertion-independent-20260929. New conditional continuations, not independent preparations or untouched holdout.",
+  "scope": "Specified safe-reference decomposition, not unique causal allocation between imported birth geometry and subsequent safe growth; not original U, continuum scaling or an explained fraction of the previous realization.",
+  "compute": "One32-prefix old-seed cost probe (scientific weight zero), then one14x10000 bounded cloud job. No additional precision run. Benchmark anticipated ideal time at most1800 seconds and within80% measured memory.",
+  "frozen_before_continuation": true
+}
diff --git a/analysis/safe-transport-selection-20260930/data/batch-00.json b/analysis/safe-transport-selection-20260930/data/batch-00.json
new file mode 100644
index 000000000..e5b799ce4
--- /dev/null
+++ b/analysis/safe-transport-selection-20260930/data/batch-00.json
@@ -0,0 +1,24 @@
+{
+  "batch": 0,
+  "permutation_seed": "9722445355900932119",
+  "continuation_seed": "7229524146119961523",
+  "samples": 10000,
+  "original_file": "production-square-L512-b00.csv.gz",
+  "original_sha256": "dfea3bdaa73f93c05078c87b00bfcc1f65f262a353ea9623c58f0c339b904f5f",
+  "dependency_group": "safe-insertion-independent-20260929/square/L512/batch0",
+  "command": [
+    "/workspace/Matching-One-TgFr7R/safe-transport-selection-20260930/analysis/safe-transport-selection-20260930/build/engine",
+    "512",
+    "10000",
+    "9722445355900932119",
+    "7229524146119961523",
+    "155385",
+    "/workspace/Matching-One-TgFr7R/safe-transport-selection-20260930/analysis/safe-transport-selection-20260930/data/selection-b00.csv"
+  ],
+  "returncode": 0,
+  "stdout": "prefixes=10000 seconds=651.772\n",
+  "stderr": "",
+  "wall_seconds": 651.8640293800272,
+  "file": "selection-b00.csv.gz",
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new file mode 100644
index 000000000..4e5b86127
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index 000000000..da2104396
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new file mode 100644
index 000000000..d7488d955
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new file mode 100644
index 000000000..b174f746c
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new file mode 100644
index 000000000..28cdcff0b
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index 000000000..f08c37d18
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index 000000000..9206a9476
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index 000000000..192dfe2b6
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index 000000000..5151620dc
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index 000000000..23d4f6666
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index 000000000..1ab10d1e0
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index 000000000..8dc864af9
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index 000000000..53aee817c
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diff --git a/analysis/safe-transport-selection-20260930/data/started.json b/analysis/safe-transport-selection-20260930/data/started.json
new file mode 100644
index 000000000..f0047296f
--- /dev/null
+++ b/analysis/safe-transport-selection-20260930/data/started.json
@@ -0,0 +1,29 @@
+{
+  "schema": "matching-one.safe-transport-selection-run.v1",
+  "status": "running",
+  "pid": 209,
+  "started_utc": "2026-09-30T03:05:32.320816+00:00",
+  "environment": {
+    "platform": "Linux-5.10.0-182.0.0.95.r3582_286.hce2.aarch64-aarch64-with-glibc2.34",
+    "machine": "aarch64",
+    "python": "3.9.9",
+    "cpu_quota": 14.5,
+    "memory_bytes": 26843545600,
+    "hostname": "83750ac4eae34bbbbc64b894b854dd8c"
+  },
+  "source_commit": "5054c02383242473834045fcbddc7379d83acfb9",
+  "binary_sha256": "9cd09ffeafc7bda7f4e2468e39b30266fa0f56830dcf61b5c14d9cad97bd60ef",
+  "workers": 14,
+  "source_sha256": {
+    "analysis/safe-transport-selection-20260930/engine.cpp": "4d5d1ed48beffc0e13f8db12da67024118a4b1ea75fa7f8e68b3c6153c072c0b",
+    "analysis/safe-transport-selection-20260930/run.py": "a5bf404c70bf65cd28dc18a4aefa2b59729f65c64b74c11b0f0f74a548e9f31c",
+    "analysis/safe-transport-selection-20260930/score.py": "50e681e2d1422e0d2b1c4aa089bdc462d1a5b7767de2f585821c50a1db2d7b74",
+    "analysis/safe-transport-selection-20260930/contract.json": "22ca3b823989df75eeb1fe6091bdeee38eaf6229950c4cc441421fcbd3850bc2",
+    "analysis/exact-completion-pairs-20260929/engine.cpp": "58610c299052b8ceded1606e4a7618363b90eef60a6ad9efa2b8256aee72f6e3",
+    "analysis/completion-hazard-production-20260929/engine.cpp": "da45ae8a2682f9edb24eaf87a6733c55013f1fb95b3622bf2faf97208c405c0b",
+    "analysis/geometric-source-window-20260930/run.py": "314d4929d8ff45c6e74464cad5a65061510fb331f4fb6521b3000e9d94f7d0f0"
+  },
+  "original_manifest_sha256": "8096d3d0a327604dd921e9c82d022daa685e2c7d6c4bf6e5b87c195b1d3dbf89",
+  "new_independent_prefixes": 0,
+  "batches": []
+}
diff --git a/analysis/safe-transport-selection-20260930/engine.cpp b/analysis/safe-transport-selection-20260930/engine.cpp
new file mode 100644
index 000000000..6e079eff7
--- /dev/null
+++ b/analysis/safe-transport-selection-20260930/engine.cpp
@@ -0,0 +1,59 @@
+// First-birth prefix replay, followed by a uniform-safe reference chain.
+// First-proposal survival flags exactly couple it to the killed uniform chain.
+#define MATCHING_ONE_PAIR_LIBRARY
+#include "../exact-completion-pairs-20260929/engine.cpp"
+
+int main(int argc,char**argv){
+    try{
+        if(argc!=7)throw std::invalid_argument("L samples permutation_seed continuation_seed b output.csv");
+        int L=std::stoi(argv[1]),N=L*L,b=std::stoi(argv[5]);
+        auto samples=std::stoull(argv[2]),seed=std::stoull(argv[3]),continuation=std::stoull(argv[4]);
+        if(L<3 || b<1 || b>N-2 || !samples)throw std::invalid_argument("bounds");
+        Geometry g(L,false);ProjectedPairs counter(L,false);
+        std::vector<int>order(N),reference,vacancies;reference.reserve(N);vacancies.reserve(N);
+        std::mt19937_64 rng(seed),future(continuation);
+        std::ofstream out(argv[6]);if(!out)throw std::runtime_error("cannot open output");
+        out<<"J1,entry_rank,dx_entry,dy_entry,reference_rank_b,natural_alive,nu_b,synergy_edges,first_rejection_count,proposal_count\n";
+        auto start=std::chrono::steady_clock::now();
+        for(std::uint64_t sample=0;sample<samples;++sample){
+            std::iota(order.begin(),order.end(),0);
+            for(int v=N-1;v>0;--v)std::swap(order[v],order[bounded(rng,v+1)]);
+            g.reset();int j1=0,rank=0;
+            for(int k=1;k<=b;++k){rank=g.insert(order[k-1]);if(rank){j1=k;break;}}
+            if(rank!=1){out<<j1<<','<<rank<<",0,0,"<<rank<<",0,-1,-1,0,0\n";continue;}
+            auto direction=g.direction();int alive=1,first_rejection=0;
+            std::uint64_t proposals=0;
+            reference.assign(order.begin(),order.begin()+j1);
+            vacancies.assign(order.begin()+j1,order.end());
+            for(int k=j1;k<b;++k){
+                const int m=vacancies.size();bool accepted=false;
+                // A fresh random ordering, without replacement. Rejected
+                // vacancies remain vacant and eligible on the next count.
+                for(int r=m-1;r>=0;--r){
+                    std::swap(vacancies[r],vacancies[bounded(future,r+1)]);
+                    const int v=vacancies[r];++proposals;
+                    const bool completes=g.completes_if_inserted(v);
+                    if(r==m-1 && completes && alive){alive=0;first_rejection=k+1;}
+                    if(completes)continue;
+                    if(g.insert(v)!=1)throw std::logic_error("unsafe accepted site");
+                    reference.push_back(v);
+                    std::swap(vacancies[r],vacancies.back());vacancies.pop_back();
+                    accepted=true;break;
+                }
+                if(!accepted){rank=-1;alive=0;break;} // exact cemetery, no retry cap
+            }
+            if(rank==-1){
+                out<<j1<<",1,"<<direction.first<<','<<direction.second<<",-1,0,-1,-1,"
+                   <<first_rejection<<','<<proposals<<'\n';continue;
+            }
+            if(int(reference.size())!=b || g.direction()!=direction)throw std::logic_error("reference endpoint");
+            auto pairs=counter.count(reference,b,direction);
+            out<<j1<<",1,"<<direction.first<<','<<direction.second<<",1,"<<alive<<','<<pairs.nu<<','<<pairs.edges<<','
+               <<first_rejection<<','<<proposals<<'\n';
+        }
+        out.close();if(!out)throw std::runtime_error("write failed");
+        std::cout<<"prefixes="<<samples<<" seconds="
+                 <<std::chrono::duration<double>(std::chrono::steady_clock::now()-start).count()<<'\n';
+        return 0;
+    }catch(const std::exception&e){std::cerr<<e.what()<<'\n';return 1;}
+}
diff --git a/analysis/safe-transport-selection-20260930/execution.json b/analysis/safe-transport-selection-20260930/execution.json
new file mode 100644
index 000000000..570a45a8a
--- /dev/null
+++ b/analysis/safe-transport-selection-20260930/execution.json
@@ -0,0 +1,79 @@
+{
+  "status": "completed_and_resources_released",
+  "source_commit": "5054c02383242473834045fcbddc7379d83acfb9",
+  "remote_root": "/workspace/Matching-One-TgFr7R/safe-transport-selection-20260930",
+  "instance": "DevEnvC_TgFr7R",
+  "instance_uuid": "83750ac4eae34bbbbc64b894b854dd8c",
+  "local_ssh_port": 10027,
+  "initial_state": "Ready",
+  "startup": "Running verified after power-on, one successful ssh-key-reset, local key mode600; only this instance's tunnel started",
+  "tunnel_pid": 2132,
+  "runner_pid": 209,
+  "resources": {
+    "architecture": "aarch64",
+    "python": "3.9.9",
+    "compiler": "GCC10.3.1 in previously owned persistent installroot, new safe-transport-selection-project compilation subtree",
+    "reused_toolchain": "/workspace/Matching-One-TgFr7R/geometric-source-window-20260930/toolchain",
+    "old_source_or_results_overwritten": false,
+    "system_packages_changed": false,
+    "cpu_quota_cores": 14.5,
+    "memory_limit_bytes": 26843545600,
+    "workers": 14,
+    "worker_observed_rss_kib_approx": 79000,
+    "thread_environment": "OMP_NUM_THREADS=OPENBLAS_NUM_THREADS=1"
+  },
+  "benchmark": {
+    "old_prefixes": 32,
+    "scientific_weight": 0,
+    "wall_seconds": 1.3772434319835156,
+    "all_original_births_match": true,
+    "predicted_ideal_seconds": 430.38857249484863,
+    "actual_production_longer_than_probe_prediction": true,
+    "note": "children_peak_rss includes compilation, not an engine-only peak; no sample extension"
+  },
+  "continuation": {
+    "started_utc": "2026-09-30T03:05:32.320816+00:00",
+    "finished_utc": "2026-09-30T03:16:36.417712+00:00",
+    "wall_seconds": 664.0940968480427,
+    "batches": 14,
+    "original_prefixes": 140000,
+    "new_independent_prefixes": 0,
+    "new_conditional_continuations": 97666,
+    "exit_status": "all14engines0; data/run.json completed",
+    "command": "python3 analysis/safe-transport-selection-20260930/run.py run --source-commit 5054c02383242473834045fcbddc7379d83acfb9",
+    "binary_sha256": "9cd09ffeafc7bda7f4e2468e39b30266fa0f56830dcf61b5c14d9cad97bd60ef"
+  },
+  "transfer": {
+    "input_bundle_sha256": "6bfeb5c374e1f1fbadcecded27d3548d873d49a420a6db78315586c3f1a37882",
+    "return_bundle_sha256": "c6c3df008516b7325b6dcf25e56f6e11c124df56d4f795be85202a2c14b0a248",
+    "local_and_remote_sha_match": true,
+    "run_manifest_sha256": "94db3a4ae91b46f93c162a0979a3d7a4479525cb1809b62fa80ab7e1daa2d5d2",
+    "result_sha256": "07c800c52df4fd60b4e6568751308c440ab7c01a564d256b4c0b19d02b3b0a65",
+    "raw_remote_files_preserved": true
+  },
+  "analysis": {
+    "local_python": "research-py311 Python3.11",
+    "full_score_runs": 1,
+    "all140000_original_births_match": true,
+    "reference_cemetery_count": 0,
+    "no_safe_endpoint_count": 0,
+    "natural_survivors": 54850,
+    "common_natural_survivors": 54295,
+    "common_reference_endpoints": 94123,
+    "common_cells": 1198,
+    "natural_contrast": -0.00220506528552,
+    "safe_reference_contrast": -0.00253705879386,
+    "selection_contrast": 0.000331993508343,
+    "full_covariance_and_deletions": "results/result.json",
+    "support_review": "verification.md: one fixed L4 query control and one exact analytic coupling; source/covariance review; no production rerun"
+  },
+  "cleanup": {
+    "pre_shutdown_no_other_research_jobs_verified": true,
+    "own_tunnel_stopped": true,
+    "power_off_requested": true,
+    "last_verified_cloud_state": "Ready",
+    "final_cloud_state_verified_utc": "2026-09-30T03:20:51Z",
+    "other_machine_operations": 0
+  },
+  "scope": "New conditional Monte Carlo continuations from archived first-birth prefixes with their original dependency groups. One specified reference-process decomposition, not independent confirmation, unique birth/evolution causal attribution, original-U identification or continuum proof."
+}
diff --git a/analysis/safe-transport-selection-20260930/results/RESULT.md b/analysis/safe-transport-selection-20260930/results/RESULT.md
new file mode 100644
index 000000000..653709dfe
--- /dev/null
+++ b/analysis/safe-transport-selection-20260930/results/RESULT.md
@@ -0,0 +1,14 @@
+# Safe transport versus natural survival selection
+
+Same archived entrances, new conditional continuations; not an independent prefix block.
+
+| Contrast | Estimate | Whole-batch SE |
+|---|---:|---:|
+| natural_contrast | -0.00220506528552 | 0.00078709169974 |
+| safe_reference_contrast | -0.00253705879386 | 0.000636492673968 |
+| selection_contrast | 0.000331993508343 | 0.000287375294045 |
+
+1198 common natural-cohort cells; one set of endpoint weights for all components.
+Natural support [24123, 30172] / eligible [24216, 30634]; reference support [53828, 40295].
+Natural = safe reference + selection exactly. Full singular3x3 covariance and14 aligned deletions in result.json.
+This decomposes a specified reference process; it does not distinguish imported birth geometry from subsequent safe growth, or identify original U/continuum fields.
diff --git a/analysis/safe-transport-selection-20260930/results/result.json b/analysis/safe-transport-selection-20260930/results/result.json
new file mode 100644
index 000000000..69ab80990
--- /dev/null
+++ b/analysis/safe-transport-selection-20260930/results/result.json
@@ -0,0 +1,29129 @@
+{
+  "schema": "matching-one.safe-transport-selection.v1",
+  "names": [
+    "natural_contrast",
+    "safe_reference_contrast",
+    "selection_contrast"
+  ],
+  "estimate": [
+    -0.002205065285515362,
+    -0.002537058793857894,
+    0.00033199350834253174
+  ],
+  "batch_delete_one_se": [
+    0.0007870916997397121,
+    0.0006364926739678066,
+    0.00028737529404467075
+  ],
+  "covariance": [
+    [
+      6.195133437991492e-07,
+      4.7102585409328845e-07,
+      1.4848748970586082e-07
+    ],
+    [
+      4.7102585409328845e-07,
+      4.051229240146886e-07,
+      6.590293007859984e-08
+    ],
+    [
+      1.4848748970586082e-07,
+      6.590293007859984e-08,
+      8.258455962726098e-08
+    ]
+  ],
+  "deletions": [
+    [
+      -0.002389001883660518,
+      -0.0027178325980468255,
+      0.0003288307143863075
+    ],
+    [
+      -0.0023397765460692354,
+      -0.0025648180137156857,
+      0.00022504146764645025
+    ],
+    [
+      -0.0022710704665725085,
+      -0.0025818208924605988,
+      0.00031075042588809025
+    ],
+    [
+      -0.002447164516780324,
+      -0.0027656644182167825,
+      0.00031849990143645833
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+    [
+      -0.0024228057705667763,
+      -0.0027368966871279313,
+      0.00031409091656115504
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+    [
+      -0.00231149128774032,
+      -0.0025151570713367053,
+      0.00020366578359638527
+    ],
+    [
+      -0.002242220147139466,
+      -0.0025731325221072356,
+      0.00033091237496776955
+    ],
+    [
+      -0.0019851774179876686,
+      -0.002452663165962812,
+      0.0004674857479751433
+    ],
+    [
+      -0.0017516066229812507,
+      -0.0022066544832428387,
+      0.00045504786026158797
+    ],
+    [
+      -0.0019623070860447893,
+      -0.0022331990530696545,
+      0.00027089196702486523
+    ],
+    [
+      -0.0020540682384420227,
+      -0.0024623270941015224,
+      0.00040825885565949974
+    ],
+    [
+      -0.002022452672590986,
+      -0.002472670655978616,
+      0.00045021798338763016
+    ],
+    [
+      -0.0025467239181118265,
+      -0.002822748462065834,
+      0.0002760245439540074
+    ],
+    [
+      -0.0021255872951161776,
+      -0.0024533831014744443,
+      0.0003277958063582667
+    ]
+  ],
+  "support": {
+    "common_cells": 1198,
+    "weight_sum": 13033.026451291296,
+    "eligible_natural_counts": [
+      24216,
+      30634
+    ],
+    "common_natural_counts": [
+      24123,
+      30172
+    ],
+    "common_safe_counts": [
+      53828,
+      40295
+    ],
+    "weighted_natural_means": [
+      0.09610311807735657,
+      0.09830818336287192
+    ],
+    "weighted_safe_means": [
+      0.09540936877362605,
+      0.09794642756748394
+    ],
+    "cells": [
+      {
+        "cell": [
+          0,
+          1,
+          0
+        ],
+        "weight": 643.2188577999234,
+        "safe_counts": [
+          1151,
+          1458
+        ],
+        "natural_counts": [
+          1151,
+          1458
+        ],
+        "safe_means": [
+          0.14587281116755504,
+          0.1410708253815939
+        ],
+        "natural_means": [
+          0.14587281116755504,
+          0.1410708253815939
+        ]
+      },
+      {
+        "cell": [
+          0,
+          1,
+          1
+        ],
+        "weight": 6.0,
+        "safe_counts": [
+          12,
+          12
+        ],
+        "natural_counts": [
+          12,
+          12
+        ],
+        "safe_means": [
+          0.1901434396797742,
+          0.07643923640382923
+        ],
+        "natural_means": [
+          0.1901434396797742,
+          0.07643923640382923
+        ]
+      },
+      {
+        "cell": [
+          0,
+          1,
+          2
+        ],
+        "weight": 10.69767441860465,
+        "safe_counts": [
+          20,
+          23
+        ],
+        "natural_counts": [
+          20,
+          23
+        ],
+        "safe_means": [
+          0.11452832132787545,
+          0.11422690539384242
+        ],
+        "natural_means": [
+          0.11452832132787545,
+          0.11422690539384242
+        ]
+      },
+      {
+        "cell": [
+          0,
+          1,
+          3
+        ],
+        "weight": 15.954545454545455,
+        "safe_counts": [
+          27,
+          39
+        ],
+        "natural_counts": [
+          27,
+          39
+        ],
+        "safe_means": [
+          0.10840157479222264,
+          0.13193902030126692
+        ],
+        "natural_means": [
+          0.10840157479222264,
+          0.13193902030126692
+        ]
+      },
+      {
+        "cell": [
+          0,
+          1,
+          4
+        ],
+        "weight": 14.40677966101695,
+        "safe_counts": [
+          25,
+          34
+        ],
+        "natural_counts": [
+          25,
+          34
+        ],
+        "safe_means": [
+          0.13776741136246545,
+          0.14226417277603748
+        ],
+        "natural_means": [
+          0.13776741136246545,
+          0.14226417277603748
+        ]
+      },
+      {
+        "cell": [
+          0,
+          1,
+          5
+        ],
+        "weight": 14.140350877192983,
+        "safe_counts": [
+          26,
+          33
+        ],
+        "natural_counts": [
+          26,
+          31
+        ],
+        "safe_means": [
+          0.12692444599445743,
+          0.13147587685309925
+        ],
+        "natural_means": [
+          0.12692444599445743,
+          0.1366034784826375
+        ]
+      },
+      {
+        "cell": [
+          0,
+          1,
+          6
+        ],
+        "weight": 16.484848484848484,
+        "safe_counts": [
+          32,
+          35
+        ],
+        "natural_counts": [
+          32,
+          34
+        ],
+        "safe_means": [
+          0.14955083229511115,
+          0.1309099376263765
+        ],
+        "natural_means": [
+          0.14955083229511115,
+          0.13234784419889564
+        ]
+      },
+      {
+        "cell": [
+          0,
+          1,
+          7
+        ],
+        "weight": 18.746666666666666,
+        "safe_counts": [
+          38,
+          37
+        ],
+        "natural_counts": [
+          38,
+          37
+        ],
+        "safe_means": [
+          0.10147276915309857,
+          0.12081652245770948
+        ],
+        "natural_means": [
+          0.10147276915309857,
+          0.12081652245770948
+        ]
+      },
+      {
+        "cell": [
+          0,
+          1,
+          8
+        ],
+        "weight": 18.31168831168831,
+        "safe_counts": [
+          30,
+          47
+        ],
+        "natural_counts": [
+          30,
+          47
+        ],
+        "safe_means": [
+          0.07973820698010635,
+          0.15571771015349503
+        ],
+        "natural_means": [
+          0.07973820698010635,
+          0.15571771015349503
+        ]
+      },
+      {
+        "cell": [
+          0,
+          1,
+          9
+        ],
+        "weight": 20.702380952380953,
+        "safe_counts": [
+          41,
+          47
+        ],
+        "natural_counts": [
+          37,
+          47
+        ],
+        "safe_means": [
+          0.1364839207174273,
+          0.13534505954457124
+        ],
+        "natural_means": [
+          0.13440749414519906,
+          0.13534505954457124
+        ]
+      },
+      {
+        "cell": [
+          0,
+          1,
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+        ],
+        "weight": 19.42168674698795,
+        "safe_counts": [
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+        ],
+        "natural_counts": [
+          31,
+          52
+        ],
+        "safe_means": [
+          0.1181027732454306,
+          0.1519783647766111
+        ],
+        "natural_means": [
+          0.1196475663895318,
+          0.1519783647766111
+        ]
+      },
+      {
+        "cell": [
+          0,
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+        ],
+        "weight": 20.558139534883722,
+        "safe_counts": [
+          36,
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+        ],
+        "natural_counts": [
+          34,
+          52
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+          355
+        ],
+        "weight": 0.5,
+        "safe_counts": [
+          3,
+          4
+        ],
+        "natural_counts": [
+          1,
+          1
+        ],
+        "safe_means": [
+          0.07547335313208776,
+          0.07610146235103943
+        ],
+        "natural_means": [
+          0.06386977933160408,
+          0.08069245517085824
+        ]
+      },
+      {
+        "cell": [
+          1,
+          1,
+          366
+        ],
+        "weight": 0.5,
+        "safe_counts": [
+          5,
+          2
+        ],
+        "natural_counts": [
+          1,
+          1
+        ],
+        "safe_means": [
+          0.08173094094536296,
+          0.08529696502589457
+        ],
+        "natural_means": [
+          0.0640267686783905,
+          0.08105796433975919
+        ]
+      },
+      {
+        "cell": [
+          1,
+          1,
+          381
+        ],
+        "weight": 0.6666666666666666,
+        "safe_counts": [
+          5,
+          2
+        ],
+        "natural_counts": [
+          2,
+          1
+        ],
+        "safe_means": [
+          0.08361127300757676,
+          0.06851040628701423
+        ],
+        "natural_means": [
+          0.07917990561958299,
+          0.07683919607437628
+        ]
+      }
+    ]
+  },
+  "counts": {
+    "prefixes": 140000,
+    "rank_one_entrance_early": 56286,
+    "reference_endpoint_early": 56286,
+    "natural_endpoint_early": 24216,
+    "rank_one_entrance_late": 41380,
+    "reference_endpoint_late": 41380,
+    "natural_endpoint_late": 30634,
+    "no_entrance_by_b": 42329,
+    "direct_rank_two_entrance": 5
+  },
+  "batch_counts": [
+    {
+      "batch": 0,
+      "counts": {
+        "prefixes": 10000,
+        "rank_one_entrance_early": 3986,
+        "reference_endpoint_early": 3986,
+        "natural_endpoint_early": 1740,
+        "rank_one_entrance_late": 2989,
+        "reference_endpoint_late": 2989,
+        "natural_endpoint_late": 2168,
+        "no_entrance_by_b": 3025
+      },
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch0"
+    },
+    {
+      "batch": 1,
+      "counts": {
+        "prefixes": 10000,
+        "rank_one_entrance_early": 3996,
+        "reference_endpoint_early": 3996,
+        "natural_endpoint_early": 1722,
+        "no_entrance_by_b": 3046,
+        "rank_one_entrance_late": 2957,
+        "reference_endpoint_late": 2957,
+        "natural_endpoint_late": 2191,
+        "direct_rank_two_entrance": 1
+      },
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch1"
+    },
+    {
+      "batch": 2,
+      "counts": {
+        "prefixes": 10000,
+        "rank_one_entrance_late": 2967,
+        "reference_endpoint_late": 2967,
+        "natural_endpoint_late": 2207,
+        "no_entrance_by_b": 2997,
+        "rank_one_entrance_early": 4036,
+        "reference_endpoint_early": 4036,
+        "natural_endpoint_early": 1740
+      },
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch2"
+    },
+    {
+      "batch": 3,
+      "counts": {
+        "prefixes": 10000,
+        "rank_one_entrance_late": 2941,
+        "reference_endpoint_late": 2941,
+        "natural_endpoint_late": 2151,
+        "rank_one_entrance_early": 4022,
+        "reference_endpoint_early": 4022,
+        "natural_endpoint_early": 1689,
+        "no_entrance_by_b": 3037
+      },
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch3"
+    },
+    {
+      "batch": 4,
+      "counts": {
+        "prefixes": 10000,
+        "rank_one_entrance_early": 4017,
+        "reference_endpoint_early": 4017,
+        "natural_endpoint_early": 1694,
+        "rank_one_entrance_late": 2921,
+        "reference_endpoint_late": 2921,
+        "natural_endpoint_late": 2140,
+        "no_entrance_by_b": 3062
+      },
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch4"
+    },
+    {
+      "batch": 5,
+      "counts": {
+        "prefixes": 10000,
+        "rank_one_entrance_late": 3006,
+        "reference_endpoint_late": 3006,
+        "natural_endpoint_late": 2247,
+        "rank_one_entrance_early": 3986,
+        "reference_endpoint_early": 3986,
+        "natural_endpoint_early": 1771,
+        "no_entrance_by_b": 3008
+      },
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch5"
+    },
+    {
+      "batch": 6,
+      "counts": {
+        "prefixes": 10000,
+        "rank_one_entrance_late": 2931,
+        "reference_endpoint_late": 2931,
+        "natural_endpoint_late": 2158,
+        "rank_one_entrance_early": 4034,
+        "reference_endpoint_early": 4034,
+        "natural_endpoint_early": 1725,
+        "no_entrance_by_b": 3035
+      },
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch6"
+    },
+    {
+      "batch": 7,
+      "counts": {
+        "prefixes": 10000,
+        "rank_one_entrance_early": 3928,
+        "reference_endpoint_early": 3928,
+        "natural_endpoint_early": 1715,
+        "rank_one_entrance_late": 3006,
+        "reference_endpoint_late": 3006,
+        "natural_endpoint_late": 2230,
+        "no_entrance_by_b": 3066
+      },
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch7"
+    },
+    {
+      "batch": 8,
+      "counts": {
+        "prefixes": 10000,
+        "no_entrance_by_b": 3051,
+        "rank_one_entrance_early": 4027,
+        "reference_endpoint_early": 4027,
+        "natural_endpoint_early": 1764,
+        "rank_one_entrance_late": 2920,
+        "reference_endpoint_late": 2920,
+        "natural_endpoint_late": 2173,
+        "direct_rank_two_entrance": 2
+      },
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch8"
+    },
+    {
+      "batch": 9,
+      "counts": {
+        "prefixes": 10000,
+        "rank_one_entrance_early": 4104,
+        "reference_endpoint_early": 4104,
+        "natural_endpoint_early": 1769,
+        "rank_one_entrance_late": 2939,
+        "reference_endpoint_late": 2939,
+        "natural_endpoint_late": 2236,
+        "no_entrance_by_b": 2956,
+        "direct_rank_two_entrance": 1
+      },
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch9"
+    },
+    {
+      "batch": 10,
+      "counts": {
+        "prefixes": 10000,
+        "rank_one_entrance_early": 4023,
+        "reference_endpoint_early": 4023,
+        "natural_endpoint_early": 1683,
+        "no_entrance_by_b": 3051,
+        "rank_one_entrance_late": 2926,
+        "reference_endpoint_late": 2926,
+        "natural_endpoint_late": 2182
+      },
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch10"
+    },
+    {
+      "batch": 11,
+      "counts": {
+        "prefixes": 10000,
+        "rank_one_entrance_early": 3972,
+        "reference_endpoint_early": 3972,
+        "natural_endpoint_early": 1674,
+        "rank_one_entrance_late": 3001,
+        "reference_endpoint_late": 3001,
+        "natural_endpoint_late": 2204,
+        "no_entrance_by_b": 3027
+      },
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch11"
+    },
+    {
+      "batch": 12,
+      "counts": {
+        "prefixes": 10000,
+        "no_entrance_by_b": 2986,
+        "rank_one_entrance_late": 2891,
+        "reference_endpoint_late": 2891,
+        "natural_endpoint_late": 2138,
+        "rank_one_entrance_early": 4122,
+        "reference_endpoint_early": 4122,
+        "natural_endpoint_early": 1756,
+        "direct_rank_two_entrance": 1
+      },
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch12"
+    },
+    {
+      "batch": 13,
+      "counts": {
+        "prefixes": 10000,
+        "no_entrance_by_b": 2982,
+        "rank_one_entrance_early": 4033,
+        "reference_endpoint_early": 4033,
+        "natural_endpoint_early": 1774,
+        "rank_one_entrance_late": 2985,
+        "reference_endpoint_late": 2985,
+        "natural_endpoint_late": 2209
+      },
+      "dependency_group": "safe-insertion-independent-20260929/square/L512/batch13"
+    }
+  ],
+  "all_original_births_match": true,
+  "new_independent_prefixes": 0,
+  "dependency_group": "safe-insertion-independent-20260929/square/L512",
+  "contract": {
+    "schema": "matching-one.safe-transport-selection-contract.v1",
+    "question": "Does the early-minus-late completion-creation contrast persist in the uniform-safe reference chain without natural survival selection?",
+    "L": 512,
+    "N": 262144,
+    "a": 154646,
+    "b": 155385,
+    "batches": 14,
+    "samples_per_batch": 10000,
+    "source": "Replay every archived safe-insertion permutation up to first homology birth or b. For rank-one entrances J1<=b, use one fresh uniform-safe continuation to b, retaining a coupled natural survival flag. Do not restrict to original rank_b=1.",
+    "seed_rule": "first 64 SHA256 bits of matching-one/safe-transport-selection-20260930/continuation/square/L512/{batch}",
+    "coupling": "At each count, sample vacancies in uniform random order without replacement until first safe site. Kill the natural flag permanently if the first proposal completes. Insert the first safe site. All-completing states go to a separately counted cemetery. No retry cap.",
+    "endpoint": "Y=2*synergy_edges/(N-b-nu_b), within C=(primitive direction,integer nu_b)",
+    "groups": "early J1<=a; late a<J1<=b; rank-one entrance required",
+    "eligibility": "reference reaches b at rank one and nu_b<N-b. Direct rank-zero-to-two entrances, no entrance, reference cemetery and no safe endpoint are separately counted.",
+    "weights": "Common cells with both natural-alive cohorts: w=n_natural_early*n_natural_late/(n_natural_early+n_natural_late). Same normalized weights and support for all three contrasts.",
+    "primary_vector": [
+      "natural_contrast",
+      "safe_reference_contrast",
+      "selection_contrast=natural-safe"
+    ],
+    "prediction": "A selection-only explanation for this reference decomposition requires safe_reference_contrast=0. Persistence requires entrance/within-safe transport information as well; zero does not establish all-history closure.",
+    "inference": "Recompute cells, means, eligibility, common support and weights after each aligned whole-batch deletion. Save full 3x3 covariance and all14 deletions. No adaptive subgroup, feature, cutoff, sample extension or size ladder.",
+    "dependency": "Same original first-birth prefixes and batch groups as safe-insertion-independent-20260929. New conditional continuations, not independent preparations or untouched holdout.",
+    "scope": "Specified safe-reference decomposition, not unique causal allocation between imported birth geometry and subsequent safe growth; not original U, continuum scaling or an explained fraction of the previous realization.",
+    "compute": "One32-prefix old-seed cost probe (scientific weight zero), then one14x10000 bounded cloud job. No additional precision run. Benchmark anticipated ideal time at most1800 seconds and within80% measured memory.",
+    "frozen_before_continuation": true
+  }
+}
diff --git a/analysis/safe-transport-selection-20260930/run.py b/analysis/safe-transport-selection-20260930/run.py
new file mode 100644
index 000000000..32e1218de
--- /dev/null
+++ b/analysis/safe-transport-selection-20260930/run.py
@@ -0,0 +1,149 @@
+#!/usr/bin/env python3
+"""One bounded cloud coupling experiment; standard-library runtime."""
+import argparse
+from concurrent.futures import ThreadPoolExecutor, as_completed
+import csv
+from datetime import datetime, timezone
+import gzip
+import hashlib
+import importlib.util
+import json
+import math
+import os
+from pathlib import Path
+import resource
+import shutil
+import subprocess
+import time
+
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parent.parent
+OLD = HERE.parent / "safe-insertion-production-20260929/data"
+SPEC = importlib.util.spec_from_file_location("window_runtime", HERE.parent / "geometric-source-window-20260930/run.py")
+UTIL = importlib.util.module_from_spec(SPEC)
+SPEC.loader.exec_module(UTIL)
+sha, save, env = UTIL.sha, UTIL.save, UTIL.env
+SOURCES = [HERE / name for name in ["engine.cpp", "run.py", "score.py", "contract.json"]] + [
+    HERE.parent / "exact-completion-pairs-20260929/engine.cpp",
+    HERE.parent / "completion-hazard-production-20260929/engine.cpp",
+    HERE.parent / "geometric-source-window-20260930/run.py"]
+
+
+def seed(batch):
+    text = f"matching-one/safe-transport-selection-20260930/continuation/square/L512/{batch}"
+    return int.from_bytes(hashlib.sha256(text.encode()).digest()[:8], "big")
+
+
+def command(batch, samples, output):
+    return [str(HERE / "build/engine"), "512", str(samples), str(batch["seed"]),
+            str(seed(batch["batch"])), "155385", str(output)]
+
+
+def main():
+    parser = argparse.ArgumentParser()
+    parser.add_argument("mode", choices=["benchmark", "run"])
+    parser.add_argument("--source-commit", required=True)
+    parser.add_argument("--toolchain", type=Path)
+    args = parser.parse_args()
+    original = json.loads((OLD / "run.json").read_text())
+    batches = sorted(original["batches"], key=lambda x: x["batch"])
+    assert original["status"] == "completed" and len(batches) == 14
+    assert all(b["samples"] == 10000 for b in batches)
+    environment = env()
+    if args.mode == "benchmark":
+        build = HERE / "build"
+        build.mkdir(exist_ok=False)
+        compiler = shutil.which("g++")
+        if compiler:
+            compile_cmd = [compiler, "-O3", "-std=c++17", str(HERE / "engine.cpp"), "-o", str(build / "engine")]
+            version_cmd = [compiler, "--version"]
+        else:
+            if args.toolchain is None:
+                raise ValueError("explicit task-owned toolchain required")
+            project = args.toolchain / "safe-transport-selection-project"
+            project.mkdir(exist_ok=False)
+            for src in SOURCES:
+                if src.suffix == ".cpp":
+                    dst = project / src.relative_to(ROOT)
+                    dst.parent.mkdir(parents=True, exist_ok=True)
+                    shutil.copyfile(src, dst)
+            compile_cmd = ["chroot", str(args.toolchain), "/usr/bin/g++", "-O3", "-std=c++17",
+                           "/safe-transport-selection-project/analysis/safe-transport-selection-20260930/engine.cpp",
+                           "-o", "/safe-transport-selection-project/engine"]
+            version_cmd = ["chroot", str(args.toolchain), "/usr/bin/g++", "--version"]
+        subprocess.run(compile_cmd, check=True)
+        if not compiler:
+            shutil.copyfile(project / "engine", build / "engine")
+            (build / "engine").chmod(0o755)
+        output = build / "old-seed-32.csv"
+        cmd = command(batches[0], 32, output)
+        begin = time.monotonic()
+        proc = subprocess.run(cmd, check=True, capture_output=True, text=True)
+        seconds = time.monotonic()-begin
+        with output.open() as f, gzip.open(OLD / batches[0]["file"], "rt") as g:
+            rows = list(csv.DictReader(f))
+            assert len(rows) == 32
+            previous = csv.DictReader(g)
+            for row in rows:
+                assert row["J1"] == next(previous)["J1"]
+        rss = resource.getrusage(resource.RUSAGE_CHILDREN).ru_maxrss
+        workers = min(14, math.floor(environment["cpu_quota"]), int(environment["memory_bytes"]*.8//(rss*1024)))
+        assert workers >= 1
+        predicted = seconds/32*10000*math.ceil(14/workers)
+        result = dict(environment=environment, command=cmd, build_command=compile_cmd,
+                      compiler=subprocess.check_output(version_cmd, text=True), stdout=proc.stdout,
+                      wall_seconds=seconds, children_peak_rss_kib=rss, workers=workers,
+                      predicted_ideal_seconds=predicted, within_cost_budget=predicted<=1800,
+                      rows=32, original_births_match=True, scientific_weight=0,
+                      source_commit=args.source_commit, binary_sha256=sha(build / "engine"))
+        save(HERE / "benchmark.json", result)
+        print(json.dumps(result), flush=True)
+        return
+    bench = json.loads((HERE / "benchmark.json").read_text())
+    assert bench["within_cost_budget"] and sha(HERE / "build/engine") == bench["binary_sha256"]
+    workers = min(bench["workers"], math.floor(environment["cpu_quota"]), 14)
+    data = HERE / "data"
+    data.mkdir(exist_ok=False)
+    record = dict(schema="matching-one.safe-transport-selection-run.v1", status="running", pid=os.getpid(),
+                  started_utc=datetime.now(timezone.utc).isoformat(), environment=environment,
+                  source_commit=args.source_commit, binary_sha256=bench["binary_sha256"], workers=workers,
+                  source_sha256={str(p.relative_to(ROOT)): sha(p) for p in SOURCES},
+                  original_manifest_sha256=sha(OLD / "run.json"), new_independent_prefixes=0, batches=[])
+    save(data / "started.json", record)
+    def one(batch):
+        output = data / ("selection-b%02d.csv" % batch["batch"])
+        cmd = command(batch, 10000, output)
+        begin = time.monotonic()
+        proc = subprocess.run(cmd, capture_output=True, text=True)
+        row = dict(batch=batch["batch"], permutation_seed=batch["seed"], continuation_seed=str(seed(batch["batch"])),
+                   samples=10000, original_file=batch["file"], original_sha256=batch["sha256"],
+                   dependency_group=batch["dependency_group"], command=cmd, returncode=proc.returncode,
+                   stdout=proc.stdout, stderr=proc.stderr, wall_seconds=time.monotonic()-begin)
+        if not proc.returncode:
+            compressed = output.with_suffix(".csv.gz")
+            with output.open("rb") as f, compressed.open("xb") as g:
+                with gzip.GzipFile(filename="", mode="wb", fileobj=g, mtime=0) as z:
+                    shutil.copyfileobj(f, z)
+            row.update(file=compressed.name, sha256=sha(compressed))
+        save(data / ("batch-%02d.json" % batch["batch"]), row)
+        if proc.returncode:
+            raise RuntimeError(row)
+        print(json.dumps(dict(batch=row["batch"], status="completed", seconds=row["wall_seconds"])), flush=True)
+        return row
+    begin = time.monotonic()
+    try:
+        with ThreadPoolExecutor(max_workers=workers) as pool:
+            for task in as_completed([pool.submit(one, b) for b in batches]):
+                record["batches"].append(task.result())
+        record["batches"].sort(key=lambda b: b["batch"])
+        record["status"] = "completed"
+    except Exception as exc:
+        record.update(status="failed", error=str(exc))
+        raise
+    finally:
+        record.update(wall_seconds=time.monotonic()-begin, finished_utc=datetime.now(timezone.utc).isoformat())
+        save(data / "run.json", record)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/analysis/safe-transport-selection-20260930/score.py b/analysis/safe-transport-selection-20260930/score.py
new file mode 100644
index 000000000..d17ac2573
--- /dev/null
+++ b/analysis/safe-transport-selection-20260930/score.py
@@ -0,0 +1,134 @@
+#!/usr/bin/env python3
+"""Same-cell natural/reference/selection contrast, with one batch covariance."""
+from collections import Counter, defaultdict
+import csv
+import gzip
+import hashlib
+from itertools import zip_longest
+import json
+import math
+from pathlib import Path
+
+HERE = Path(__file__).resolve().parent
+OLD = HERE.parent / "safe-insertion-production-20260929/data"
+A, B, M = 154646, 155385, 262144-155385
+NAMES = ["natural_contrast", "safe_reference_contrast", "selection_contrast"]
+
+
+def evaluate(batches, omit=None, detail=False):
+    # Cell vectors: [early safe count,sum,natural count,sum, late ...].
+    cells = defaultdict(lambda: [0., 0., 0., 0., 0., 0., 0., 0.])
+    eligible = [0, 0]
+    for batch in batches:
+        if batch["batch"] == omit:
+            continue
+        for key, row in batch["cells"].items():
+            for j, value in enumerate(row):
+                cells[key][j] += value
+            eligible[0] += int(row[2])
+            eligible[1] += int(row[6])
+    common = []
+    for key, row in sorted(cells.items()):
+        se, ye, ne, ze, sl, yl, nl, zl = row
+        if ne and nl:
+            weight = ne*nl/(ne+nl)
+            common.append(dict(cell=list(key), weight=weight, safe_counts=[int(se), int(sl)],
+                               natural_counts=[int(ne), int(nl)],
+                               safe_means=[ye/se, yl/sl], natural_means=[ze/ne, zl/nl]))
+    weight = math.fsum(c["weight"] for c in common)
+    if not weight:
+        raise ValueError("not_scoreable: no natural-cohort overlap")
+    natural = math.fsum(c["weight"]*(c["natural_means"][0]-c["natural_means"][1]) for c in common)/weight
+    safe = math.fsum(c["weight"]*(c["safe_means"][0]-c["safe_means"][1]) for c in common)/weight
+    result = [natural, safe, natural-safe]
+    if not detail:
+        return result
+    return result, dict(common_cells=len(common), weight_sum=weight,
+                        eligible_natural_counts=eligible,
+                        common_natural_counts=[sum(c["natural_counts"][g] for c in common) for g in range(2)],
+                        common_safe_counts=[sum(c["safe_counts"][g] for c in common) for g in range(2)],
+                        weighted_natural_means=[math.fsum(c["weight"]*c["natural_means"][g] for c in common)/weight for g in range(2)],
+                        weighted_safe_means=[math.fsum(c["weight"]*c["safe_means"][g] for c in common)/weight for g in range(2)],
+                        cells=common)
+
+
+def main():
+    manifest = json.loads((HERE / "data/run.json").read_text())
+    assert manifest["status"] == "completed"
+    batches = []
+    for batch in manifest["batches"]:
+        path = HERE / "data" / batch["file"]
+        oldpath = OLD / batch["original_file"]
+        assert hashlib.sha256(path.read_bytes()).hexdigest() == batch["sha256"]
+        assert hashlib.sha256(oldpath.read_bytes()).hexdigest() == batch["original_sha256"]
+        cells = defaultdict(lambda: [0., 0., 0., 0., 0., 0., 0., 0.])
+        counts = Counter()
+        with gzip.open(path, "rt") as f, gzip.open(oldpath, "rt") as g:
+            for new, old in zip_longest(csv.DictReader(f), csv.DictReader(g)):
+                assert new is not None and old is not None and new["J1"] == old["J1"]
+                counts["prefixes"] += 1
+                j1, entry, ref, alive, c, e = (int(new[k]) for k in ["J1", "entry_rank", "reference_rank_b", "natural_alive", "nu_b", "synergy_edges"])
+                if entry == 0:
+                    assert j1 == 0 and ref == 0 and alive == 0
+                    counts["no_entrance_by_b"] += 1
+                    continue
+                if entry == 2:
+                    assert 1 <= j1 <= B and ref == 2 and alive == 0
+                    counts["direct_rank_two_entrance"] += 1
+                    continue
+                assert entry == 1 and 1 <= j1 <= B and alive in (0, 1)
+                group = 0 if j1 <= A else 1
+                label = "early" if group == 0 else "late"
+                counts["rank_one_entrance_"+label] += 1
+                if ref == -1:
+                    assert alive == 0
+                    counts["reference_cemetery_"+label] += 1
+                    continue
+                assert ref == 1 and 0 <= c <= M and e >= 0
+                counts["reference_endpoint_"+label] += 1
+                counts["natural_endpoint_"+label] += alive
+                if c == M:
+                    counts["no_safe_endpoint_"+label] += 1
+                    continue
+                d = (int(new["dx_entry"]), int(new["dy_entry"]))
+                assert d != (0, 0)
+                y = 2*e/(M-c)
+                row = cells[d+(c,)]
+                off = 4*group
+                row[off] += 1
+                row[off+1] += y
+                row[off+2] += alive
+                row[off+3] += alive*y
+        assert counts["prefixes"] == batch["samples"] == 10000
+        batches.append(dict(batch=batch["batch"], dependency_group=batch["dependency_group"],
+                            counts=dict(counts), cells=dict(cells)))
+    assert sorted(b["batch"] for b in batches) == list(range(14))
+    estimate, support = evaluate(batches, detail=True)
+    deletions = [evaluate(batches, omit=i) for i in range(14)]
+    center = [math.fsum(d[j] for d in deletions)/14 for j in range(3)]
+    covariance = [[13/14*math.fsum((d[i]-center[i])*(d[j]-center[j]) for d in deletions) for j in range(3)] for i in range(3)]
+    se = [math.sqrt(covariance[i][i]) for i in range(3)]
+    counts = sum((Counter(b["counts"]) for b in batches), Counter())
+    result = dict(schema="matching-one.safe-transport-selection.v1", names=NAMES, estimate=estimate,
+                  batch_delete_one_se=se, covariance=covariance, deletions=deletions, support=support,
+                  counts=dict(counts), batch_counts=[dict(batch=b["batch"], counts=b["counts"], dependency_group=b["dependency_group"]) for b in batches],
+                  all_original_births_match=True, new_independent_prefixes=0,
+                  dependency_group="safe-insertion-independent-20260929/square/L512",
+                  contract=json.loads((HERE / "contract.json").read_text()))
+    target = HERE / "results"
+    target.mkdir(exist_ok=False)
+    (target / "result.json").write_text(json.dumps(result, indent=2, allow_nan=False)+"\n")
+    lines = ["# Safe transport versus natural survival selection", "",
+             "Same archived entrances, new conditional continuations; not an independent prefix block.", "",
+             "| Contrast | Estimate | Whole-batch SE |", "|---|---:|---:|"]
+    lines += [f"| {name} | {value:.12g} | {error:.12g} |" for name, value, error in zip(NAMES, estimate, se)]
+    lines += ["", f"{support['common_cells']} common natural-cohort cells; one set of endpoint weights for all components.",
+              f"Natural support {support['common_natural_counts']} / eligible {support['eligible_natural_counts']}; reference support {support['common_safe_counts']}.",
+              "Natural = safe reference + selection exactly. Full singular3x3 covariance and14 aligned deletions in result.json.",
+              "This decomposes a specified reference process; it does not distinguish imported birth geometry from subsequent safe growth, or identify original U/continuum fields.", ""]
+    (target / "RESULT.md").write_text("\n".join(lines))
+    print("\n".join(lines))
+
+
+if __name__ == "__main__":
+    main()
diff --git a/analysis/safe-transport-selection-20260930/verification.md b/analysis/safe-transport-selection-20260930/verification.md
new file mode 100644
index 000000000..ed66919fb
--- /dev/null
+++ b/analysis/safe-transport-selection-20260930/verification.md
@@ -0,0 +1,142 @@
+# Narrow verification: safe transport and survival selection
+
+2026-09-30. **No actionable correctness error found in the reviewed source.**
+The random-order rejection/first-proposal coupling, endpoint eligibility,
+common-cell decomposition and aligned batch covariance implement the stated contract.
+This review supplies no new precision estimate and does not certify a cloud run.
+
+Reviewed at source commit `5054c02383242473834045fcbddc7379d83acfb9`;
+the six reviewed files were unchanged through observed HEAD
+`a58643d1f787cd73e3f12610d517b2fc5d0e3b91`.
+Only this verification file was written; no commits or pushes.
+
+## Algebra and source review
+
+| Item | Finding |
+|---|---|
+| First-birth source | engine.cpp:18–27 generates the original complete Fisher–Yates permutation from its archived seed, then stops at the first positive rank or b. It includes every rank-one entrance J1≤b, including prefixes whose **old** endpoint later reached rank two. Direct 0→2 entrances and no entrance by b are separately recorded. Restricting to old rank_b=1 would have selected the source twice; the code does not do that. |
+| Separate continuation stream | The old permutation RNG and new `future` RNG are separate (engine.cpp:14). run.py derives one continuation seed per original batch from the exact contracted SHA256 namespace, first eight bytes in big-endian order. Conditional branches share their original batch dependency; they are not new independent preparations. Birth-only alignment to the old CSV is appropriate: the new endpoints are deliberately different realizations. |
+| Random-order rejection | At descending position r, swapping with a uniform index in [0,r] draws uniformly from the as-yet unexamined vacancies. Previously rejected sites remain in the frozen suffix for this step. On acceptance, swapping that site with `back()` and popping removes exactly the accepted site, preserving every rejected vacancy. Starting the next partial shuffle from any resulting array again gives a uniform random order; its arbitrary ordering cannot bias the next accepted site. |
+| Local completion query | Geometry.completes_if_inserted:94–107 uses the same lifted-neighbour potential conflict as completion_count:70–90. It checks neighbours in a common old component, detects a winding transverse to the current rank-one direction, and never inserts a site. `find` may compress union-find paths; this is a representation change, not a change in occupation or homology. The query is called only on current vacancies of a rank-one geometry. |
+| First-proposal flag | For each safe v, P(accepted v)=1/s and P(first proposal safe, accepted v)=1/m. Consequently the first-safe indicator has probability s/m independently of accepted v, conditional on the geometry. engine.cpp:36 updates the permanent flag only on the first proposal; later rejected proposals are not additional natural insertion times. If the flag was already zero it remains zero. `first_rejection_count` records the first killing **insertion count** k+1, rather than a tally of all rejections. |
+| Whole-path coupling | The above conditional probabilities hold for any vacancy-array ordering and past flag history. Thus, conditional on the accepted safe path, E[I_end]=∏s_i/m_i=W. Alive accepted transitions have probability 1/m, exactly the killed original uniform kernel; dead flags do not stop the reference chain. No completion-census estimator or product of estimated probabilities enters this construction. |
+| Cemetery | If s=0, all m vacancies are examined once, no site is inserted, the code sets rank=-1 and alive=0, and exits the count loop. There is no retry cap or infinite rejection loop. This branch was source-reviewed, not separately simulated. |
+| Endpoint and direction | Successful paths contain exactly b distinct occupied sites. Safe insertions preserve the primitive direction; the explicit endpoint direction assertion justifies keying endpoint cells by `dx_entry,dy_entry`. The pair counter is invoked once at b and supplies c and e; Y=2e/(N−b−c). No entrance, direct rank-two entrance, reference cemetery and c=N−b are counted separately, with c=N−b excluded because Y is undefined there. |
+| Fixed parameters | run.py passes L=512 and b=155385; score.py uses a=154646, b=155385 and m=106759. These match contract.json. Source replay, cohort definitions and endpoint eligibility refer to the same fixed count clock. |
+
+For each eligible cell C=(D,c), write nS, sumY, nN=sum I and sumIY
+for the reference count, reference outcome sum, natural-alive count and
+natural-alive outcome sum. score.py:95–101 accumulates precisely these
+four quantities separately for each cohort. In that cell,
+
+```text
+mu_safe = sumY/nS
+mu_natural = sumIY/nN
+Cov_emp(Y,I)/mean_emp(I)
+  = [sumIY/nS - (sumY/nS)(nN/nS)] / (nN/nS)
+  = mu_natural - mu_safe.
+```
+
+The covariance here uses the empirical probability convention (denominator nS),
+not an nS−1 unbiased sample-covariance convention. The scorer implements the
+difference directly, so no erroneous finite-sample covariance factor is introduced.
+
+score.py:31–43 retains only cells with nN_early>0 and nN_late>0;
+these automatically also have positive reference counts. It uses the **same**
+weight w=nN_early*nN_late/(nN_early+nN_late), support and normalization
+for natural and reference contrasts, then sets selection=natural−reference.
+Thus the decomposition is of a natural-overlap-weighted endpoint-cell contrast,
+not an unconditional population contrast or a unique causal attribution.
+Reference observations with dead flags remain in their reference cell means.
+Natural conditional means have the original survivor law in distribution;
+they need not reproduce the previously observed archive endpoints row by row.
+
+On each deletion, evaluate() reconstructs cell aggregates from the remaining
+eligible batch rows, redetermines natural-cohort overlap, and recomputes all
+means and weights. Precomputed per-row eligibility is fixed by the endpoint
+contract, so excluding the deleted batch's eligible rows is the same operation
+as reevaluating that fixed eligibility rule. If no natural overlap remains,
+the code raises `not_scoreable` rather than inventing a zero contrast.
+
+score.py:107–110 uses the 14 aligned three-vectors and
+`(13/14)*sum((theta_minus_batch - deletion_mean) outer itself)`.
+The full 3×3 covariance, including cross terms, is correct; its singularity
+is expected because the third coordinate is the first minus the second on
+every deletion. The scorer saves all 14 deletion vectors. It does not add
+component variances as if the natural/reference estimators were independent.
+
+## Actual check 1: one deterministic L4 vacancy-query control
+
+Used one fixed square-L4 geometry, v=x+4y:
+
+```text
+A={0,1,2,4,5,8,12}; rank=1; D=(0,1); vacancies=9; c=1.
+```
+
+For each of its nine vacancies, the **actual current C++** Geometry was copied
+and `copy.insert(v)==2` compared with `g.completes_if_inserted(v)` on the
+original geometry. All **9/9 agreed**; only site 3 completes. The original
+direction and completion_count remained unchanged after all queries.
+This single geometry therefore exercises both safe and completing answers.
+
+To keep the sole-file write boundary, the Geometry source was read into memory,
+followed by a tiny C++ harness; `/usr/bin/clang++ -std=c++17 -O1 -S -emit-llvm
+-x c++ -o - -` took stdin and returned LLVM IR on stdout. Research Python
+3.11.15 and llvmlite/LLVM 20.1.8 executed it in memory; no test source or
+build artifact was saved. The first JIT setup aborted before executing the
+control because Apple Clang 21 emitted the unsupported backend attribute
+`"probe-stack"="__chkstk_darwin"`. Removing only that attribute from the
+in-memory IR allowed the single fixed control to execute successfully.
+No Geometry logic, repository source or production binary was modified.
+This is a local semantic control, not a verification of the cloud compiler.
+
+## Actual check 2: one exact analytic coupling/decomposition control
+
+Used a minimal two-step monotone abstract kernel with vacancies
+`{a,b,c,d,x}`: x always completes and {a,b} is a synergy pair. Thus initially
+m=5,s=4; after a or b there are two safe sites among four vacancies, and
+after c or d there are three. This is an analytic kernel control, not an
+extra lattice preparation or Monte Carlo replica.
+
+A Python standard-library Fraction recursion enumerated the descending
+partial Fisher–Yates choices, first-proposal flag update and swap/pop removal
+literally as in the engine; no RNG was used. Every branch deleted only its
+accepted site and retained all rejected vacancies. Exact results:
+
+- Each initial safe site has P(accepted v)=1/4 and
+  P(first-safe, accepted v)=1/5.
+- For every complete accepted path, E[I_end | path] equals W:
+  2/5 for paths starting a or b; 3/5 for paths starting c or d.
+- With f=1{endpoint={c,d}}, E_safe[f]=1/6, E[I]=1/2,
+  E[If]=1/10 and E[f | I=1]=1/5.
+- `natural−safe = 1/5−1/6 = 1/30 = Cov(f,I)/E[I]` exactly.
+
+No other physical configuration, lag grid, census, archive score or production
+was run. Support and batch covariance were reviewed algebraically from the
+current scorer; no synthetic batch-replica suite was added.
+
+## Scope and handoff
+
+Read the requested note, engine, contract, runner, scorer and added Geometry
+query, with their directly relevant reuse interfaces. No SSH, cloud operation,
+production, archive replay, extra precision calculation, commit or push.
+No substantive issue requires delaying the main investigator's bounded job.
+Cloud execution, runtime row correspondence, empirical support and numerical
+results remain the main thread's responsibility; they were not claimed checked.
+
+Reviewed SHA256 values:
+
+```text
+notes/safe-path-selection-coupling-20260930.md
+4001b5127d4ceb7b362d6f772a000e9c8a4cccbab102004e0af133f82c42fc96
+analysis/safe-transport-selection-20260930/engine.cpp
+4d5d1ed48beffc0e13f8db12da67024118a4b1ea75fa7f8e68b3c6153c072c0b
+analysis/safe-transport-selection-20260930/contract.json
+22ca3b823989df75eeb1fe6091bdeee38eaf6229950c4cc441421fcbd3850bc2
+analysis/safe-transport-selection-20260930/run.py
+a5bf404c70bf65cd28dc18a4aefa2b59729f65c64b74c11b0f0f74a548e9f31c
+analysis/safe-transport-selection-20260930/score.py
+50e681e2d1422e0d2b1c4aa089bdc462d1a5b7767de2f585821c50a1db2d7b74
+analysis/completion-hazard-production-20260929/engine.cpp
+da45ae8a2682f9edb24eaf87a6733c55013f1fb95b3622bf2faf97208c405c0b
+```
diff --git a/analysis/stochastic-parity-block-encoding-20260929/check.py b/analysis/stochastic-parity-block-encoding-20260929/check.py
new file mode 100644
index 000000000..f3981ef6c
--- /dev/null
+++ b/analysis/stochastic-parity-block-encoding-20260929/check.py
@@ -0,0 +1,197 @@
+#!/usr/bin/env python3
+"""One small-k exact control of parity encoding, tensor rank and readout TV.
+
+Standard library only. k=2,3,4; no physical graph enumeration, old-census
+rerun, Monte Carlo, cloud access, or full future-query rank claim.
+Print JSON to stdout; the geometric and all-k proofs are in the note.
+"""
+
+from collections import Counter
+from fractions import Fraction
+from itertools import product
+import json
+from math import comb
+
+
+LOGICAL = ((0, 0), (1, 0), (0, 1), (1, 1))
+EPSILONS = (Fraction(1, 10), Fraction(1, 4))
+
+
+def qstr(value):
+    value = Fraction(value)
+    return f"{value.numerator}/{value.denominator}"
+
+
+def exact_rank(rows):
+    basis = []
+    for source in rows:
+        row = [Fraction(x) for x in source]
+        for pivot, vector in basis:
+            factor = row[pivot]
+            if factor:
+                row = [x - factor*y for x, y in zip(row, vector)]
+        pivot = next((i for i, value in enumerate(row) if value), None)
+        if pivot is not None:
+            leading = row[pivot]
+            basis.append((pivot, [x/leading for x in row]))
+    return len(basis)
+
+
+def monotone_truth_tables(k):
+    # A monotone function on k variables consists uniquely of monotone
+    # lower/upper sections f0 <= f1 on k-1 variables. Avoid enumerating
+    # all 2^(2^k) Boolean tables or all encoders.
+    tables = [0, 1]
+    for dimension in range(1, k+1):
+        shift = 1 << (dimension-1)
+        tables = [low | (high << shift)
+                  for low in tables for high in tables if low & ~high == 0]
+    return tables
+
+
+def parity_difference(table, k):
+    signed = sum((-1)**x.bit_count() for x in range(1 << k)
+                 if table >> x & 1)
+    return Fraction(signed, 1 << (k-1))
+
+
+def analyze(k):
+    full = (1 << k)-1
+    classes = [[x for x in range(1 << k) if x.bit_count() % 2 == bit]
+               for bit in (0, 1)]
+    code_supports = [set(product(classes[u], classes[v])) for u, v in LOGICAL]
+    assert all(len(support) == 1 << (2*k-2) for support in code_supports)
+    assert sum(map(len, code_supports)) == len(set.union(*code_supports)) == 4**k
+
+    marginal_checks = 0
+    for support in code_supports:
+        for subset in range(full):  # every proper subset of physical blocks
+            counts = Counter((x & subset, y & subset) for x, y in support)
+            assert len(counts) == 4**subset.bit_count()
+            assert set(counts.values()) == {4**(k-subset.bit_count()-1)}
+            marginal_checks += 1
+
+    queries = list(product(range(3), repeat=k))
+    rows = []
+    moment_checks = 0
+    for (u, v), support in zip(LOGICAL, code_supports):
+        row = []
+        for query in queries:
+            a = sum(1 << i for i, q in enumerate(query) if q == 1)
+            b = sum(1 << i for i, q in enumerate(query) if q == 2)
+            actual = Fraction(sum(x & a == a and y & b == b for x, y in support),
+                              len(support))
+            predicted = Fraction(1, 1 << (a.bit_count()+b.bit_count()))
+            if a == full:
+                predicted *= 1 + (-1)**(k+u)
+            if b == full:
+                predicted *= 1 + (-1)**(k+v)
+            assert actual == predicted
+            row.append(actual)
+            moment_checks += 1
+        rows.append(row)
+    rank = exact_rank(rows)
+    assert rank == 3
+    assert all(rows[0][j]-rows[1][j]-rows[2][j]+rows[3][j] == 0
+               for j in range(len(queries)))
+    selected = [queries.index((q,)*k) for q in range(3)]
+    product_tv = max(abs(rows[0][j]-rows[1][j]) for j in range(len(queries)))
+    assert product_tv == Fraction(1, 1 << (k-1))
+
+    tables = monotone_truth_tables(k)
+    differences = [parity_difference(table, k) for table in tables]
+    bound = Fraction(comb(k-1, (k-1)//2), 1 << (k-1))
+    assert max(map(abs, differences)) == bound
+    threshold_results = []
+    for threshold in range(1, k+1):
+        table = sum(1 << x for x in range(1 << k)
+                    if x.bit_count() >= threshold)
+        delta = parity_difference(table, k)
+        assert delta == Fraction((-1)**threshold * comb(k-1, threshold-1),
+                                 1 << (k-1))
+        if abs(delta) == bound:
+            threshold_results.append(threshold)
+
+    noise_results = []
+    for epsilon in EPSILONS:
+        attenuation = (1-2*epsilon)**k
+        # Direct finite BSC summation, not the Fourier formula under test.
+        noisy = []
+        for parity in (0, 1):
+            distribution = []
+            for y in range(1 << k):
+                mass = sum(epsilon**((x ^ y).bit_count()) *
+                           (1-epsilon)**(k-(x ^ y).bit_count())
+                           for x in classes[parity]) / len(classes[parity])
+                formula = Fraction(1, 1 << k) * (
+                    1 + (-1)**(parity+y.bit_count())*attenuation)
+                assert mass == formula
+                distribution.append(mass)
+            assert sum(distribution) == 1
+            noisy.append(distribution)
+        full_tv = sum(abs(a-b) for a, b in zip(*noisy))/2
+        assert full_tv == attenuation
+        parity_error = sum(noisy[0][y] for y in range(1 << k)
+                           if y.bit_count() % 2)
+        assert parity_error == (1-attenuation)/2
+        noisy_tvs = []
+        for table, delta in zip(tables, differences):
+            noisy_delta = sum(noisy[0][y]-noisy[1][y]
+                              for y in range(1 << k) if table >> y & 1)
+            assert noisy_delta == attenuation*delta
+            noisy_tvs.append(abs(noisy_delta))
+        assert max(noisy_tvs) == attenuation*bound
+        noise_results.append({
+            "epsilon": qstr(epsilon),
+            "all_noisy_shares_TV": qstr(full_tv),
+            "parity_decoder_error": qstr(parity_error),
+            "max_single_monotone_TV": qstr(max(noisy_tvs)),
+        })
+
+    return {
+        "k": k,
+        "parity_class_size": len(classes[0]),
+        "joint_support_size_per_logical_state": len(code_supports[0]),
+        "four_supports_pairwise_disjoint": True,
+        "proper_block_marginal_checks": marginal_checks,
+        "tensor_columns": len(queries),
+        "exact_moment_checks": moment_checks,
+        "exact_tensor_response_rank": rank,
+        "alternating_row_relation": [1, -1, -1, 1],
+        "constant_allU_allV_columns": [[qstr(row[j]) for j in selected] for row in rows],
+        "max_single_product_query_TV_for_u": qstr(product_tv),
+        "monotone_boolean_functions_checked": len(tables),
+        "max_single_monotone_TV": qstr(bound),
+        "best_single_monotone_equal_prior_error": qstr((1-bound)/2),
+        "maximizing_monotone_functions": sum(abs(delta) == bound for delta in differences),
+        "maximizing_weight_thresholds": threshold_results,
+        "hypothetical_independent_share_BSC": noise_results,
+    }
+
+
+def main():
+    cases = [analyze(k) for k in (2, 3, 4)]
+    result = {
+        "schema": "matching-one.stochastic-parity-block-encoding.v1",
+        "contract": {
+            "logical_state_order": [list(state) for state in LOGICAL],
+            "encoding": "independent uniform U,V subject to xor(U)=u and xor(V)=v",
+            "encoder_randomness": "private; no side channel revealing the sampled shares",
+            "response_space": "full tensor of local (1,U_i,V_i)",
+            "general_physical_future_query_rank_upper_bound_proved_here": False,
+            "monotone_TV_scope": "one Boolean output; not a transcript or adaptive controller",
+            "BSC_scope": "independent errors on observed shares, not derived physical instruction errors",
+        },
+        "arithmetic": "exact integers and fractions.Fraction",
+        "cases": cases,
+        "total_tensor_moments_checked": sum(case["exact_moment_checks"] for case in cases),
+        "total_monotone_functions_checked": sum(case["monotone_boolean_functions_checked"] for case in cases),
+        "failures": 0,
+        "not_run": ["old deterministic census", "physical graph enumeration", "Monte Carlo",
+                    "cloud jobs", "whole-machine audit"],
+    }
+    print(json.dumps(result, ensure_ascii=False, indent=2, sort_keys=True))
+
+
+if __name__ == "__main__":
+    main()
diff --git a/analysis/stochastic-parity-block-encoding-20260929/result.json b/analysis/stochastic-parity-block-encoding-20260929/result.json
new file mode 100644
index 000000000..60e270514
--- /dev/null
+++ b/analysis/stochastic-parity-block-encoding-20260929/result.json
@@ -0,0 +1,224 @@
+{
+  "arithmetic": "exact integers and fractions.Fraction",
+  "cases": [
+    {
+      "alternating_row_relation": [
+        1,
+        -1,
+        -1,
+        1
+      ],
+      "best_single_monotone_equal_prior_error": "1/4",
+      "constant_allU_allV_columns": [
+        [
+          "1/1",
+          "1/2",
+          "1/2"
+        ],
+        [
+          "1/1",
+          "0/1",
+          "1/2"
+        ],
+        [
+          "1/1",
+          "1/2",
+          "0/1"
+        ],
+        [
+          "1/1",
+          "0/1",
+          "0/1"
+        ]
+      ],
+      "exact_moment_checks": 36,
+      "exact_tensor_response_rank": 3,
+      "four_supports_pairwise_disjoint": true,
+      "hypothetical_independent_share_BSC": [
+        {
+          "all_noisy_shares_TV": "16/25",
+          "epsilon": "1/10",
+          "max_single_monotone_TV": "8/25",
+          "parity_decoder_error": "9/50"
+        },
+        {
+          "all_noisy_shares_TV": "1/4",
+          "epsilon": "1/4",
+          "max_single_monotone_TV": "1/8",
+          "parity_decoder_error": "3/8"
+        }
+      ],
+      "joint_support_size_per_logical_state": 4,
+      "k": 2,
+      "max_single_monotone_TV": "1/2",
+      "max_single_product_query_TV_for_u": "1/2",
+      "maximizing_monotone_functions": 2,
+      "maximizing_weight_thresholds": [
+        1,
+        2
+      ],
+      "monotone_boolean_functions_checked": 6,
+      "parity_class_size": 2,
+      "proper_block_marginal_checks": 12,
+      "tensor_columns": 9
+    },
+    {
+      "alternating_row_relation": [
+        1,
+        -1,
+        -1,
+        1
+      ],
+      "best_single_monotone_equal_prior_error": "1/4",
+      "constant_allU_allV_columns": [
+        [
+          "1/1",
+          "0/1",
+          "0/1"
+        ],
+        [
+          "1/1",
+          "1/4",
+          "0/1"
+        ],
+        [
+          "1/1",
+          "0/1",
+          "1/4"
+        ],
+        [
+          "1/1",
+          "1/4",
+          "1/4"
+        ]
+      ],
+      "exact_moment_checks": 108,
+      "exact_tensor_response_rank": 3,
+      "four_supports_pairwise_disjoint": true,
+      "hypothetical_independent_share_BSC": [
+        {
+          "all_noisy_shares_TV": "64/125",
+          "epsilon": "1/10",
+          "max_single_monotone_TV": "32/125",
+          "parity_decoder_error": "61/250"
+        },
+        {
+          "all_noisy_shares_TV": "1/8",
+          "epsilon": "1/4",
+          "max_single_monotone_TV": "1/16",
+          "parity_decoder_error": "7/16"
+        }
+      ],
+      "joint_support_size_per_logical_state": 16,
+      "k": 3,
+      "max_single_monotone_TV": "1/2",
+      "max_single_product_query_TV_for_u": "1/4",
+      "maximizing_monotone_functions": 1,
+      "maximizing_weight_thresholds": [
+        2
+      ],
+      "monotone_boolean_functions_checked": 20,
+      "parity_class_size": 4,
+      "proper_block_marginal_checks": 28,
+      "tensor_columns": 27
+    },
+    {
+      "alternating_row_relation": [
+        1,
+        -1,
+        -1,
+        1
+      ],
+      "best_single_monotone_equal_prior_error": "5/16",
+      "constant_allU_allV_columns": [
+        [
+          "1/1",
+          "1/8",
+          "1/8"
+        ],
+        [
+          "1/1",
+          "0/1",
+          "1/8"
+        ],
+        [
+          "1/1",
+          "1/8",
+          "0/1"
+        ],
+        [
+          "1/1",
+          "0/1",
+          "0/1"
+        ]
+      ],
+      "exact_moment_checks": 324,
+      "exact_tensor_response_rank": 3,
+      "four_supports_pairwise_disjoint": true,
+      "hypothetical_independent_share_BSC": [
+        {
+          "all_noisy_shares_TV": "256/625",
+          "epsilon": "1/10",
+          "max_single_monotone_TV": "96/625",
+          "parity_decoder_error": "369/1250"
+        },
+        {
+          "all_noisy_shares_TV": "1/16",
+          "epsilon": "1/4",
+          "max_single_monotone_TV": "3/128",
+          "parity_decoder_error": "15/32"
+        }
+      ],
+      "joint_support_size_per_logical_state": 64,
+      "k": 4,
+      "max_single_monotone_TV": "3/8",
+      "max_single_product_query_TV_for_u": "1/8",
+      "maximizing_monotone_functions": 2,
+      "maximizing_weight_thresholds": [
+        2,
+        3
+      ],
+      "monotone_boolean_functions_checked": 168,
+      "parity_class_size": 8,
+      "proper_block_marginal_checks": 60,
+      "tensor_columns": 81
+    }
+  ],
+  "contract": {
+    "BSC_scope": "independent errors on observed shares, not derived physical instruction errors",
+    "encoder_randomness": "private; no side channel revealing the sampled shares",
+    "encoding": "independent uniform U,V subject to xor(U)=u and xor(V)=v",
+    "general_physical_future_query_rank_upper_bound_proved_here": false,
+    "logical_state_order": [
+      [
+        0,
+        0
+      ],
+      [
+        1,
+        0
+      ],
+      [
+        0,
+        1
+      ],
+      [
+        1,
+        1
+      ]
+    ],
+    "monotone_TV_scope": "one Boolean output; not a transcript or adaptive controller",
+    "response_space": "full tensor of local (1,U_i,V_i)"
+  },
+  "failures": 0,
+  "not_run": [
+    "old deterministic census",
+    "physical graph enumeration",
+    "Monte Carlo",
+    "cloud jobs",
+    "whole-machine audit"
+  ],
+  "schema": "matching-one.stochastic-parity-block-encoding.v1",
+  "total_monotone_functions_checked": 194,
+  "total_tensor_moments_checked": 468
+}
diff --git a/docs/RESEARCH-FRONTIER.md b/docs/RESEARCH-FRONTIER.md
index 5e3c52cec..a039b9c66 100644
--- a/docs/RESEARCH-FRONTIER.md
+++ b/docs/RESEARCH-FRONTIER.md
@@ -1,157 +1,274 @@
-# Research frontier: one active paper, two visible reserves
-
-**2026-09-13, owner-delegated reset reconciled with the latest #650.**
-The single active deliverable is a probability-theorem manuscript and independent
-proof audit: #735's arbitrary-period root theorem, with #613/#736's sharp axial
-full-law boundary. Return one acceptance package to #735. The other directions
-below remain scientifically visible but are not parallel active assignments.
-
-[ROADMAP](ROADMAP.md) controls execution; live Issue updates override this dated
-snapshot. [STATUS](STATUS.md) distinguishes current scope from the
-[verbatim historical ledger](history/STATUS-before-20260913.md). The baseline
-read was main at `eb89e9422791d9e3c3a78f0e65d56912b815a7bd` (#702--#705 merged).
-Navigation was integrated by #738; the research PRs below remain unmerged.
-
-## 1. Active probability paper: correct root does not imply a concentrated law
-
-For independent NN square-site percolation on an honest torus,
-`M=P2-P0`, `F=E[r]/2=(1+M)/2`, `Q=F^{-1}`. The balance root is Q(1/2).
-The conditional rare-sector function `H=P2/(P0+P2)` is not F.
-
-#735 supplies arbitrary-integer-period root consistency as the genuine shortest
-period ell grows, with no aspect or area-versus-systole restriction. At fixed
-subcritical p it compares rates: `P2/P0 <= exp[-kappa(p) N/ell]`.
-#736 adds the exact axial contrast: for 2<=w<=m and wm->infinity,
-all fixed interior Q(u) converge to p_c **iff** `log(m)/w -> 0`.
-These are proofs under named probability/topology inputs, not a new numerical
-critical point, a fitted root-shift exponent or an established priority claim.
-
-The new contribution in this reset is #736's necessity argument. Critical
-square-site RSW and finite-product continuity give a seam-closed occupied ring
-with probability >=exp(-eta*w) at a FIXED p_eta<p_c. Independent transverse
-bands amplify it. On a subsequence with log(m)>=d*w this keeps every fixed
-lower quantile away from p_c, while the median can remain consistent. Failure
-of a union upper bound alone would never have proved this implication.
-
-**Current acceptance work:** consolidate the theorem statement and imported
-inputs, independently audit #735's local support/injectivity, oblique entry
-count, backtracking-path extraction and disjoint-band independence; audit #736's
-seam and quantifier order; compare the exact statements with their closest
-strip-percolation, RSW and homological-percolation predecessors. Deliver one
-manuscript, not another growing list of review tickets. Finite code checks and
-verbatim quotations are not substitutes for the all-size argument.
-
-**Potential strengthening, not a new prerequisite or second dispatch:** build
-uniform fixed-subcritical winding corridors around an arbitrary shortest
-integer period u, with probability >=exp(-eta*|u|), then pack disjoint corridors.
-This could establish necessity of `log N/ell -> 0` beyond axial tori. It is NOT
-proved here. Orientation and ambient nonprimitivity must be handled on the
-physical NN lattice; changing a period basis does not rotate its interaction.
-A counterexample could instead reveal a finer geometric invariant. Neither
-outcome invalidates the already supplied narrower results by implication.
-
-#736's executed controls cover 21,760 exhaustive strip configurations and 2,365
-deterministic uneven-cell masks, with exact Fraction inequalities, three local
-mathematical tests and compilation. Full repository CI was not run. The theorem
-rests on its argument and imported inputs, not on extrapolating these checks.
-
-## 2. Representation reserve: sources determine which states can be merged
-
-#708 gives lifted rank-future states and 509 deterministic continuation classes
-at width four. #710 gives the all-p scalar spectrum and exact cancellation of
-common P0/P2 modes in M. #733 gives source-compatible quotients, physical two-row
-site responses and exact final-rank-conditioned backward sampling. Two adjacent
-independently addressed columns require all 509 classes within the investigated
-common-strong-lumping class. These completed calculations are not assignments
-to repeat merely because their PRs are absent from main.
-
-The distinction worth pursuing after reactivation is an actual all-width
-closure-weight theorem or a source-faithful response separating concrete
-candidate representations. Deterministic classes, strong lumpings, scalar
-Hankel order, positive order and noise-limited effective order are different.
-A full-operator spectral gap need not be the gap visible to M. P398 is the
-known periodic IC O(1) TL calibration process (#718), not square-site percolation.
-
-### Concurrent #737: a concrete bridge to spatial structure factors
-
-The spatial-Hessian and invisible-mark notes were read at the pinned head below.
-For a mean-zero logit source h, H=sum_i h_i n_i and occupation count K, the
-finite logit-root curvature is
-
-    z_*''(h) = -[Var(H|r=2)-Var(H|r=0)] / [E(K|r=2)-E(K|r=0)].
-
-This identifies a difference of rank-conditioned spatial structure factors,
-not another uniform thermal derivative. Additive probability and additive logit
-fields have different second-order corrections. The reported 4x4 additive-root
-Hessian has both signs; no infinite-volume disorder relevance follows.
-
-The invisible-mark example preserves the entire uniform (r,K) law but separates
-under spatial sources. It uses positive DEPENDENT marked measures, not an
-ambiguity about the already specified Bernoulli law or the two original-U fields.
-A separate orbit calculation in this reset reproduced sizes 16/128/64, contrast
-matrix `[[-512,-128],[0,-256]]` and determinant 131072. This checks that finite
-algebra, not the full Hessian or complete CI. #737 discloses three missing
-delivery artifacts; resolve actual `delivery/` paths and reproduction inputs
-before treating archive execution logs as a fresh-checkout installation.
-
-### Exact class growth need not be robust at finite precision
-
-#549's k+1 branching classes have identical full unbranched survival. In its
-specified single-fork readout, adjacent separation is `1/[2k(8k-1)^2]`, while
-the whole family's span is `1/[2(8k-1)^2]`. More exact classes coexist with a
-shrinking observable range. This is not a bound for every branching experiment.
-The reserve question is whether budgeted admissible interventions amplify the
-separation, not whether larger k gives a larger exact class count.
-
-## 3. Theory-blocked flagship: original U, not a substitute observer
-
-#275 remains P1 with UNIDENTIFIABLE_WITH_CURRENT_ASSETS. This does not disprove
-Jordan or equate two physical theories. The missing input is two candidate maps
-through the SAME source, six-coordinate thermal jet, physical normalizer,
-rank-one denominator and pooled moving-root counterterm. Establish baseline
-representability, then compare nuisance-profiled prediction images using existing
-covariance. More precision cannot supply unspecified theoretical columns.
-
-#735's positive irreducible stochastic controls share all ordinary trace jets
-while differing in Jordan structure. Actual visibility depends on
-`C N_lambda^k P_lambda B`. A derivative-jet lift can have its own nilpotent part;
-its repeated pole does not identify the physical operator. Spin, a power and a
-logarithm are not substitutes for the source/readout map. #737 does not supply
-the two missing continuum candidates.
-
-## Pinned unmerged assets
-
-| PR | Head | Role |
+# 当前科学前沿：过程、预测与物理映射
+
+**2026-09-30；基线 main `d31fa5fe9584b77595e4a78b9560aa2ff77cf0b3`。**
+本页替代 9 月 13 日单论文快照。阅读了 #739/#771/#773、#828 的修订及 #829–#837；
+不是对所有历史脚本的重新认证。开放 PR 的结果仍是开放 PR；来源的固定 commit/path
+统一放在 [当前索引](../analysis/research_frontier.json)，防止 issue 尾部评论成为唯一入口。
+
+## 1. 联合出生：无聚合推导已给，当前区分预测状态
+
+当前最具体的分叉是：仅 rank、rank＋方向、rank＋方向＋完成数，哪一个能预测未来？
+无聚合和时钟比较已有作者证明，不能把后面的历史推导误当作仍待启动的任务。
+
+**当前主导工作已从精度转为机制干预。** [保时钟几何源](../notes/birth-selection-intervention-20260930.md)
+保留全部占据数到达时钟与即时退出概率，却具有非零的延迟 rank 响应；两步响应由
+安全协同图的度数方差精确给出。三个物理构型的全滞后响应已计算，不是又一轮相关性精化。
+同一推导把出生记忆表示为向后同调保留与向前完成的协方差，并用安全路径/生存权重分开
+几何演化和筛选。精度与独立块复核由一个支持工作者承担，不主导路线。
+该支持项[已经完成](../notes/completion-creation-independent-replication-20260930.md)：
+另一14万档案首次 e 读出为 −.003199±.000620，与前块相近，未合并且不称全新前瞻验证。
+主线[735步几何源传输](../notes/geometric-source-window-readout-20260930.md)也已完成：
+源导数 −.011042±.000158；当前时钟/退出概率不变，延迟rank明确响应。
+[单侧第二出生源](../notes/one-sided-birth-source-normal-response-20260930.md)保持首次出生边际，
+满足deltaM=deltaE_top；给出对称L4的严格正moving-root normal响应，不替代原norm-4/U合同。
+[自然存活flag与安全路径耦合](../notes/safe-transport-selection-readout-20260930.md)现也已完成：
+安全参考差 **−.002537±.000636**，自然存活差 −.002205±.000787；筛选项 +.000332±.000287 未分辨。
+同1198格与权重、14批完整协方差，97666条原出生前缀的条件续接。去掉自然重加权后差仍在，
+主问题转为入口/安全演化的源区分，而不是继续确认旧效应。不得把未分辨筛选项写成严格零或已证抵消。
+
+**先解决识别结构，不追精度：** 同(k,D,c)的Y与两步生存严格仿射；两个保持首次出生边际的源，
+在同p的M/E_top及移动根读出上也共线。热p＋一个几何源的秩二不等于两个几何源已可区分。
+下一项应是明确入口源和转移源在非冗余、同合同读出上的响应比较，而非增加等价坐标。
+
+| 候选状态 | 当前信息 | 仍需区分什么 |
 |---|---|---|
-| [#708](https://github.com/LightChainr/Matching-One/pull/708) | `f782061c1a592ed2f9fd0e9dabaa45f0e54bc4e7` | Finite rank closure |
-| [#710](https://github.com/LightChainr/Matching-One/pull/710) | `d543ba1afe052a36682d2c1723c8a07281c28950` | Parametric spectrum and geometric onsets |
-| [#718](https://github.com/LightChainr/Matching-One/pull/718) | `72d2ee6b8e5d25bd96aaef586c113bc94201f64a` | Axial root theorem and IC TL identification |
-| [#733](https://github.com/LightChainr/Matching-One/pull/733) | `525c8e98d99c44d4c76280a1f0f83064df9bfa72` | Physical sources, sampler and dictionary correction |
-| [#734](https://github.com/LightChainr/Matching-One/pull/734) | `bb41df7da8021d922a48ad353709f35912977c37` | Prior-art sweep, not novelty certification |
-| [#735](https://github.com/LightChainr/Matching-One/pull/735) | `9d29d014df28af7c635e6859d98a95ffe2b34d06` | Arbitrary-period root proof and Jordan controls |
-| [#736](https://github.com/LightChainr/Matching-One/pull/736) | `64d809b4404f80ff3f9adf9713337cc76008e92d` | New axial full-law iff proof |
-| [#737](https://github.com/LightChainr/Matching-One/pull/737) | `7b459c4809baee9dfc2b910091e00a3a8d508d1c` | Spatial-source Hessian and invisible marks |
-
-## Corrections that must travel with the result
-
-| Old inference | Current reading |
+| rank | 原始窗口生存差较弱；预定 pooled 完成数差可测 | 过程历史与几何混合，而非更多边际精度 |
+| rank＋方向 | 旧事后即时差 +1.629±.391；新独立复核 −.040±.570，未复现 | 大尺寸闭合仍开放；不能把旧差当已稳定排除模型 |
+| rank＋方向＋完成数 | L4 有精确反例；L512 同一块精确降噪后 −.003449±.000772，分辨出历史差 | 几何源的有限窗口传播、出生/演化/筛选分离；独立档案复核仅作为支持 |
+
+均匀随机排列的两个秩出生为 `J1<=J2`，iid 标签过滤的出生为 `T1<=T2`。
+#773 的有限流量恒等式已经给出：
+
+- 静态 rank law 决定两出生边际，配合直接双跳流量决定相邻时刻转移表；
+- 这些并不决定全路径、copula 或 gap 分布；
+- 条件于 `D=J2-J1=d>0`，标签间隔为 `Beta(d,N+1-d)`；
+  `d=0` 是直接双跳原子。
+
+#823 给出白 matching 分隔的作者证明及 L3–L5 全枚举；#828 已独立复算这些有限数字。
+它们不自动给方格臂指数，也不排除不同插入在缩放后合并。L6 **静态秩**表已在，
+但 L6 **jump 分层**新表仍没有交付（#825/#816）；不把两者混写。
+
+本轮 [新分析](../notes/birth-gap-noncoalescence-20260929.md) 给出了更尖锐的反例：
+两个过程拥有相同的全部静态 law、相同相邻转移表、零直接双跳，但极限对角原子为 0 与 1/2。
+因此下一目标是真正的两时间信息：
+
+    Z_L(delta) = E[(1-G_L/delta)_+],
+    G_L = 缩放后的两出生间隔。
+
+在出生对紧性下，`lim_delta↓0 limsup_L Z_L(delta)=0` 等价于所有子序列极限无聚合。
+它还等于秩一驻留长度与 lagged overlap 的归一化差；这给理论和可复用 paired 档案同一接口。
+本轮 [pilot](../notes/birth-gap-pilot-20260929.md) 直接采集这类联合信息，而不是再拟一个自由指数。
+判据本身是一般概率结果；下述后续推导开始补上渗流概率输入。
+
+本轮 [云端独立块](../notes/birth-gap-cloud-20260929.md) 已完成 L=64..512、
+56 万条新过滤，分辨率下探到 delta=.00625。L=512 的宽度归一化 Z(.00625) 为
+方格 .00336±.00015、三角格 .00378±.00014（batch SE）；依次将 delta 加倍时 Z
+近似加倍，未见已分辨的平台。不同原始几何不作同模参数普适性比较，有限网格也不证明双极限。
+后续 [两时间密度推导](../notes/two-time-pivotal-density-bound-20260929.md)
+已给出更强的局部结果：条件于两个不同站点的标签值，在不交空间环带保留四臂必要事件，
+得出生联合密度 `f_L(s,t)<=C_Lambda`。这不假设两出生独立。
+加上六臂直接原子界和紧性，三角格得到所有子序列极限无聚合的作者证明。
+统一 soft-gap 界带有窗口尾项，不能将局部常数当作全局常数。
+
+[方格转接续篇](../notes/square-birth-transfer-and-iqr-clock-20260929.md) 已补上两条原缺口：
+
+- 黑 NN / 白 matching 的交替臂与 planarised-white 分隔约定相容；具名文献提供
+  四臂幂小于 2、六臂幂大于 2、近临界比较和互补尾界。方格也得到自身 pivotal 时钟上的
+  子序列无聚合作者证明，**不需要**假定精确 5/4 或 conformal invariance。
+- 方格四臂幂大于 1 控制标签次序统计量的误差；结合紧性与局部密度界，证明出生混合
+  **总体分布**的计数 IQR 与 `N*a_eta` 同阶。因此这种归一化不会制造或移除极限零间隔原子。
+  这不是已测定的转换常数，也不是有限样本经验分位数的保证。
+
+下一缺口现在是连续环面观察量、联合 law / copula 的识别及唯一性。
+四臂幂的两侧界在证明中作用不同；不要重新把已补上的时钟比较或方格转接派成空白任务。
+
+最新[出生历史分析](../notes/birth-history-memory-readout-20260929.md)把这个缺口转成一个
+具体可区分约束。定义 `H(s,t)=P(T1<=s,T2>t)`；仅以当前秩为状态的时间非齐次 Markov
+模型必须满足 `H(a,c)H(b,b)=H(a,b)H(b,c)`，全三元组成立也给充分性。
+同一秩一风险集中比较早/晚首次出生组，正好检验这个等式，不需要增加自由指数。
+
+复用旧 L3 完整表已算出精确标签核：
+方格 `H(p,q)=6p^3(1-q^2)^3` 可分离，三角格为
+`9p^3(1-q)^3[A(p)+B(q)]`，其中 `A(x)=2x^3-6x^2+3x`、`B(x)=-2x^3+3x+1`。
+由此方格 L3 是 Markov 正控制、三角格 L3 是非 Markov 反例；这是精确有限结论，不向大尺寸外推。
+插入时钟的全三元组也已用旧表求值，没有重跑枚举。
+
+既有 56 万档案只用一个预选分位数三元组：L512 的早/晚组后续生存差为
+方格 `+1.422±0.511`、三角格 `+0.702±0.515` 个百分点（14 批 delete-one SE）。
+这是较小的探索性历史关联，不是新独立证据或连续极限非 Markov 的证明。
+“加上 birth age 就能描述过去”本身是两出生观察历史的编码事实，不是微观机制发现。
+
+新[完成几何计算](../notes/birth-completion-geometry-20260929.md)已经补上一条具体机制：
+定义 `nu_2(A)` 为秩一构型中、加上后立即达到秩二的空站点数。一般精确关系为
+插入时钟退出概率 `nu_2/(N-k)`、标签时钟退出强度 `nu_2/(1-t)`；
+[配套推导](../notes/completion-pivotal-hazard-kernel-20260929.md)把两队列的平均完成数差
+积分为 H 的对数 Markov 缺陷。
+
+- 三角 L3 在 k=5 的快型（nu_2=4）比例，早出生组为 1/4、晚出生组为 3/7，
+  其余慢型（nu_2=3）下一步存活率为 1/4；精确给出 `3/16-1/7=5/112`。
+  三个方向的核完全相同；方向不能消除历史。完成相态却递归闭合，给出从均匀空初态出发的
+  四状态插入时钟模型，完整复现出生联合表。不宣称最小维数或任意 rank0 准备的强闭合。
+- 方格 L3 每个秩一层的 `nu_2=k-3`，解释插入时钟 Markov 性。
+  标签时钟中，每个轴向核为 `3p^3(1-q)^3[(1+q)^3-p^3]`，每个对角核为
+  `3p^6(1-q)^3`。相加恢复未标记 Markov 核，但轴向自身的三时刻行列式严格为负。
+  **增加方向观察反而暴露隐藏记忆**，不与未标记结论矛盾。
+
+这次仅枚举每个格点模型的 512 个占据集合并做前缀 DP/精确多项式变换，未重跑 9!。
+原先的大尺寸出生对没有这些构型标记，不能事后虚构；也不能把 L3 闭合外推为连续机制。
+
+此后[独立 L512 完成几何实验](../notes/completion-hazard-independent-block-20260929.md)
+已真正执行：采样前固定合同及采样代码 `9520c9bd`，TV2N0X 14 workers、14 万条过滤，
+326 秒完成，未根据结果追加样本。沿用旧方格三时刻，但随机块全新：
+
+| 比较 | 生存差，百分点 ± batch SE |
+|---|---|
+| 原始早减晚 | +0.686 ± 0.444 |
+| 同精确方向，overlap 加权 | −0.415 ± 0.451 |
+| 同方向、同整数完成数，overlap 加权 | +0.122 ± 0.476 |
+
+随机区间时刻的无偏累计 hazard 估计给 pooled 差 `+1.550±0.351` 个百分点；
+实际生存减 hazard 预测为 `−0.865±0.639`。不是独立的第二份证据，也不把此较强辅助读出
+偷换成原始主读出。三种比较权重不同，不能据此声称“解释了百分之多少”。
+早/晚队列的轴向比例为 92.295% / 86.094%；方向混合是具体候选机制。
+
+新 raw 生存关联比旧块小，同方向及同方向同完成数比较均未分辨出剩余。
+这曾将方向混合推到候选首位；它的必要预测随后已真正计算，而不是直接接受。
+[同方向即时 hazard 检查](../notes/directional-hazard-contrast-20260929.md)沿用全部原 cutoff、
+方向和权重，只将终点生存换成已保存的完成数。未分方向时早减晚为 `−2.185±.479`，
+同方向为 `+1.629±.391` 个站点，对应下一插入退出概率差
+`(1.526±.366)×10^-5`。方向模型若 Markov，这个总体差必须为零。
+这是同一块的**事后**定向结果，保留全协方差，不算独立验证，也不改写原窗口零差。
+
+[方向 Markov 完成](../notes/directional-markov-completion-20260929.md)给出实际流量生成的
+唯一风险行转移：它保留所有 marked 单时刻和相邻联合表，却不自动保留真实路径。
+全时刻、全进入 cutoff 的风险平均完成数独立于历史，才给严格 Markov 充要条件。
+一次长窗口零差不能替代它；一般 toy 可因时序抵消为零，但本数据未证明发生抵消。
+
+方向仍有信息，但其独自充分性受到挑战。完成几何的**后继动态**随后已先做精确计算：
+[L4 方格与三角格](../notes/completion-pair-synergy-L4-20260929.md)都存在同 `(k,D,nu)`
+出生队列，一步生存恰好都为 1、两步生存却相差 `−4/4131` 与 `2/121`。
+这是从原均匀空初态的弱 Markov 反例，不仅是特殊准备之间的强 lumping 失败。
+每模型只算 65,536 个占据集合及整数前缀 DP，未枚举 16!、未做新生产。
+
+[协同完成对推导](../notes/completion-pair-synergy-20260929.md)定义空安全站点对的图：
+两个站点各自不完成第二方向，但一起加入会完成。其边数 e 给出精确两步生存
+`[(m-nu)(m-nu-1)-2e]/[m(m-1)]`，安全插入后的完成数增量均值为 `2e/(m-nu)`。
+这把同当前 hazard、不同后继几何直接接到未来 rank，而非新增一个无消费者的描述量。
+下一问题是这个传递在大尺寸的权重；随机安全插入可无偏测其漂移，不必枚举 O(m²) 对。
+L3 闭合仍正确、L4 已有反例；L512 的同条件有限窗口剩余及连续极限仍开放。
+原始批次、所有共同支持格和完整协方差已落库；本轮不自动扩尺寸或重跑生产。
+
+随后[独立安全插入实验](../notes/safe-insertion-independent-block-20260929.md)已落地：
+在同一 L512、固定旧 a,b 下另采14万前缀，每个合格前缀4个安全插入副本，先在前缀内平均。
+同 (D,nu) 新增完成数差 `+0.0066818±0.0120871`；由精确公式换算的两步生存差
+`−6.254×10^-8±1.132×10^-7`，不是额外独立测量。
+预定独立复核的同方向当前 nu 差为 `−0.03987±0.56988`，**没有复现**旧事后 `+1.629±.391`。
+因此旧挑战现在标记为未独立复现，不能继续用作已接受的大尺寸机制约束；这也不证明 Markov 性。
+
+55,028个 rank1 前缀的220,112次探测中只有1,233次非零；同条件共同支持仍保留98.973%风险。
+下一数值增益优先来自同一完成创建目标的无偏降噪，例如条件平均/明确加权采样；
+不直接把高接触点的便利样本换成原均匀安全源。连续核理论仍并行推进。
+本次519.843秒完成，所有批次保留，不补样、不合并旧新块、不用未分辨结果覆盖 L4 精确反例。
+
+[投影连通分量判据](../notes/projected-component-pair-criterion-20260929.md)现已给出精确计算途径：
+两个安全空站点对共享分量的横向势约束发生冲突，当且仅当共同插入会完成第二方向。
+相邻站点的直接边额外给一个约束。分量对分桶只产生真实协同边；固定格点度数下预计 O(N+e)，
+不再扫描所有空站点对。这样可直接求同一目标的条件均值 `2e/(m-nu)`，而不改变均匀安全源。
+
+[完整精确重放](../notes/exact-completion-pairs-readout-20260930.md)现已真正落地：14万行与原前缀逐一对应。
+同原 (D,nu) 权重的早减晚增量为 **−0.00344939±0.00077225**，约4.47个批次SE；
+标准误缩小15.65倍。早/晚均值为 .09557456/.09902395，差约为晚组的−3.48%。
+同一几何公式给两步生存差 `+3.22818e−8±7.23066e−9`；不是额外测得的独立终点。
+这支持有限 L512 扩展观察者的历史依赖，改变“该目标尚未分辨”的状态；但它是同块事后
+条件平均精化，不是新独立确认，也不改变方向-only 次要复核未复现的结论。
+单一支持工作者固定这一负向预测，在可正确重放的独立块上复核一次；主线推进上面的机制干预，不等待精度结果。
+8批保留本地结果、6批按用户新偏好在华为完成，云端420.381秒；完整8维协方差已保存。
+
+[标签时间曲率续篇](../notes/completion-pair-label-curvature-20260930.md)把同一个 e 接上连续核：
+固定进入 cutoff 的 `H_tt=E[I(c(c-1)-2e)]/(1-t)^2`；风险集中
+`hazard'=[mean c+2 mean e−Var(c)]/(1-t)^2`。这明确分出**创建与生存选择**：
+高完成数构型先退出，可能抵消安全插入造成的正创建。计数时钟也有精确风险集漂移
+`(2 mean e−Var(c))/(m−mean c)`，不同于在每个原前缀强制一次安全插入后的未重加权平均。
+这是有限公式，不把固定 k 数据换成固定标签时间，也不将两步差外推成735步差。
+下一理论预测应针对这一具名组合在近临界缩放中的行为；尚无极限系数或统一 Taylor 余项。
+
+## 2. 预测状态已从“有多少态”推进到源合同与噪声代价
+
+| 来源 | 新结果 / 必须保留的合同 |
 |---|---|
-| #628 bond duality fails | Code/convention defects, corrected #631/#646/#653 |
-| Raw M(p)+M(1-p) diagnoses normalized shape | Wrong centre/gauge; corrected anchored quantiles #702/#706 |
-| #675 rules out any single-operator representation | Unrestricted no-go withdrawn; efficiency and existence differ |
-| #715 P398 is unnamed | Periodic IC TL via the explicit map #718/#729 |
-| #717 finite 2D/0D categorically differs from rank2/rank0 | Same-site event dictionary #733; no all-width intertwiner follows |
-| #724/#731 linear split implies semisimplicity | Counterexample #735; m*lambda^m alone is not either diagnostic |
-| Primitive Gaussian C3 H8 label identifies local spin | Later unit-rotation/H0 correction in #275 supersedes the near-alias |
-| N580 compatibility identifies bare-aspect scaling | Same-block nominal compatibility after #703/#704, not a physical law |
-
-N is site count; square-period linear size is sqrt(N). Do not compare N=425
-with a published L=425 lattice. Site/bond, rank-two cross/rank-one spiral,
-fixed-p/moving-root and physical/jet operators must remain typed.
-
-Global H4 finite-size evidence, #537's proof obligations, #622's completed #706
-analysis, cut networks, finite terminal algebra and publication units remain in
-the [atlas](RESEARCH-ATLAS.md). No new Monte Carlo, GPU campaign, width or
-angle ladder, generic venue survey, or census expansion is started by exposing
-a reserve. Preserve their historical data and failures. Allocation should reduce
-open claims, not automatically generate a fresh branch of tasks from every result.
+| #829 | 宽四末端预测合同的 62/94/274/509 随均匀、逐行、逐列、逐点源而变；62 不是完整递归阶数 |
+| #830 | p=1/2 的单时刻可见维数 30；全部两点信息达到 94 并决定所述轨迹。混合准备与纯历史不相同 |
+| #831/#832 | 特殊参数的线性盲点；包含当前读出、相同全历史合同下正阶数 94 > 线性 93。有延迟的合同另算 |
+| #834/#835 | 宽五对称源正=线性=385；定向受控正3438>线性3328。两者不是矛盾 |
+| #836/#837 | 人工张量准备下，足够小固定噪声的固定错误读位高度为 Theta(log w)；不是自然临界典型态复杂度 |
+
+#837 的**五点补件**将 rank-only 误差下界加强为
+`[1-(1-epsilon)^5]/2`；第一步完整边界的四次幂界不再是 rank-only 可达到的目标。
+五次幂也尚未证明可达到。查询高度可以改善宏观导线，不能恢复已被接口擦掉的位。
+
+最新 [#837 讨论](https://github.com/LightChainr/Matching-One/pull/837#issuecomment-5814562555)
+又排除了一个具体捷径：直接复制逻辑块产生 `uv` 查询，响应秩从 `3^t` 变为 `4^t`。
+本轮进一步交付[任意块数的确定性四态编码分类](../notes/deterministic-block-encoding-obstruction-20260929.md)：
+完整 `(1,u,v)` 张量空间下，秩 3 的单射编码恰好只有一个携带信息的块，其余全恒定；
+任何真正分布式编码都为秩 4。双块 65,536 个编码一次精确枚举与分类一致。
+
+同一笔记给出实际两行掩码及黑绕路/白分隔证明，物理 rank 直接返回每块至多一位的任意乘积；
+不是把多个实验结果事后相乘。固定噪声的完整指令族经可逆翻转矩阵仍保留列空间，
+因此障碍适用于原五列块合同；它不决定有限样本代价或最优误差。
+
+这轮还完成了[随机奇偶编码](../notes/stochastic-parity-block-encoding-20260929.md)：
+四个混合准备支持不交，任意真子集的块不带逻辑信息，完整 `(1,U_i,V_i)` 张量响应秩仍为 3。
+但实际乘积查询的单次 TV 仅 `2^(1-k)`；任意单调 Boolean 读出的尖锐上界为
+`binom(k-1,floor((k-1)/2))/2^(k-1) ~ sqrt(2/(pi*k))`。
+单次非自适应物理最终 rank 也受此界约束；整个轨迹或反馈控制不在此结论内。
+
+更重要的是，这组乘积查询已有显式三标签归一化正模型：**正初始记忆=线性秩=3**，
+并未保留原正/线性差距。更大的实际物理查询族目前只有秩下界 3、上界 4；
+不能把张量上界当作所有未来查询的上界。一次 k=2,3,4 精确控制检查了 468 个期望和
+194 个单调函数；全 k 结论靠展示的证明，不靠外推。
+
+下一候选应针对具体允许查询的误差—记忆权衡，仍同时报告准备数、响应秩、误差、宽度和高度。
+这不是一般无编码定理，也不禁止接受秩 4 以换取可靠性；“随机编码是否可能”已不再是空白问题。
+
+## 3. 几何底座比 main 入口所说的更完整
+
+#739 已提供任意整数周期的 full-law iff 作者证明，不再仅限轴向：
+
+    ell -> infinity: balance root -> p_c，不限制面积/长宽比；
+    全部固定内部分位数 -> p_c iff log N / ell -> 0。
+
+必要性采用真实 NN 格点上短周期的固定亚临界绕行见证与不交平移打包，
+不是旋转坐标后直接套轴向结果。定理范围、输入和作者证明状态必须随结果保留；
+本页不宣称完成独立全证明审阅或原创性认证。
+
+#771 继续给出固定亚临界的同调自由能：
+`rho=min_{lambda!=0} tau_p(lambda)`，`log P(r>0)=-(rho-log N)_+ + o(rho)`，
+以及稀有出生/共同标签桥接。固定-p、移动 near-critical、任意变化方向的均匀性是不同层次；
+不要把固定-p PPP 自动当作临界过程。
+
+## 4. 原始 U 与全局 H4：相容的信号，不是同一个识别实验
+
+#275 的生产和 E_top 分析不是空白。当前缺口是两个候选在原 source、六坐标 thermal jet、
+normalizer、rank-one denominator、pooled moving-root 下的**normal forward columns**。
+原始 U 的结果不可由一个 aggregate rank 观察量替换。
+
+#771 的有限 same-model 结果支持显著 thermal-tangent H4，且后续控制发现高阶 dressing/H8；
+不能再把 post-H4 residual 称作纯标量律。若某项只平移热坐标，moving-root 正会将它消去，
+所以 root H4 强、normalized-U 弱并不矛盾。热/度量效应与真正新 normal 方向要分开。
+
+两个与下一步直接相关的纠正：
+
+- 水平黑/白 pair 源满足 `H_W=N-2K+H_B`。保留行配分正规化后，
+  它们不是两个独立的 normal 源；不要各启动一套生产。
+- 动量零不等于局域共形 spin0；公共磁扇区相消也尚未完整推出 E4 振幅。
+  #822/#828 指出的 actual rank projection、Ward/null、接触项接口仍值得做。
+
+## 5. 研究资产、执行与证明分开记
+
+[STATUS](STATUS.md) 记录结论边界；[ROADMAP](ROADMAP.md) 只排序注意力。
+[RESEARCH-MAP](RESEARCH-MAP.md) 区分观察者；[历史 atlas](RESEARCH-ATLAS.md) 保留长尾资产。
+旧 ledger 的证据块仍有效地描述其历史交付，不代表 2026-09-29 的全项目判决。
+
+全局 matching-odd、charged sector、边界 Q-tangent、跨微观 log-pair、cut-network、
+精确模型消除等都保留；本轮选择联合过程不构成否定它们。
+不锁任务、不批量关闭、不为了导航合并研究 PR，也不把新工具数量作为科研进度。
diff --git a/docs/RESEARCH-MAP.md b/docs/RESEARCH-MAP.md
index ccf1f8bea..7b834d1c7 100644
--- a/docs/RESEARCH-MAP.md
+++ b/docs/RESEARCH-MAP.md
@@ -1,44 +1,25 @@
-# Scientific map
-
-Updated 2026-09-13. The [frontier](RESEARCH-FRONTIER.md) gives evidence and
-corrections; the [roadmap](ROADMAP.md) sets current allocation.
-
-## The central chain
-
-    microscopic occupation and honest torus geometry
-        -> ambient rank and topology-resolved probabilities
-        -> observable closure under a specified update/source language
-        -> exact response or covariance-aware finite evidence
-        -> distinguishable candidate forward maps
-        -> only then a continuum-field identification
-
-## Three main objects
-
-**Geometric threshold laws (#613/#718/#735/#736).** Root consistency, full-law
-concentration and normalized profile convergence are three different questions.
-The axial full-law criterion is now sharp under named RSW/sharpness inputs;
-uniform oblique necessity is the next missing geometric step.
-
-**Intervention-dependent predictive state (#636/#708/#710/#733; #549).**
-Sufficient topology must survive the allowed future experiments. Linear scalar
-order can be far smaller than deterministic state count. Exact class growth
-can coexist with small observable separation at fixed precision. Physical
-square-site interventions and calibration-process interventions stay distinct.
-
-**Original-U identification (#275/#537).** Same spin and compatible exponents
-are not an observable map. A thermal jet, a normalizer and a moving-root
-counterterm belong to the prediction, not to a later plotting choice. The
-current candidate pair remains unidentifiable until its forward columns exist.
-
-## Evidence retained outside the primary queue
-
-Global matching-odd H4 finite-size tests; whole-curve and shape failures;
-cut-network continuation; finite terminal algebra; bounded exact threshold
-relation exclusion; covariance/projective inference; primitive homology
-characters; local sources and contact asymptotics. None is deleted because it
-is not today's primary. [RESEARCH-ATLAS.md](RESEARCH-ATLAS.md) retains the wider
-historical map, including branch-only and closed-unmerged science.
-
-The old map is [preserved verbatim](history/RESEARCH-MAP-before-20260913.md).
-Its Q4/Jordan and primitive-character framing is historical, not a current
-claim of physical identification.
+# 研究对象与观察者地图
+
+2026-09-30。结果及固定来源见 [前沿](RESEARCH-FRONTIER.md)，注意力见 [路线](ROADMAP.md)。
+
+| 对象 | 状态 / 几何 | 源与读出 | 已知信息的边界 | 最有用的新增信息 |
+|---|---|---|---|---|
+| 几何阈值律 | 任意整数周期 honest torus | Bernoulli p；M、F、Q | 根一致不要求 law 集中；#739 的 iff 已有作者证明 | 已给证明整合；固定-p 向 near-critical 的统一性 |
+| 两次拓扑出生 | 固定环面的排列/共同标签过滤，时钟分开 | 出生对、rank path、秩一核 H(s,t) | 无聚合及时钟比较已给；方向即时必要条件已有定向挑战 | 完成几何后继动态与连续核识别 |
+| 方向标记的出生过程 | 同一 rank1 构型的持久同调方向 | H_d、完成站点数 nu_2；源不变、观察者改变 | 旧事后 L512 nu 差 +1.629±.391 未被独立 −.040±.570 复现 | 大尺寸方向闭合仍开放，不把旧选择信号当稳定约束 |
+| 完成数的后继过程 | 相同 (k,D,nu)；安全空站点协同图 | 一次安全插入后的 nu；两步 rank 生存 | 同块精确读出−.003449±.000772；另档案−.003199±.000620，均已交付、未合并 | 已转入源机制区分；不再派复核或精度块 |
+| 保时钟几何干预 | 相同占据到达过程和当前退出概率 | 安全站点 exp(theta*degree) 转移源；未来 rank | L512735步导数 −.011042±.000158；人为修改源，不是自然H4归因 | 比较入口/转移源的非冗余响应；不再重复证实传输存在 |
+| 安全路径与自然生存筛选 | 原首次出生前缀的新条件续接；同(k,D,c)端点 | 同Y、同权重的自然/参考/筛选三项 | 参考差−.002537±.000636仍在，筛选项+.000332±.000287未分辨；同一依赖组 | 入口和安全生长源的同合同响应；Y/两步生存及单p M/E_top的结构冗余先排除 |
+| 单侧第二出生源 | 首次出生边际及占据时钟不变 | deltaM=deltaE_top；moving-root E_top | 对称L4构造给严格正normal响应与秩2的真实源Jacobian | 原U仍须其同源、六坐标、正规化映射；不能用动态正控制代替 |
+| 完成创建与生存选择 | iid 标签时间的固定进入队列，不是固定占据数 | 核 H 的二阶导数、退出 hazard 的导数 | 精确组合为 mean c+2 mean e−Var(c)；强创建可以被风险集选择抵消 | 相同时间/观察合同下的连续核定量预测；不将 t=b/N 当作数据换钟 |
+| 预测状态 | 生长圆柱的临时闭合前缀 | 均匀/逐列/逐站点源，末端/全轨迹 rank | 强 lumping、线性维数、正阶数、初始记忆不同 | 在同一源合同下比较可靠性与表示成本 |
+| 噪声查询 | 人工三行张量/随机奇偶准备 | 有界概率 site 源、单次 rank-only query | 确定性分布式秩 4；随机乘积查询秩=正初始记忆=3，但单次区分力衰减 | 可实现查询下的误差—记忆权衡；全轨迹与反馈另立合同 |
+| 全局 matching-odd | 方格/斜向同周长 | 根位移、thermal tangent | 有限 H4 不直接识别 CFT 模；高阶有 dressing/alias | 实际投影下混合响应和模参数预测 |
+| 原始 U | 原 norm-4/norm-5 合同 | 指名 source、六坐标 jet、normalizer、pooled moving root | rank-only 或 root H4 不替代 U 的 normal 列 | 两候选在同一映射下的分离 |
+| 带电/边界/跨微观 | Gaussian cover、annulus、三角格等 | deck character、Q tangent、边界/局部源 | 观察者依赖是信息，不应压成统一 H4/H8 故事 | 真实共同源—读出接口而非标签类比 |
+
+共同的研究链是：**微观模型 → 允许源/过滤 → 可测量量 → 预测差异**。
+可以大胆提出新机制；代价在于给它一条可计算或可证明的映射，而不是再添加名字。
+
+旧 [atlas](RESEARCH-ATLAS.md) 与 [ledger](../analysis/research_ledger.yaml) 保留历史索引。
+[当前结构化入口](../analysis/research_frontier.json) 指向具体提交；旧地图的执行优先级不再适用。
diff --git a/docs/ROADMAP.md b/docs/ROADMAP.md
index f6868c5c9..b761b9401 100644
--- a/docs/ROADMAP.md
+++ b/docs/ROADMAP.md
@@ -1,106 +1,184 @@
-# Research execution roadmap
-
-**Owner-delegated reset: 2026-09-13; reconciled with the latest #650.**
-One active deliverable: a consolidated probability-theorem manuscript and
-independent proof audit. Representation and original-U work remain visible
-research reserves, not simultaneous active assignments. This replaces the stale acquisition list; it does not
-rewrite frozen experimental decisions. Live Issue comments/state take
-precedence over this dated snapshot.
-
-## One active package: #735 with #613/#736 — root versus full-law geometry
-
-Delivered, not to be reassigned: #613/#670 supplied the full-law sufficient
-condition; #718/#735 removed aspect restrictions for the balance root; #736
-now supplies the missing axial full-law necessity. #276 remains completed.
-
-First consolidate and independently audit #735's arbitrary-period root proof,
-including oblique entry counts, support injectivity and disjoint bands. Return
-one theorem manuscript and a closest-prior-theorem comparison to #735. Use
-#736's axial iff as the precise full-law boundary chapter, with its own RSW
-input and independent seam/quantifier check. This is one probability paper,
-not two new dispatches.
-
-A possible strengthening INSIDE this programme, not a second active task, is
-the **uniform oblique-corridor lemma**:
-for every eta>0, find a fixed p_eta<p_c and bounded-thickness occupied winding
-corridors around an actual shortest period u, with probability >=exp(-eta|u|),
-uniformly in its orientation and ambient primitivity. Pack disjoint corridors
-in an arbitrary integer-period torus, or exhibit the geometric obstruction.
-The physical NN interaction must not be rotated by a change of period basis.
-
-As part of that same acceptance package, independently examine #736's
-seam construction and quantifier order once, then compare its exact statement
-with strip-percolation, RSW finite-size criteria and homological-percolation
-prior art. A negative search is not an originality certificate.
-
-**Deliverable:** one consolidated theorem manuscript, independent lemma-level
-audit and precise novelty comparison. Treat a proved oblique-corridor extension
-as a strengthening, or record its exact missing lemma; do not make it a new
-prerequisite for the already supplied narrower results. **Not useful:** another enormous
-thin-torus simulation, another proof that a fixed-width root is not p_c,
-or re-enumeration of the tiny checks already delivered.
-
-## Representation reserve: #636 — source-visible topology and structural closure
-
-This lane is exposed for scientific value but paused for new extension during
-the one-package allocation in #650. Reactivate only for a concrete need of the
-primary theorem or a later explicit allocation decision. #737's spatial
-Hessian and invisible marks are also already delivered finite assets, not new
-assignments.
-
-Treat #708's rank-future closure, #710's all-p width-four scalar spectrum,
-and #733's source-compatible quotients and exact rank-conditioned sampler as
-available research inputs at their pinned, unmerged commits. Do not rebuild
-those jobs because they are absent from main.
-
-Upon reactivation, the next result is either an all-width description of the actual site
-transfer's topological closure weights and observable cancellations, or a
-source-faithful response that distinguishes two concrete surviving state
-representations. Keep deterministic continuation classes, strong lumpings,
-scalar Hankel order, positive realization order and noise-limited effective
-order distinct. A new width is informative only when it can refute a specific
-structural conjecture named before the computation.
-
-Physical two-row site sources are the preferred exact intervention laboratory.
-P398 is the known periodic IC O(1) TL calibration process, not square-site
-percolation. Its pulse kernels do not transfer without a microscopic map.
-
-## Flagship, theory-blocked: #275 — original U, not a substitute observer
-
-Scientific upside stays high; active allocation moves from P0 to P1 because
-the next missing object is a pair of forward maps, not data precision.
-Preserve the named source, six-coordinate thermal jet, physical normalizer,
-rank-one denominator and pooled moving-root counterterm.
-
-For two named candidates: establish baseline representability of every used
-coordinate; specify nuisance amplitudes/phase transport; compare the profiled
-prediction images on existing dictionaries and covariance. Equal images keep
-UNIDENTIFIABLE_WITH_CURRENT_ASSETS. Separated images permit one frozen score.
-A new acquisition needs one explicit separating coordinate absent from the
-archive. More generic PCA coordinates, angle scans or a log-looking curve do
-not supply that map. Unanchored model cones meet at zero; design information
-must be priced at a stated nonzero signal or fixed SNR.
-
-## Retained, but not the default compute queue
-
-- #537: the three existing thermal/contact proof obligations remain; N145 is
-  unresolved under its frozen rule. A third-size fit does not close them.
-- #549/#550: write the cut-network statement and exact-versus-noisy complexity
-  boundary. Do not enlarge the parallel family merely to increase k+1.
-- P2/P3/P4 manuscripts: reconcile corrected claims and evidence dependencies.
-  No larger algebraic census, N580 replay, or unidentifiable N650 design by default.
-- #622: use the delivered #706 same-observable shape analysis. Do not rerun it
-  as a new result or infer a limiting shape from three sizes.
-
-These are allocation choices, not a prohibition on a genuinely informative
-new idea. A proposal should say which uncertainty it removes, the smallest
-calculation/proof that decides it, and what would make that route stop.
-No servers, credentials, paid compute or autonomous production were activated
-by this reset. No new recurring task was scheduled.
-
-## Integration is not scientific promotion
-
-This roadmap and the entry-point rewrite are independent of the unmerged
-research stack. Existing results, result-producing code and frozen designs
-are untouched. The old text is in [history](history/README.md); the compact
-[frontier](RESEARCH-FRONTIER.md) carries the current correction map.
+# 下一步：按研究信息增量安排注意力
+
+2026-09-30。替代 9 月 13 日单论文队列，并吸收 #828 与 #829–#837 及本轮 #838。
+这里的 P0/P1/P2 **不是许可、锁或对正确性的投票**；任何新反例或可区分预测都可以改变分配。
+旧的 freeze 保留其统计含义，不扩展成禁止探索的规则。
+
+GitHub 注意力标签同步为：#778 P0；#833/#768/#802/#780 P1；
+#816/#801 P2；#650 保留总控 P0。#801 是运维支持，不是第三条科研主线。
+没有关闭、锁定这些题目，也没有从标签推断任何旧作业被停止。
+
+## P0 — 联合出生的预测状态与连续核
+
+**当前问题：** 筛选不能单独解释已测历史差后，入口几何与安全生长的源响应如何区分？
+最新[直接耦合分解](../notes/safe-transport-selection-readout-20260930.md)已完成：
+去自然重加权的参考差 −.002537±.000636，筛选贡献 +.000332±.000287 未分辨。
+这支持入口/安全传输参与，不等于已分离出生与演化。计算使用旧出生前缀的新条件续接，不是独立确认。
+同一 L512 块的精确条件平均现给 −.003449±.000772（约4.47批次SE），支持扩展观察者的有限历史依赖。
+旧方向-only 信号仍未获独立复现，不能把两个观察者的结论混为一谈。
+按用户最新分工，精度与独立档案复核交给**一个支持工作者**；主线不等待它，也不继续以加精度替代机制。
+该支持项现已完成：[独立档案复核](../notes/completion-creation-independent-replication-20260930.md)
+给 −.003199±.000620，与前块相近，未合并；不再派作未完成任务。
+连续核唯一性仍是并行理论目标。前面的无聚合/时钟比较推导已给，不重复派工。
+
+### 机制主线：传播和筛选分解已给，下一步区分源
+
+[新几何源](../notes/birth-selection-intervention-20260930.md)在安全站点内按
+`exp(theta*degree)` 重分配到达概率，保持总占据时钟和即时退出 hazard 不变。
+两步生存 susceptibility 为 `−(m−c) Var_safe(degree)/[m(m−1)]`；三个实际方格构型的
+全滞后响应也已算出，一步为零、第三步在这些控制中最强、最终饱和为零。下一项不是重复测这个恒等式。
+
+1. **已完成的传输与真正下一问：** 固定L512735步源导数已为 −.011042±.000158；
+   不再把“局部作用能否传到这个窗口”派作未完成任务。下述筛选/传输分解也已交付；
+   若要连续尺度结论，再为具体消费者研究缩放增益，不自动开启尺寸阶梯。
+   仅改变到达时钟的模型不能解释“相同时钟、零即时响应、非零延迟响应”的这项干预；
+   这不自动排除不同合同中的热投影，也不把两步负号当作任意长窗口的定理。
+2. **传输还是筛选，已回答本合同：** 1198共同端点格中，安全参考差 −.002537±.000636
+   仍非零；自然差 −.002205±.000787；筛选差 +.000332±.000287 未分辨。入口/安全演化需要进入解释，
+   不能再优先把负差归因于纯筛选。不要把点值反号当成已经分辨的抵消，不追加复核或精度块。
+3. **当前主问：区分入口与安全生长源。** 写出实际可实现的入口源和行和为零的安全转移源，
+   用同端点、同正规化的非冗余读出比较响应列；源位置的传播公式已给。
+   同(k,D,c)下Y与两步生存是同一量；两个保持首次出生边际的源在单p的M/E_top上也必共线。
+   应使用真实长窗口/联合时间读出，先证明所选两列能分开，不以更高精度填补结构缺秩。
+   固定出生count再交换早晚标签不是入口干预；当前数据不能唯一拆出生与生长。
+
+这是 P0 的注意力顺序，不是任务许可。中大计算优先华为；复核不扩成第二条准备流水线。
+
+第一项已落实为[固定735步几何源读出](../notes/geometric-source-window-readout-20260930.md)：
+对原窗口的无序到达集合取条件平均，直接算原均匀源下的一次干预导数，不用多组theta扫描。
+合同和代码在计算前落库，复用原safe-insertion排列；这是新机制消费者，不是独立样本或旧差的精度升级。
+TgFr7R14workers、519.145秒完成并已Ready。下一项已有[直接耦合算法](../notes/safe-path-selection-coupling-20260930.md)：
+均匀安全拒绝采样继续路径，以首个提议是否安全记录自然存活flag；无需每步扫描全格计算W。
+该算法现已在TgFr7R14workers、664.094秒完成；140000个J1均与原档案对应，97666个入口完成续接。
+同一端点Y的分解已给，不再作为未完成生产任务派发，也不追加描述变量。
+
+[单侧出生源的normal映射](../notes/one-sided-birth-source-normal-response-20260930.md)也已交付：
+首次出生边际不变，deltaM=deltaE_top；移动根后的系数为−2P0'/M'，非零源不能被全消去。
+实际对称L4控制的Jacobian秩为2、normal导数严格为正；这是动态源正控制，不是原U候选列已齐。
+
+本轮已交付：
+
+1. [反例和非合并判据](../notes/birth-gap-noncoalescence-20260929.md)：明确需要控制哪种近对角质量，
+   哪些边际量不能决定它。
+2. [同过滤 paired-birth pilot](../notes/birth-gap-pilot-20260929.md)：方格与三角格并行，保存批次联合表，
+   同时看直接原子、近对角带及不预设方格临界指数的宽度归一化。
+3. [已完成的云端独立块](../notes/birth-gap-cloud-20260929.md)：新种子、56 万条过滤、L≤512，
+   更细 delta=.00625 网格；同样保留协方差，不与本地 pilot 混成独立坐标票。
+4. [两时间 pivotal 密度界](../notes/two-time-pivotal-density-bound-20260929.md)：
+   不交环带将两时间问题化成单时间臂概率，给紧窗联合密度上界；三角格无聚合作者推导已写出，
+   不以 Monte Carlo 作为证明。
+5. [方格 NN/matching 转接与总体 IQR 时钟](../notes/square-birth-transfer-and-iqr-clock-20260929.md)：
+   标准臂/长度估计接上方格约定，补上子序列无聚合；次序统计量误差及局部密度给出
+   `W_population asymp N*a_eta`。不宣称 IQR 比值收敛或有限样本分位数已获保证。
+6. [Markov 核与出生历史分析](../notes/birth-history-memory-readout-20260929.md)：
+   一条全三元组乘法判据；旧 L3 表上的方格 Markov/三角格非 Markov 精确控制（包含连续标签核）；
+   既有大尺寸档案的单一三时刻比较。L512 关联较小，未识别极限或证明潜在机制。
+7. [完成几何机制](../notes/birth-completion-geometry-20260929.md)：
+   完成站点数 nu_2 的精确 hazard 传递；三角 L3 的混合权重解释 `5/112`，并给出四状态
+   插入模型。方向本身不消除三角记忆；方格标签时钟反而因方向细化暴露轴向记忆。
+8. [独立方格 L512 完成几何块](../notes/completion-hazard-independent-block-20260929.md)：
+   预定14万条过滤已完成。原始差 +0.686±0.444、同方向 −0.415±0.451、
+   同方向同完成数 +0.122±0.476 个百分点（批次 SE）；条件剩余未分辨，完整协方差保留。
+9. [方向模型的即时必要条件](../notes/directional-hazard-contrast-20260929.md)：
+   同方向早减晚完成数 +1.629±.391；明确是同块事后读出，不是新验证。
+   [流量 Markov 完成与充要判据](../notes/directional-markov-completion-20260929.md)区分相邻表兼容和真实路径闭合。
+10. [协同完成对与 L4 反例](../notes/completion-pair-synergy-L4-20260929.md)：
+    原均匀源中同 (方向,完成数) 的早晚历史，两步生存仍差 `−4/4131`（方格）与 `2/121`（三角）。
+    [精确后继公式](../notes/completion-pair-synergy-20260929.md)给出图度数→新完成数→两步 rank 的映射。
+11. [独立 L512 安全插入块](../notes/safe-insertion-independent-block-20260929.md)：
+    新14万前缀、合格前缀4探测，主要增量差 +.00668±.01209；同方向 nu 复核 −.040±.570。
+    旧事后方向信号未复现，主要差未分辨。共同支持98.973%，只有0.560%探测非零。
+12. [投影分量判据](../notes/projected-component-pair-criterion-20260929.md)：
+    精确枚举协同完成对，取代稀疏探针的条件随机平均；不是新增观察量或新随机块。
+    [同前缀精确读出](../notes/exact-completion-pairs-readout-20260930.md)已完成14万行，
+    −.003449±.000772，标准误缩小15.65倍；旧正点估计在其大误差内，与精化结果不矛盾。
+    [标签曲率接口](../notes/completion-pair-label-curvature-20260930.md)同时给出
+    `hazard'=[mean c+2 mean e−Var(c)]/(1-t)^2`，区分创建和风险集选择；计数数据不直接代入标签公式。
+
+下一理论产物不再是重复请求同一小球界、方格转接或时钟比较：
+
+- **极限识别：** 把固定环面的两出生接到连续 near-critical 观察量，解决 copula 唯一性或其可区分预测。
+  已有局部密度不推出 Markov 性、出生独立或普适振幅。
+- **具体预测对象：** 用已给的 `nu_2 -> completion hazard -> H` 公式研究历史如何传递。
+  L3 的方向假说和二元相态、L512 的首次独立几何比较都已完成，不再作为空白派工。
+  方向混合 H_d 的即时条件已检查，但事后挑战没有独立复现；不要把它当作已排除的模型。
+  匹配 (D,nu_b) 后的历史依赖在 L4 已精确出现，L512 精化估计也已有信号，且有两步消费者。
+  相同负向目标的独立档案读出交由支持工作者；不将旧档案再分析称为新前瞻生产，也不作为主理论的等待条件。
+  当前 nu 固定时的即时 hazard 相等是定义结果，不是递归充分性的成功检验。
+  大尺寸的现有固定三元组不扩成显著性 cutoff 搜索；有限插入结果也不静默替换成标签时间结果。
+- **几何信息：** 明确联合分布中哪些量超出已知静态边际；需要比较格点时先对齐模参数和时钟合同。
+  不把现有两个原始几何的数据直接宣称为同模参数普适性检验。
+- **具体连续系数：** 用上述创建减选择组合给出竞争核的不同定量预测。
+  近临界导数带 `a_L^2` 因子；有限公式不授权交换极限与导数，需要控制矩与时间余项。
+  这比再命名一个状态变量更直接；允许并行探索，不设置研究许可。
+
+首次新几何块已用独立随机区间时刻测累计 hazard，不再只是中点代理；旧出生对档案没有的
+标记已在新数据保存。实际生存与积分预测的差未显示明确不闭合，但同块派生读出不是第二票。
+当前块仅增加了上述单一定向必要条件检查，不扩成自适应年龄/特征扫描。
+匹配 (D,nu_b) 的安全插入及方向即时 contrast 独立复核现在**均已执行**，不要重复派成未做任务。
+新增完成数均值直接为 `2e/(m-nu)`，但实际均匀探测非常稀疏。
+该条件平均方法现已实现并给出实际结果，不再作为缺失工具派工。固定同一预测的独立读出
+是一次支持性工作，不取代上面的干预/传播研究；现块不算第二票，也不自动扩大尺寸或采样量。
+同一 prefix 的多个 continuation 始终属于同一依赖组；连续核理论可并行。
+
+上一阶段按 pilot 决定计算：若固定近对角带已有稳定存活质量，量化它对更窄带的变化；若质量太小，
+优先独立配对样本，不同时扩大 L、模型、几何、源。有限 pilot 是探索，不预称验证极限。
+
+本轮这个计算选择已经执行：云端将 L 和 delta 分辨率各推进四倍，仍见近似随 delta
+成比例的 soft mass，没有已分辨的平台。随后推进的是上述概率推导，
+不在本轮末尾自动开 L=1024。有限数据本身并未排除更小原子或缓慢交叉。
+
+**可改变路线的结果：** 两时间可积小球界支持无聚合；缩放后向零集中的联合质量则反对当前
+“简单单位跳极限”图景，要求包含聚合机制。直接 jump-2 下降与否不是这二者的判决。
+
+## P1 — 有噪声的预测状态：保留什么优势、付出什么代价
+
+#837 已有 `0<epsilon<=10^-5` 下 `Theta(log w)` 高度读位的作者证明。
+不要重新开“是否存在”的任务。rank-only 微观错误下界采用五次幂补件，不再采用已被加强的四次幂目标。
+
+任意块数的[确定性四态编码障碍](../notes/deterministic-block-encoding-obstruction-20260929.md)
+已完成，并由显式两行乘积查询接上原五列块物理合同。不要再指派“换一种确定性分块试试”作为未做问题。
+在构造的张量查询族中，秩 3 恰好对应单一信息块；真正分布式单射编码的响应秩为 4，
+包括非复制和奇偶编码。任何包含该族的更大查询合同仍保留这个秩 4 下界；
+单信息块在任意额外查询下的秩上界不由这项构造单独保证。
+
+具体[随机奇偶构造](../notes/stochastic-parity-block-encoding-20260929.md) 已完成：
+乘积查询秩=正初始记忆=3；单次乘积 TV 为 `2^(1-k)`，任意单调二值读出的尖锐 TV 界为
+`binom(k-1,floor((k-1)/2))/2^(k-1)`。抽象阈值达到后者，不等于物理实现已给。
+这条构造既绕开确定性障碍，也暴露了可靠性损失和正/线性间隙消失。
+
+下一项有价值的问题是**具体可实现读出下的误差—记忆权衡**：若扩大到整段轨迹或反馈，
+先写清多次观察合同；若保留单次读出，给出不同随机构造或可实现的匹配界。
+不再只问是否存在随机秩 3 编码。仍同时报告
+`(准备数 A, 响应秩 L, 误差 eta, 宽度 w, 查询高度 h)`；
+若只要更可靠，接受秩 4 也是有效路线。固定噪声精确秩不等于有限精度可用维数。
+自然临界先验的平均复杂度另算，不能把人工极稀有准备族的最坏情形结论搬过去。
+
+## P1 — 原始 U / 模参数机制：产出实际映射
+
+#275 不需要再被写成“E_top 一直没做”。历史生产与识别失败都保留；目前缺的是实际候选列。
+对两个候选写出 `raw = a * thermal_tangent + normal`，应用原 moving-root、normalizer 和 nuisance
+处理，比较真正留下的 normal 预测。根 H4 很强而 U 很弱可以相容。
+
+#771 的 odd thermal Q4 × even correction 混合响应，以及 #822 的 actual rank projection/Ward 接口，
+是值得做的定向理论/小矩阵工作。不能把动量零当成排除局域 spin4，不能把 Jacobsen 的公共磁扇区
+相消称为已完整推导 `L^-4` 或 E4 模参数振幅。允许新猜想，但明确哪些矩阵元、接触项还未计算。
+
+## P1 — 几何论文有界收尾
+
+#739 已提供任意整数周期 full-law 的 `log N/ell -> 0` iff 作者证明；#771 给固定-p 的 rho/logN 相图。
+把定理、输入和最邻近先例整合为一个可读稿；审查具体争议引理即可，不再启动完整重审或薄环面模拟。
+固定-p 误差常数不能无说明地搬到随尺寸移动的 near-critical 窗口。
+
+## P2 — 仍可做，但不自动占用本轮大算力
+
+- #816 L6 jump 直方图：是尚未交付的有效支持计算，不是“已有 L6 秩表”或无聚合定理。
+  #828 的短 C 控制使用 32 位 occupation；扩到 36 位前必须修正表示并实测成本。L7 是另一种 DP 工作。
+- #819 `p_pol` 代数普查、#821 修正指数分类：按尚未有落库产物处理；完成它们不等于缩小核心机制空间。
+- 新宽度/角度/高精度根、更多等价源和通用 certificate：有具体消费者时推进，不作为默认增长指标。
+
+## 本轮执行方式
+
+已有成果以原 PR/commit/path 引用，不批量合并、改名或关闭。一个新 Draft PR 承载本轮导航与新增研究；
+协调 Issue #650 留一次摘要即可。保留分支来源和随机块依赖，检查与科学风险成比例；不跑文档措辞测试、
+全仓测试或重复的“通过”仪式。云服务若不可用，小计算继续本地，不让机器登录变成研究前置条件。
diff --git a/docs/STATUS.md b/docs/STATUS.md
index a0cfec02d..2bdfd8fa2 100644
--- a/docs/STATUS.md
+++ b/docs/STATUS.md
@@ -1,6 +1,6 @@
 # Current scientific adjudications and historical claim ledger
 
-**Scope reset: 2026-09-13.** This is a compact correction-aware entry point, not
+**Current scope: 2026-09-30; sources through #837 and this open #838.** This is a compact correction-aware entry point, not
 a re-score of old experiments. The full earlier ledger is retained
 [byte-for-byte](history/STATUS-before-20260913.md), with its original snapshot
 date, numbers and claims. Use the original commit linked in
@@ -15,18 +15,84 @@ date, numbers and claims. Use the original commit linked in
 | N580 establishes bare-aspect scaling | Not established | Merged #703/#704: retrospective nominal uncertainty-region compatibility, same original block; independent rungwise H8 nuisance, not a measured H8 law |
 | P3 manuscript is submission-ready | No | Its merged correction addendum must be reconciled with historical prose |
 | M= P2-P0 and F=(1+M)/2 | Exact within the honest-torus scope | Digital Alexander input and merged #702; not a small-size conjecture |
-| Arbitrary-period balance roots converge as the shortest period grows | Proof supplied in open #735 | Fixed-off-critical site sharpness and matching inputs; no rate or numerical p_c; unmerged |
-| All axial fixed interior quantiles converge iff log(m)/w -> 0 | Proof supplied in open #736 | Critical square-site RSW plus existing sharpness/duality inputs; local controls executed, independent full proof review and novelty comparison pending |
-| Arbitrary-oblique full-law necessity | Open | A uniform short-period winding-corridor construction is not supplied by rotating the axial proof |
-| Width-four topology/source closure | Exact finite results in open #708/#710/#733 | Deterministic classes, lumpings and scalar recurrence orders are different objects; not an all-width theorem |
+| Arbitrary-period balance roots converge as the shortest period grows | Consolidated author proof in open #739 | Fixed-off-critical site sharpness and matching inputs; no aspect restriction, rate or new numerical p_c; unmerged |
+| Arbitrary-period fixed interior quantiles converge iff log N / ell -> 0 | Author proof supplied in open #739, no longer only axial | Actual NN oblique corridors and disjoint packing; this navigation update does not certify the entire proof or novelty |
+| Fixed-p homological free energy and rare-birth clocks | Author results in open #771 | Fixed-p probability inputs and directional uniformity must not silently be promoted to moving near-critical estimates |
+| Single-insertion jump-two has a six-arm-type separator | Author argument and exact finite census in #823; finite numbers independently reproduced in #828 | Does not supply square-lattice universality, nonzero amplitude or exclusion of adjacent-birth aggregation |
+| No direct double jumps implies no double birth in a scaling limit | False for general monotone two-birth processes | New explicit counterexample has equal static laws AND adjacent transition tables but limit diagonal atoms 0 and 1/2; not a percolation counterexample |
+| Joint near-diagonal criterion | New general probability derivation in this branch | Under pair tightness, vanishing double-limit soft gap mass is equivalent to no coalescence |
+| Two-time birth density and noncoalescence | Author proof in this branch, now with triangular and square NN/matching input transfer | Locally bounded joint density, vanishing same-site atom and tightness; standard arm/length estimates named; no unique copula, square exact exponent or universality claim |
+| Population insertion-IQR versus pivotal clock | New author proof: W_population asymp N*a_eta | Order-statistic coupling uses four-arm power above one; tightness plus local density bound controls population width; not convergence of the ratio or a finite-sample quantile guarantee |
+| Joint-gap finite-size evidence | New 128k local + independent 560k cloud filtrations, L up to 512 | Soft mass decreases approximately with resolution down to delta=.00625; no resolved plateau, not an exclusion of a limiting atom; no cross-modulus universality claim |
+| Rank-one kernel characterizes rank-only Markov memory | Author derivation in this branch | All triples of H(a,c)H(b,b)=H(a,b)H(b,c), on positive risk, characterize the ordinary time-inhomogeneous Markov property in rank history; one zero triple is insufficient |
+| Exact L3 rank-memory distinction | Existing complete permutation tables yield square Markov and triangular non-Markov | All count-time triples evaluated; exact multinomial label kernels give the same two verdicts for label time; finite engine convention only, no new enumeration or continuum extrapolation |
+| Large-size birth-history association | Retrospective L512 insertion survival differences: square +1.422 +/- .511 pp, triangular +.702 +/- .515 pp (batch SE) | One mixture-quantile triple; same previously inspected cloud block; modest evidence, no independent model-elimination certificate or continuum non-Markov proof |
+| Completion geometry transmits to the rank-one kernel | Exact finite hazard identity and new L3 geometric computation | Exit probability nu_2/(N-k), label intensity nu_2/(1-t); cohort-mean differences integrate to the logarithmic Markov defect; not an original-U or causal-intervention claim |
+| Triangular L3 birth-history mechanism | Fast completion-type weights 1/4 versus 3/7 explain the exact survival difference 5/112; direction does not remove it | One binary rank-one phase closes recursively at this size; four-state weak count-clock realization from uniform empty start reproduces the entire paired law, not a minimality or continuum claim |
+| Square L3 Markovness is preserved by revealing direction | False in iid-label time | Axial sector kernel has strictly negative determinant for every interior triple although the pooled rank kernel is Markov; diagonal sectors and all count-clock sectors remain Markov. Observer and hidden count matter |
+| Independent L512 geometric transmission block | Completed: 140k new square filtrations at fixed old count cutoffs | Raw early-minus-late survival +.686 +/- .444 pp; exact-D weighted -.415 +/- .451 pp; exact-(D,nu2) weighted +.122 +/- .476 pp, batch SE. Conditional contrasts unresolved; not a proof of sufficiency |
+| Interval completion geometry predicts pooled survival | Exact expectation identity, now measured on the independent block | Random-time hazard prediction +1.550 +/- .351 pp; actual-minus-predicted -.865 +/- .639 pp. Correlated readouts, not independent confirmations; no causal or original-U identification |
+| Direction alone closes the L512 count process | Unresolved: earlier retrospective challenge not independently reproduced | Old same-D nu2 contrast +1.629 +/- .391; new predeclared independent replication -.040 +/- .570. Neither independent exclusion nor closure proof; no old/new pooling |
+| Matching all marked adjacent flows identifies the Markov path law | No: it defines a unique on-risk Markov completion, not identification of the actual process | Full-count, full-entry-cutoff independence of mean completion count characterizes actual marked Markovness; an exact toy has zero long-lag but nonzero next-step contrast |
+| Direction plus completion count universally closes the count process | False already at L4, even weakly from the uniform empty start | Exact matched birth cohorts have identical immediate rank survival but two-step differences -4/4131 square and 2/121 triangular. L3 closure retained; new L512 refined finite estimate supports failure too, but no continuum claim |
+| Microscopic completion creation transmits to future rank | Exact safe-pair graph identity | Safe insertion creates degree(v) new completion sites; mean increment 2e/(m-nu), two-step survival [(m-nu)(m-nu-1)-2e]/[m(m-1)]. Does not establish recursive closure after adding e or the graph |
+| Synergy pairs can be found without scanning all vacancy pairs | Author exact projected-component criterion, implemented | Conflicting integer transverse constraints detect both nonlocal shared-component and adjacent pairs. Expected O(N+e) at fixed lattice degree; explicit output can still be quadratic when e is quadratic |
+| Label-time kernel curvature has a microscopic map | Exact finite author identity | Fixed-entry H_tt=E[I(c(c-1)-2e)]/(1-t)^2; hazard derivative=[mean c+2 mean e-Var(c)]/(1-t)^2. Surviving-cohort selection matters; fixed-count data are not label-time data and no continuum limit is asserted |
+| A geometric source can preserve the clock but change future rank | New finite derivation and three physical controls in this branch | Safe choices tilted by exp(theta*degree) preserve the entire count-arrival law and instantaneous exit hazard. Two-step susceptibility is -s Var_safe(degree)/[m(m-1)]; all-lag responses calculated for three specified L4 starts. Modified source, not original-U or continuum identification |
+| The geometric source transmits through the original L512735-step window | Completed: derivative=-.01104203 +/-.00015767, 14-batch SE | Same140k safe-insertion permutations,55028 risk prefixes; source likelihood score averaged over the actual unordered future set. Not a finite-theta effect, new independent block, natural-history mediation fraction or continuum result |
+| A rank-one-only source is erased by moving the balance root | False when its second-birth response g is nonzero | First-birth marginal unchanged, deltaM=deltaE_top=g; moving-root Etop derivative=g*(-2P0'/M'). Explicit translation/dihedral-invariant L4 source has positive g for all interior p, rank-two response Jacobian and normal derivative~.00016853. Constructed dynamic source, not original norm4/U identification |
+| Static M/E_top at one p distinguish two first-birth-preserving geometric sources | No: exact response rank at most one | Both columns are g_r*(1,1); moving-root (root position,Etop) columns stay proportional. The earlier rank-two positive control compares thermal p with ONE geometric source. Identifying two geometric mechanisms needs a nonredundant same-contract readout; this is not a statement about every original-U source |
+| Natural survival selection alone explains the specified L512 matched Y history contrast | Disfavoured by completed safe-reference coupling | Safe-reference=-.00253706+/-.00063649; natural=-.00220507+/-.00078709; selection=+.00033199+/-.00028738 unresolved. Same1198cells/weights and full14-batch covariance. New conditional futures of old entrances, not independent confirmation. Entrance/safe transport required; birth versus growth not uniquely separated |
+| Birth association identifies an intrinsic ageing force | No: backward/forward geometric coupling is the exact finite interpretation | Matched-cell future contrast equals Cov(retention,future prediction)/[p(1-p)]. Equal birth laws can have unequal futures. Safe-path weights separate transport from selection, but endpoint association alone cannot uniquely attribute birth versus evolution |
+| Matched completion-creation history effect at L512 | Resolved by exact conditional averaging on the SAME140k block | Refined same-(D,nu) increment -.00344939 +/- .00077225, about4.47 batch SE; derived survival2 +3.22818e-8 +/-7.23066e-9. Original four-probe +.0066818 +/- .0120871 preserved; SE15.65x smaller, no new independent prefixes; post-readout refinement supports finite enlarged-observer closure failure, not unmarked/continuum non-Markovness |
+| Fixed negative prediction on the other independent L512 archive | Reproduced by one support worker | DeltaY=-.00319937 +/-.00062009; second-minus-first +.00025002 +/-.00099040; not pooled. Other outcomes in that archive had already been read: first e measurement, not untouched holdout or new production. No change to the direction-only nonreplication |
+| L6 jump-stratified census | Not delivered by #825 | Revalidation of older static rank tables is not new jump enumeration; 36-site extension of #828's 32-bit control requires a representation repair |
+| Width-four source/trajectory closure | Exact finite results in #829--#832 | Terminal classes, strong lumping, linear order, positive order and delayed readout have different contracts |
+| Width-five positive and linear orders | Symmetric source: 385=385 (#834); controlled directed source: 3438>3328 (#835) | Source language changes; no contradiction and no unrestricted all-width claim |
+| Fixed small-noise query height | Author upper/lower bounds Theta(log w), #836/#837 | Artificial preparations, fixed error, epsilon<=10^-5 for the stated upper bound; not natural critical typical histories |
+| Rank-only microscopic erasure floor | At least [1-(1-epsilon)^5]/2, #837 supplement | Stronger than the complete first-boundary fourth-power bound; attainability of the fifth-power bound is open |
+| Direct copying preserves the old tensor response-rank saving | False for the stated full query family | #837 latest discussion adds uv and restores 4^t; not a universal prohibition on coding |
+| Distributed deterministic four-state block coding preserves rank three | Excluded for the full tensor language, with physical product queries now constructed | All-k classification; one exact 65,536-code census; rank-four obstruction transfers to original five-column blocks and complete noisy command family, not arbitrary codes or finite-error complexity |
+| Distributed stochastic parity coding | Tensor response rank exactly three, with disjoint supports and uninformative proper block marginals | Explicit physical product family has positive initial memory three too; no retained positive/linear gap; full physical-language rank only bounded between three and four |
+| Reliable single readout from the parity code as k grows | Excluded for the stated monotone Boolean readout contract | Product TV=2^(1-k); sharp abstract monotone TV=binom(k-1,floor((k-1)/2))/2^(k-1); applies to one nonadaptive final physical rank, not a transcript or feedback controller |
 | m*lambda^m or linear eigenvalue splitting identifies Jordan | False as a standalone diagnostic | #735 gives explicit counterexamples, including positive stochastic matrices with identical trace jets |
 | Full unbranched survival is a complete branching state | False in the declared cut-network family | #549's exact growing-family witness; no universal square-NN or finite-noise dimension claim |
 
-PR descriptions report their own validation. This reset does not certify all
+PR descriptions report their own validation. This update does not certify all
 of their code or promote every supplied proof to publication-ready status.
-No original result file or frozen scorer is changed. Full repository CI was
-not executed for the new #736 files in this delivery; its three local tests
-and exact controls are not a substitute.
+No original result file or frozen scorer is changed. This round adds
+[paired-birth analysis](../notes/birth-gap-pilot-20260929.md), whose actual run
+and uncertainty are recorded separately; finite Monte Carlo is not a proof of
+noncoalescence. No full-repository CI or document-wording tests were run.
+The subsequent [two-time proof](../notes/two-time-pivotal-density-bound-20260929.md)
+does not infer its bound from the pilot. The
+[encoding proof](../notes/deterministic-block-encoding-obstruction-20260929.md)
+separately records its sole algebraic census and all-width physical construction.
+The [square/clock continuation](../notes/square-birth-transfer-and-iqr-clock-20260929.md)
+adds probability input transfer, not new samples. The
+[stochastic continuation](../notes/stochastic-parity-block-encoding-20260929.md)
+adds displayed all-k proofs and one exact k=2,3,4 control (468 tensor moments,
+194 monotone functions), not a rerun of the older census or physical graph enumeration.
+The latest [completion geometry](../notes/birth-completion-geometry-20260929.md)
+adds a new 512-subset computation per lattice, prefix dynamic programming,
+phase transitions and exact marked-label kernels; it does not rerun 9!
+permutations or add Monte Carlo. The [general hazard note](../notes/completion-pivotal-hazard-kernel-20260929.md)
+separates instantaneous sufficiency from recursive state closure.
+The subsequent [independent completion block](../notes/completion-hazard-independent-block-20260929.md)
+is genuinely new production, not an archive relabel: 14 x 10k square-L512
+filtrations, 325.512 s on TV2N0X, with precommitted acquisition contract.
+Its smaller raw association and unresolved conditional contrasts are retained
+without post-result sampling extension or promotion to a Markov theorem.
+
+The source/normalizer corrections in #771 and the thermal-tangent discussion in
+#275 matter for interpretation: strong root H4 and weak normalized U need not
+conflict. Historical E_top production is retained. The missing original-U
+candidate maps are a theoretical information gap, not evidence that the
+experiment has never been performed.
+
+Pinned branch sources and observer/dependency boundaries are in
+[research_frontier.json](../analysis/research_frontier.json). The old atlas and
+August ledger remain historical discovery indexes, not the current allocation.
 
 Current execution choices are in [ROADMAP.md](ROADMAP.md). Older claim-level
 labels remain historical; a change of allocation is not a claim-strength upgrade.
diff --git a/notes/birth-completion-geometry-20260929.md b/notes/birth-completion-geometry-20260929.md
new file mode 100644
index 000000000..a39b30070
--- /dev/null
+++ b/notes/birth-completion-geometry-20260929.md
@@ -0,0 +1,229 @@
+# Birth-history memory comes from completion geometry: an exact L3 mechanism
+
+2026-09-29. New finite geometric computation, in Draft #838. The general
+[hazard identity](completion-pivotal-hazard-kernel-20260929.md) links this
+calculation to the [rank-one kernel theorem](birth-kernel-markov-characterization-20260929.md).
+
+**Outcome.** On triangular L3, direction does not remove birth-history
+memory. The one-step difference `5/112` is exactly explained by different
+mixtures of configurations with three or four completion sites. One binary
+rank-one phase closes the count-clock process from its uniform empty start.
+On square L3, unmarked label-time rank is Markov, but revealing its rank-one
+direction exposes non-Markov axial sectors. Observer and clock are part
+of the scientific statement, not metadata.
+
+These are finite N=9 results, not large-size extrapolations. They do not
+identify the original-U mechanism or a continuum state dimension.
+
+## 1. A microscopic quantity with an exact transmission law
+
+Use the existing birth engine's occupied-graph convention: square
+nearest-neighbour steps, and triangular steps with additional
+`+(1,1),-(1,1)` edges, on `Z^2/(3Z)^2`. Rank is the dimension over Q
+of the ambient homology image, including all occupied components.
+D is its primitive unoriented line while rank is one. Inclusion preserves
+D until rank two; a direct `0 -> 2` path has no rank-one D.
+
+For a rank-one occupied set A define
+
+    nu_2(A) = number of vacant sites v with rank(A union {v})=2.
+
+Under a uniform permutation or iid Uniform site labels, respectively,
+
+    next-insertion exit probability = nu_2(A)/(N-|A|),
+    label-time exit intensity       = nu_2(A)/(1-t).
+
+The latter conditions on current geometry and revealed past; each remaining
+label has hazard `1/(1-t)`. Occupied count is hidden and random in label
+time; substituting Nt for it is not valid. The resulting transmission is
+
+    H_count(a,k+1)=H_count(a,k)
+       - E[1{J1<=a,J2>k} nu_2(A_k)]/(N-k),
+    partial_t H_label(s,t)
+       = -E[1{T1<=s,T2>t} nu_2(A_t)]/(1-t).
+
+The companion note proves these formulas and expresses the logarithmic
+Markov defect as the integral of the two birth-by-cutoff cohorts' mean
+completion-count difference. This is a microscopic-to-observable map,
+not another descriptive covariate fit.
+
+## 2. Exact occupied-set computation
+
+[analyze.py](../analysis/birth-completion-geometry-20260929/analyze.py)
+visits all **512 subsets per lattice**, computes lifted cycles by graph
+traversal, and counts completion sites. An integer dynamic programme
+counts ordered prefixes by first birth. Each rank-one set at layer k has
+k! such prefixes; a completion edge contributes its prefix count times
+`(N-k-1)!` to the direction-marked birth table.
+
+This is new geometric computation, not a rerun of 9! permutations.
+Collapsing directions exactly recovers the old paired-birth tables.
+The old files and engine remain unchanged. Their hashes, every rank-one
+mask, completion sites, prefix counts and transition rows are in
+[result.json](../analysis/birth-completion-geometry-20260929/result.json).
+
+| k | Square: nu_2 (number of configurations) | Triangular: nu_2 (number of configurations) |
+|---|---|---|
+| 3 | 0 (6) | 0 (9) |
+| 4 | 1 (36) | 2 (54), 3 (27) |
+| 5 | 2 (72) | 3 (54), 4 (27) |
+| 6 | 3 (48) | 3 (9) |
+
+There are no other rank-one layers. Square has `nu_2=k-3` throughout:
+at fixed insertion count, exit cannot depend on birth history. This
+explains its count-clock rank-only Markov property geometrically.
+Its direction set is `(1,0),(0,1),(1,1),(1,-1)`; dropping diagonals
+changes the observable. Triangular L3 has only `(1,0),(0,1),(1,1)`.
+
+## 3. Triangular memory is a completion-mixture difference
+
+At k=5 compare early `J1<=4` with late `J1=5`, both with `J2>5`.
+Early and late ordered-prefix counts are 5184 and 4536; multiply by 4!
+for full-permutation counts. Four sites remain.
+
+| Completion type | Next-step survival | Fraction among early histories | Fraction among late histories |
+|---|---|---|---|
+| nu_2=3 (slow) | 1/4 | 3/4 | 4/7 |
+| nu_2=4 (fast) | 0 | 1/4 | 3/7 |
+
+Thus
+
+    E[nu_2 | early]=13/4,    E[nu_2 | late]=24/7,
+    S_early=3/16,           S_late=1/7,
+    S_early-S_late=-(13/4-24/7)/4=5/112.
+
+This reconstructs the previous `(4,5,6)` witness. History selects different
+current completion geometries; this is not an intervention claiming that
+changing birth age causes exit.
+
+Direction does **not** repair the memory. All three marked birth tables
+are equal; each retains the two count-time violations `(3,4,5)` and
+`(4,5,6)`, with conditional covariance `1/90`. The unimodular map
+`(x,y)->(x-y,x)` preserves the steps, period lattice and uniform source
+and cycles the three directions. Hence `H_d=H/3` for either clock.
+Direction is not the missing coordinate in this finite model.
+
+## 4. Completion types close recursively here, not just for one step
+
+Equal exit counts do not generally imply equal non-exit successor laws.
+Those laws were computed too. Every triangular configuration with the
+listed `(k,nu_2)`, across directions, has exactly this successor row;
+probabilities divide the counts by `9-k`.
+
+| k, current nu_2 | Non-exit successor counts | Exit sites |
+|---|---|---|
+| 3, 0 | 6 of nu_2=2 | 0 |
+| 4, 2 | 2 of nu_2=3, 1 of nu_2=4 | 2 |
+| 4, 3 | 2 of nu_2=3 | 3 |
+| 5, 3 | 1 of nu_2=3 | 3 |
+| 5, 4 | none | 4 |
+| 6, 3 | none | 3 |
+
+At fixed external count k, call the smaller available completion count
+slow and the other fast. This yields four states:
+
+    rank0, rank1_slow, rank1_fast, rank2.
+
+Rank-one rows close **strongly** across configurations. Before first birth,
+the observed past is all zero; from the uniform-permutation empty start,
+rank-zero configurations are uniform within their layer. Averaging their
+rows therefore gives the correct **weak** rank-zero transition. Rank two
+is absorbing. Together these give a time-inhomogeneous Markov realization
+from that start, not merely a fit to one-step marginals.
+
+The saved nine 4-by-4 matrices reproduce the entire paired-birth law
+exactly. Direct double jumps bypass rank-one phases; off-risk identity
+rows are placeholders. This is an upper construction, not minimality,
+nor strong lumpability of arbitrary rank-zero preparations. Its phases
+depend on external k: it is not a four-state label-clock closure with
+occupied count hidden.
+
+## 5. Square label-time Markovness depends on the observer
+
+[exact_marked_clock.py](../analysis/birth-completion-geometry-20260929/exact_marked_clock.py)
+uses exact multinomial label thinning:
+
+    H_d(p,q)=sum_(k<=ell) Multinomial(N;k,ell-k,N-ell;p,q-p,1-q)
+                           * H_count,d(k,ell).
+
+These are **unconditional sector masses**, normalized by 9!, not by
+sector visitation. For either individual axial or diagonal direction,
+respectively, on square L3:
+
+    H_axis(p,q)=3 p^3 (1-q)^3 [(1+q)^3-p^3],
+    H_diag(p,q)=3 p^6 (1-q)^3.
+
+Their sum recovers the previous unmarked, separable Markov kernel:
+
+    2 H_axis+2 H_diag=6 p^3 (1-q^2)^3.
+
+The axial sector instead has determinant
+
+    H_axis(a,c)H_axis(b,b)-H_axis(a,b)H_axis(b,c)
+      = -9 a^3 b^3 (1-b)^3 (1-c)^3
+          * (b^3-a^3) * [(1+c)^3-(1+b)^3] < 0
+
+for `0<a<b<c<1`. The diagonal determinant is zero. At
+`(a,b,c)=(1/3,1/2,2/3)` the axial early-minus-late survival difference
+is `-1084/250187`; unmarked and diagonal differences are zero.
+
+The marked process observes rank zero before entry, `(rank1,D)` while
+rank one, then an absorbing rank-two state; it does not reveal future D
+before birth. Its current axial state is insufficient for its observed
+history, although coarser unmarked rank is Markov. **Weak Markovness need
+not survive adding an observation.**
+
+The geometric identity makes this explicit. In the pooled square cohort,
+
+    E[nu_2 | T1<=p,T2>q] = 6q/(1+q),
+
+independent of p. Within an axial cohort it is
+
+    3 - 3(1-q)(1+q)^2 / [(1+q)^3-p^3],
+
+which depends on p. Within a diagonal cohort it is 3. Since square
+`nu_2=|A|-3`, this is hidden count selection after changing clocks,
+not a contradiction of constant hazard at fixed k.
+
+Triangular marked kernels are each one third of its unmarked kernel and
+retain its nonzero label determinant. All closed forms, determinants and
+completion-mean identities are checked by exact polynomial arithmetic in
+[marked-clock.json](../analysis/birth-completion-geometry-20260929/marked-clock.json).
+The coefficient ranks (one or two) are not positive hidden-state counts.
+
+## 6. What changes next
+
+The finite explanation now has a concrete chain:
+
+    birth-history selection -> completion-geometry mixture -> exit hazard -> H.
+
+It replaces the vague proposal to try direction or birth age as a fix.
+For larger sizes, the question is whether cohort hazard differences
+survive scaling and what recursively controls completion geometry.
+L3 closure is not assumed to extend.
+
+A useful new independent block would capture nu_2 and persistent D at
+specified rank-one times alongside paired births. First choose one clock
+and geometry, then compare early/late completion hazards within direction.
+The exact transmission formula defines the consumer; adding unrelated
+descriptors would not answer it. A finite-lag defect requires the hazard
+over an interval, not merely at its midpoint. No larger-size run is
+launched here. Existing L512 pair archives lack the geometries and cannot
+be relabelled as this experiment. No causal or original-U claim follows.
+
+## Reproduce / actual work
+
+From the repository root, with Python 3.11 and SymPy:
+
+```bash
+python3 analysis/birth-completion-geometry-20260929/analyze.py
+python3 analysis/birth-completion-geometry-20260929/exact_marked_clock.py
+```
+
+The first uses the standard library and two existing exact L3 tables
+for a source-convention comparison; it took about .021 seconds on local
+Python 3.11.15. The polynomial transform took about .252 seconds with
+SymPy 1.14.0. Runtime fields naturally vary. The geometry script was
+rerun while extending its saved phase realization, not to repeat physical
+permutation acquisition. No cloud, Monte Carlo, full suite or document
+wording tests ran in this step.
diff --git a/notes/birth-gap-cloud-20260929.md b/notes/birth-gap-cloud-20260929.md
new file mode 100644
index 000000000..2310b34d2
--- /dev/null
+++ b/notes/birth-gap-cloud-20260929.md
@@ -0,0 +1,161 @@
+# Independent fine-resolution birth-gap block
+
+2026-09-29. This block follows the local pilot; its design is informed by that
+pilot, not retrospectively described as an untouched research programme.
+
+## Question and design chosen before cloud outputs
+
+The local L=16--128 pilot has a rapidly falling direct atom and soft
+near-diagonal mass approximately proportional to the examined resolution.
+The useful extension is finer resolution and larger L, not a rerun of the
+small exact census or a free-exponent fit.
+
+- Square and triangular site-percolation tori, paired births in one uniform
+  site-permutation filtration.
+- Planned L=64,128,256,512, 14 independent batches of 10,000 samples per cell.
+  A largest-cell benchmark decides whether this fits the bounded run. A
+  benchmark-only stop is not production.
+- New seed namespace `cloud-independent-20260929`; no local-pilot samples
+  are pooled into its estimates.
+- Primary descriptive grid: delta=.00625,.0125,.025,.05 for width-normalised
+  label-gap CDF and soft mass Z. Also retain .1,.2 and the prescribed L^(3/4)
+  reference scale, with the same covariance.
+- Report Z(delta)/delta and resolution-halving behaviour without fitting an
+  intercept or exponent. A visible plateau challenges a continuous-density
+  picture on the resolved scales; a roughly proportional decrease does not
+  prove there is no unresolved atom or logarithmically slow crossover.
+- No numerical hypothesis-rejection threshold was selected. This is an
+  independent exploratory block, not model-elimination certification.
+
+## Execution context
+
+Only DevEnvC_TV2N0X, UUID `4a8d1d443419434889e49148ed0a7ba6`, is used.
+The cloud agreement initially blocked listing. The owner restored the original
+account through the web interface; authenticated listing then showed Ready.
+This round powered on that instance, performed one successful SSH key reset,
+checked key mode 600 and connected through its dedicated port 10026.
+No other instance was started, reset or stopped.
+
+Measured: aarch64, 16 visible CPUs, CPU quota 14.5 cores, memory quota 25 GiB,
+Python 3.9.9, about 14 GiB free on the 40 GiB /workspace disk. No research job
+was running in the process snapshot after this startup. The base image lacked
+a compiler; GCC C++ 10.3.1 and its dependencies were installed from the
+configured official Huawei Cloud EulerOS repositories.
+
+Task files stay in the new directory
+`/workspace/Matching-One-TV2N0X/birth-gap-20260929`.
+The prior local L=3 exact control is reused, not repeated for each worker.
+Actual run outcome and interpretation follow below.
+
+The first largest-cell benchmark took 3.188 s (square) and 3.300 s
+(triangular) per 200 L=512 samples. It predicted about 431 s at 14 workers
+for the originally planned block, exceeding the 300 s acquisition budget;
+the runner exited before producing any research samples. Its benchmark-only
+manifest is retained. Before seeing any cloud production outputs, the sample
+count was reduced to **5,000 per batch**, preserving all four sizes, both
+lattices, 14 independent batches and the same finer-resolution grid:
+**560,000 planned production filtrations**. This changes precision, not the
+readout or hypothesis after seeing a result.
+
+## Completed result
+
+All **560,000** cloud filtrations completed: 112 batches, 70,000 samples per
+lattice/size. Acquisition including build and benchmark took **329.71 s**;
+the preflight 300 s limit was a benchmark-estimate bound, not a process kill
+timer. Actual concurrent runtime exceeded the extrapolation. The subsequent
+standard-library analysis took **17.02 s**. No production was repeated and no
+exact control was rerun on the server.
+
+With W the pooled birth-mixture IQR, G=(N+1)(T2-T1)/W:
+
+| Lattice | L | Direct atoms / 70,000 | Z(.00625) | Z(.0125) | Z(.025) | Z(.05) |
+|---|---:|---:|---:|---:|---:|---:|
+| square | 128 | 8 | .00348 ± .00016 | .00695 ± .00028 | .01390 ± .00041 | .02769 ± .00049 |
+| square | 256 | 4 | .00356 ± .00015 | .00707 ± .00022 | .01421 ± .00033 | .02807 ± .00049 |
+| square | 512 | 1 | .00336 ± .00015 | .00683 ± .00024 | .01399 ± .00036 | .02792 ± .00047 |
+| triangular | 128 | 5 | .00373 ± .00014 | .00734 ± .00022 | .01443 ± .00036 | .02839 ± .00060 |
+| triangular | 256 | 2 | .00391 ± .00015 | .00763 ± .00022 | .01479 ± .00026 | .02883 ± .00037 |
+| triangular | 512 | 0 | .00378 ± .00014 | .00742 ± .00022 | .01454 ± .00038 | .02843 ± .00068 |
+
+Errors are complete-batch delete-one SE, with W recalculated in each deletion.
+CDF, Z, moments and widths from a cell are correlated views, not independent
+confirmations. The full covariance and every delete-one estimate are saved.
+
+At L=512, Z(delta)/delta on the four columns above is approximately
+(.538,.546,.560,.558) for square and (.605,.594,.582,.569) for triangular.
+These are descriptive ratios of the prespecified readouts, **not** fitted
+slopes or independent tests. There is no resolved resolution-independent
+plateau down to delta=.00625; soft mass continues approximately proportionally
+to resolution. This extends the local pilot by a factor four in both L and
+resolved delta, using new streams rather than a new view of its old samples.
+
+The statement is finite: it does not rule out a smaller unresolved atom, a
+slow crossover, or a different scaling limit. At zero direct counts, the
+one-sided 95% bound is only 1-0.05^(1/70000), about 4.28e-5.
+Square and triangular primitive tori have different conformal moduli; their
+nearby numbers after width normalisation are not a universality test.
+
+## What should follow, rather than another automatic size ladder
+
+The evidence now favours spending the next main effort on a **two-time
+anti-concentration bound**, not on L6's single-site slope or merely larger
+Monte Carlo. A useful sufficient target is
+
+    Z_L(delta) <= C delta^alpha + e_L,  alpha>0, e_L->0,
+
+uniformly for fixed small scaled delta, together with tight birth locations.
+The observed approximately linear resolution behaviour makes alpha=1 a
+working conjecture to investigate, not a law established by this experiment;
+any positive alpha would settle noncoalescence under the stated tightness.
+The two-time pivotal/arm estimate must be derived in its actual source and
+torus geometry. Triangular inputs and square universality assumptions stay
+separate. Neither further static rank precision nor the direct atom supplies
+this missing bound.
+
+## Reproduction and retained artifacts
+
+Source commit: `abc78c6a1c69e83c9a2daf4d92911833bd6f3937`.
+Only engine.cpp, run_pilot.py and analyze.py were transferred from that commit;
+the runner records their source hashes, GCC version, Python version, seeds,
+individual commands and runtime. No full repository or credentials were sent.
+
+Executed in the task directory:
+
+```bash
+TMPDIR=/workspace/Matching-One-TV2N0X/birth-gap-20260929/tmp \
+OPENBLAS_NUM_THREADS=1 OMP_NUM_THREADS=1 python3 run_pilot.py \
+  --compiler g++ --role cloud-independent-20260929 \
+  --sizes 64 128 256 512 --batches 14 --samples 5000 --workers 14 \
+  --budget-seconds 300 --skip-exact-control \
+  --exact-control-reference abc78c6a:analysis/birth-gap-20260929/data/run.json \
+  --git-base-commit abc78c6a1c69e83c9a2daf4d92911833bd6f3937 \
+  --output cloud-data-5k
+python3 analyze.py --data cloud-data-5k \
+  --epsilons .00625 .0125 .025 .05 .1 .2 \
+  --output-json cloud-summary.json --output-md cloud-SUMMARY.md
+```
+
+Use a fresh output directory to reproduce; use a fresh role to obtain an
+independent block. The same seed namespace reproduces, not independently
+confirms, this block.
+
+- [Cloud tables](../analysis/birth-gap-20260929/cloud-SUMMARY.md)
+- [Cloud JSON, including covariance](../analysis/birth-gap-20260929/cloud-summary.json)
+- [Production manifest](../analysis/birth-gap-20260929/cloud-data-5k/run.json)
+- [Initial benchmark-only stop](../analysis/birth-gap-20260929/cloud-data/run.json)
+- [Analysis timing](../analysis/birth-gap-20260929/cloud-analysis-time.txt)
+
+Raw per-batch paired histograms are the .json.gz files beside the production
+manifest (about 2.2 MiB total). The initial and production benchmarks use the
+same excluded benchmark streams; they are not additional independent data.
+
+Returned archive SHA256:
+`f775b25b884b26a9841d98af9ecde87bb85bba0a86a06da46cc5a1538fbd16a2`;
+local transfer check matched. Cloud JSON SHA256:
+`6a4b1c1e0f2e223436b822cab44225c831dc7d78b2b16b66afccbffa002bc4a0`.
+The archive itself is not duplicated in Git because its contents are committed.
+
+After transfer, a process snapshot showed no remaining research process.
+The task's managed tunnel was stopped and TV2N0X was powered off; authenticated
+listing confirmed **Ready**. Other instances were not operated. Raw task files
+remain on the instance's data disk as well as in this research branch.
diff --git a/notes/birth-gap-noncoalescence-20260929.md b/notes/birth-gap-noncoalescence-20260929.md
new file mode 100644
index 000000000..283059186
--- /dev/null
+++ b/notes/birth-gap-noncoalescence-20260929.md
@@ -0,0 +1,335 @@
+# Distinct microscopic births can still coalesce: a persistence criterion
+
+Date: 2026-09-29. New finite/probabilistic analysis for the paired-birth route.
+This note proves statements about ordered two-birth processes; it does **not**
+prove a percolation scaling limit or square-site universality.
+
+Follow-up: [Two-time pivotal density](two-time-pivotal-density-bound-20260929.md)
+derives the needed windowed small-ball bound from arm inputs and supplies a
+triangular noncoalescence author corollary. The general identities and
+counterexample below remain unchanged; uniqueness is a different question.
+
+## The useful new target
+
+The question is not only whether a single insertion can create both homology
+directions. It is whether two **different** insertions can become one birth
+after near-critical rescaling. Measure the shrinking neighbourhood of the
+diagonal, not just its microscopic zero atom.
+
+For a nonnegative, dimensionless birth gap `G`, define
+
+```text
+Z(delta) = E[(1-G/delta)_+],                     delta > 0.       (1)
+```
+
+This soft near-diagonal mass has two exact representations: one from the paired
+gap histogram, another from one- and two-time rank-one occupation. It provides
+a criterion for excluding limit double births without fitting a free exponent.
+
+The relevant prior assets are unmerged #773, commit
+`9ef22d1cc531c09dedf9b1195ab6aad8ef79668f`:
+
+- [`notes/rank-birth-flux-conservation-20260914.md`](https://github.com/LightChainr/Matching-One/blob/9ef22d1cc531c09dedf9b1195ab6aad8ef79668f/notes/rank-birth-flux-conservation-20260914.md):
+  direct jump flux, static marginals and the missing copula.
+- [`notes/canonical-two-birth-scaling-20260914.md`](https://github.com/LightChainr/Matching-One/blob/9ef22d1cc531c09dedf9b1195ab6aad8ef79668f/notes/canonical-two-birth-scaling-20260914.md):
+  canonical clock and the proposed insertion scale.
+- [#778](https://github.com/LightChainr/Matching-One/issues/778): paired archive
+  semantics and exact order-statistic mixture. Its latest September 14
+  comments also retain the optional primitive rank-one direction mark.
+
+In particular, `P(D=0)->0` alone does not close the continuity/no-aggregation
+step of #773. This note supplies the missing distinction, not a refutation of
+the proposed scaling programme.
+
+## 1. A counterexample with identical entire one-time rank laws
+
+For every integer `n>=2`, consider two equally weighted birth pairs in each
+of two models:
+
+| Model | First pair `(S,T)` | Second pair `(S,T)` |
+|---|---|---|
+| A | `(0, 1+1/n)` | `(1, 2)` |
+| B | `(0, 2)` | `(1, 1+1/n)` |
+
+Set `R(t)=1{S<=t}+1{T<=t}`. Both are monotone rank processes starting at zero
+and ending at two. In **both models, at every finite n**, `S<T`: neither ever
+has a direct jump of size two. They have exactly the same birth marginals,
+hence the same full one-time law of `R(t)` for every t, and the same mean gap
+`E(T-S)=1+1/(2n)`.
+
+But their limit gaps are
+
+```text
+A: G_n => 1,
+B: G_n => (delta_0 + delta_2)/2.                              (2)
+```
+
+Thus even all static rank probabilities **plus an identically zero direct
+jump flux** cannot decide whether distinct births aggregate. This is a
+counterexample in the class of monotone two-birth processes, not an alternative
+percolation model. It can also be put on insertion levels by taking
+`J_i=1+n*(S or T)` and `N=2n+1`; all resulting indices are valid integers.
+In that discrete version the **complete adjacent-time rank transition tables**
+are identical too: the static differences determine entry/exit fluxes once
+the common direct-jump flux is zero. Even that one-step kernel is therefore
+insufficient for the aggregation question; a Markov reconstruction would
+silently discard relevant path dependence.
+
+There is an additional moment trap. A positive gap that equals `1/n` with
+probability `1-1/n` and `n` with probability `1/n` converges to zero in
+probability while its mean converges to one. Tightness alone does not license
+passing occupation means to the limit: uniform integrability is needed.
+
+## 2. The occupation-defect theorem
+
+Let `(S,T)` be any almost surely finite ordered birth pair, `G=T-S>=0`, and
+assume `E G<infinity`. Define
+
+```text
+q(t)       = P(S<=t<T),
+H_delta(t) = P(S<=t, t+delta<T),
+A(delta)   = integral_R H_delta(t) dt.
+```
+
+Then, exactly,
+
+```text
+integral q(t) dt = E G,
+A(delta)         = E[(G-delta)_+],
+E G - A(delta)   = E[min(G,delta)],
+Z(delta)         = 1 - [E G-A(delta)]/delta.                   (3)
+```
+
+**Proof.** For each realised interval `[S,T)`, the set of t for which both
+t and `t+delta` lie in it has Lebesgue length `(G-delta)_+`. Integrate the
+indicators and use Tonelli; the remaining equalities are pointwise identities.
+No Markov assumption is used.
+
+The last equality is useful conceptually; numerically evaluate (1) directly
+from gaps, rather than subtract two nearly equal large occupation integrals.
+Formula (1) is meaningful even when `E G` is infinite, in which case the
+subtraction in (3) must not be used.
+
+For `0<c<1`, the following bounds convert the observable into a probability:
+
+```text
+P(G=0) <= Z(delta) <= P(G<delta),
+P(G<=c*delta) <= Z(delta)/(1-c).                              (4)
+```
+
+Hence `Z(delta)` is not the zero atom at a fixed finite resolution. It combines
+that atom with short but positive bars. For a fixed distribution,
+`lim_{delta down to 0} Z(delta)=P(G=0)` by bounded convergence.
+
+### Noncoalescence criterion for a sequence
+
+Suppose the rescaled ordered pairs `(S_n,T_n)` are tight in `R^2`. The following
+are equivalent:
+
+1. Every subsequential weak limit satisfies `P(S=T)=0`.
+2. `lim_{epsilon down to 0} limsup_n P(T_n-S_n<=epsilon)=0`.
+3. `lim_{delta down to 0} limsup_n Z_n(delta)=0`.
+
+Equivalence of 2 and 3 follows from (4), for example with `c=1/2`.
+For 2 implies 1, use the open sets `{t-s<epsilon}` and Portmanteau, then let
+epsilon decrease to zero. Conversely, failure of 2 gives a subsequence with
+uniform positive mass in shrinking diagonal neighbourhoods; tightness gives
+a weakly convergent further subsequence. For every fixed epsilon its mass in
+the closed set `{t-s<=epsilon}` is bounded below, so the limit has positive
+diagonal mass. This proves 1 implies 2.
+
+Tightness of the gaps alone gives the analogous statement about gap limits;
+it does not locate the birth pair in a finite thermal window. Pair tightness
+or a separate location control is needed for that stronger statement.
+
+The order of limits is essential. Taking `delta->0` first returns only
+`P(G_n=0)` and misses model B in (2). In measurements, take an increasing size
+sequence at several **fixed scaled** delta values, then examine smaller delta.
+A single finite-size grid cannot prove the double limit.
+
+### Exact discrete version
+
+For insertion births `1<=J1<=J2<=N`, put `D=J2-J1` and
+`R_k=1{J1<=k}+1{J2<=k}`. Extend the rank-one indicator by zero outside
+`k=0,...,N`. For every nonnegative integer m,
+
+```text
+A_m = sum_k P(R_k=1, R_{k+m}=1) = E[(D-m)_+],
+A_0 = sum_k P(R_k=1) = E D,
+1-(A_0-A_m)/m = E[(1-D/m)_+],                     m>=1.       (5)
+```
+
+At microscopic resolution `m=1`, this equals `P(D=0)`. At a mesoscopic
+resolution `m approximately delta * insertion_window`, it tests aggregation.
+The whole distribution can be recovered by discrete curvature:
+
+```text
+P(D=d) = A_{d-1} - 2 A_d + A_{d+1},                d>=1.      (6)
+```
+
+Thus the new object is a lagged rank-one overlap, not a larger catalogue of
+one-time observables. An existing paired histogram already determines it.
+
+## 3. Exactly what marginal information can establish
+
+Positive mean rank-one occupation is useful, but it does not exclude a partial
+diagonal atom; models A and B have the same mean and all the same marginals.
+With `m=E G` and `v=E G^2<infinity`, Cauchy--Schwarz does give
+
+```text
+P(G>epsilon) >= (m-epsilon)_+^2/v.                          (7)
+```
+
+Indeed `m <= epsilon + E[G 1{G>epsilon}]`. If `G<=B` almost surely, the sharper
+bounded-support version is `(m-epsilon)_+/(B-epsilon)`, for `epsilon<B`.
+These bound the separated fraction; they do not establish that it is one.
+
+There is also a **sharp marginal-only upper bound on diagonal mass**. Let
+`mu_1,mu_2` be the two birth marginal measures and let
+`TV(mu_1,mu_2)=sup_A |mu_1(A)-mu_2(A)|`. Assume the marginals admit an ordered
+coupling. Then
+
+```text
+sup_{ordered couplings} P(S=T) = 1-TV(mu_1,mu_2).             (8)
+```
+
+**Proof.** Any diagonal part is a common submeasure of both marginals, so its
+mass is at most the overlap `nu=mu_1 wedge mu_2`. Remove this maximal common
+submeasure from both marginals. Their cumulative distribution difference is
+unchanged and nonnegative, so the equal-mass residuals still admit an ordered
+quantile coupling. The two residual measures are mutually singular and hence
+put no mass on equality. Combine that residual coupling with the diagonal
+coupling of nu. This attains the bound. The zero-residual case is immediate.
+
+For ordinary density marginals the overlap is `integral min(f1,f2)`, but the
+measure formulation includes atoms. Also `q(t)=F_1(t)-F_2(t)`, giving the weaker
+bound `P(S=T)<=1-sup_t q(t)` directly from rank-one occupancy.
+
+Consequently, overlapping limiting birth marginals generally leave a real
+diagonal ambiguity. The new two-time statistic removes information that cannot
+be restored by making the same static curves arbitrarily precise. Equation
+(8) is an upper bound over all ordered couplings, not a claim that percolation
+can realise every such coupling.
+
+Apply this observation to the limiting marginals with care: finite-n total
+variation need not converge under weak convergence. In the counterexample,
+the finite marginals have disjoint supports and (8) is zero at every n, but
+their limit overlap is `1/2`. Thus a finite marginal-overlap bound of zero
+does not repair the missing uniform near-diagonal control either.
+
+## 4. Paired permutation archives: exact continuous-gap conversion
+
+For iid continuous Uniform site labels, their random ordering is independent
+of their sorted values. If the ranks are determined by that ordering, then
+conditional on `J1=i,J2=j`,
+
+```text
+(T1, T2-T1, 1-T2) ~ Dirichlet(i, j-i, N+1-j),               (9)
+```
+
+with a zero middle spacing when `i=j`. In particular, for `d>=1`,
+
+```text
+T2-T1 | D=d ~ Beta(d,N+1-d),
+E[T2-T1|d]  = d/(N+1),
+Var[T2-T1|d] = d*(N+1-d)/[(N+1)^2*(N+2)].                  (10)
+```
+
+For an affine scaled gap `G=a_N*(T2-T1)`, let `h_d=P(D=d)` and
+`x=min(delta/a_N,1)`. With `I_x(alpha,beta)` the regularised beta CDF,
+
+```text
+P(G<=delta) = h_0 + sum_{d>=1} h_d I_x(d,N+1-d),
+
+Z(delta) = h_0 + sum_{d>=1} h_d [
+    I_x(d,N+1-d)
+    - (a_N/delta) * d/(N+1) * I_x(d+1,N+1-d)
+].                                                        (11)
+```
+
+This is exact conditional integration of label spacings, not new samples and
+not a Gaussian approximation. A D histogram suffices for (11); a full `(J1,J2)`
+histogram additionally retains location and midpoint-gap dependence. A pair of
+separate marginal histograms, or a few joint moments, does not supply `h_d`.
+Non-uniform labels require their known CDF transformation. A conditional or
+biased permutation sampler requires its sampling weights and cannot silently
+be read as an iid permutation ensemble.
+
+For a nonlinear canonical clock `b_N(p)`, the gap
+`b_N(T1)-b_N(T2)` needs the joint `(i,j)` law and the joint spacing integral in
+(9), not only D. An estimated clock introduces shared estimation error; either
+propagate it within the same batches or use a separately calibrated clock.
+
+### When insertion and label gaps have the same limit
+
+Write `G_tilde=a_N*D/(N+1)`. Conditional on D,
+
+```text
+E[G-G_tilde|D] = 0,
+E[(G-G_tilde)^2|D]
+    = a_N*G_tilde*(1-D/(N+1))/(N+2).                       (12)
+```
+
+If `G_tilde` is tight and `a_N/N -> 0`, then `G-G_tilde -> 0` in probability:
+restrict to `G_tilde<=K`, apply conditional Chebyshev to (12), then let K
+increase. A bounded mean yields the stronger mean-square bound
+`E[(G-G_tilde)^2] <= a_N E[G_tilde]/(N+2)`.
+
+Thus the two gap variables have the same subsequential laws under these
+conditions, including their limiting diagonal atom, even though their finite
+CDFs differ. Uniform closeness of CDFs at a shrinking delta is **not** implied;
+use (11) for a finite-size near-diagonal report.
+
+For an `L x L` lattice (`N=L^2`), the proposed thermal scaling is
+`a_N=L^(3/4)`, hence the insertion scale is `(N+1)/a_N`, asymptotic to
+`L^(5/4)`. The spacing condition in (12) is then favourable. For centred birth
+**locations**, each order statistic has variance at most `1/[4(N+2)]`, so
+`a_N=o(sqrt(N))` also makes the two individual clock errors vanish; this
+stronger condition holds for the proposed scaling as well.
+
+The exponent `3/4` is a square-site universality hypothesis/control here, not
+a theorem supplied by this note. A triangular comparison should use a specified
+near-critical normalisation and torus modulus; known triangular ingredients do
+not by themselves prove the required torus hitting-time continuity. The exact
+identities (1)--(12) do not depend on choosing this exponent.
+
+## 5. A small prospective comparison, not another permission system
+
+Use one paired archive or one independently seeded production block, retain
+aligned batch identities, and report the same summaries at every size. The
+following are competing interpretations, not claims already established.
+
+| Question | Prespecified readout | Result that changes the mechanism picture |
+|---|---|---|
+| Separate macroscopic births? | `Z_L(delta)` and beta-mixture `P(G<=delta)` at `delta=1/2,1/4,1/8,1/16`, then smaller fixed resolutions if informative | A size-stable positive small-delta floor challenges simple-birth noncoalescence; decreasing values support it without proving the double limit. |
+| Microscopic direct fusion versus aggregation? | `h_0` alongside `Z_L(delta)-h_0` on the same block | Falling `h_0` with persistent soft mass identifies the aggregation alternative that an arm count alone misses. |
+| Clock or location escape rather than aggregation? | Birth marginal quantiles and tails, `G_tilde` quantiles; raw thermal scale plus a separately specified canonical clock when available | Escaping locations/gaps invalidate an asserted tight pair limit; clock-dependent near-diagonal conclusions call for clock control, not a new fitted exponent. |
+| What has one-time analysis already told us? | `E D=sum q_1(k)` and marginal overlap bound, versus the paired `Z` curve | Equal static summaries but distinct `Z` directly isolate process information, as in the counterexample. |
+
+These rows can be computed together; they are not sequential approval gates.
+The numerical delta grid is a reporting convention, not a universal resolution
+or significance boundary. Fix the clock and grid before inspecting a new block.
+Save the cross-delta/cross-readout batch covariance. Shared random streams or
+reanalysed archives remain one dependency group; do not add the several rows as
+independent evidence. If the paired archive is missing, report the marginal
+bound (8) and the precise missing object instead of inventing a copula.
+
+No GPU or massive trajectory storage is intrinsic to this test: a joint birth
+histogram, batch IDs and optional direction marks are sufficient. The next
+theory target is a **uniform near-diagonal bound** for this paired object,
+potentially through multi-time pivotal geometry. A single-site six-arm bound
+only addresses its direct-jump part unless an additional argument controls
+aggregation of separate events.
+
+## 6. Executed check and claim boundary
+
+`python3 scripts/check_birth_gap_counterexample.py` performs one small exact
+Fraction calculation: equal static marginals in the two counterexample
+families; the discrete overlap/curvature identities; and (11) against direct
+integration of integer-parameter beta density polynomials. It enumerates no
+percolation configurations and draws no Monte Carlo samples.
+
+This note adds the counterexample, occupation-defect criterion, sharp
+marginal-overlap bound, and exact finite archive interface. It does not provide
+a measured large-size diagonal mass, a new percolation exponent, a proof of
+canonical copula universality, or an identification of the original-U operator.
diff --git a/notes/birth-gap-pilot-20260929.md b/notes/birth-gap-pilot-20260929.md
new file mode 100644
index 000000000..a3a56bfa0
--- /dev/null
+++ b/notes/birth-gap-pilot-20260929.md
@@ -0,0 +1,185 @@
+# Joint births: an executed finite-size pilot, not another readiness exercise
+
+Date: 2026-09-29. Claim level: new exploratory independent Monte Carlo block.
+
+**The useful result is a separation of scales.** Single-insertion double births
+become rare, but a fixed small near-critical neighbourhood of the diagonal
+still contains substantial paired-birth mass. The new soft statistic behaves
+approximately linearly in its resolution on this pilot's grid, rather than
+showing a resolved positive intercept. This supports prioritising quantitative
+near-diagonal control; it does not prove that the scaling limit has no double
+births.
+
+## What actually ran
+
+- NN square-site and triangular-site tori, L=16,32,64,128; N=L² sites.
+- Eight independent pseudorandom batches × 2,000 uniform site permutations per
+  lattice/size: **128,000 production filtrations** in total.
+- Both births are recorded in the **same** monotone filtration:
+  `J1=first rank>=1`, `J2=first rank=2`, `D=J2-J1`.
+- The rank is the integer winding-image rank of the union of all occupied
+  components. It is not two directional wrap Booleans. A diagonal rank-one
+  cycle is still rank one.
+- Triangular edges add only the consistent `(1,1),(-1,-1)` diagonal to the
+  four square neighbours. This is triangular site percolation on a primitive
+  rhombic torus, **not** square matching-lattice percolation with both diagonals.
+- Local ARM64, Apple clang 21, C++17 and Python 3.13.7 standard library;
+  maximum four simultaneous processes. Acquisition including compilation,
+  control and benchmark took **6.89 seconds**. Sum of individual production
+  process wall times was 17.43 seconds; that is not measured CPU time.
+- Before production, all 9! permutations were enumerated once for each L=3
+  lattice. Square reproduced `P(D=0)=3/35`, `E D=3/2`, `E D²=43/14`;
+  triangular reproduced `2/35`, `3/2`, `81/28`. The largest requested cell was
+  then benchmarked with 200 separate samples per lattice. Control/benchmark
+  samples are excluded from production estimates.
+
+The displacement convention is adapted from
+[`src/threshold_rank_axis_mc.cpp`](../src/threshold_rank_axis_mc.cpp).
+The new engine accumulates a global basis from every non-tree cycle, using
+the winding-vector determinant. All edges of an insertion are processed before
+checking rank, so an actual `0->2` step is not artificially split by edge order.
+
+## Results that change the next question
+
+| Lattice | L | Direct atoms / 16,000 | E[D]/L^(5/4) | Z(.025), width-normalised labels | Z(.05), width-normalised labels |
+|---|---:|---:|---:|---:|---:|
+| square | 16 | 97 | .4230 ± .0032 | .01815 ± .00123 | .02996 ± .00217 |
+| square | 32 | 24 | .4206 ± .0029 | .01389 ± .00055 | .02616 ± .00089 |
+| square | 64 | 8 | .4273 ± .0026 | .01402 ± .00052 | .02684 ± .00111 |
+| square | 128 | 1 | .4278 ± .0030 | .01365 ± .00054 | .02718 ± .00080 |
+| triangular | 16 | 38 | .4032 ± .0018 | .01439 ± .00043 | .02634 ± .00065 |
+| triangular | 32 | 13 | .4031 ± .0027 | .01372 ± .00031 | .02665 ± .00049 |
+| triangular | 64 | 4 | .4075 ± .0030 | .01360 ± .00079 | .02709 ± .00127 |
+| triangular | 128 | 1 | .4056 ± .0031 | .01435 ± .00103 | .02813 ± .00137 |
+
+Errors are eight-batch delete-one SE, not confidence limits. Here
+`W=Q.75-Q.25` of the equal-weight empirical J1/J2 mixture,
+`G=(N+1)(T2-T1)/W`, and `Z(delta)=E[(1-G/delta)_+]`. W is re-estimated in each
+delete-one sample. Its discrete quantile can jump at small L; the unusually
+large L=16 square width-normalised uncertainties are retained, not smoothed
+away or interpreted as a new physical effect.
+
+At L=128, the exponent-free label-gap CDF at `.025,.05,.1,.2` is
+
+```text
+square:     .02728, .05426, .10843, .20801
+triangular: .02832, .05534, .10771, .20563
+```
+
+The corresponding Z values are
+
+```text
+square:     .01365, .02718, .05440, .10634
+triangular: .01435, .02813, .05487, .10610
+```
+
+These values are compatible with a finite near-origin gap density over the
+resolved range: halving the resolution roughly halves Z. This is a descriptive
+finite-grid observation, **not** a fitted exponent, a hypothesis test, or an
+exclusion of an unresolved smaller-scale diagonal mass. The two lattices have
+different conformal torus shapes; their numerical proximity after width
+normalisation is not a same-modulus universality test.
+
+One observed direct atom out of 16,000 means frequency `0.0000625`, with
+two-sided 95% Wilson interval approximately `[0.0000110,0.0003540]`. Even zero
+observations would only give a one-sided 95% upper bound `0.0001872`; neither
+would be a theorem of zero probability.
+
+The full table also reports the prescribed `L^(5/4)` insertion scale and
+`L^(3/4)` label scale. They are useful fixed reference scales, not an established
+square-site critical-exponent theorem. The width-normalised readout avoids
+assuming that exponent. No free power, lattice metric factor or cutoff was
+fitted to improve agreement.
+
+## Exact use of the paired histogram
+
+Uniform random ordering is independent of the order statistics of iid uniform
+labels. Conditional on `D=d>0`, `T2-T1 ~ Beta(d,N+1-d)`; d=0 is an atom.
+The analyzer integrates this conditional law exactly up to floating arithmetic,
+using integer-parameter beta/binomial identities. No extra label Monte Carlo
+noise is added. Thus the same histogram supplies:
+
+1. the direct atom and discrete near-diagonal CDF;
+2. continuous-label near-diagonal CDF at the same four resolutions;
+3. both discrete and label-integrated Z(delta);
+4. birth correlation, gap mean and self-normalised widths.
+
+Z was added after acquisition, following the theory agent's exact statistic
+request; its grid is the original `.025,.05,.1,.2`. It is not an independently
+preregistered second experiment. All these coordinates share one dependency
+group per lattice/size/batch. Complete delete-one covariance and all deleted
+estimates are saved, not treated as independent evidence votes.
+
+The link to a genuine noncoalescence target is proved in
+[`birth-gap-noncoalescence-20260929.md`](birth-gap-noncoalescence-20260929.md):
+under pair tightness, every limit has distinct births iff
+`lim_delta->0 limsup_L Z_L(delta)=0`. The pilot does not supply pair tightness
+or this double limit. A stable mean gap alone would not exclude a partial
+diagonal atom either.
+
+## A useful independent next block, if the parent allocates it
+
+Do not use a decreasing direct atom as the success criterion. Two competing
+behaviours to distinguish are a stable intercept of Z at shrinking resolution
+versus near-origin continuous mass with Z decreasing with resolution. The
+smallest useful extension is independent L=128/256 (512 if its benchmark is
+cheap), at a declared finer width-normalised grid such as
+`.00625,.0125,.025,.05`. Repeated large samples at L=16 are less relevant to
+that distinction. This is a suggestion, not an executed cloud campaign or
+a requirement that other research wait for it.
+
+For an independent run, use a **new random-seed namespace**, keep its batches
+separate, and carry both CDF and Z under the same paired covariance. Reject
+the continuous-only interpretation at the examined scales if a clearly
+resolved stable intercept develops; retain the unresolved range if precision
+or size drift prevents a distinction. Do not invent a numerical rejection
+threshold after seeing that new block, and do not claim a finite intercept
+distinguishes all possible logarithmically slow approaches to zero.
+
+## Reproduction and handoff
+
+The exact executed command was
+
+```bash
+python3 analysis/birth-gap-20260929/run_pilot.py \
+  --sizes 16 32 64 128 --batches 8 --samples 2000 --workers 4
+python3 analysis/birth-gap-20260929/analyze.py
+```
+
+The acquisition runner refuses to overwrite an existing run; for reproduction
+choose `--output analysis/birth-gap-20260929/reproduction-data` and pass that
+directory to the analyzer with `--data`, plus distinct output paths. Repeating
+the original seeds reproduces the block; it is not new independent data.
+
+Files:
+
+- [`engine.cpp`](../analysis/birth-gap-20260929/engine.cpp): transparent C++17 engine;
+- [`run_pilot.py`](../analysis/birth-gap-20260929/run_pilot.py): bounded local
+  acquisition (this run used four); subsequently extended for an explicitly
+  named independent cloud block with a portable compiler and up to 14 workers;
+- [`analyze.py`](../analysis/birth-gap-20260929/analyze.py): stdlib readout;
+- [`data/run.json`](../analysis/birth-gap-20260929/data/run.json): per-batch
+  seeds, commands, timings, compiler, base commit, source/data SHA256;
+- [`SUMMARY.md`](../analysis/birth-gap-20260929/SUMMARY.md) and
+  [`summary.json`](../analysis/birth-gap-20260929/summary.json): full CDF/Z grid,
+  raw counts, uncertainty and covariance;
+- `data/pilot-*.json.gz`: full sparse `(J1,J2,count)` histogram for every
+  batch; all production data together are below one megabyte.
+
+For ARM Linux the engine is directly portable, with no OpenMP dependency:
+
+```bash
+g++ -O3 -std=c++17 -DNDEBUG analysis/birth-gap-20260929/engine.cpp -o /task/path/birth-gap
+/task/path/birth-gap square 256 10000 2026092900256000 /task/path/square-L256-b00.json sample
+```
+
+The example seed is a **new** cloud block seed, not a local batch seed. The
+extended runner now accepts `--compiler g++ --role NEW_NAMESPACE --workers 14`,
+`--skip-exact-control --exact-control-reference ...`, and an explicit
+`--git-base-commit` for copied source trees. See
+[`CLOUD-HANDOFF.md`](../analysis/birth-gap-20260929/CLOUD-HANDOFF.md) for the full
+bounded invocation and finer prospective epsilon grid. The local manifest and
+generated results were not rewritten after extending the runner; its saved
+runner SHA records the acquisition-time revision, while the engine is unchanged.
+No local Monte Carlo was rerun. No cloud access, commits, pushes or navigation
+edits were performed by this pilot subtask.
diff --git a/notes/birth-history-memory-readout-20260929.md b/notes/birth-history-memory-readout-20260929.md
new file mode 100644
index 000000000..0b1d394cb
--- /dev/null
+++ b/notes/birth-history-memory-readout-20260929.md
@@ -0,0 +1,218 @@
+# A concrete process question: does rank one forget its birth history?
+
+2026-09-29. #778 continuation in Draft #838. This is a new analysis of
+existing joint-birth archives, not new sampling or independent validation.
+
+**Outcome.** A single preselected three-time comparison finds a small positive
+history association in the large-size insertion data. It does not settle the
+continuum Markov question. The existing exact L3 archives provide a stronger,
+finite conclusion: the square rank process is Markov under both insertion
+and iid-label clocks, while the triangular rank process is not. Closed-form
+two-time label kernels certify the distinction without repeating enumeration.
+
+This separates three objects which should not be conflated: an exact finite
+counterexample, an exploratory large-size signal, and an unidentified limit.
+
+## 1. One null with an actual consequence
+
+For the two births T1<=T2, define
+
+```text
+R(t) = 1{T1<=t} + 1{T2<=t},
+H(s,t) = P(T1<=s, T2>t),          s<=t.
+```
+
+If current rank alone is a time-inhomogeneous Markov state, then for every
+a<b<c with positive rank-one risk,
+
+```text
+H(a,c)*H(b,b) = H(a,b)*H(b,c).                           (1)
+```
+
+The [kernel characterization](birth-kernel-markov-characterization-20260929.md)
+proves the converse when (1) holds for all triples. Direct double jumps are
+allowed; they never enter the rank-one risk set. This concerns the ordinary
+Markov property in the natural rank history, not the full configuration,
+an arbitrary hidden-state process or a strong Markov assertion.
+
+Here is a directly interpretable violation. Condition on R(b)=1 and compare
+future survival to c between paths whose first birth occurred by a and paths
+whose first birth occurred later:
+
+```text
+Delta = P(T2>c | T1<=a, T2>b)
+      - P(T2>c | a<T1<=b<T2).                           (2)
+```
+
+If both groups have positive mass, Delta=0 is equivalent to (1). A nonzero
+value rules out every current-rank-only Markov model at this time triple,
+including models with arbitrarily time-dependent rates. It does not rule
+out models that retain connectivity, direction or another hidden state.
+Conditioning on rank one is part of this probabilistic null, not a causal
+intervention on birth time.
+
+Static rank curves and adjacent fluxes admit a Markov completion. That
+completion can reproduce those inputs while violating the actual longer
+history. Equation (1) tests a constraint that fitting the static curve
+cannot certify.
+
+## 2. Existing large-size block: modest evidence, no cutoff hunt
+
+The calculation uses the 112 production batches in
+[`cloud-data-5k/run.json`](../analysis/birth-gap-20260929/cloud-data-5k/run.json):
+14 independent batches x 5000 filtrations for each of two lattices and
+L=64,128,256,512. No benchmark rows, local-pilot pooling or new samples.
+
+Before this scoring, choose exactly one triple in each cell: the .25, .50,
+.75 **lower quantiles of the equal mixture of J1,J2**. Time here is occupied
+insertion count, not iid-label time. L512 is the primary available-size
+readout; smaller sizes are context. These choices are fixed in the script,
+but the block was already examined for gap statistics: this is retrospective,
+not a prospective model-elimination certificate.
+
+| L512 insertion process | Early-group survival | Late-group survival | Delta, percentage points |
+|---|---:|---:|---:|
+| Square | 0.44914 | 0.43491 | +1.422 +/- 0.511 |
+| Triangular | 0.43248 | 0.42546 | +0.702 +/- 0.515 |
+
+Errors are 14-batch delete-one standard errors, **not confidence intervals**.
+Every deletion recomputes the three empirical cutoffs and both risk groups.
+The JSON retains raw four-cell counts, all deleted estimates, cutoffs,
+source batch identities and the full six-metric covariance. A standard error
+does not account for the wider adaptive research programme.
+
+The four size-specific differences are all positive in each lattice, but
+we do not fit a limiting constant or exponent, combine eight signs into a
+new significance claim, or treat related gap readouts as extra evidence.
+At L512 the square difference is suggestive and the triangular one less
+resolved. No continuum non-Markov theorem follows from this table.
+
+The source geometries are primitive-coordinate square and triangular tori
+with different moduli. Differences between the rows are not a same-modulus
+universality test. The insertion/label clock comparison proved earlier is
+asymptotic, not a finite-N license to replace this observable by a label-time
+one.
+
+## 3. Exact finite result from the saved complete tables
+
+The existing L3 archives each contain all 9! permutations. Their paired
+birth histogram determines the entire rank-only insertion path. Evaluating
+all 120 triples 0<=a<b<c<=9 with exact integer counts gives:
+
+- Square: all Markov kernel identities hold. By the characterization,
+  its **L3 insertion rank process is Markov**.
+- Triangular: two identities fail. At (a,b,c)=(4,5,6), the early/late by
+  survive/exit table is `[[23328,101088],[15552,93312]]`. Survival probabilities
+  are 3/16 and 1/7, so Delta=5/112 and conditional covariance=1/90.
+  The other violating triple, (3,4,5), has Delta=2/25 and covariance=1/90.
+
+This is an exact finite positive control alongside an exact finite negative
+control, **conditional on the stored complete engine tables**. No 9!
+enumeration was repeated. The all-triples check is a finite identity
+calculation, not a search for a significant production cutoff. It also shows
+why nontrivial microscopic connectivity does not by itself prove a rank
+projection is non-Markov at every size.
+
+## 4. Continuous-label kernels, computed exactly rather than substituting Np
+
+The uniform permutation is independent of the ordered iid uniform labels.
+For p<=q, the numbers of sites with label at most p, between p and q, and
+above q have Multinomial(N; p,q-p,1-q) law. Thus the exact clock bridge is
+
+```text
+H_label(p,q) = sum_(0<=k<=ell<=N)
+  [N!/(k!(ell-k)!(N-ell)!)]
+  p^k (q-p)^(ell-k) (1-q)^(N-ell) H_J(k,ell).          (3)
+```
+
+This is not H_J evaluated at expected counts. Applying (3) to the two
+complete L3 tables and expanding over the rationals gives:
+
+```text
+H_square(p,q) = 6 p^3 (1-q^2)^3,
+
+H_triangular(p,q) = 9 p^3 (1-q)^3 [A(p)+B(q)],
+A(x) = 2x^3-6x^2+3x,
+B(x) = -2x^3+3x+1,                  0<=p<=q<=1.        (4)
+```
+
+The square kernel separates as a function of p times a function of q.
+It therefore satisfies (1) identically, not just at the rational check
+points: **the square L3 rank process is Markov in iid-label time too**.
+On its rank-one interval its survival transition is
+
+```text
+P(R(t)=1 | R(s)=1) = [(1-t^2)/(1-s^2)]^3,
+q_12(t) = 6t/(1-t^2),                         0<s<t<1. (5)
+```
+
+This is the rank-one exit rate, not the 0->1/0->2 rate or a large-L claim.
+
+For the triangular kernel the determinant in (1) is exactly
+
+```text
+81 a^3 b^3 (1-b)^3 (1-c)^3
+   [A(b)-A(a)] [B(c)-B(b)].                            (6)
+```
+
+It is not identically zero. For example at label times (1/3,1/2,2/3),
+Delta=-1156/716639. This certifies finite non-Markovianity under the actual
+iid-label clock. The sign need not be positive at every triple; the
+determinant's explicit factors show how it changes. Do not attribute the
+sign difference from the insertion example solely to the clock: the tested
+time triples are different too.
+
+The coefficient matrices of the two polynomials have ranks one and two.
+These are kernel separation ranks, **not** assertions about a minimum
+positive latent-state realization or the topology's full predictive memory.
+Neither L3 identity is extrapolated to large size or to a continuum limit.
+
+## 5. What this changes next
+
+The useful question is no longer whether one can invent a history-sensitive
+readout. We now have an exact null identity, finite controls of both signs
+of the Markov verdict, and a small large-size contrast on the actual archive.
+
+The next discriminating theoretical object is the continuum rank-one kernel
+H(s,t), not more equivalent gap descriptors. A rank-only mechanism must
+predict its multiplicativity; a proposed geometric mechanism should predict
+the defect's dependence on entry history or a specified homology mark.
+The current birth archive has no such direction marks, so it cannot test
+conditional sufficiency of a direction state by retrospective invention.
+
+Adding first-birth time as a state variable is not by itself a mechanism
+discovery. When the current rank is one, the whole observed rank past is
+already encoded by that one birth time. A time-dependent rank-plus-age
+description is therefore a complete encoding of this observation history;
+it does not explain how microscopic geometry produces the conditional
+completion law, nor imply a time-homogeneous semi-Markov law.
+
+No larger-size production or finer-cutoff search is launched here. If a new
+block is used later to assess the small large-size contrast, its clock,
+fixed time triple and microscopic mechanism prediction should be stated
+before scoring. That is attention guidance, not a research lock.
+
+## Reproduce and evidence boundaries
+
+From the repository root:
+
+```sh
+python3 analysis/birth-markov-kernel-20260929/analyze.py
+python3 analysis/birth-markov-kernel-20260929/exact_clock.py
+```
+
+The first script uses the standard library; the second uses SymPy exact
+polynomial arithmetic. Executed with managed research Python 3.11 on arm64;
+runtime/version details are in the outputs. The analysis skill's validation
+pass checks source identities, full batch counts, weighted risk denominators,
+exact determinant algebra and complete-batch covariance; it does not certify
+a continuum conclusion or turn this reused block into independent evidence.
+
+- [Readable archive result](../analysis/birth-markov-kernel-20260929/RESULT.md)
+- [Raw counts, all delete-one estimates and covariance](../analysis/birth-markov-kernel-20260929/result.json)
+- [Exact label-kernel polynomial certificate](../analysis/birth-markov-kernel-20260929/exact-clock.json)
+- [General kernel characterization and Markov completion](birth-kernel-markov-characterization-20260929.md)
+
+These are author analysis/proof outputs, not independent certification of
+the older engine or a novelty claim. Original archives and previous results
+are unchanged. No cloud job, original-U contract change or PR merge.
diff --git a/notes/birth-kernel-markov-characterization-20260929.md b/notes/birth-kernel-markov-characterization-20260929.md
new file mode 100644
index 000000000..27c6e1932
--- /dev/null
+++ b/notes/birth-kernel-markov-characterization-20260929.md
@@ -0,0 +1,471 @@
+# Two-birth kernels: an exact rank-only Markov characterization
+
+2026-09-29. Bounded author derivation for the joint-birth programme and its
+three-time rank-one memory statistic. No claim of novelty, general
+percolation Markov/non-Markov behaviour, continuum identification, or unique
+scaling limit is made. Section 7 links the companion exact finite archive
+analysis; this theorem derivation is not an independent reproduction of
+those calculations.
+
+**Main statement.** For an ordered two-birth process, the identities
+
+    H(a,c) H(b,b) = H(a,b) H(b,c),    all a <= b <= c,          (M)
+
+characterize ordinary time-inhomogeneous Markov behaviour in its natural
+rank filtration. Direct `0 -> 2` jumps and arbitrary birth-time atoms are
+allowed. A single three-time identity is necessary, not sufficient.
+The continuous marginal completion below additionally assumes absolutely
+continuous birth marginals and no direct double birth; those assumptions
+are not needed for the characterization itself.
+
+## 1. Process, filtration and the precise Markov contract
+
+Let `T1<=T2` be random variables in `R union {+infinity}` and, for every
+deterministic real t, set
+
+    R(t) = 1{T1<=t} + 1{T2<=t},
+    F_t^R = sigma(R(r): r<=t),
+    H(s,t) = P(T1<=s, T2>t),    s<=t.
+
+One can equivalently work on a fixed observation interval, extending the
+initial all-zero process to earlier times when appropriate. We use the
+raw natural filtration, or its completion by null sets; no enlargement by
+hidden geometry, future information, or an encoder/random-clock state is
+implicit. The right-continuous path convention is fixed by `T_i<=t`.
+
+Write `F_i(t)=P(T_i<=t)` and
+
+    P0(t)=1-F_1(t),   P1(t)=F_1(t)-F_2(t)=H(t,t),
+    P2(t)=F_2(t).
+
+Ordinary Markov means that for each deterministic `b<=c` there is a
+stochastic kernel `K_(b,c)` on `{0,1,2}` such that
+
+    E[g(R(c)) | F_b^R] = K_(b,c)g(R(b))   almost surely       (1.1)
+
+for every bounded g. Null sets may depend on the deterministic times.
+This note does not assert a strong Markov property at arbitrary stopping
+times, or silently substitute a right-continuous enlarged filtration.
+
+The complete theorem is: (1.1) holds if and only if (M) holds for **every**
+deterministic triple with `P1(b)>0`. When `P1(b)=0`, all terms of (M)
+vanish and it supplies no restriction. Endpoint cases `a=b` or `b=c`
+are identities, so the substantive triples have `a<b<c`.
+
+## 2. Necessity and sufficiency, without density assumptions
+
+### Necessity
+
+Fix `a<=b<=c`. On `B_b={R(b)=1}={T1<=b<T2}`, rank at c is still one
+exactly when `T2>c`. If `P1(b)>0`, Markovness therefore gives the survival
+probability
+
+    K_11(b,c) = H(b,c)/P1(b).
+
+The early-entry event
+
+    E_a = {T1<=a, T2>b} = {R(a)=1, R(b)=1}
+
+belongs to `F_b^R`. Integrating (1.1) over `E_a` yields
+
+    H(a,c) = H(a,b) H(b,c)/P1(b),
+
+which is (M). This uses neither a density nor a claim that births are
+independent. A path that jumped directly from zero to two is simply not
+in `B_b`.
+
+### Sufficiency on state one: CDF events generate the observed history
+
+Assume (M) for all the stipulated triples and fix b,c with `P1(b)>0`.
+On `B_b`, for every r<=b,
+
+    R(r)=1{T1<=r},
+
+because `T2>b`. Thus the trace of `F_b^R` on `B_b` is generated precisely
+by the sets `B_b intersect {T1<=a}`, a<=b. This is the censored first-birth
+CDF information, not an additional hidden-state assumption.
+
+Put `s_bc=H(b,c)/P1(b)`. Identity (M) says
+
+    P(B_b, T1<=a, T2>c) = s_bc P(B_b, T1<=a)            (2.1)
+
+for every a<=b. The CDF sets form a pi-system containing `B_b` (take a=b).
+The two finite measures in (2.1) therefore agree on their generated
+sigma-field, by the pi-lambda theorem/monotone class argument. Consequently
+
+    E[1{T2>c} | F_b^R] = s_bc   on B_b, a.s.           (2.2)
+
+It follows that the entire conditional law of `R(c)` on state one is
+`(0,s_bc,1-s_bc)`. This proof includes atomic and singular birth laws;
+it does not condition on an event `T1=t` of probability zero.
+
+On `B_b^0={R(b)=0}` the whole observed history is identically zero.
+The trace of `F_b^R` on this event is trivial, so the conditional future
+law depends on b and the current state zero only. On `{R(b)=2}` the future
+is identically two, irrespective of the possibly informative past. These
+facts together with (2.2) prove (1.1). The null-risk state-one case is
+irrelevant to conditional expectations.
+
+For clarity, the on-risk kernel is explicitly
+
+    K_00(b,c) = P0(c)/P0(b),
+    K_01(b,c) = [P1(c)-H(b,c)]/P0(b),
+    K_02(b,c) = 1-K_00(b,c)-K_01(b,c),                  if P0(b)>0;
+
+    K_11(b,c) = H(b,c)/P1(b),
+    K_12(b,c) = 1-K_11(b,c),   K_10(b,c)=0,            if P1(b)>0;
+
+    K_22(b,c)=1,   K_20(b,c)=K_21(b,c)=0.
+
+Rows at zero-probability current states may be filled arbitrarily as
+probability rows; they do not affect the law started from the specified
+process. Chapman-Kolmogorov follows on reachable rows. Arbitrary off-risk
+fillings are not claimed to produce a canonical kernel for every artificial
+starting state.
+
+This is why the all-triples condition suffices here, although a generic
+process is not certified Markov by a few three-point checks: on rank one,
+the entire relevant past is generated by just the first-birth CDF events.
+
+## 3. A readable survival contrast and its four-cell determinant
+
+For a fixed `a<b<c`, let
+
+    p = H(b,b),   e = H(a,b),   x = H(a,c),   z = H(b,c).
+
+Within the state-one risk set at b, distinguish
+
+    early: T1<=a, T2>b;
+    late:  a<T1<=b<T2.
+
+Their unconditional masses are e and p-e. If both are positive, the
+conditional survival difference is exactly
+
+    Delta(a,b,c)
+      = P(T2>c | T1<=a, T2>b)
+        - P(T2>c | a<T1<=b<T2)
+      = x/e - (z-x)/(p-e)
+      = [x*p-e*z]/[e*(p-e)].                          (3.1)
+
+Here the four-cell table contains **unconditional** masses:
+
+| Entry group within R(b)=1 | Survives: T2>c | Completes: b<T2<=c |
+|---|---:|---:|
+| Early | x | e-x |
+| Late | z-x | p-e-z+x |
+
+Its determinant, with exactly this row/column ordering, is
+
+    D = x*(p-e-z+x) - (e-x)*(z-x)
+      = x*p-e*z
+      = H(a,c)H(b,b)-H(a,b)H(b,c).                    (3.2)
+
+After conditioning the whole table on `R(b)=1`, its determinant is
+`D/p^2`, also the covariance of the early-entry and survival indicators
+under that conditional law. On positive early and late risks, `D=0`,
+`Delta=0`, and conditional independence of these two indicators are
+equivalent. A positive Delta means earlier entrants among survivors at b
+are more likely to remain rank one at c; a negative value means the reverse.
+
+If p=0 there is no rank-one comparison. If e=0 or p-e=0, Delta is undefined,
+the determinant is automatically zero, and the triple is uninformative
+about entry-history dependence. Do not replace an undefined contrast by
+a measured zero or rely on an odds ratio with empty cells.
+
+Conditioning on `R(b)=1` is **selection of a risk set**, not a causal
+intervention setting rank to one. The criterion tests sufficiency of the
+current rank for the ordinary predictive law. It does not identify a
+causal effect of early birth, lifetime, geometry, or a microscopic source.
+
+## 4. Continuous marginals: the unique no-double-jump Markov completion
+
+For this section only, suppose T1,T2 are finite almost surely,
+`P(T1<T2)=1`, and both CDFs F_1,F_2 are absolutely continuous, with densities
+f_1,f_2. They are assumed compatible with such an ordered strict pair.
+Their joint law need not have a two-dimensional density.
+
+The marginal occupancies are the P_i above. On positive risks the only
+possible absolutely continuous transition rates of a Markov completion are
+
+    q_01(t)=f_1(t)/P0(t),
+    q_12(t)=f_2(t)/P1(t),
+    q_02(t)=0.                                       (4.1)
+
+With `q_00=-q_01`, `q_11=-q_12`, and state two absorbing, these rates
+produce the marginally matching time-inhomogeneous pure-birth completion.
+Its rank-one kernel is
+
+    H_M(s,t) = P1(s) exp[-integral_s^t q_12(r) dr].    (4.2)
+
+The formula is first read within a connected positive-P1 interval.
+It is zero when P1(s)=0, or when a zero-P1 time is crossed. Extended
+integrals and risk exhaustion give the same convention, as detailed below.
+
+### Uniqueness uses only ordinary deterministic-time Markovness
+
+Here is a direct justification, avoiding a strong Markov assumption at T1.
+On a compact interval `[s,t]` with P1 bounded below by m>0, take partitions
+`s=t_0<...<t_n=t` of vanishing mesh. Define
+
+    epsilon_i = P(t_i<T1<=T2<=t_(i+1)),
+    r_i = [F_2(t_(i+1))-F_2(t_i)-epsilon_i]/P1(t_i).
+
+The second numerator is precisely the mass that leaves rank one between
+the two deterministic times having already entered by t_i. Therefore
+`K_11(t_i,t_(i+1))=1-r_i`. Markovness gives
+
+    K_11(s,t)=product_i (1-r_i).
+
+There is no diagonal birth mass, so
+
+    sum_i epsilon_i <= P(0<T2-T1<=mesh) -> 0.
+
+Also `max_i r_i->0` by continuity of F_2 and the risk lower bound, while
+`sum_i r_i` is bounded. Thus the logarithm of the product differs from
+`-sum_i r_i` by a term tending to zero. Continuity and positivity of P1
+give the Riemann-Stieltjes limit
+
+    sum_i r_i -> integral_s^t dF_2(r)/P1(r)
+                 = integral_s^t f_2(r)/P1(r) dr.
+
+This proves (4.2) for any ordinary Markov process with the stipulated
+marginals and no double jumps. Similarly the state-zero survival is
+`P0(t)/P0(s)=exp[-integral_s^t f_1/P0]` on positive state-zero risk.
+The on-risk kernels in section 2 are consequently fixed. Thus the
+Markov **path law**, not merely an arbitrarily chosen off-risk generator,
+is unique with these marginals and the no-double-jump convention.
+
+### Existence and zero-risk boundaries
+
+Let `O={t:P1(t)>0}`, an open set since the CDFs are continuous. On every
+component `(l,r)` of O, the rate `q=f_2/P1` is locally integrable and
+
+    P1' = f_1-f_2 = f_1-q*P1   almost everywhere.
+
+Variation of constants, with zero occupancy at the left boundary, yields
+
+    P1(t) = integral_l^t f_1(x)
+              exp[-integral_x^t q(y)dy] dx.           (4.3)
+
+For l=-infinity, use `P1(l)=0` as the limiting value. More formally start
+at l+delta and let delta decrease to zero; the initial survival term is
+bounded by P1(l+delta) and vanishes.
+
+An ordered strict pair puts its births inside these components almost
+surely. To see the only boundary issue, choose a countable dense subset
+of `O^c`. Each of its points has zero rank-one probability. Hence almost
+surely the open random interval `(T1,T2)` meets none of those points and
+therefore none of `O^c`; it lies inside one component of O. Its endpoints
+can lie outside O only at endpoints of these countably many components.
+Absolute continuity gives those endpoints probability zero. Thus f_1 and
+f_2 vanish almost everywhere off O; no unmodelled birth mass is placed
+at a zero-risk boundary.
+
+Construct T1 with density f_1. Given its sampled value x in O, draw its
+second birth using survival `exp[-integral_x^t q]` within that component.
+This defines a new process; it is not an invocation of a stopping-time
+property for the original one. Local integrability gives T2>T1 almost
+surely. At a right endpoint r where P1 tends to zero, the inequality
+
+    exp[-integral_x^t q] <= P1(t)/P1(x),    x<t<r,
+
+follows from `P1' >= -q*P1`. Thus survival falls to zero there and the
+integral diverges. This also applies at r=+infinity, because both births
+are finite and `P1(t)->0`. No residual atom or direct double jump needs
+to be inserted at the boundary.
+
+Equation (4.3) shows that the constructed occupancy is exactly P1(t);
+its completion density is `q(t)P1(t)=f_2(t)`. Conditional survival within
+state one multiplies at deterministic intermediate times, so its H is
+(4.2) and satisfies (M). Section 2 then proves ordinary Markovness.
+
+Set ratios with zero numerator and zero risk to zero for notation, or
+choose any off-risk row convention; their values do not affect the
+specified law. Rates may diverge near risk exhaustion, and the extended
+survival integral, not a finite rate assigned at the endpoint, is what
+matters. If absolute continuity, strict birth order, or compatibility
+fails, this construction is not asserted. In particular an unknown
+direct-double-jump flow is not determined by the two marginals alone.
+
+**Interpretation:** fitting all static rank laws by this Markov completion
+does not show that the actual process is Markov. The observed joint H
+may differ from H_M while both marginals, and hence every static rank
+law, agree exactly.
+
+## 5. Finite insertion counts: marginals plus direct-jump mass
+
+For a finite count clock, take `1<=J1<=J2<=N`,
+`R_k=1{J1<=k}+1{J2<=k}`, `k=0,...,N`. Define F_i(k), P_i(k) analogously.
+At step k+1 let
+
+    m_1=P(J1=k+1),   m_2=P(J2=k+1),
+    d_(k+1)=P(J1=J2=k+1).
+
+The unconditional one-step flow table is fixed exactly:
+
+| From / to | 0 | 1 | 2 |
+|---|---:|---:|---:|
+| 0 | P0(k)-m_1 | m_1-d_(k+1) | d_(k+1) |
+| 1 | 0 | P1(k)-m_2+d_(k+1) | m_2-d_(k+1) |
+| 2 | 0 | 0 | P2(k) |
+
+Divide each positive-risk row by P_i(k). The resulting time-inhomogeneous
+Markov chain, started at R_0=0, is the unique Markov completion matching
+these marginals **and** these per-step direct-jump masses, on reachable
+states. Row/column sums reproduce P_i(k) and P_i(k+1), respectively, so
+existence follows directly by forward multiplication. It also reproduces
+the specified d_(k+1). Its survival kernel is
+
+    H_M(k,l)=P1(k) product_(j=k)^(l-1)
+                 [1-(P(J2=j+1)-d_(j+1))/P1(j)]       (5.1)
+
+along positive risks; it is zero once the intervening rank-one risk is
+exhausted. Arbitrary off-risk rows cannot change this path law.
+
+For compatible data all flows are nonnegative. Explicitly the allowed
+direct-jump mass lies in
+
+    max(0,m_2-P1(k)) <= d_(k+1) <= min(m_1,m_2).
+
+If P0(k)=0 then m_1=d_(k+1)=0. If P1(k)=0 then compatibility forces
+`m_2=d_(k+1)`, so no state-one exit is divided by a fictitious risk.
+Static marginals without d generally leave a family of completions.
+Conversely, supplying both marginals and d fixes only adjacent-time
+flows of the actual process; it still does not establish its Markovness.
+The CDF proof in section 2 applies on this finite time grid as well.
+
+## 6. Minimal exact four-cell example
+
+Take T1 in `{1,2}`, T2 in `{3,4}`, and assign the following joint masses:
+
+| | T2=4 | T2=3 |
+|---|---:|---:|
+| T1=1 | 3/8 | 1/8 |
+| T1=2 | 1/8 | 3/8 |
+
+For `(a,b,c)=(3/2,5/2,7/2)`, every path is rank one at b and
+
+    p=1,  e=1/2,  z=1/2,  x=3/8;
+    D=1/8,   Delta=3/4-1/4=1/2.
+
+The past first-birth information changes the future survival law even
+though the current rank is the same. This process is not Markov in the
+specified filtration. Replacing all four masses by 1/4 preserves both
+birth marginals, all static rank laws and the absence of direct jumps.
+The resulting independent pair (whose supports already enforce order)
+has `H(s,t)=F_1(s)[1-F_2(t)]`, satisfies every identity (M), and is Markov.
+This is hand algebra, not an enumeration or a percolation example.
+
+## 7. Exact L3 archive controls from the companion analysis
+
+The [companion analysis](birth-history-memory-readout-20260929.md) reports
+an integer evaluation of all 120 strict triples
+`0<=a<b<c<=9` from the already existing full `9!` paired-birth tables:
+
+- [Square L3 paired table](../analysis/birth-gap-20260929/data/exact-control-square-L3-b00.json.gz):
+  zero violations of (M) on the complete insertion-time grid.
+- [Triangular L3 paired table](../analysis/birth-gap-20260929/data/exact-control-triangular-L3-b00.json.gz):
+  exactly two violations, at `(3,4,5)` and `(4,5,6)`; the conditional
+  early-entry/survival covariance is `1/90` for each.
+
+These are supplied exact-archive results, not an independent re-reading of
+the files in this note. The [reproducible archive script](../analysis/birth-markov-kernel-20260929/analyze.py)
+does not rerun the `9!` configuration/permutation enumeration.
+Applying the sufficient direction of section 2, the square result is an
+**exact finite insertion-clock rank-only Markov positive control** for this
+L3 ensemble. The triangular result is an exact negative control for its
+corresponding ensemble. This is not a presumption that every microscopic
+projection must be non-Markov, nor an extrapolation to other sizes or clocks.
+
+For the triangular triple `(4,5,6)`, the reported count table, in the same
+early/late and survives/completes ordering as section 3, is
+
+| Entry group within R(5)=1 | J2>6 | 5<J2<=6 |
+|---|---:|---:|
+| Early: J1<=4 | 23,328 | 101,088 |
+| Late: 4<J1<=5 | 15,552 | 93,312 |
+
+The risk count is 233,280 out of 362,880 total permutations. The reported
+conditional survivals are `3/16` and `1/7`, giving `Delta=5/112`; the
+conditional covariance is `1/90`. This last number is the determinant of
+the **risk-normalized** table, not the unnormalized count determinant or
+the determinant of counts divided by the full 362,880 population. Production
+three-quantile estimates and their uncertainties belong in the separate
+results note, not in this proof's exact-control claims.
+
+The [exact clock script](../analysis/birth-markov-kernel-20260929/exact_clock.py)
+supplies a separate **exact iid-label clock transformation** of these same
+tables, using multinomial order-count
+weights, not new configuration enumeration. For `0<=p<=q<=1` the reported
+closed kernels are
+
+    H_square(p,q) = 6 p^3 (1-q^2)^3,
+    H_tri(p,q) = 9 p^3 (1-q)^3 [A(p)+B(q)],
+    A(x)=2x^3-6x^2+3x,   B(x)=-2x^3+3x+1.
+
+The square kernel is separated and satisfies (M) for **all continuous
+label-time triples**, so it is also a label-clock Markov positive control.
+This conclusion comes from the explicit transformed kernel, not from
+assuming that count-clock Markovness survives a random time change.
+For the triangular kernel the exact factorization is
+
+    D(a,b,c) = 81 a^3 b^3 (1-b)^3 (1-c)^3
+                 [A(b)-A(a)] [B(c)-B(b)],
+
+which is not identically zero. At `(a,b,c)=(1/3,1/2,2/3)` its reported
+survival contrast is `-1156/716639`, giving a continuous-label negative
+control as well. These closed forms are recorded in the companion
+[polynomial certificate](../analysis/birth-markov-kernel-20260929/exact-clock.json)
+and are not independently recomputed here. Their polynomial coefficient
+matrices have reported algebraic ranks one and two, respectively; those
+ranks are not automatically dimensions of a hidden positive state model.
+The finite L3 conclusions do not establish a general lattice or scaling
+limit theorem.
+
+## 8. What a three-quantile archive comparison can and cannot say
+
+The birth-mixture CDF is `(F_1+F_2)/2`. Its **population** .25, .5 and .75
+quantiles provide deterministic times a,b,c for one application of (3.1).
+They must be distinct with positive early and late risk to give an
+informative survival contrast. Empirical plug-in quantiles are estimated
+times; sampling uncertainty and their dependence on the same archive do
+not disappear by calling them fixed quantiles.
+
+A nonzero population determinant excludes rank-only ordinary Markovness
+for this time/observer/ensemble contract. A finite estimate requires its
+statistical uncertainty to be assessed. A zero value at one triple cannot
+establish the all-triples condition; it may miss dependence at other
+times. A finite-N violation need not persist in a scaling limit, and
+neither outcome proves uniqueness of a continuum birth copula. Selection
+of rank one remains predictive conditioning, not an intervention.
+
+**Birth age is a history coordinate, not a discovered mechanism.** On
+rank one, the observed past is already encoded by T1. Thus augmenting
+the state by the observed birth time, or by age `t-T1` while retaining
+calendar time t, supplies a complete predictive state for this two-birth
+rank process. On rank zero there is no extra observed history; on rank
+two the future is absorbing. This gives a time-inhomogeneous ordinary
+Markov description of the augmented process, regardless of whether rank
+alone is Markov. It does not automatically give a time-homogeneous
+semi-Markov law: the completion holding-time distribution may still
+depend on the calendar entry time (with direct-jump routing handled
+separately). Finding prediction improves after adding age is therefore
+not itself a microscopic mechanism result or a reason to search more
+age descriptors. A mechanism would explain how T1, through geometry,
+direction or other specified marks, changes the completion law.
+
+Finally, the count clock k is not the finite-N iid-label clock p. For iid
+continuous labels, let `K(p)` be the number of sites with labels <=p;
+the ordering permutation is independent of the order statistics. Then
+
+    R_label(p)=R_count(K(p)),
+    H_label(s,t)=P(J1<=K(s), J2>K(t)).
+
+At one time, static laws are binomial mixtures over k; at several times,
+the count increments have the corresponding joint multinomial law.
+The random hidden count can affect transition probabilities even when
+the count-clock rank has a Markov completion. Neither the measured
+three-time contrast nor the Markov property transfers automatically by
+substituting k/N for p. A deterministic invertible time reparameterization
+is a different matter and preserves the ordinary Markov property on its
+time image. Clock and observer definitions must accompany the result.
diff --git a/notes/birth-selection-intervention-20260930.md b/notes/birth-selection-intervention-20260930.md
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+# A clock-preserving geometric source: from history correlation to intervention
+
+2026-09-30. Author finite-model derivation in Draft #838. The main research
+advance is a **source with zero immediate rank response but a specified
+delayed response**, not a more precise estimate of the old contrast.
+One separate worker handles the old target's independent-archive replication.
+
+**Outcome.** We can change which safe site arrives while preserving both the
+entire occupied-count clock and the current completion hazard. This changes
+the two-step rank law with an exact, nonpositive susceptibility. Birth-history
+association, meanwhile, factors into a backward homology-retention functional
+and a forward completion functional. They are not the same geometric object.
+An exact path-weight decomposition separates safe geometric transport from
+survivor selection. These give a concrete next mechanism experiment.
+
+All statements below are for the specified finite occupied-site torus and
+uniform source at theta=0. They do not identify original U, a conformal
+operator, a continuum memory coefficient or the sign of a long-window effect.
+
+## 1. The intervention, not another passive descriptor
+
+Fix a rank-one occupied set A with k sites, m=N-k vacancies, c completing
+vacancies and s=m-c safe vacancies. In its existing synergy graph, let
+d_v be the degree of safe vacancy v and e the edge count. Then
+
+    mean_safe(d) = 2e/s,
+    c(A+v) = c+d_v.
+
+For s>0, define a new insertion source:
+
+    P_theta(v | A) = 1/m                         if v completes,
+                    (s/m) exp(theta*d_v)/Z_A(theta)  if v is safe,
+    Z_A(theta) = sum_safe exp(theta*d_v).                    (1)
+
+Use the original uniform source outside rank one or when s=0. At theta=0
+this is exactly the original source, not a sampled approximation. For every
+theta the total probability of completing on the next insertion is c/m.
+The rank-one direction also stays fixed on every safe insertion. Thus this
+source changes continuation geometry while leaving the immediate coarse
+rank response **identically zero**, configuration by configuration.
+
+Let S2(theta|A) be rank-one survival after two insertions, with the kick (1)
+on the first insertion and the ordinary uniform source on the second. Then
+
+    S2(theta|A) = (s/m) [1 - (c+E_theta[d])/(m-1)],
+    d S2/d theta |0 = -s Var_safe(d) / [m(m-1)].             (2)
+
+Proof: after the first safe choice v, exactly c+d_v vacancies complete;
+the second insertion exits with probability (c+d_v)/(m-1). Differentiate
+the normalized exponential mean. Its derivative is Var_safe(d).
+The result also holds when (1) is active on both insertions: the second
+step's total exit mass is still unchanged. Assume m>=2; s=0 has zero
+survival and zero source response, with no need to define a degree variance.
+
+Consequently the two-step response is strictly negative whenever the safe
+degrees vary. It is zero for a regular synergy graph, even if e is large.
+The number e controls the baseline two-step law; degree variance controls
+this *specified perturbation's* gain. This is a consumer for graph structure,
+not a proposal to rank arbitrary extra features by explained variance.
+
+### Continuous clock: the occupation process itself is unchanged
+
+In the iid-label baseline at time t, every remaining site has arrival rate
+1/(1-t). Replace safe-site rates by
+
+    r_theta(v|A,t) = [s/(1-t)] exp(theta*d_v)/Z_A(theta),
+
+and leave completing-site rates equal to 1/(1-t). Total arrival rate stays
+m/(1-t) for every A. The entire occupied-count process K(t) therefore has
+the same law for all theta, and can be coupled with identical arrival times.
+The current rank-exit intensity c/(1-t) is also unchanged. At nonzero theta
+these are **modified geometry-dependent arrivals, not iid site labels**.
+
+The second survival derivative at a fixed starting geometry is
+
+    S_theta''(0|A) = [c(c-1)-s E_theta[d]]/(1-t)^2,
+    partial_theta S_theta''(0|A)|0
+                  = -s Var_safe(d)/(1-t)^2.                 (3)
+
+The killed-generator calculation replaces the old 2e creation term by
+s E_theta[d]; its loss term is still c^2, and differentiating 1/(1-t)
+supplies the -c term. Hence the geometric source first reaches rank at
+order h^2, while changing neither K(t) nor the first-order exit rate.
+This distinguishes it from a mere change of the arrival clock. It does not
+by itself identify a CFT normal direction or rule out thermal projections
+in a different readout contract. Remainders here are finite-N, not uniform
+near-critical bounds.
+
+## 2. Birth memory is a backward/forward coupling, not an ageing force
+
+For a rank-one k-site set A define its backward retention probability
+
+    R_a(A) = #{B subset A: |B|=a, rank(B)=1} / binom(k,a).
+
+Under the original uniform permutation, conditional on A_k=A the first a
+sites form a uniform a-subset. Monotonicity ensures no rank-two subset is
+possible. Therefore
+
+    P(J1<=a | A_k=A) = R_a(A).                              (4)
+
+The remaining permutation is independent of the internal ordering of A.
+Let C be any positive-probability current-state cell, for example exact
+(rank=1,D,c), and p_C=E[R_a|C] in (0,1). For any future readout Z, write
+z(A)=E[Z|A_k=A]. Conditional past/future independence at full A gives
+
+    E[Z|J1<=a,C] - E[Z|J1>a,C]
+        = Cov_C(R_a,z) / [p_C(1-p_C)].                      (5)
+
+For two-step survival, all c-dependent terms are constant within C, so
+
+    Delta S2 = -2 Cov_C(R_a,e) / [m(m-1)p_C(1-p_C)],
+    Delta E[Y] = 2 Cov_C(R_a,e) / [s p_C(1-p_C)],
+    Y(A)=2e/s.                                             (6)
+
+The current L512 matched negative Y contrast thus estimates a negative
+overlap-weighted sum of these normalized covariances, not a universal
+negative sign in every cell. More first-direction robustness under thinning
+is associated, in that weighted readout, with less second-direction creation.
+This interpretation is exact for the *population target*; finite estimates
+and replication retain their own uncertainty. Merely finding e variation,
+or saying that older paths are different, does not establish this coupling.
+
+In label time, conditional on A_t=A, retained earlier sites are independent
+with probability u=s0/t. Set R_A(u)=P(rank(thin_u A)=1). Equation (4) becomes
+P(T1<=s0|A_t=A)=R_A(s0/t), and the analogous covariance identity holds under
+the correct label-time law of A_t. This is not permission to substitute
+t=k/N into a fixed-count archive.
+
+### Three specified physical controls
+
+On square L4, v=x+4y, take
+
+    A={0,1,4,5,8,12}, B={0,2,4,5,8,12}, C={0,1,5,8,9,12}.
+
+All have k=6, D=(0,1), c=0. A/B are the existing physical pair; C is a
+specified six-site bent winding loop. Only these three preparations and
+their subsets/extensions are evaluated, not a new whole-lattice census.
+
+| Preparation | e | R_A(u) | Var_safe(d) | S2(0) | S2'(0) | S2(log 2) |
+|---|---:|---:|---:|---:|---:|---:|
+| A | 2 | u^4 | 6/25 | 43/45 | -2/75 | 59/63 |
+| B | 3 | u^4 | 16/25 | 14/15 | -16/225 | 71/81 |
+| C | 2 | u^6 | 6/25 | 43/45 | -2/75 | 59/63 |
+
+A and B have the **same entire conditional first-birth law** but different
+future and intervention laws. A and C have the same c,e and this two-step
+response but different birth laws. Thus backward robustness and forward
+completion are distinct axes; neither can generally be reconstructed from
+the other. A 50:50 A/B preparation has zero birth-history association at any
+nondegenerate cutoff despite heterogeneous futures, because its two R_a
+values coincide. These are specified-preparation controls, not population
+estimates for natural L512 prefixes.
+
+The longer-window response has also been **calculated**, not left as a
+request to extend the local formula. Exact continuation over subsets
+reachable from these starts gives single-kick survival derivatives:
+
+| Further insertions | A | B | C |
+|---|---:|---:|---:|
+| 1 | 0 | 0 | 0 |
+| 2 | -2/75 | -16/225 | -2/75 |
+| 3 | -7/180 | -173/1800 | -17/450 |
+| 4 | -17/525 | -307/4200 | -6/175 |
+| 5 | -2/105 | -47/1260 | -2/105 |
+| 6 | -1/150 | -67/6300 | -2/315 |
+| 7 through 10 | 0 | 0 | 0 |
+
+Mean remaining steps-to-completion derivatives are -779/6300, -454/1575
+and -391/3150 respectively. A/C first diverge at lag three despite equal
+initial c,e and degree variance. The response peaks in magnitude at lag
+three in these controls and vanishes when all continuations have completed;
+it is not a constant curvature accumulated over time. All nonzero responses
+here are negative; these three controls do not prove that sign for arbitrary
+geometries or windows. No whole-lattice census was rerun.
+
+## 3. Separating geometric transport from survivor selection
+
+The previous covariance identifies what is coupled now, not how that
+coupling developed. There is an exact decomposition with a stated reference
+process; it is not a unique causal attribution from endpoint observations.
+
+Start with a rank-one entrance distribution nu_j at count j. Let P^S_i
+choose uniformly among safe vacancies at count i, keeping one trajectory
+per starting prefix instead of discarding those with large c. Along this
+reference safe path put
+
+    W_{j:k} = product_{i=j}^{k-1} (1-c(A_i)/(N-i)).
+
+The natural killed transition is Q_i(A,A+v)=1/(N-i) for safe v, hence
+Q_i=(1-c/(N-i))P^S_i. Multiplying along a path proves
+
+    E_natural[1{rank_k=1} f(A_k)] = E_safe[W_{j:k} f(A_k)],
+    E_natural[f | rank_k=1]
+        = E_safe[f] + Cov_safe(f,W)/E_safe[W].               (7)
+
+If a reference path has no safe vacancy before k, send it to a cemetery
+and set its subsequent W to zero. The same identity holds. To condition
+on an endpoint cell C, first condition the reference law on A_k in C and
+use its conditional covariance and weight mean; exclude the cemetery.
+
+For actual birth cohorts let alpha_j(A) be the unnormalized original
+probability of first entering rank one at count j in configuration A.
+Direct rank0-to-rank2 jumps are not in this flux. Then, for example,
+
+    H(a,k) = sum_{j<=a} sum_A alpha_j(A) E_safe,A[W_{j:k}].  (8)
+
+Late cohorts use a<j<=k. Thus the first-birth ensemble, safe transport and
+selection weights enter separately. Apply (7) to each normalized cohort's
+mixture over j to decompose its endpoint f, and subtract the two identities.
+Different cohort weights and endpoint-cell normalizers must be kept; a
+single uniform average of safe paths is not the original survivor law.
+
+This is the precise reason that a large current e-history association
+cannot alone decide “imported at birth” versus “selected during survival.”
+Post-hoc relabelling ages on fixed geometries is not a physical intervention.
+Nor is this a request for a larger descriptor catalogue: the reference
+process in (7) directly switches off survival reweighting while retaining
+explicit within-rank-one geometric evolution.
+
+## 4. The next decision, with distinct predictions
+
+The main lane is now **mechanism intervention and propagation**, not further
+precision rounds. Independent archival replication runs on a support lane.
+
+1. **Geometric continuation kick.** Use (1) on the same starting prefixes,
+   sharing the arrival clock with the unperturbed branch. One-step rank
+   response must be zero; two-step susceptibility is (2), and for the three
+   controls the finite responses are already calculated above. For a
+   specified longer window, predict the response by the actual continuation
+   law rather than extending the two-step sign. A model that represents the
+   perturbation only as a clock change misses this zero-first/nonzero-delayed
+   response. The useful large-size question is whether this channel has a
+   nonvanishing *scaled finite-window gain*, not another confirmation of (2).
+2. **Selection versus transport.** For specified first-birth ensembles and
+   a common destination count, compare the unweighted safe reference to
+   its W-weighted counterpart in (7). “Selection-only for the chosen
+   contrast” predicts zero unweighted contrast and a nonzero weight
+   covariance contribution. A nonzero unweighted contrast excludes that
+   explanation: entrance/transport already carries it. Both nonzero means
+   a mixed mechanism; cancellation is also allowed. This decomposition
+   does not by itself split entrance geometry from subsequent safe evolution.
+3. **Source at entrance, if that split is needed.** At fixed j,D,c, tilt an
+   explicitly specified entrance ensemble by exp(theta*g(A)), then use the
+   original continuation. Its two-step derivative is
+   -2 Cov_nu(g,e)/[m(m-1)]. Choosing g=e gives a definite negative result
+   when e varies (equal A/B: -1/180). This is a separate *preparation*
+   source, unlike (1)'s *transition* source. In a longer-window comparison,
+   keep j and source normalization fixed, not an age swap across sizes.
+
+For an instantaneous transition kick at i, the finite-window derivative is
+
+    nu Q_k ... Q_{i-1} B_i Q_{i+1} ... Q_{l-1} 1,
+    B_i(A,A+v) = [d_v-2e/s]/(N-i)  for safe v.              (9)
+
+Other entries of B_i are zero; its row sum is zero. Summing (9) gives the
+response to a field active over several counts. This is the explicit
+source -> geometric transition -> future-rank map to evaluate, not a
+generic request for a new certificate. A long-window sign and a continuum
+coefficient are unresolved predictions, not consequences of local negativity.
+
+No new large production is launched by this note. First fix which of these
+mechanisms the longer-window calculation distinguishes; medium/large runs
+go to Huawei. These are attention priorities, not locks on other research.
+
+## Artifacts
+
+[Exact controls](../analysis/birth-selection-mechanism-20260930/RESULT.md),
+[fractions and subset counts](../analysis/birth-selection-mechanism-20260930/result.json),
+[short calculation](../analysis/birth-selection-mechanism-20260930/derive.py).
+The script reuses only the old lifted topology routine; no old census,
+Monte Carlo, full test suite or numerical differentiation is repeated.
diff --git a/notes/completion-creation-independent-replication-20260930.md b/notes/completion-creation-independent-replication-20260930.md
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+# Independent archival reproduction of the completion-creation history contrast
+
+2026-09-30. One support worker completed the fixed negative prediction on
+the other existing 140k square-L512 permutation block. This is a support
+result for finite mechanism research, not the next main precision project.
+
+| Same exact-(D,c), overlap-weighted early-minus-late target | Estimate | 14-batch SE |
+|---|---:|---:|
+| Original safe-insertion block, exact conditional average | -0.0034493857 | 0.0007722528 |
+| Independent completion-hazard archive, first e readout | -0.0031993684 | 0.0006200906 |
+| Second minus first, no pooling | +0.0002500173 | 0.0009903973 |
+
+The prescribed negative direction is reproduced. The second block is about
+5.16 batch SE from zero. Derived mean-pair difference is -170.6490 +/-33.0941
+and two-step survival difference +2.99453e-8 +/-5.80732e-9; these are correlated
+transforms of the same e readout, not extra outcomes. Common support is
+54120/54723 risk prefixes (98.8981%). All 140000 rows match the original
+archive under its documented first-five-field mapping.
+
+The independent *random block* is genuine, but its other outcomes were
+already inspected. This was a fixed-sign first readout of e, not untouched
+holdout data or new prospective acquisition. The blocks were not pooled.
+The conclusion remains finite L512 history dependence of the enlarged
+(rank,D,c) observer. It does not restore the old unreplicated direction-only
+claim or establish unmarked/continuum non-Markovness.
+
+The worker retained integer cell statistics, all aligned deletions and the
+5x5 covariance. One 32-row cost/correspondence check and one full score were
+run; no extra samples, old census or full suite. Huawei TV2N0X used14workers,
+520.876 seconds; its data were retrieved, own tunnel stopped and Ready
+verified at 2026-09-30T02:25:19Z. Other jobs were not operated. The main
+investigator did not repeat that verification.
+
+[Result](../analysis/exact-pair-independent-archive-20260930/results/RESULT.md),
+[full record](../analysis/exact-pair-independent-archive-20260930/results/result.json),
+[execution](../analysis/exact-pair-independent-archive-20260930/execution.json)
+are pinned in commit d9824b5bf478fc8acad238be2b9f31ce99787966, Draft #838.
+
+The next main calculation is the [finite-window geometric-source response](geometric-source-window-identity-20260930.md),
+not another precision extension of these two contrasts. It asks a different
+mechanism question and remains in the original safe-insertion dependency group.
diff --git a/notes/completion-hazard-independent-block-20260929.md b/notes/completion-hazard-independent-block-20260929.md
new file mode 100644
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+# Independent L512 completion-hazard experiment
+
+2026-09-29. Draft #838; continuation of the
+[finite completion-geometry mechanism](birth-completion-geometry-20260929.md).
+
+**Outcome: completed, 140,000 new filtrations.** The new block resolves
+differences in completion geometry more clearly than the raw finite-lag
+survival contrast. After exact direction conditioning, or direction plus
+integer completion count, the predeclared weighted survival differences
+are not resolved. This shifts the next candidate toward a direction-mixture
+model; it does not establish direction sufficiency or license adding an
+unlimited sequence of descriptors to the same block.
+
+**Subsequent interpretation update:** the [directional next-step check](directional-hazard-contrast-20260929.md)
+uses the already saved completion counts and challenges direction-only
+Markovness (+1.629 +/- .391 sites within direction). It is explicitly
+retrospective and leaves every prospective result below unchanged; the
+working candidate suggested by this report must not be read as accepted.
+The subsequent [independent safe-insertion block](safe-insertion-independent-block-20260929.md)
+did not reproduce that retrospective challenge; direction-only closure
+remains unresolved rather than accepted or independently excluded.
+
+| Fixed readout | Estimate, percentage-point scale | Aligned batch SE, same scale |
+|---|---:|---:|
+| Early-minus-late survival | +0.686 | 0.444 |
+| Within exact direction, overlap-weighted survival | -0.415 | 0.451 |
+| Within exact (direction, nu_b), overlap-weighted survival | +0.122 | 0.476 |
+| Integrated-hazard prediction of pooled survival difference | +1.550 | 0.351 |
+| Actual minus integrated prediction | -0.865 | 0.639 |
+| Instantaneous completion contrast, window-scaled, hazard sign | -1.504 | 0.330 |
+
+The last row is dimensionless with a percentage-point display convention,
+**not** an observed finite-window exit probability. The two conditional
+comparisons have different target weights from the pooled row. All outputs
+share this block and its full covariance. The uncertainty is a 14-batch
+delete-one estimate, not an exact finite-sample confidence guarantee.
+
+There are 24,218 early and 30,505 late rank-one histories. Their mean
+completion counts are 99.402 and 101.587. The early cohort is 92.295% axial,
+the late cohort 86.094%; axial directions also have greater subsequent
+survival than diagonals in the descriptive table. Composition is therefore
+a concrete candidate explanation for pooled memory, not a measured causal
+or mediated fraction. Exact-D comparison retains 99.973% of the risk set;
+exact-(D,nu_b) comparison retains 98.898%, across 1,202 shared cells.
+
+The random-time prediction is an unbiased estimator of the **same
+population pooled contrast**, but not the same sample statistic as raw
+survival. Its stronger estimate-to-SE ratio is not another independent
+replication. Actual-minus-predicted closure is about 1.35 batch SE from
+zero; groupwise residuals are -0.009411 +/- 0.006315 (early) and
+-0.000766 +/- 0.004129 (late). No resolved inconsistency with the exact
+hazard map appears here; this is not proof of correctness or state closure.
+
+The old 70k square-L512 contrast was +1.422 +/- .511 percentage points,
+with cutoffs re-estimated on its own batch deletions. The new raw estimate
+is smaller and retains only about 1.5 estimate/SE. We do not hide that
+weakening, pool the blocks, or redefine the primary outcome around the
+stronger ancillary hazard statistic. The old exact L3 results are unchanged.
+
+Full [numerical readout](../analysis/completion-hazard-production-20260929/results/RESULT.md),
+[covariance and cells](../analysis/completion-hazard-production-20260929/results/result.json),
+and [raw batch manifest](../analysis/completion-hazard-production-20260929/data/run.json)
+are retained.
+
+## Question and before-data choices
+
+The L3 calculation supplied an exact microscopic hazard and, at that size,
+a binary rank-one phase realization. The existing L512 archive showed a
+small birth-history association but stored no geometry. The new block asks
+whether current direction, or current direction plus the **exact integer**
+completion count, removes a specified finite-lag history contrast at L512.
+It is new acquisition, not a reconstruction of unavailable old snapshots.
+
+The [contract](../analysis/completion-hazard-production-20260929/contract.json),
+engine and runner were committed and pushed at
+`9520c9bdf702a62435f1054e64184b1df4fc0ab3` **before** new sampling.
+The contract specifies:
+
+- Square NN torus, L=512, N=262144, count clock only.
+- Fixed `(a,b,c)=(154646,155385,156120)`, taken from the previous independent
+  block's birth-mixture quartiles. No re-estimation on this new block.
+- Fourteen independent batches of 10,000 permutations. Separate namespace
+  from all earlier runs; 32 benchmark permutations excluded.
+- Early cohort `J1<=a,J2>b`; late cohort `a<J1<=b,J2>b`.
+- Current direction D and completion count nu_b at b. An independent
+  uniform integer audit time tau in `[b,c-1]` supplies nu_tau when rank one.
+- Three primary contrasts: pooled survival difference, direction-stratified
+  difference, and exact `(D,nu_b)`-stratified difference. Common-support
+  strata use the predeclared overlap weight `n_E*n_L/(n_E+n_L)`.
+- Full aligned delete-one-batch covariance; no cutoff scan, feature search,
+  significance stopping, old/new pooling or added size/geometry.
+
+Each predictive-sufficiency null implies zero for its corresponding
+contrast. Zero is not a proof of full Markovness. The stratified targets
+use different weights from the pooled target; their ratio is **not** a
+fraction of memory explained or a causal mediation effect.
+
+## Efficient exact completion geometry
+
+The new [engine](../analysis/completion-hazard-production-20260929/engine.cpp)
+retains the existing lifted union-find convention. It scans vacant sites
+only at b and tau, without changing the permutation or topology.
+
+To see why a trial site's completion count can be evaluated without a
+full rebuild, contract each old occupied component. A newly added vertex
+and its incident edges form a star with possibly repeated component ends.
+Every new independent cycle is generated by a pair of neighbours in the
+**same** old component. For each occupied neighbour, its union-find root
+and lifted potential specify the new vertex's implied displacement from
+that root. The difference between two such displacements is a torus-period
+cycle. Existing cycles span the current rank-one line; rank two appears
+iff one new cycle is not parallel to it. Distinct old components linked
+once each do not create an additional cycle.
+
+This gives an `O(N d^2 alpha(N))` snapshot scan for fixed degree d, using
+the existing union-find arrays. The direction is primitive and unoriented.
+Path compression changes representation, not geometry or future sampling.
+
+One narrow preproduction control used the 512 L3 subsets of each lattice:
+648 square and 810 triangular explicit copied-state insertion probes
+matched the fast count. All 162 square and 180 triangular rank-one
+direction/count rows also matched the previously saved independent
+lifted-DFS census. This checks the new measurement convention; it is not
+another 9! enumeration or a test of the scientific null.
+
+## A random-time integral, not a midpoint proxy
+
+For a trajectory in either cohort at b, define
+
+    I=(c-b) 1{J2>tau} nu_tau/(N-tau).
+
+The audit time is uniform and independent of the permutation. Conditioning
+on a cohort g and averaging over tau, then using the exact insertion hazard,
+
+    E[I | g]
+      = sum_(k=b)^(c-1) E[1{J2>k}nu_2(A_k)/(N-k) | g]
+      = P(b<J2<=c | g).
+
+Thus `E[I|late]-E[I|early]` predicts the early-minus-late survival difference
+without assuming that the midpoint hazard persists over the whole window.
+An individual I can exceed one; it is an unbiased integral estimator, not
+a pathwise probability. The actual/predicted closure residual uses their
+**paired** covariance. Agreement checks the implemented microscopic map,
+not the claim that nu_b is recursively sufficient: only the conditional
+future-history contrasts address that latter question.
+
+The scaled midpoint difference
+`(c-b)/(N-b)*(mean(nu_b|early)-mean(nu_b|late))` is also retained, explicitly
+as an instantaneous readout, not an alternative finite-lag prediction.
+
+## Execution provenance
+
+Only DevEnvC_TV2N0X, UUID `4a8d1d443419434889e49148ed0a7ba6`, is used.
+The target was Ready, then started for this job; one authorized SSH-key
+rotation after Running succeeded and the key mode was 600. No other
+environment was started, reset or inspected for jobs.
+
+Observed container limits: aarch64, nproc=16, CPU quota 14.5 cores, memory
+25 GiB. At entry there were no other research processes. The task uses
+14 workers in the new, previously absent directory
+`/workspace/Matching-One-TV2N0X/completion-hazard-20260929`.
+After the container restart, g++ was absent; gcc-c++ 10.3.1 and its package
+dependencies were installed on this target. Python is 3.9.9; the runner
+uses only its standard library. No SSH key or credentials are transferred.
+
+The three uploaded source files match their committed SHA256 hashes.
+The 32-sample excluded benchmark took .661 seconds of engine time and
+predicted about 209 seconds under ideal 14-worker scaling. The runner
+records actual command, source/binary hashes, seeds, platform, cgroup,
+batch completion and timings. Scientific results and terminal status are
+reported only after the fixed block is complete: all 14 batches succeeded,
+with runner wall time **325.512 seconds** (individual engines 308.9--323.2 s).
+All 140,000 rows were scored once locally using the existing Python 3.11.15
+research environment. Every compressed batch matched its recorded SHA256.
+No production was extended, discarded or repeated after reading its result.
+
+The manifest SHA256 is
+`7c63af9d9c7dcf5eff4b0f14644fe83560d73190532d8528f3e0cf0c5f0af2df`;
+the result JSON SHA256 is
+`35cfa9ae64e97b4ef81e4fb339e4c37fa1f6615c8968510e32711c79c3f88abf`.
+After download and scoring, the target had no remaining research process;
+its task tunnel was stopped and power-off requested. No remote data were
+deleted. Terminal Ready status is recorded in the execution receipt.
+
+## Next scientific choice, not a new production queue
+
+The most economical next hypothesis is a **persistent-direction mixture**
+of rank-one survival kernels. In an ideal marked-Markov model each sector
+would satisfy `H_d(a,c)H_d(b,b)=H_d(a,b)H_d(b,c)`, while their sum need not.
+The present finite-lag conditional contrasts are compatible with this
+candidate; they neither prove sectorwise equality nor determine all times.
+Opposite cell effects or changing hazard differences can cancel at a
+single triple. The exact L3 square label-time counterexample remains an
+explicit warning against promoting count-clock compatibility to label time.
+
+Next theory should derive a directional exit kernel or a second specific
+prediction that could contradict that mixture, under a declared clock.
+If additional data are later justified, reserve that prediction for a new
+block; do not turn this completed experiment into an adaptive age/feature
+scan. No new larger size or extra sampling is automatically dispatched.
+The original-U forward-map question remains a separate unresolved lane.
+
+## Reproduce the analysis without new acquisition
+
+```bash
+python3 analysis/completion-hazard-production-20260929/analyze.py \
+  --data-dir analysis/completion-hazard-production-20260929/data \
+  --output-dir analysis/completion-hazard-production-20260929/results
+```
+
+The [estimand note](../analysis/completion-hazard-production-20260929/ESTIMANDS.md)
+gives the exact signs, conditional targets and covariance formula. The
+scorer requires the complete fixed block. Its source was developed in
+parallel with acquisition from the precommitted contract, before scoring
+the data; the acquisition source and contract, not this later scorer,
+were in the pre-run commit. No full-repository or repeated test suite ran.
diff --git a/notes/completion-pair-label-curvature-20260930.md b/notes/completion-pair-label-curvature-20260930.md
new file mode 100644
index 000000000..59ff1077f
--- /dev/null
+++ b/notes/completion-pair-label-curvature-20260930.md
@@ -0,0 +1,137 @@
+# Completion pairs determine label-time survival curvature
+
+2026-09-30. Finite-model author derivation, connecting the
+[completion-pair identities](completion-pair-synergy-20260929.md) to the
+[label-time kernel](completion-pivotal-hazard-kernel-20260929.md).
+No new production, fitted exponent, literature-novelty claim or continuum
+limit assertion. This is a consumer for the same geometric quantity e,
+not an additional independent experimental outcome.
+
+## 1. Conditional short-time survival
+
+At label time t<1, fix the entire revealed past with current rank-one set A.
+The m remaining site labels are independent Uniform(t,1). Let c be the number
+of individually completing vacancies and e the number of safe synergy pairs.
+During (t,t+h], each vacancy arrives independently with probability
+q=h/(1-t). For this fixed finite graph,
+
+    S_A(h) = P(T2>t+h | revealed past)
+           = 1 - c*q + [c*(c-1)/2 - e]*q^2 + O_N(q^3).       (1)
+
+Proof: survival requires that no completing singleton arrives, contributing
+(1-q)^c. Among safe sites, each completing unordered pair contributes q^2
+to failure at second order. Intersections of distinct pairs involve at least
+three sites, as do completing sets with no completing singleton or pair.
+Their contribution starts at degree three. Thus the remaining factor is
+1-e*q^2+O_N(q^3), proving (1) by multiplication. This is a finite polynomial
+expansion; its remainder bound need not be uniform as N grows.
+
+In particular,
+
+    S_A'(0)  = -c/(1-t),
+    S_A''(0) = [c*(c-1)-2e]/(1-t)^2.                         (2)
+
+An explicit hand control is one completing singleton plus two safe sites
+whose pair completes: survival is exactly (1-q)*(1-q^2). Its second-degree
+coefficient is -1, as (1) gives for c=e=1. This is a monotone finite control,
+not a claimed torus witness or a new census.
+
+For two revealed-past cohorts with the same t and exact c (and any matched
+persistent direction D), their initial hazards coincide. Their curvature
+difference is nevertheless
+
+    Delta S''(0) = -2*Delta E[e]/(1-t)^2.                    (3)
+
+Condition on fixed initial cohorts, then let them evolve; do not continually
+reselect a fixed-c cohort as time advances. A difference in (3) is a local
+history-dependent survival response despite identical initial hazard.
+
+## 2. The fixed-entry kernel, without conditioning on c
+
+Fix s<t and write I_t=1{T1<=s,T2>t}, H(s,t)=E[I_t]. No new member enters
+this cohort as t advances. On positive risk, let
+
+    mu_s(t)=E[c(A_t)|I_t=1],   eps_s(t)=E[e(A_t)|I_t=1],
+    v_s(t)=Var(c(A_t)|I_t=1).
+
+Set the killed count to zero after completion. Summing over all m potential
+insertions, each with rate 1/(1-t), gives
+
+    d/dt E[I_t*c(A_t)] = E[I_t*(2e-c^2)]/(1-t).              (4)
+
+Indeed a completing site loses c, giving -c^2 in total; safe sites add their
+synergy degrees, summing to 2e. This is the existing killed count drift with
+the actual label clock, not with k replaced by Nt. Since
+partial_t H=-E[I_t*c]/(1-t), differentiation yields
+
+    partial_t^2 H(s,t) = E[I_t*(c*(c-1)-2e)]/(1-t)^2,       (5)
+    mu_s'(t) = [2*eps_s(t)-v_s(t)]/(1-t).                   (6)
+
+The hazard lambda_s=-partial_t log H=mu_s/(1-t) therefore obeys
+
+    lambda_s'(t) = [mu_s + 2*eps_s - v_s]/(1-t)^2.          (7)
+
+The variance term is selection among surviving geometries: high-c
+configurations leave sooner. It must not be omitted. Safe completion
+creation is nonnegative at each fixed geometry, while the risk-set mean c
+can decrease because v_s can exceed 2*eps_s. Neither (6) nor (7) requires
+rank-only Markovness or a recursively closed pair graph. Fixed disjoint
+early/late entry cohorts satisfy the same formulas after their common
+starting time. Direct rank0-to-rank2 births do not enter these rank-one cohorts.
+
+There is also an exact count-clock counterpart. For a fixed rank-one cohort
+at count k, put m=N-k, mu=E[c], eps=E[e], v=Var(c). If m-mu>0, its mean
+completion count after the next insertion, conditional on surviving, obeys
+
+    mu_next - mu = (2*eps-v)/(m-mu).                          (7a)
+
+The surviving numerator is E[(m-c)*c+2e], and its normalizer is m-mu;
+subtracting mu gives (7a). The negative variance term is again selection.
+Forcing one safe insertion in every original prefix would instead average
+c+2e/(m-c) without survival reweighting. These agree on an exact-c cell,
+not generally on a mixed-c cohort. This distinction is why the current
+exact-(D,c) target is well-defined but is not a whole-cohort hazard derivative.
+
+## 3. What this changes, and what it does not
+
+This provides the next theoretical map in explicit coefficients:
+
+    completion count -> instantaneous exit;
+    pair creation minus survival selection -> hazard change;
+    their cohort contrasts -> history-dependent kernel curvature.
+
+At a deterministic near-critical rescaling t=t0+a_L*tau, a candidate limiting
+kernel must account for the scaled quantity
+
+    d lambda_scaled/d tau
+      = a_L^2*[mu_s+2*eps_s-v_s]/(1-t)^2.                   (8)
+
+Equation (8) is just a finite-chain-rule identity, with the entry cutoff held
+fixed when differentiating. Passing it to a limit needs moment control and
+justified differentiation/convergence. A small fixed-N Taylor remainder
+does not control a near-critical interval containing many microscopic arrivals.
+The present notes supply no limit of eps_s or v_s, no cross-lattice exponent,
+and no nonzero continuum memory coefficient.
+
+The current L512 replay fixes an **insertion count**, not a label time. It
+exactly scores its original two-insertion/safe-step target. It cannot be
+substituted into (5)--(8) by taking t=b/N. A label-time numerical consumer
+needs the correct count/history mixture under iid labels, or an explicitly
+matched label-time acquisition. Nor may a two-step contrast be extrapolated
+to the old 735-insertion endpoint by assuming constant curvature.
+
+The useful next theory question is thus quantitative: do competing continuum
+descriptions predict the same or different scaled pair/selection combination
+in (8), under the same observer and entry-history contract? Merely appending
+another geometric feature does not answer this question. This is attention
+guidance, not a restriction on parallel research.
+
+One immediate necessary condition is worth distinguishing. A rank-only
+Markov kernel has the same mu_s(t) for every surviving entry cutoff s;
+where differentiation is valid, (6) then requires **2*eps_s-v_s** to be
+independent of s too. It does not require eps_s and v_s separately to lose
+their history dependence: cancellation is compatible with the coarser
+observer. Conversely, matching exact c and D removes this initial variance
+term and asks about closure of the enlarged observer. A failure there is
+not by itself a proof that the unmarked rank-only law is non-Markov. This
+is the same observer distinction already exhibited by the finite L3 kernels.
diff --git a/notes/completion-pair-synergy-20260929.md b/notes/completion-pair-synergy-20260929.md
new file mode 100644
index 000000000..faea6eb27
--- /dev/null
+++ b/notes/completion-pair-synergy-20260929.md
@@ -0,0 +1,213 @@
+# Completion-pair synergy and the successor law of completion count
+
+2026-09-29. Bounded author derivation for the finite uniform-insertion model.
+This continues the [hazard identity](completion-pivotal-hazard-kernel-20260929.md)
+and [directional closure note](directional-markov-completion-20260929.md).
+It gives no literature-novelty assessment, empirical validation or scaling claim.
+The next discriminating quantity is creation of completion sites after a safe
+insertion: its distribution, not just the already fixed immediate exit hazard.
+
+## 1. The completion graph gives the exact next-state row
+
+Let V contain N sites and let `r:2^V -> {0,1,2}` be monotone, with
+`r(empty)=0`, `r(V)=2`. Fix a rank-one prefix A of size k and put `m=N-k`.
+Conditional on the revealed prefix, remaining sites arrive in uniform order.
+Write `A+v=A union {v}` and define
+
+    C(A) = {v in V\A : r(A+v)=2},       c=nu_2(A)=|C(A)|,
+    S(A) = (V\A)\C(A),                 s=|S(A)|=m-c.
+
+Define the simple undirected graph G_A on S(A) by
+
+    {v,w} in E(G_A) iff v != w and r(A+v+w)=2.
+
+Its edges are pairs that complete together although neither completes alone;
+they need not be adjacent sites of the physical lattice. Put `e=|E(G_A)|`.
+For a safe v, every old completion site remains vacant and completing by
+monotonicity. Among the other safe sites, exactly its graph neighbours become
+completion sites. Thus the stronger set identity, and its count version, are
+
+    C(A+v)=C(A) union N_G(v),           nu_2(A+v)=c+deg_G(v).       (1)
+
+Let `a_j(A)=#{v in S(A):deg_G(v)=j}`. Then `sum_j a_j=s` and
+`sum_j j*a_j=2e`. Assume D is a persistent rank-one mark: inclusion between
+rank-one sets preserves D. For ambient homology its unoriented rational line
+is persistent because homology images are nested; arbitrary marks need not be.
+Observe `X_k=0`, `(1,D(A_k),nu_2(A_k))`, or `2`, according to rank.
+For `m>0`, the complete next-state row from this fixed A is
+
+    P(X_(k+1)=2 | A)=c/m,
+    P(X_(k+1)=(1,D(A),c+j) | A)=a_j(A)/m.                        (2)
+
+If `s>0`, survival has probability s/m and the conditional completion-count
+law is `P(nu_2(A_(k+1))=c+j | A,survival)=a_j/s`. Direction is unchanged.
+If `s=0`, exit is certain and this conditional law is undefined, not zero.
+
+## 2. Two drifts and an exact two-insertion survival formula
+
+Extend nu_2 by zero after rank two, solely as a killed observable. Starting
+from A let `Z'=1{r(A_(k+1))=1}nu_2(A_(k+1))`. Summing (1) over safe sites,
+
+    E[Z' | A]=(s*c+2e)/m,
+    E[Z'-c | A]=(2e-c^2)/m.                                     (3)
+
+This killed drift includes a loss of c on each completing insertion, whose
+probability is c/m. It can be negative. In contrast, the ordinary count
+increment conditioned on remaining rank one satisfies, for `s>0`,
+
+    E[nu_2(A_(k+1))-c | A,survival]=2e/s >= 0.                   (4)
+
+For `m>=2` and `s>0`, the conditional next exit probability after that safe step is
+`(c+2e/s)/(m-1)`. Its difference from c/m is
+`c/[m(m-1)]+2e/[s(m-1)]`: removal of a vacancy and creation of completion
+sites both matter. This conditional increase is not the killed drift (3).
+
+For two insertions, survival means choosing an ordered pair of safe sites
+that is not an edge. There are `s(s-1)-2e` such pairs out of `m(m-1)`:
+
+    P(J2>k+2 | A)=[s(s-1)-2e]/[m(m-1)],                         (5)
+    P(J2=k+2 | A)=[s*c+2e]/[m(m-1)].                            (6)
+
+Here `J2=min{j:r(A_j)=2}`. Equation (6) concerns first completion at the
+second insertion; first-insertion completion has probability c/m. These
+three probabilities sum to one. Equivalently, surviving the first step
+and then using its mean completion count gives (5), with no Markov assumption.
+The undivided formulas remain valid when s=0 or s=1: two-step survival is
+zero in either case. For s=0 there is no surviving first-step cohort; for
+s=1, e=0 and its sole safe insertion makes every remaining site completing.
+When m=1 only the one-step formulas apply; a second insertion does not exist.
+
+## 3. Exact contrast between matched birth cohorts
+
+At the same count k, match the exact direction d and integer count c. For
+a fixed cutoff `a<k`, let E and L be the positive-probability cells
+
+    E={J1<=a, X_k=(1,d,c)},       L={a<J1<=k, X_k=(1,d,c)},
+    J1=min{j:r(A_j)>=1},         Delta_e=E[e(A_k)|E]-E[e(A_k)|L].
+
+The future remains uniform conditional on each microscopic prefix, so (5)
+can be averaged directly. For `m>=2`, the early-minus-late contrast is
+
+    P(J2>k+2 | E)-P(J2>k+2 | L)=-2*Delta_e/[m(m-1)].             (7)
+
+Both first-step survivals are exactly s/m. If `s>0`, this probability is
+constant across configurations in the cell, so conditioning on first-step
+survival does not reweight its distribution of A_k. Consequently
+
+    E[nu_(k+1)|E,J2>k+1]-E[nu_(k+1)|L,J2>k+1]=2*Delta_e/s,
+    P(J2>k+2|E,J2>k+1)-P(J2>k+2|L,J2>k+1)
+        =-2*Delta_e/[s(m-1)].                                  (8)
+
+Here nu_(k+1) denotes nu_2(A_(k+1)); the second line requires m>=2.
+The killed successor-mean difference is `2*Delta_e/m`, by (3).
+For s=0, e=0 in both cohorts and (7) is zero, while (8) is undefined.
+Absent cohort cells have no conditional contrast. Exact matching matters:
+mixing different c also mixes the baseline `s(s-1)` and survival weights.
+For common weights w_(d,c) summing to one at fixed k, the standardized
+contrast is exactly `-2 sum_(d,c) w_(d,c) Delta_e(d,c)/[m(m-1)]`.
+Cell effects can cancel; a zero weighted contrast need not mean zero per cell.
+
+For `s>0`, the full next-count contrast, conditional on survival, is
+`[E[a_j|E]-E[a_j|L]]/s` at value c+j. Equal mean e tests only one moment
+of this law. Equation (7) supplies an exact two-step consumer for that moment.
+
+## 4. What closure would actually require
+
+**Strong rank-one closure** means one transition kernel works for every
+microscopic preparation having the same `(k,D,c)`. By (2), this holds iff
+the entire vector `(a_j(A))_j` is identical across every such rank-one cell.
+Necessity follows by preparing each A separately; sufficiency follows by
+averaging the common row. Equal e alone gives only a common two-step survival.
+Different degree vectors refute this strong claim, even if their means agree.
+
+For a **full strong closure** including arbitrary rank-zero preparations,
+one must additionally have equal rank-zero counts of insertion sites leading
+to each next observed state, including each `(1,d,c')` and direct rank two.
+The rank-one graph criterion says nothing about those entry rows. Rank two
+is absorbing. Direct `0 -> 2` jumps have no intervening rank-one direction.
+
+For **weak closure from the uniform empty start**, let
+`F_k^X=sigma(X_0,...,X_k)` under that particular source law. The exact
+rank-one condition is, for every k and every j on positive-risk cells,
+
+    E[a_j(A_k) | F_k^X]=E[a_j(A_k) | X_k].                       (9)
+
+The fixed external k is implicit on the right. The exit row c/m is already
+fixed by X_k; (9) supplies all the other probabilities in (2). On rank zero
+the observed past is all zero, so its averaged transition row automatically
+depends only on k and current state. Uniform empty preparation makes A_k
+uniform over rank-zero sets of size k there. With absorption in rank two,
+(9) is necessary and sufficient for the full weak Markov property. In
+particular, strong rank-one rows suffice for this weak full process without
+strong rank-zero lumpability, as in the [L3 construction](birth-completion-geometry-20260929.md).
+
+The observed past now contains **the entire nu history**, as well as J1 and
+D. Unlike the direction-only observer, that past is not encoded by entry
+time alone. A population discrepancy in the matched birth-cohort successor
+law, or in (7), refutes weak Markovness because birth cohort is observed past.
+Equality for one split does not prove it; even equality of full successor
+laws for all birth cutoffs need not establish (9) after conditioning on
+different nu histories. Conversely, distinct microscopic degree vectors
+need not refute weak closure: the source-conditioned history mixtures may
+still satisfy (9). Reachability of two configurations alone is insufficient.
+
+The parent's separate [exact L4 companion](completion-pair-synergy-L4-20260929.md)
+reports matched birth-cohort discrepancies for square and triangular models.
+These establish actual weak failure from the uniform empty start, as well as
+strong failure; the finite witnesses and census evidence belong to that note.
+
+## 5. Why the present graph is not automatically a recursive state
+
+After a safe insertion v, (1) gives
+
+    S(A+v)=S(A)\({v} union N_G(v)).
+
+Edges of G_A induced on this new safe set persist by monotonicity. Additional
+edges `{w,z}` can appear if `r(A+v+w+z)=2` although `r(A+w+z)=1`.
+Thus graph incidence can matter when vertices and their neighbours are
+removed, and triple-completion information can matter for newly created
+edges. A degree histogram does not specify the former; even the whole
+current graph does not in general specify the latter under monotonicity alone.
+
+For example, three currently safe sites may have no completing pair, while
+their triple completes; after the first safe insertion the other two form
+an edge. Monotonicity also permits their triple to remain rank one with the
+same initial pair data. These are abstract local possibilities, not asserted
+lattice witnesses. They explain why even adding e or G_A is no automatic
+closure proof. The bounded question here is the next-count law (2) and its
+two-step implication (7), not an unending hierarchy of proposed descriptors.
+
+## 6. A physical prediction and the next scientific action
+
+For occupied-graph homology, a safe insertion can join components or extend
+a path without adding a second homology direction. Its graph neighbours are
+then newly completing vacancies: together with that insertion they supply
+the second direction. The increment `nu_2(A+v)-c` counts these newly enabled
+opportunities. This gives a concrete interpretation of pair synergy as
+creation of future completion sites while rank and D remain unchanged.
+
+A prospective comparison can fix the lattice, count k, birth cutoff and
+common-support weights, then retain independent naturally sampled rank-one
+prefixes matched on exact `(D,c)`. In each prefix with s>0, choose a uniformly
+random safe site, insert it in a copy, and measure `Y=nu_2(A+v)-c`.
+Conditional on that prefix, `E[Y]=2e/s`; at matched c, this samples the actual
+next-step law conditioned on survival. Within each cell, the early-minus-late
+two-step survival contrast is exactly `-s/[m(m-1)]` times its mean Y contrast.
+Forced safe insertion is a conditional continuation experiment, not the
+unconditional process; its survival factor s/m restores the latter prediction.
+
+Multiple continuations of one prefix share its geometry and birth history.
+They can reduce continuation noise but are not independent configurations;
+average within prefix and retain prefix/batch dependence in cohort uncertainty.
+The full Y distribution can address (2), while its mean has the fixed consumer
+(7). The reported L4 failure already redirects finite-model work from accepting
+`(D,nu)` closure to how safe insertions create completion sites. A prospective
+comparison would address that specific mechanism at its declared target size.
+Equality leaves recursive sufficiency open and does not justify a feature scan.
+
+The parent's L4 census is not recomputed here. Existing L512 evidence is
+reported only in its [source note](completion-hazard-independent-block-20260929.md)
+and [retrospective companion](directional-hazard-contrast-20260929.md); this
+derivation neither reanalyses nor upgrades it. All formulas here use the count
+clock; hidden-count label-time closure requires its own argument. No tests,
+enumeration, cloud work, web research, or data analysis were performed here.
diff --git a/notes/completion-pair-synergy-L4-20260929.md b/notes/completion-pair-synergy-L4-20260929.md
new file mode 100644
index 000000000..126a41d32
--- /dev/null
+++ b/notes/completion-pair-synergy-L4-20260929.md
@@ -0,0 +1,182 @@
+# Completion count does not close the L4 process: the two-site transmission
+
+2026-09-29. Exact finite calculation under the uniform-permutation **count
+clock**, on square NN and triangular (+diagonal) occupied-site tori.
+
+The later [independent L512 safe-insertion block](safe-insertion-independent-block-20260929.md)
+has now measured the proposed mean successor contrast without resolving a
+history difference. That empirical outcome does not alter the exact L4
+counterexamples below or promote them to large-size conclusions.
+
+**Result.** Current rank, persistent direction and completion count do not
+give a Markov state even from the uniform empty preparation at L4. There
+are positive-probability early/late first-birth groups with identical
+current `(k,D,nu_2)` and identical next-step exit probability, but different
+next completion-count distributions and two-step rank survival. This is
+stronger than finding two specially prepared configurations with different
+futures. It is not a large-size or continuum non-Markov theorem.
+
+The microscopic transmission is explicit: two empty sites that are each
+safe alone can jointly complete the second homology direction. Their
+pair count, rather than another fitted descriptor, gives the exact next
+term after the already known one-site completion hazard.
+
+## 1. A fixed two-step law
+
+For rank-one A with k occupied sites, put `m=N-k`, `c=nu_2(A)`.
+Let C be its completing vacancies and S its safe vacancies. On S draw an
+edge between v and w exactly when `r(A union {v,w})=2`; let e be the number
+of these unordered edges. The [general derivation](completion-pair-synergy-20260929.md)
+gives, for m>=2 in the two-step formula,
+
+    nu_2(A union {v}) = c + degree(v),  v in S,
+
+    P(rank remains 1 after two insertions | A)
+      = [(m-c)(m-c-1)-2e] / [m(m-1)].
+
+Existing completion sites cannot cease completing after a safe insertion,
+by monotonicity. New completing sites are exactly the neighbours of the
+inserted safe site in this graph. The graph degree distribution therefore
+gives the full one-step successor law of the completion count.
+
+At fixed k,D,c, averaging the last formula over early and late histories
+produces the **exact** identity
+
+    Delta(survival for two insertions)
+      = -2 Delta(mean e) / [m(m-1)].
+
+This is an actual source-to-transition map, not a fitted association. It
+does not say that adding e closes all subsequent steps. Even the whole
+current degree distribution need not specify how the graph evolves.
+
+## 2. What was computed
+
+One local run enumerated the `2^16=65,536` occupied subsets of each L4
+lattice. The existing lifted-graph rank/direction routine was reused with
+L=4; no old L3 computation was rerun. For each rank-one subset, direct
+one- and two-site extensions give C, the synergy graph and successor
+counts. Integer dynamic programming counts every possible J1 for its
+occupied prefixes; it does **not** enumerate 16! complete permutations.
+
+At a fixed k every k-site set is equally likely, and every ordering of
+its sites has the same probability. Consequently prefix-order counts are
+the exact appropriate weights for the conditional early/late comparison;
+the common remaining `(N-k)!` factor cancels. All fractions below are
+exact, not Monte Carlo estimates.
+
+| Lattice | Rank-one subsets | `(k,D,c)` cells | Cells with differing e / successor laws |
+|---|---:|---:|---:|
+| square | 19,932 | 72 | 34 / 34 |
+| triangular | 23,502 | 90 | 48 / 48 |
+
+The saved result includes all per-cell integer birth masses, edge sums,
+two-step-survival numerators and successor-count totals. Witnesses are
+selected lexicographically to demonstrate existence, not ranked by effect
+size or significance. This is an exact finite structural search, not a
+prespecified stochastic experiment.
+
+## 3. Failure under the original uniform preparation, not only strong lumping
+
+Within the following current states compare `early={J1<=a}` with
+`late={a<J1<=k}`. Both groups have positive probability.
+
+| Quantity | Square | Triangular |
+|---|---:|---:|
+| k, a, D, c | 6, 4, (0,1), 0 | 5, 4, (0,1), 0 |
+| Early / late prefix mass | 10,368 / 156,672 | 1,152 / 8,448 |
+| Early mean e | 70/27 | 4 |
+| Late mean e | 130/51 | 54/11 |
+| Both one-step rank-one survival | 1 | 1 |
+| Early mean next completion count | 14/27 | 8/11 |
+| Late mean next completion count | 26/51 | 108/121 |
+| Early two-step survival | 229/243 | 51/55 |
+| Late two-step survival | 433/459 | 551/605 |
+| Early minus late two-step survival | **−4/4131** | **2/121** |
+
+Here c=0, so the next step always survives; conditioning on its survival
+does not alter these particular successor means. Their differences are
+already a **one-step** obstruction to Markovness of the augmented
+`(rank,D,nu_2)` process. They become a **two-step** obstruction if the
+future readout is rank alone.
+
+J1 is observable in the augmented process history. If its future law were
+determined by the current augmented state, conditioning further on that
+birth event could not change the future law. Thus these are weak-Markov
+counterexamples from the uniform empty start, not merely failures of
+strong lumpability under arbitrary microscopic preparations.
+
+The square and triangular signs need not agree. These finite geometries
+are not a matched-modulus or universality experiment.
+
+## 4. A small physical witness that can be checked directly
+
+For square L4 index sites by `v=x+4y`, with x,y=0,...,3 and periodic edges.
+Two six-site preparations, shown in increasing y, are
+
+```text
+A = {0,1,4,5,8,12}    B = {0,2,4,5,8,12}
+##..                  #.#.
+##..                  ##..
+#...                  #...
+#...                  #...
+```
+
+Both have vertical direction `(0,1)`, no horizontal winding and zero
+one-site completions. For A the only safe pairs completing another
+direction are `{2,3}` and `{6,7}`. For B they are `{1,3}`, `{3,6}` and
+`{6,7}`. In B the middle pair completes the bent winding route
+
+    0 -> 4 -> 5 -> 6 -> 2 -> 3 -> 0 across the x boundary.
+
+The other pairs close the two partially occupied horizontal rows. A
+two-site route cannot close through the two other rows, which each lack
+three sites. This supplies a direct geometric check of one census
+witness, in addition to the general pair-count identity.
+
+There are 45 unordered vacancy pairs. Exactly 2 or 3 cause completion,
+giving survival `43/45` versus `42/45=14/15`. The respective next-nu counts
+are `{0:6,1:4}` and `{0:6,1:2,2:2}`, over 10 equally likely insertions.
+This particular pair proves strong-preparation failure. The distinct
+cohort calculation in section 3 is what establishes uniform-start failure.
+
+## 5. Consequence for the research queue
+
+The earlier L3 four-state completion-phase realization remains correct
+at L3. The L4 result prevents treating it as a universal closure principle.
+The [L512 conditional finite-window result](completion-hazard-independent-block-20260929.md)
+also remains unchanged: it did not resolve a same-(D,nu) residual.
+Neither a small-size exact counterexample nor a large-size unresolved
+contrast settles the strength of this channel in a scaling limit.
+
+The next useful large-size question is now **how much birth-history
+dependence enters through completion creation**, not whether identical
+current completion counts force identical immediate exit hazards (they
+do by definition). One fixed prospective target is the conditional safe-step
+increment of nu, which directly estimates `2 E[e]/(m-c)` at matched k,D,c.
+Its corresponding two-step rank effect has the displayed known factor.
+
+A single uniformly chosen safe insertion, followed by a new completion
+count, is an unbiased probe of this drift; it does not require enumerating
+all O(m^2) pairs. Independent prefixes remain the sampling units. Multiple
+continuations of one prefix share its dependency group. At large N the
+two-step rank difference can be tiny, so a meaningful acquisition should
+compare this mechanistic drift or a specified finite-window consequence,
+not blindly count rare exits or request the next full subset census.
+
+Existing L512 files save nu at b and an independent random later count,
+not a one-safe-step successor. They cannot be relabelled as this new
+measurement. No such large-size production was launched in this calculation.
+
+## 6. Reproduction
+
+```bash
+python3 analysis/completion-pair-synergy-20260929/analyze.py
+```
+
+[Result JSON](../analysis/completion-pair-synergy-20260929/result.json)
+and [readable output](../analysis/completion-pair-synergy-20260929/RESULT.md).
+The one standard-library Python 3.11 run took 0.540 s square and 0.693 s
+triangular. The new enumeration checks the graph successor formula and
+direct two-insertion count on its rank-one rows; the displayed square pair
+is a separate hand-geometric check. No full suite, 9!/16! rerun, package
+installation, sampling or cloud operation.
diff --git a/notes/completion-pivotal-hazard-kernel-20260929.md b/notes/completion-pivotal-hazard-kernel-20260929.md
new file mode 100644
index 000000000..f593df209
--- /dev/null
+++ b/notes/completion-pivotal-hazard-kernel-20260929.md
@@ -0,0 +1,351 @@
+# Completion-pivotal geometry transmits to the two-birth kernel
+
+2026-09-29. Bounded author derivation, continuing the
+[birth-kernel Markov characterization](birth-kernel-markov-characterization-20260929.md).
+This note supplies an exact finite-model link
+
+    occupied geometry -> completion-pivotal count -> exit hazard -> H.
+
+It does not certify a large-size limit, independent births, an original-U
+operator mechanism, or a recursively sufficient compressed geometry state.
+The last section quotes the [L3 geometric calculation](birth-completion-geometry-20260929.md);
+no archive, subset enumeration, simulation, or test is run here.
+
+## 1. Finite geometry and the two clocks
+
+Let V be a fixed set of N sites, and let
+`r:2^V -> {0,1,2}` be a monotone rank function, with `r(empty)=0` and
+`r(V)=2`. In the intended application this is the occupied graph's ambient
+homology rank on a fixed torus. The argument uses monotonicity and the
+specified sampling law, not a critical exponent or a topology-specific
+formula. A single insertion may increase rank directly from zero to two.
+
+For an occupied set A of rank one, define
+
+    nu_2(A) = #{v in V\A : r(A union {v})=2}.           (1.1)
+
+These are the currently empty sites whose individual insertion completes
+rank two. Set nu_2=0 outside rank one when writing indicator-weighted
+expectations; no exit interpretation is assigned to that extension.
+
+We distinguish two experiments throughout:
+
+- **Count clock.** A uniformly random site permutation gives occupied
+  prefixes A_k, |A_k|=k, and births `J_i=min{k:r(A_k)>=i}`.
+- **Label clock.** Independent Uniform[0,1] site labels give
+  `A_t={v:U_v<=t}` and `T_i=inf{t:r(A_t)>=i}`.
+
+Write
+
+    H_count(a,k)=P(J1<=a, J2>k),     a<=k,
+    H_label(s,t)=P(T1<=s, T2>t),     s<=t.
+
+We use H without its subscript only when the clock is explicit. The count
+denominator `N-k` is deterministic. In label time the number of remaining
+sites is random, and each has its own conditional hazard `1/(1-t)`.
+One does not obtain the label formulas by substituting `k=Nt`.
+
+## 2. Count-clock exit and the exact H recursion
+
+Let G_k contain the complete revealed permutation prefix, not merely its
+rank. Conditional on G_k, the next site is uniform on the N-k empty sites.
+For k<N, on `{r(A_k)=1}`,
+
+    P(J2=k+1 | G_k) = nu_2(A_k)/(N-k).                (2.1)
+
+Fix a<=k. The event `C_a(k)={J1<=a,J2>k}` is G_k-measurable and lies
+inside rank one. It loses mass at the next step exactly when a completion
+site is chosen. Therefore
+
+    H_count(a,k+1) = H_count(a,k)
+      - E[1{J1<=a,J2>k} nu_2(A_k)]/(N-k).            (2.2)
+
+This remains an undivided identity when the cohort has zero probability.
+For positive cohort risk, put
+
+    m_a(k)=E[nu_2(A_k) | J1<=a,J2>k].
+
+Its one-step conditional survival is
+
+    H_count(a,k+1)/H_count(a,k) = 1-m_a(k)/(N-k).      (2.3)
+
+Thus two entry cohorts at the same k can have different exit probabilities
+only through their different conditional averages of the current
+completion count. The full geometry determines that count pointwise;
+the entry history changes the distribution of geometries in the risk set.
+
+For disjoint early and late groups `J1<=a` and `a<J1<=b`, both surviving
+to b, their **next-step** survival contrast is exactly
+
+    survival_early - survival_late
+      = [E(nu_2 | late,J2>b)-E(nu_2 | early,J2>b)]/(N-b).   (2.4)
+
+This is a predictive risk-set comparison, not an intervention on birth
+time or geometry. Equality of these cohort averages can occur even when
+nu_2 varies substantially across individual rank-one configurations.
+
+When every relevant survival factor is positive, the finite-clock
+multiplicative defect also has the exact sum representation
+
+    log[H(a,c)H(b,b)/(H(a,b)H(b,c))]
+      = sum_(k=b)^(c-1)
+          log[(1-m_a(k)/(N-k))/(1-m_b(k)/(N-k))].     (2.5)
+
+If a factor is zero, use (2.2) and the undivided determinant rather than
+assigning a finite logarithm. No interpolation of (2.5) is being assumed.
+
+## 3. Uniform-label completion intensity and the kernel derivative
+
+Let G_t contain the occupied history and the labels already revealed by t,
+but no future labels. Conditional on G_t, all empty sites have independent
+Uniform(t,1) residual labels. Each empty site therefore arrives in
+`(t,t+h]` with probability `h/(1-t)` and has instantaneous hazard
+`1/(1-t)`. The independence here concerns the remaining site clocks,
+**not** the two topological births.
+
+For a rank-one configuration, completion in the next short interval occurs
+to first order exactly when one of its nu_2 completion sites arrives.
+Any discrepancy involves at least two arrivals and is
+`O_N(h^2/(1-t)^2)`. Consequently, conditional on G_t and rank one,
+
+    P(T2<=t+h | G_t)
+      = nu_2(A_t) h/(1-t) + O_N(h^2/(1-t)^2).        (3.1)
+
+Fix `s<t<1`. The old cohort `C_s(t)={T1<=s,T2>t}` is already in rank
+one at t; it cannot gain new members as t increases. Conditioning (3.1)
+and taking expectations gives
+
+    partial_t H_label(s,t)
+      = -E[1{T1<=s,T2>t} nu_2(A_t)]/(1-t).           (3.2)
+
+This derivative holds throughout the interior s<t<1, not just formally:
+for a finite model the two-time configuration probabilities are finite
+sums of terms in s, t-s and 1-t, so H is polynomial on that triangle.
+Alternatively the uniform short-interval bound gives local absolute
+continuity and the same derivative. A diagonal component `T1=T2` in the
+joint birth law does not invalidate this off-diagonal argument.
+
+For positive H(s,t), the cohort's logarithmic completion hazard is
+
+    -partial_t log H_label(s,t)
+      = E[nu_2(A_t) | T1<=s,T2>t]/(1-t).             (3.3)
+
+Equivalently this is the hazard of the conditional survival
+`H(s,t)/H(s,s)`, when H(s,s)>0. The cutoff s is held fixed.
+
+### Do not differentiate the moving diagonal as a fixed cohort
+
+Along `H(t,t)=P1(t)`, new rank-one entrants arrive while old ones complete.
+Let gamma_01(t), gamma_12(t), gamma_02(t) be the unconditional probability
+fluxes for the indicated rank transitions. The formula above gives
+
+    gamma_12(t)=E[1{r(A_t)=1}nu_2(A_t)]/(1-t),
+    P1'(t)=gamma_01(t)-gamma_12(t),
+    F_2'(t)=gamma_12(t)+gamma_02(t).                  (3.4)
+
+The other fluxes similarly count the individual empty sites causing their
+specified transition, multiplied by `1/(1-t)`. Thus neither
+`-d log P1(t)/dt` nor, when direct jumps occur, `F_2'(t)/P1(t)` is the
+rank-one completion hazard. The latter coincides with the mean rank-one
+exit intensity only when the direct `0 -> 2` flux is absent.
+
+Continuous independent site labels are distinct almost surely, but a
+single site can still cause `T1=T2`. Such direct double jumps never enter
+the rank-one cohorts in (2.2) or (3.2); they must not be silently included
+in their completion count.
+
+## 4. The Markov defect is an integrated cohort hazard difference
+
+Take `0<=a<b<c<1` with all four H values below positive, and define
+
+    L(a,b,c)=log[H(a,c)H(b,b)/(H(a,b)H(b,c))],
+    m_s(t)=E[nu_2(A_t) | T1<=s,T2>t].
+
+Both conditional survival ratios start at one at t=b. Integrating (3.3)
+therefore proves the exact identity
+
+    L(a,b,c) = -integral_b^c [m_a(t)-m_b(t)]/(1-t) dt.   (4.1)
+
+In particular, higher early-cohort completion counts throughout a time
+interval lower its relative survival. A single zero defect can also
+result from cancellation of hazard differences over that interval; it
+does not establish history independence at all intermediate times.
+
+**The second cohort in (4.1) is birth-by-b, not all rank-one sites at t.**
+Its event is `T1<=b,T2>t`, with b fixed. It excludes paths that first enter
+rank one between b and t. Replacing it by `{T1<=t<T2}` changes the
+comparison and breaks the identity.
+
+To relate it to the early/late contrast, let
+
+    alpha(t)=H(a,t)/H(b,t),
+    m_late(t)=E[nu_2(A_t) | a<T1<=b,T2>t].
+
+When the late cohort also has positive risk,
+
+    m_b(t)=alpha(t)m_a(t)+(1-alpha(t))m_late(t),
+    m_a(t)-m_b(t)=(1-alpha(t))[m_a(t)-m_late(t)].      (4.2)
+
+The weights are survivor weights at t, not the original weights at b.
+This identifies exactly how different geometric mixtures are transmitted
+into the observable two-time kernel.
+
+The earlier Markov theorem states that the undivided H identity must hold
+for **all** deterministic triples to give ordinary rank-only Markovness.
+Equation (4.1) is its finite-geometry mechanism interface, not a replacement
+by one fitted hazard or one triple. In label time, positive cohort risk at
+b remains positive for every c<1: the conditional event of no further site
+arrivals has probability at least `((1-c)/(1-b))^N`. Thus interior logs do
+not hide an exhausted positive cohort. At a zero initial risk or at the
+endpoint 1 use the undivided equations or justified one-sided limits.
+
+## 5. Persistent direction marks and symmetry
+
+Let D(A) be a mark on rank-one configurations, taking values in a finite
+set of sectors. Assume it stays unchanged along every occupied-set
+inclusion path that remains rank one. This is a real hypothesis: a generic
+geometric descriptor need not be persistent. For an ambient homology rank,
+the unoriented one-dimensional homology image over Q is one natural
+persistent direction, because those images are nested under inclusion.
+An arbitrary chosen winding representative or other geometric statistic
+must not be assumed constant for this reason.
+
+Define the **unconditional sector mass**
+
+    H_d(s,t)=P(T1<=s,T2>t,D(A_t)=d).
+
+Its birth-by-s cohort remains in sector d until completion, so exactly the
+same argument gives
+
+    partial_t H_d(s,t)
+      = -E[1{T1<=s,T2>t,D(A_t)=d}nu_2(A_t)]/(1-t),  (5.1)
+
+and, for the count clock,
+
+    H_d(a,k+1)=H_d(a,k)
+      -E[1{J1<=a,J2>k,D(A_k)=d}nu_2(A_k)]/(N-k).   (5.2)
+
+Conditional sector logarithms and the defect integral follow by replacing
+H and m with H_d and the corresponding within-sector conditional means.
+If D can change while rank remains one, inter-sector transfer terms are
+needed; (5.1)--(5.2) cannot then be used without modification.
+
+Suppose a group acts on the sites and directions, preserves r and the
+sampling law of complete labelled/permuted histories, and satisfies
+`D(gA)=gD(A)`. The same bijection gives `nu_2(gA)=nu_2(A)`. If the action
+is transitive on m admissible directions, then, for either clock,
+
+    H_d(s,t)=H(s,t)/m,
+    E[1{cohort,D=d}nu_2]=E[1{cohort}nu_2]/m.         (5.3)
+
+Consequently each sector has the same conditional completion mean as
+the pooled cohort, the same early/late survival contrast, and the same
+logarithmic Markov defect. The sector's **unnormalized** determinant is
+the pooled determinant divided by m^2; its risk-normalized conditional
+covariance is unchanged. Under these exact symmetry hypotheses, splitting
+by pure direction cannot remove the entry-history defect.
+
+When there are several symmetry orbits, equality is guaranteed only within
+an orbit. Anisotropic sources or a geometry without the required symmetry
+need not satisfy (5.3). Also H_d is not a probability conditional on ever
+visiting sector d: that conditional kernel would divide by the visitation
+probability. Direct double-jump paths visit no rank-one sector, so that
+normalization must not count them as though they had a direction.
+
+In particular, unmarked Markovness does **not** imply Markovness after
+revealing a persistent direction. The pooled kernel is `H=sum_d H_d`,
+but its determinant is not the sum of sector determinants: cross-sector
+terms matter. Without transitivity on the whole direction set, averaging
+can hide entry-history dependence that remains within a revealed sector.
+Conversely, adding the mark supplies more information, not less; it is
+the Markov sufficiency property, rather than prediction quality, that need
+not be preserved by this change of observer.
+
+## 6. What nu_2 does and does not summarize
+
+At a fixed occupied set it is exactly sufficient to determine the
+**immediate completion hazard** under these clocks. It is not thereby a
+recursively sufficient state for future predictions. Two configurations
+with the same nu_2 can have different counts of non-completing successor
+configurations with each possible future nu_2, or different mark transitions.
+Those successor laws matter for longer survival and are absent from (1.1).
+A claimed `(k,nu_2,D)` Markov compression would require a separate closure
+argument for these transition distributions.
+
+If nu_2 is a function of k alone on every accessible rank-one configuration
+in the count model, then rank-one exit is indeed independent of all past
+history. Together with the all-zero past in state zero and absorption of
+state two, this suffices for count-clock rank-only Markovness. The converse
+does not require pointwise constancy: suitable cohort averages can agree
+even when microscopic counts differ. In label time, even a function of
+the hidden count |A_t| need not reduce to a function of t and current rank.
+
+These identities therefore provide a genuine predictive transmission law,
+but not a causal intervention, a proof that the births are independent,
+or an explanation of the original normalized U observable with its own
+source, normalization and moving-root contract. No large-L behaviour or
+scaling exponent follows from them alone.
+
+Uniformity of the sampling law is also substantive. A biased permutation
+requires its actual conditional probabilities of choosing each empty
+site. For independent nonuniform site-label laws with hazards h_v(t),
+the instantaneous completion intensity is the sum of h_v(t) over the
+completion sites, not nu_2/(1-t). Correlated or history-controlled sources
+require the corresponding conditional intensities and may invalidate the
+remaining-site independence used here.
+
+## 7. Supplied L3 illustration: geometric mixtures, not an age mechanism
+
+The parent reports the following exact results from a new **512-subset
+geometric dynamic programme** for N=9. The designated companion is
+[birth-completion-geometry-20260929.md](birth-completion-geometry-20260929.md),
+where the parent supplies the finite-mechanism derivation, exact code and
+count-clock realization. These numbers are quoted, not recomputed or
+independently certified in this note.
+
+- On the reported square rank-one layers `k=3,4,5,6`, every configuration
+  has `nu_2=k-3`. Thus entry cohorts on each such layer have identical
+  instantaneous count-clock exit probabilities. The general all-risk-layer
+  sufficiency statement is the one in section 6.
+- Triangular rank-one configurations have `nu_2 in {2,3}` at k=4 and
+  `nu_2 in {3,4}` at k=5. Mere variation does not by itself establish a
+  rank-only memory defect; the cohort mixture is the decisive quantity.
+- The parent finds both previous triangular Markov-defect triples still
+  present separately in each of the three reported direction sectors
+  `(1,0),(0,1),(1,1)`. This is consistent with the symmetry mechanism in
+  section 5; pure direction is not the missing predictive coordinate there.
+
+For the triangular comparison at b=5, early entry `J1<=4` has reported
+mean `E nu_2=13/4`, whereas late entry `J1=5` has mean `24/7`, conditional
+on survival to b in each case. The weight of the faster `nu_2=4` class
+is respectively `1/4` and `3/7`; the other class has nu_2=3. With four
+empty sites left, (2.4) directly gives
+
+    survival_early = 1-(13/4)/4 = 3/16,
+    survival_late  = 1-(24/7)/4 = 1/7,
+    survival_early-survival_late = 5/112.
+
+This is the already reported one-step contrast explained through a
+specified microscopic exit mechanism: the later-entry risk set contains
+more configurations with four completion sites. It is not evidence that
+changing birth age itself causes an exit change. It does not establish
+that nu_2 alone closes subsequent dynamics; the parent's separate finite
+state construction addresses that additional question.
+
+The parent's subsequent exact marked-kernel calculation gives a particularly
+useful **clock/observer boundary**. For square L3, all direction sectors
+are Markov in insertion time. In iid-label time the unmarked rank kernel
+is still Markov, but each axial sector has a strictly negative determinant
+for `0<a<b<c<1`; each diagonal sector remains Markov. Axial and diagonal
+directions are different symmetry orbits, so (5.3) never licensed dividing
+the unmarked square kernel equally among all directions. For triangular
+L3 the three marked kernels instead satisfy `H_d=H/3`, consistently
+retaining the history defect under pure direction conditioning.
+
+The companion note is the source of these closed forms and their exact
+certificates; they are not rederived here. Their separation/coefficient
+matrix ranks are not automatically dimensions of a positive hidden-state
+model. In particular, neither count-clock sector Markovness nor unmarked
+label-clock Markovness can substitute for checking the actual marked
+label-clock contract. None of these finite L3 classifications is a
+large-L conclusion.
diff --git a/notes/deterministic-block-encoding-obstruction-20260929.md b/notes/deterministic-block-encoding-obstruction-20260929.md
new file mode 100644
index 000000000..965864181
--- /dev/null
+++ b/notes/deterministic-block-encoding-obstruction-20260929.md
@@ -0,0 +1,341 @@
+# Deterministic four-state redundancy in the full tensor-query algebra
+
+2026-09-29. Consumers: #809/#833 and the
+[#837 copying counterexample](https://github.com/LightChainr/Matching-One/pull/837#issuecomment-5814562555).
+This note proves a bounded algebraic classification. It does not claim a new
+general no-copying principle, a percolation theorem, or a literature novelty result.
+
+**Result.** Encode four different logical states deterministically into any
+number of the existing four-corner blocks. If all products of the local
+responses `(1,u,v)` are admitted, response rank can remain below four **only
+when exactly one physical block carries all four states and every other block
+is constant**. Every genuinely distributed injective encoding has rank four.
+This rules out much more than direct repetition. Section 5 now supplies an
+explicit two-row physical rank query for every tensor monomial in the original
+#836 five-column-block preparation. Thus the rank-four obstruction transfers
+to that physical source/observer contract, not to every possible contract.
+
+## 1. Contract and theorem
+
+Work over the real numbers, equivalently over the rationals for these matrices.
+Let `C={(0,0),(1,0),(0,1),(1,1)}`, and let `S` be four labelled logical states.
+For `k>=1` a deterministic encoder is
+
+    e:S -> C^k,    e(s)=(e_1(s),...,e_k(s)).
+
+The local feature row is `v(u,v)=(1,u,v)`; the overloaded letter `v` is only
+notation for the row and its second bit. Define the four-row response matrix
+
+    R_e(s,alpha) = product_i f_(alpha_i)(e_i(s)),
+    alpha in {0,1,2}^k,    (f_0,f_1,f_2)=(1,u,v).
+
+Thus each block contributes at most one of its two bit functions to a column.
+The full linear span of these `3^k` columns is admitted. No within-block `uv`
+column is assumed: obtaining its logical equivalent from distinct blocks is
+precisely what can destroy the old relation. Row rank means exact ordinary
+linear response rank, not nonnegative rank, recursive state count, mutual
+information, or finite-error predictive complexity.
+
+**Theorem (all k).** If `e` is injective, then
+
+    rank R_e = 3
+        iff there exists j such that e_j:S -> C is bijective
+            and e_i is constant for every i != j;
+    rank R_e = 4 otherwise.
+
+In particular, rank below four cannot be one or two for an injective encoder.
+
+**Noninjective classification.** Put `m=|e(S)|`. If `m<=3`, then
+`rank R_e=m`: repeated logical labels only repeat rows and do not create a
+new compression of the distinct encoded preparations. If `m=4`, the theorem
+applies. These statements classify every map from four labelled states.
+
+## 2. Proof
+
+### 2.1 At most three used corners give all local indicators
+
+The rows `(1,u,v)` on any set of at most three distinct corners are linearly
+independent. For three corners this is the nonzero determinant of a triangle
+in the unit square (each determinant is `+1` or `-1`); smaller cases follow.
+Consequently, on any local used set `A_i=e_i(S)` of size at most three, every
+function on `A_i`, in particular each singleton indicator, is a linear
+combination of `1,u,v`.
+
+Suppose every `|A_i|<=3`. For each distinct codeword `c in e(S)`, the function
+
+    product_i 1{e_i(s)=c_i}
+
+is the indicator of the logical states assigned that entire codeword.
+Expanding the product puts it in the admitted tensor span. The `m` resulting
+indicators are independent, while only `m` distinct rows exist. Hence rank is
+exactly `m`. This proves the noninjective assertion and the injective case in
+which no block uses all four corners.
+
+### 2.2 A four-corner block fixes the only possible relation
+
+Now suppose block `j` uses all four corners. Since `S` has exactly four states,
+it is a bijection. Order its rows as `00,10,01,11`. The three columns
+`1,u_j,v_j` have rank three and their left kernel is generated by
+
+    lambda=(1,-1,-1,1).
+
+Every entry of `lambda` is nonzero. If the full matrix has rank below four,
+its left kernel is this same one-dimensional space.
+
+Take another block `i!=j`, and either coordinate function `g=u_i` or `g=v_i`,
+viewed as a four-vector indexed by `S`. The full tensor language contains
+
+    g,    u_j*g,    v_j*g.
+
+Annihilation of these three columns says that `lambda*g` (entrywise product)
+lies in the left kernel of `(1,u_j,v_j)`. Thus `lambda*g=c*lambda` for a
+scalar `c`. Since all entries of `lambda` are nonzero, `g` must be constant.
+Both coordinates of every other block are therefore constant.
+
+Conversely, if every other block is constant, its local tensor row has first
+coordinate one and is nonzero. Tensoring by these fixed rows preserves the
+rank-three row space of the one bijective block. This proves both directions.
+
+The proof also identifies a short witness when another block varies: at
+least one of `g,u_j*g,v_j*g` fails to annihilate `lambda` and, with the three
+columns from block `j`, yields a nonsingular four-column minor.
+
+## 3. Consequences that are actually within scope
+
+1. **No deterministic spatial redundancy with the old rank-three advantage.**
+   A rank-three injective code has block Hamming distance exactly one between
+   every pair of distinct codewords. It has only one informative block.
+   Any injective code with two or more nonconstant blocks has rank four,
+   even if no block is a literal copy of another. Constants are not redundant
+   carriers of the logical information.
+2. **Split and parity codes are included.** Splitting `(u,v)` into `(u,0)`
+   and `(v,0)` admits the cross-block product `uv`. Appending `(u xor v,0)`
+   to `(u,v)` already adds `u+v-2uv`. Both give rank four without direct copying.
+3. **Independent logical groups.** If each of `t` four-state logical groups
+   is encoded into its own disjoint block set and the query language tensors
+   across groups, the total response rank is the product of their individual
+   ranks. Distributed encoding of each group therefore gives `4^t`, not
+   `3^t`. This is not a theorem about arbitrary joint encoders of `4^t`
+   logical states; the four-state proof does not classify those codes.
+4. **Invertible input noise, conditionally.** For a complete finite binary
+   instruction cube, independent instruction flips transform the response
+   matrix by `Q_epsilon=Q_0 W_epsilon^(tensor n)`, where
+   `W_epsilon=[[1-epsilon,epsilon],[epsilon,1-epsilon]]`. At
+   `epsilon!=1/2` this is invertible, so it preserves exact rank. This needs
+   the complete instruction cube, not merely a chosen subset of queries;
+   it does not preserve numerical conditioning or finite-error advantage.
+
+## 4. Exact k=2 census, including repeated codewords
+
+The accompanying [standard-library program](../analysis/deterministic-block-encoding-20260929/check.py)
+enumerates all `16^4=65,536` assignments of four labelled logical states to
+two physical blocks. Each `4 x 9` matrix is ranked by integer Gaussian
+elimination over the rationals, not by floating point or a modular surrogate.
+The theorem's structural classification is compared with every computed rank.
+The [result](../analysis/deterministic-block-encoding-20260929/result.json)
+also records local-image cardinalities and representative non-copying codes.
+
+The independent combinatorial counts are:
+
+| Distinct codewords | Rank | Labelled assignments |
+|---|---:|---:|
+| 1 | 1 | 16 |
+| 2 | 2 | 1,680 |
+| 3 | 3 | 20,160 |
+| 4, one informative block | 3 | 192 |
+| 4, distributed | 4 | 43,488 |
+
+There are `16*15*14*13=43,680` injective encoders. The 192 exceptions are
+`2` choices of informative block, `4` choices of the other constant corner,
+and `4!` assignments of the logical labels. Forgetting label order leaves
+eight deficient codebooks among `binom(16,4)=1,820` total codebooks. More
+generally, the theorem gives `24*k*4^(k-1)` deficient labelled injections
+into `k>=1` blocks, without enumerating them.
+
+Only this one complete k=2 census is run. Its finite output is a control on
+the all-k proof, not the reason to extrapolate it. No old geometry test suite,
+new percolation sampling, cloud job, or larger-width machine is run.
+
+## 5. Physical span: explicit two-row product queries
+
+The relevant original assets are branch results, not silently integrated
+main facts:
+
+- PR #836, commit `19b6771437276ce0522d21dd1a5cbf1b7c7c5c1c`,
+  [`analysis/p833-noise-depth-literature-20260924/NOTE.md`](https://github.com/LightChainr/Matching-One/blob/19b6771437276ce0522d21dd1a5cbf1b7c7c5c1c/analysis/p833-noise-depth-literature-20260924/NOTE.md),
+  section 1 defines the three-row, five-column-block preparation and the
+  temporary-closure rank observer. Its
+  [`check.py`](https://github.com/LightChainr/Matching-One/blob/19b6771437276ce0522d21dd1a5cbf1b7c7c5c1c/analysis/p833-noise-depth-literature-20260924/check.py)
+  lines 24–36 implement preparation and **single-bit** two-row queries.
+  For block origin `a=5g`, these remove columns `(a,a+2)` and then `a+2`
+  to read `u`, or `(a+1,a+3)` and then `a+1` to read `v`.
+- PR #837, commit `0ed7e4f7a6e68ba628ce26dd1e9cf1e65d64c9f3`,
+  [`analysis/p833-fixed-noise-wedge-20260924/NOTE.md`](https://github.com/LightChainr/Matching-One/blob/0ed7e4f7a6e68ba628ce26dd1e9cf1e65d64c9f3/analysis/p833-fixed-noise-wedge-20260924/NOTE.md),
+  sections 1 and 7 retain this preparation and assert the inherited tensor
+  response rank. The note's original four-point rank-only target is superseded
+  by its separate `INTERFACE-REFINEMENT.md`; this sidecar does not weaken the
+  current five-point lower bound.
+- The latest [#837 comment](https://github.com/LightChainr/Matching-One/pull/837#issuecomment-5814562555)
+  reports the actual constant, first-block `u`, second-block `v`, and joint
+  `uv` queries for a direct-copy preparation. It also asserts general
+  monomial queries and describes finite physical BFS checks. Those checks
+  are the comment author's reported work, not a run performed here.
+
+The earlier `input/tensor_check.py` referenced by the optional comparison mode
+is not present in the fetched trees. A dimension assertion alone would not
+identify the physical column space. The following direct construction closes
+that interface without relying on that missing asset, multiplying outcomes
+from separate preparations, or rerunning a geometry census.
+
+### 5.1 Masks and exact observer
+
+Take `k>=1`, `w>=5k`, with columns modulo `w` and block origins `a_g=5g`.
+The old three rows, indexed `y=0,1,2`, are exactly
+
+    row 0 = 0,
+    row 1 = sum_g (15 << a_g),
+    row 2 = sum_g ((9+2u_g+4v_g) << a_g).
+
+All unused old columns are empty. Choose disjoint sets of blocks `U,V`:
+read `u_g` for `g in U`, `v_g` for `g in V`, and no bit otherwise. The
+disjointness is important: at most one bit is selected in any physical block.
+With `F=(1<<w)-1`, append deterministic rows `m,n` at `y=3,4`:
+
+    D_m = union_(g in U) {a_g,a_g+2}
+          union_(g in V) {a_g+1,a_g+3},
+    D_n = union_(g in U) {a_g+2}
+          union_(g in V) {a_g+1},
+    m = F - sum_(x in D_m) (1<<x),
+    n = F - sum_(x in D_n) (1<<x).
+
+This is one experiment, independent of the unknown bit values. Observe only
+the final occupied-site nearest-neighbor ambient rank on the temporarily
+closed `w x 5` torus. No intermediate temporary-closure edges are retained.
+Write this rank as `R(U,V)`. Then, for every old preparation,
+
+    R(U,V) = product_(g in U) u_g * product_(g in V) v_g.       (P)
+
+The empty product is one. In every case `R<=1`: the entirely empty old row 0
+prevents any occupied loop from having nonzero vertical winding.
+
+### 5.2 A zero selected bit gives a white vertical separator
+
+First take a selected `u_g=0`, with block origin `a`. In the universal cover
+of the `w x 5` torus, the following sites form a white matching (8-neighbor)
+path, using `(column,row)` coordinates:
+
+    (a-1,0), (a-1,1), (a-1,2), (a,3),
+    (a+1,2), (a+2,3), (a+2,4), (a+2,5).
+
+The first three sites lie in the block's empty old separator column;
+`(a+1,2)` is empty because `u_g=0`; the three new-row sites are deleted by
+the selected-u masks; row 5 is row 0 in the next vertical period. Continue
+left along this empty row to `(a-1,5)`. The projected closed white curve has
+homology `(0,1)`. Additional selected blocks delete more new sites and cannot
+damage this witness. Extra old padding is empty, so the first block and
+`w>5k` cause no exception.
+
+For a selected `v_g=0`, the reflected witness starts
+
+    (a+4,0), (a+4,1), (a+4,2), (a+3,3),
+    (a+2,2), (a+1,3), (a+1,4), (a+1,5),
+
+then continues right along row 5 to `(a+4,5)`. Again its winding is `(0,1)`.
+
+These explicit zigzags are simple after projection. Embed their diagonal
+steps inside square faces. They cannot intersect occupied NN edges: a
+diagonal meets the face boundary only at its white endpoints, and the
+remaining path segments have white endpoints as well. Any occupied cycle
+with nonzero horizontal winding would have nonzero intersection number
+with the vertical white curve. It therefore cannot exist. Vertical occupied
+winding was already excluded by row 0, so `R=0` whenever one selected bit is
+zero. Equivalently, this is the original single-bit separator, retained when
+other queries remove additional occupied sites on the same final torus.
+
+### 5.3 All selected bits one give a black horizontal circuit
+
+Start with the horizontal circuit of row `y=4`. Its only missing sites are
+the selected holes in `D_n`. Replace each interrupted two-edge segment by
+the following NN detour. For a selected `u_g=1`, bypass the hole `(a+2,4)`:
+
+    (a+1,4), (a+1,3), (a+1,2), (a+1,1),
+    (a+2,1), (a+3,1), (a+3,2), (a+3,3), (a+3,4).
+
+For a selected `v_g=1`, bypass the hole `(a+1,4)`:
+
+    (a,4), (a,3), (a,2), (a,1),
+    (a+1,1), (a+2,1), (a+2,2), (a+2,3), (a+2,4).
+
+Every listed site is occupied: the one optional old site is precisely the
+selected bit, row 1 is full inside its block, the other old endpoint is
+fixed occupied, and neither new endpoint is deleted by its block's masks.
+Different blocks occupy disjoint four-column intervals separated by a
+fifth column; their detours and deletions cannot interfere. Stitch the
+detours together with the surviving row-4 edges, including the periodic
+seam. The resulting occupied circuit advances exactly `w` columns and
+returns to the same lifted row, hence has winding `(1,0)`. Thus `R>=1`;
+the empty row gives `R<=1`, proving (P). With no selected blocks row 4 is
+full and supplies the constant-one query directly.
+
+### 5.4 Transfer to encoded preparations and noisy instruction sources
+
+Equation (P) holds simultaneously on all `4^k` old block preparations.
+Restricting its rows to an arbitrary deterministic encoding `e(S)` therefore
+gives **exactly** the matrix `R_e` for this concrete product-query family.
+The full tensor column span is physically available; no independent copies
+or multiplication of separately measured outcomes are used. In particular,
+every injective encoding with at least two nonconstant blocks has response
+rank four already under these two-row final-rank queries. Any larger allowed
+query family containing them also has rank four, since there are only four
+preparation rows. For the exceptional one-informative-block encoding, this
+particular query family has rank three; this argument alone is not an upper
+bound for arbitrary additional observers or new block geometries.
+
+At fixed noise `0<epsilon<1/2`, a single noisy version of a selected mask
+does **not** return the literal product (P). Nevertheless, if the source
+contract admits all `2^(2w)` binary two-row commands, realized sitewise as
+probabilities `epsilon` or `1-epsilon`, then
+
+    Q_epsilon = Q_0 W_epsilon^(tensor 2w)
+
+for the complete final-rank response matrices. The independent flip matrix
+is invertible, so the noiseless tensor columns remain in the linear span
+of noisy command-response columns. Hence the rank-four obstruction also
+holds for this complete noisy instruction contract. This is a statement
+about exact rank, not a stable inversion algorithm, sample complexity,
+optimal error probability, or the rank-only five-point error floor.
+
+If only some commands are admitted, or the geometry/preparation/observer is
+changed, this invertible-cube argument need not apply. A restricted-source
+code can still have a rank deficiency. This is the remaining **contract
+boundary**, not an unresolved product-query construction for the original
+five-column preparation. This section is a directed proof, with no new
+enumeration, simulation, or rerun of the earlier census.
+
+## 6. How this changes the next question
+
+Within the full tensor-query model, do not spend another search on a
+deterministic four-state block code with multiple informative blocks while
+requiring rank three: the classification answers that search for every k.
+It does not ban work or rule out reliable coding; rank four can be an
+acceptable cost if reliability is the objective.
+
+The remaining ways to pursue redundancy **and** a rank advantage change an
+explicit assumption: stochastic encoders (mixtures of physical codewords),
+different local block geometry/response relation, restricted query sources,
+multiple independently prepared experiments with their cost counted, or
+larger joint logical alphabets with a fresh rank analysis. These are not
+established successes. The original five-column physical-span interface is
+now supplied by section 5, so it need not be dispatched as another missing
+proof or turned into a broad full-machine re-enumeration.
+
+Reproduce the sole census from the repository root:
+
+```sh
+python3 analysis/deterministic-block-encoding-20260929/check.py
+```
+
+The JSON is printed to standard output. The saved result records the
+algebraic census scope explicitly; its physical-verification flag remains
+false because that program does not check section 5's geometric proof.
+No external proof review or novelty certification is claimed.
diff --git a/notes/directional-hazard-contrast-20260929.md b/notes/directional-hazard-contrast-20260929.md
new file mode 100644
index 000000000..4c1e3f5ea
--- /dev/null
+++ b/notes/directional-hazard-contrast-20260929.md
@@ -0,0 +1,137 @@
+# Direction-only closure meets a sharper next-insertion prediction
+
+2026-09-29. Same completed 140k square-L512 block; **one retrospective
+necessity check**, not a new experiment. The
+[original prospective readout](completion-hazard-independent-block-20260929.md)
+and every original result, cutoff and contract remain unchanged.
+
+**Independent follow-up:** the [new safe-insertion block](safe-insertion-independent-block-20260929.md)
+predeclared and repeated the same-D completion-count contrast, obtaining
+`-0.03987 +/- .56988`, not reproducing the `+1.62917 +/- .39098` below.
+The original post-hoc result is preserved, but must not be read as a stable
+empirical exclusion. L512 direction-only closure remains unresolved.
+
+**Outcome.** Within the same primitive direction, early histories have
+`1.6292 +/- .3910` more completion sites than late histories, using the
+existing exact-D overlap weights and aligned batch SE. Direction-only
+Markovness requires that population contrast to vanish. This is a concrete
+challenge to that candidate, despite its unresolved longer-lag survival
+contrast. It is not an independently validated rejection or a continuum
+non-Markov theorem.
+
+## 1. The necessary prediction comes from the sampling law
+
+At fixed insertion count b, current rank one, and persistent direction d,
+the uniform-source identity gives
+
+    P(J2=b+1 | entry cohort g, D=d, J2>b)
+      = E[nu_2(A_b) | g,D=d,J2>b] / (N-b).
+
+If current `(rank,D)` is a time-inhomogeneous Markov state in its observed
+history, the right side cannot depend on the first-birth cohort g. Thus
+
+    E[nu_b | early,d] = E[nu_b | late,d]
+
+is an exact necessary condition. It is more informative than simply
+requesting another fitted transition curve. Averaging zero differences
+with positive common-support weights must still give zero.
+
+The [companion marked-model note](directional-markov-completion-20260929.md)
+supplies the full-cutoff criterion, the flow-matching Markov surrogate,
+and an exact example showing why a zero long-lag contrast cannot replace
+this next-step requirement. That surrogate preserves single-time marked
+laws and adjacent transition tables but need not preserve true history.
+
+No claim here depends on choosing a new direction subset, cutoff, size,
+binning scheme, regression feature or fitted exponent. The question was
+selected **after** the original block had been read, including its
+direction-resolved means, so its uncertainty must not be presented as a
+prespecified independent confirmation.
+
+## 2. Compute one new contrast, keep the original comparison fixed
+
+Use the same early/late cohorts, all exact primitive unoriented directions,
+and weights `w_d=e_d*l_d/(e_d+l_d)` as the original exact-D survival
+contrast. Replace only the outcome by the already saved nu_b. Recompute
+those weights after deleting each whole batch. All 140,000 records remain
+in the same source block; rank-one cohort sizes are 24,218 and 30,505.
+Shared directions retain all early histories and 30,490 late histories.
+
+| Quantity, early minus late | Estimate | Batch SE |
+|---|---:|---:|
+| Pooled mean completion count | -2.18473 sites | .47950 sites |
+| Within-D weighted mean completion count | +1.62917 sites | .39098 sites |
+| Within-D **next-insertion exit probability** | +0.0000152602 | .0000036622 |
+| Same exit contrast multiplied by c-b=735 | +0.0112163 | .0026917 |
+| Original within-D survival through c | -0.0041500 | .0045119 |
+
+The fourth row is a scaled intensity contrast, **not** a 735-insertion
+exit probability. It is included solely to put the microscopic one-step
+quantity on the existing window's scale. The old survival estimate and
+SE are reproduced by the same weights, without changing its definition.
+
+The direction-adjusted completion contrast is about 4.17 batch SE from
+zero. Fourteen-batch uncertainty is approximate, this comparison is
+retrospective, and no multiplicity-adjusted certification is claimed.
+Its positive sign means greater immediate early-cohort exit propensity.
+The longer-lag survival estimate has the consistent negative sign but
+substantially less separation from zero. The data do **not** demonstrate
+temporal cancellation; lower precision or changing hazard differences
+can both leave the finite-lag comparison unresolved.
+
+The pooled completion difference has the opposite sign. This is consistent
+with the already observed direction-composition difference between cohorts.
+It does not license a causal or mediated fraction: the pooled and
+within-direction comparisons target differently weighted populations.
+The narrow statement is that direction composition and within-direction
+completion geometry need not point in the same direction.
+
+## 3. What to retain, revise, and ask next
+
+- Retain the original result: the prespecified same-D and same-(D,nu_b)
+  finite-lag survival contrasts were unresolved. Do not rewrite them as
+  significant, or discard the experiment because a primary outcome is weak.
+- Revise the working model: a persistent-direction **Markov** mixture is
+  now a challenged candidate, not a preferred closure justified by one
+  unresolved window. Direction remains scientifically useful; only its
+  purported sufficiency is under question.
+- Keep `(D,nu_b)` predictive sufficiency unresolved. Its immediate exit
+  probability is fixed by construction, so equality of nu_b inside an
+  exact-(D,nu_b) cell is a tautology, not a successful test of recursive
+  dynamics. Its longer-lag original comparison remains the relevant datum.
+
+The next discrimination should concern **evolution of completion geometry**,
+not another list of static descriptors. A concrete future experiment could
+compare conditional continuations from matched `(D,nu_b)` preparations,
+with first-birth history as the contrasting label, and estimate a fixed
+future survival or completion-count transition. Multiple continuations of
+one prefix remain one prefix-level dependency group; they are not extra
+independent configurations. Define that prediction and its cost before any
+new acquisition. No such continuation campaign is started by this note.
+
+An independent confirmation of the present same-D immediate-hazard
+contrast would likewise have one fixed direction weighting, the existing
+cutoffs, and a declared sample budget, not a search for a more favourable
+time or sign. The present analysis alone does not determine a scaling
+limit, an original-U operator, or a minimum hidden-state dimension.
+
+## 4. Reproduction and evidence scope
+
+```bash
+python3 analysis/directional-hazard-contrast-20260929/analyze.py
+```
+
+The new small scorer reuses the unchanged production reader and batch
+sufficient-statistic code. It reads each compressed batch once, verifies
+the original hashes, and saves the three-dimensional joint covariance and
+all aligned delete-one vectors in
+[result.json](../analysis/directional-hazard-contrast-20260929/result.json).
+[RESULT.md](../analysis/directional-hazard-contrast-20260929/RESULT.md)
+provides the numeric view. Both input manifest and reader hashes are saved.
+
+The source is the `87f6644d` block, not a second random block. One local
+standard-library Python 3.11 analysis took under one second; no packages,
+cloud access, simulation, source regeneration, cutoff scan or full test
+suite was needed. A narrow arithmetic identity confirms that the retained
+old within-D survival point/SE are unchanged; this is not an independent
+validation of the entire acquisition pipeline.
diff --git a/notes/directional-markov-completion-20260929.md b/notes/directional-markov-completion-20260929.md
new file mode 100644
index 000000000..5e47c42cf
--- /dev/null
+++ b/notes/directional-markov-completion-20260929.md
@@ -0,0 +1,325 @@
+# Directional Markov completion and its exact count-clock criterion
+
+2026-09-29. Bounded author derivation for the existing finite-site model.
+This is a concrete model interface for the
+[completion-pivotal hazard identity](completion-pivotal-hazard-kernel-20260929.md)
+and the [two-birth Markov criterion](birth-kernel-markov-characterization-20260929.md),
+not a novelty claim or an analysis of production data. The designated empirical
+companion is [directional-hazard-contrast-20260929.md](directional-hazard-contrast-20260929.md).
+No new source, observable, cutoff scan, simulation, or test is introduced here.
+
+**Main statements.** Actual directional one-step probability flows always
+define a time-inhomogeneous Markov completion preserving every marked
+single-time law and every unconditional adjacent-time joint table. The actual
+marked process has that Markov law if and only if, at every count and within
+every direction, the risk-set mean completion count is independent of every
+first-entry cutoff. A zero survival contrast over just one window does not
+give this condition.
+
+## 1. Observer, probability law, and persistence
+
+Let V have N sites, and let `r:2^V -> {0,1,2}` be monotone, with
+`r(empty)=0` and `r(V)=2`. A uniform random permutation gives the occupied
+prefix A_k at count k, for `k=0,...,N`. Set
+
+    J_i = min{k:r(A_k)>=i},     i=1,2.
+
+Direct `0 -> 2` jumps, hence `J1=J2`, are allowed. On rank-one sets let
+D(A) take values in a finite set of directions, and assume persistence:
+
+    A subset B, r(A)=r(B)=1  =>  D(A)=D(B).
+
+The primitive unoriented ambient homology direction has this property when
+the rank and direction are defined by the nested homology images. A general
+geometric descriptor need not have it.
+
+The complete *current marked state* under consideration is
+
+    X_k = 0                  if r(A_k)=0,
+          (1,D(A_k))         if r(A_k)=1,
+          2                 if r(A_k)=2.
+
+Its natural observation filtration is `F_k^X=sigma(X_0,...,X_k)`. It does
+not additionally reveal the occupied set, the insertion order, or nu_2.
+The complete microscopic prefix filtration is denoted G_k. Write `d` as
+shorthand for the state `(1,d)` when indexing probabilities. Once a path
+leaves d it is absorbed in 2; different rank-one directions cannot succeed
+one another on the same path. No direction is assigned to a direct double
+jump. The current state 2 does not retain the old mark, although the
+observation history can retain it; this is harmless because its future is
+constant.
+
+For rank-one A define the existing completion count
+
+    nu_2(A) = #{v in V\A:r(A union {v})=2}.
+
+Uniform permutation sampling gives, on `{X_k=d}` and for `k<N`,
+
+    P(X_(k+1)=2 | G_k) = nu_2(A_k)/(N-k).             (1.1)
+
+All claims below concern this preparation and deterministic **count** times.
+They are not strong lumpability statements for arbitrary microscopic initial
+preparations.
+
+## 2. The flow-defined directional Markov completion
+
+Let `p_x(k)=P(X_k=x)`. Define the actual, unconditional one-step flows
+
+    alpha_d(k) = P(X_k=0, X_(k+1)=d),
+    beta_d(k)  = P(X_k=d, X_(k+1)=2),
+    gamma(k)   = P(X_k=0, X_(k+1)=2).
+
+In particular,
+
+    beta_d(k) = E[1{X_k=d} nu_2(A_k)]/(N-k).          (2.1)
+
+The entry and direct-jump flows also have microscopic counting forms:
+count empty sites taking a rank-zero A respectively to `(1,d)` or to 2,
+multiply by `1{r(A_k)=0}`, take expectations, and divide by N-k.
+There is no assumed independence between entry and completion.
+
+Persistence and monotonicity leave exactly the following nonzero entries
+of the actual adjacent joint table `F_k(x,y)=P(X_k=x,X_(k+1)=y)`:
+
+    F_k(0,0) = p_0(k) - sum_d alpha_d(k) - gamma(k),
+    F_k(0,d) = alpha_d(k),      F_k(0,2) = gamma(k),
+    F_k(d,d) = p_d(k) - beta_d(k),
+    F_k(d,2) = beta_d(k),       F_k(2,2) = p_2(k).    (2.2)
+
+These are actual probability masses, so they are nonnegative; their row
+sums are p_x(k) and their column sums are p_y(k+1). On a positive-risk row
+define
+
+    Q_k(x,y) = F_k(x,y)/p_x(k).                       (2.3)
+
+On zero-risk rows choose, for example, a self-loop; set state 2 absorbing
+in all cases. Start the new time-inhomogeneous Markov chain Y at `Y_0=0`
+and use Q_k for its step k to k+1.
+
+**Exact preservation and uniqueness.** Induction gives
+
+    P(Y_k=x)=p_x(k),
+    P(Y_k=x,Y_(k+1)=y)=p_x(k)Q_k(x,y)=F_k(x,y).
+
+Indeed the induction step is the column-sum identity for (2.2). If
+p_x(k)=0 the whole corresponding flow row is zero, so its arbitrary
+transition convention contributes no mass. Any Markov chain with the
+same starting law and these adjacent joint tables must use (2.3) on all
+positive-risk rows. Thus its path law from this preparation is unique;
+off-risk rows are not uniquely specified.
+
+For example, the completion's directional birth kernel is
+
+    H_d^M(a,k) = p_d(a) product_(j=a)^(k-1)
+                            [1-beta_d(j)/p_d(j)],    (2.4)
+
+when the displayed risks are positive. In all cases it is defined by
+the chain itself: it is zero if p_d(a)=0 or if the surviving cohort is
+exhausted. A factor at an unreachable zero-risk row is not evaluated as
+0/0. The event in this kernel is `Y_a=d,Y_k=d`; because d is persistent,
+it is precisely entry by a followed by survival in d to k.
+
+This completion is an abstract marked-state process. It is not asserted
+to be another physical uniform-site model with the same geometry.
+Matching even all adjacent tables does **not** show that X and Y have
+the same nonadjacent or full-path law. Static marked occupancies alone
+do not even supply the separate entry, exit, and direct-jump flows above.
+
+## 3. Exact characterization by completion-count cohort means
+
+For `0<=a<=k<N` define the unconditional sector kernel and its cohort mean
+
+    H_d(a,k) = P(J1<=a, X_k=d),
+    m_d(k)   = E[nu_2(A_k) | X_k=d],
+    m_(d,a)(k) = E[nu_2(A_k) | J1<=a, X_k=d].
+
+Means are defined only on positive risks; `H_d(k,k)=p_d(k)`.
+
+**Theorem.** The actual process X is ordinary time-inhomogeneous Markov
+in its natural marked filtration if and only if, for every count `k<N`,
+every direction d with `p_d(k)>0`, and every cutoff `a<=k` with
+`H_d(a,k)>0`,
+
+    m_(d,a)(k) = m_d(k).                             (C)
+
+Every count is required, including future counts beyond a selected
+measurement window. The condition is on conditional **means**, not on
+pointwise constancy of nu_2, its entire conditional distribution, or the
+whole microscopic geometry. Zero-risk cohorts impose no conditional test.
+
+### Necessity
+
+By the tower property applied to (1.1),
+
+    P(X_(k+1)=2 | J1<=a,X_k=d) = m_(d,a)(k)/(N-k),
+    P(X_(k+1)=2 | X_k=d)      = m_d(k)/(N-k).         (3.1)
+
+The first-entry cutoff is observable from `F_k^X`. If X is Markov, its
+one-step law conditioned on this additional past event cannot change,
+which proves (C). The denominator N-k is common and strictly positive.
+
+### Sufficiency
+
+On `{X_k=d}`, the entire observed path has the form
+
+    0,...,0, d,...,d,
+
+with its sole entry at J1. Thus the trace of `F_k^X` on this risk set is
+generated by the finite collection of events `{J1=j}`, or equivalently
+the entry-CDF events `{J1<=a}`. Condition (C) is equivalently the
+undivided relation
+
+    E[1{J1<=a,X_k=d}nu_2(A_k)]
+      = H_d(a,k)m_d(k).                              (3.2)
+
+It also holds when H_d(a,k)=0, since then both sides vanish. Subtract
+(3.2) at a=j and a=j-1. Every positive-mass entry atom consequently has
+mean m_d(k). It follows that
+
+    E[nu_2(A_k) | F_k^X] = m_d(k)  on {X_k=d}.
+
+Equation (1.1), another application of the tower property, and persistence
+now give the complete one-step observed law: exit to 2 has probability
+`m_d(k)/(N-k)`, otherwise the next state is d. Both depend only on k,d.
+
+On `{X_k=0}` the observed past is all zero, so its trace sigma-algebra
+is trivial. Its one-step law is already the row Q_k(0,.), without any
+extra criterion on the hidden geometry. On `{X_k=2}` the future is
+identically 2. Therefore the one-step conditional law everywhere is
+`Q_k(X_k,.)`. Repeated conditioning on successive deterministic counts
+gives the usual multi-step Markov property and the path law of section 2.
+This proves sufficiency, including direct double jumps and null risks.
+
+### Kernel form and its all-count requirement
+
+The sector hazard recursion from the companion identity is
+
+    H_d(a,k+1)=H_d(a,k)[1-m_(d,a)(k)/(N-k)].          (3.3)
+
+At positive risks, the adjacent triple determinant is exactly
+
+    H_d(a,k+1)p_d(k)-H_d(a,k)H_d(k,k+1)
+      = -H_d(a,k)p_d(k)[m_(d,a)(k)-m_d(k)]/(N-k).   (3.4)
+
+Its undivided expectation form remains valid at zero cohort risk.
+Thus checking (C) at all counts and entry cutoffs is equivalently
+checking all these adjacent triple identities. Iteration then gives,
+for **all** `a<=b<=c`,
+
+    H_d(a,c)p_d(b)=H_d(a,b)H_d(b,c).                 (3.5)
+
+The theorem is also the directional version of the earlier all-triples
+criterion. It is not a proposal that one selected triple, one aggregated
+directional contrast, or one finite-sample null result proves closure.
+
+## 4. The immediate necessary condition versus a finite-window zero
+
+Fix `a<b<N` and direction d. Within `X_b=d` let
+
+    E = {J1<=a,X_b=d},       L = {a<J1<=b,X_b=d},
+
+with both masses positive. Write m_E(b),m_L(b) for their mean nu_2.
+Condition (C) implies their equality: m_d(b) is their probability-weighted
+average, and the early cutoff mean must equal this average. Conversely
+equality for this split supplies only this cutoff's necessary condition.
+Exactly,
+
+    P(J2>b+1 | E)-P(J2>b+1 | L)
+      = [m_L(b)-m_E(b)]/(N-b).                      (4.1)
+
+A nonzero population within-direction completion-count contrast therefore
+rules out the current marked-state Markov model for this finite count
+contract. No extrapolation to another size, clock, or scaling limit is
+included. A sample estimate must retain its uncertainty; a zero estimate
+is not an all-count identity.
+
+For a longer window, retain the same initial E or L cohort and condition
+on its survival to each j. Where those survivor risks are positive, let
+
+    h_G(j)=E[nu_2(A_j) | G,J2>j]/(N-j),   G=E,L.
+
+Then, without assuming Markovness,
+
+    P(J2>c | G)=product_(j=b)^(c-1)[1-h_G(j)].        (4.2)
+
+If a cohort is exhausted its survival is zero, with no conditional mean
+assigned afterwards. The products can agree even though their first
+factors, or several intermediate factors, differ. These are survivors of
+the fixed birth-by-b groups, not new rank-one entrants after b. Averaging
+across directions can introduce further cancellation; equality of an
+overlap-weighted contrast is weaker than equality in every direction.
+
+The parent's designated saved-geometry comparison concerns only (4.1)
+at its already chosen cutoffs. It is a **post-hoc** use of the existing
+140k block, not new independent evidence, a new coordinate/cutoff scan,
+or a test of every condition in the theorem. This note neither reads that
+block nor anticipates its result; interpretation belongs in the
+[directional hazard companion](directional-hazard-contrast-20260929.md).
+
+## 5. Exact nonphysical toy: window cancellation
+
+Consider an abstract two-birth process with one persistent direction and
+the following probability masses; all unlisted pairs have zero mass:
+
+| (J1,J2) | Probability |
+|---|---:|
+| (1,3) | 1/8 |
+| (1,4) | 1/8 |
+| (1,5) | 1/4 |
+| (2,3) | 1/4 |
+| (2,5) | 1/4 |
+
+This is hand algebra for a probability toy, **not a percolation model**
+or a claimed realization of a site-completion count. At b=2 all paths
+are rank one. Early `J1=1` and late `J1=2` each have probability 1/2.
+
+At c=3 their survivals are respectively
+
+    (1/8+1/4)/(1/2)=3/4,       (1/4)/(1/2)=1/2.
+
+At c=4 they are both `(1/4)/(1/2)=1/2`. The step 2 to 3 completion
+probabilities are 1/4 early and 1/2 late. Among survivors at count 3,
+the next completion probabilities are 1/3 early and zero late. Thus
+
+    (3/4)(2/3)=1/2=(1/2)(1),
+
+which explicitly exhibits cancellation of the first-step difference.
+
+Here `H(1,2)=1/2`, `H(2,2)=1`. At c=3,
+`H(1,3)=3/8`, `H(2,3)=5/8`, so the determinant is
+`3/8-(1/2)(5/8)=1/16`. At c=4, `H(1,4)=1/4` and
+`H(2,4)=1/2`, so it is zero. The process is not Markov despite the
+second triple's exact zero.
+
+Its flow-defined Markov completion has exit probabilities 3/8 at count
+2 and 1/5 at count 3. Both entry groups in that completion consequently
+have survival 5/8 to count 3 and 1/2 to count 4. It matches the toy's
+single-time laws and adjacent joint tables but not its entry-conditioned
+survival to count 3. This is the precise distinction between constructing
+a compatible Markov model and identifying the actual path law as Markov.
+
+## 6. Clock and interpretation boundaries
+
+- **Count versus label.** All transitions and the criterion (C) above
+  use the uniform-permutation count clock. For iid Uniform site labels,
+  the instantaneous completion intensity is `nu_2(A_t)/(1-t)`, not
+  `nu_2/(N-k)`. The label observer hides the random occupied count;
+  count-clock closure does not automatically survive this random time
+  change. It requires its own sector kernels or intensity analysis.
+- **Observer.** A persistent direction makes the rank-one past exactly
+  `(J1,d)`. If the mark can change, its previous transitions remain in
+  the history and the entry-cutoff proof is no longer sufficient. Neither
+  revealing nor forgetting directions generally preserves Markovness.
+- **Immediate versus recursive.** Pointwise nu_2 determines immediate
+  completion hazard, not the successor distribution of nu_2. No Markov
+  claim for `(D,nu_2)` follows here. Conversely, marked-state Markovness
+  requires equality of the cohort means, not identical microscopic
+  configurations or identical nu_2 distributions.
+- **Predictive, not causal.** Selecting rank-one survivors and early/late
+  entries is observational conditioning, not an intervention on birth or
+  geometry. These formulas assert neither independent births nor an
+  explanation of the original normalized U observable.
+- **What is proved.** The completion and iff criterion are finite-model
+  probability deductions. They contain no large-L or continuum claim,
+  no certification of the empirical companion's outcome, and no general
+  novelty assessment.
diff --git a/notes/exact-completion-pairs-readout-20260930.md b/notes/exact-completion-pairs-readout-20260930.md
new file mode 100644
index 000000000..1149e8771
--- /dev/null
+++ b/notes/exact-completion-pairs-readout-20260930.md
@@ -0,0 +1,134 @@
+# Exact averaging resolves a large-size completion-creation history contrast
+
+2026-09-30. Same 140,000 square-L512 prefixes as the
+[safe-insertion block](safe-insertion-independent-block-20260929.md), not a
+new independent sample. The target, entry cutoff, exact (D,c) cells and
+overlap weights are unchanged. The estimator was improved after the first
+readout; this timing and the original result are retained.
+
+**Outcome:** early births create fewer new completion sites than late births
+at the same current direction and completion count. Exact conditional
+averaging gives **−0.00344939 +/- 0.00077225** (aligned 14-batch SE), versus
+the original noisy **+0.00668183 +/- 0.01208706**. The uncertainty shrinks
+by a factor **15.65**, revealing a finite-size history signal about 4.47 SE
+from zero. This is substantive large-size evidence against recursive
+closure of the enlarged (rank,D,c) observer, not a proof of its continuum
+behaviour or a second independent confirmation.
+
+## 1. What changed computationally
+
+For each current occupied set A, the new
+[projected-component criterion](projected-component-pair-criterion-20260929.md)
+enumerates the exact unordered safe synergy pairs. Shared occupied
+components impose integer transverse-potential constraints; disagreements
+between them, or with a direct physical edge, detect exactly the pairs
+that complete together. The fixed-degree expected cost is O(N+e), where e
+is the number of output pairs, not an all-vacancy-pair scan.
+
+With m=N-b, c=nu_b and e pairs, the mean increment of a uniform safe
+insertion is exactly 2e/(m-c). Replacing four random probes with this
+conditional expectation removes probe randomness. The original prefix
+sample, all cell weights and finite-prefix uncertainty remain. This is
+conditional averaging of the same target, not oversampling high-contact
+sites or changing the preparation source.
+
+All 140,000 tuples `(J1,rank_b,dx_b,dy_b,nu_b)` matched the original data
+row by row. Integer c/e values are saved. Reported cohort estimates and
+covariances are floating-point statistical calculations, not exact
+rational population probabilities.
+
+## 2. Actual complete-block result
+
+Early means J1<=154646; late means 154646<J1<=155385; rank_b=1,
+L=512, N=262144. Exact-cell weights are n_E*n_L/(n_E+n_L).
+
+| Early minus late, unless otherwise marked | Estimate | Batch SE |
+|---|---:|---:|
+| Exact conditional mean safe increment | −0.00344939 | 0.00077225 |
+| Original four-probe increment | +0.00668183 | 0.01208706 |
+| Paired exact-minus-probe estimator difference | −0.01013121 | 0.01239566 |
+| Exact mean synergy-pair count contrast | −183.96398 | 41.20529 |
+| Derived two-next-insertion survival contrast | +3.22818e−8 | 7.23066e−9 |
+| Exact weighted early increment mean | 0.09557456 | 0.00045022 |
+| Exact weighted late increment mean | 0.09902395 | 0.00052572 |
+| Unchanged secondary: same-D current c contrast | −0.03986512 | 0.56987795 |
+
+The mean increment difference is about 3.48% of the weighted late mean;
+this is a relative response, not an explained or mediated fraction.
+The original and refined point estimates differ by only about 0.82 SE
+of their paired difference: the old positive point was poorly resolved,
+not an established opposite physical effect.
+
+The original primary and secondary points and SEs are reproduced by the
+aligned data, without rerunning their original scoring pipeline. This is
+an identity check, not independent scientific replication. Every deletion
+removes the same entire batch from both measurements and recomputes support
+and weights. The result retains the full 8x8 covariance and 14 deletion
+vectors, including the exact dependence among the derived quantities.
+
+There are 55,028 rank-one prefixes. The same 1,190 common (D,c) cells retain
+54,463 of them, 98.973%; no binning or replacement definition was introduced.
+The empirical SE ratio is 0.063891 and variance ratio 0.004082. Those numbers
+describe this estimator comparison; they are not an increase in independent
+sample count or a claim that every future jackknife must improve equally.
+
+## 3. Scientific interpretation and next distinction
+
+At fixed c the immediate exit hazard is identical by definition. The
+negative edge-count contrast gives a positive two-insertion survival
+contrast by the exact transfer −2*Delta(mean e)/[m(m-1)]. Thus this signal
+has a concrete future-rank consumer; it is not just a geometric feature
+correlated with age. A nonzero population contrast would violate a
+necessary condition for weak Markov closure of (rank,D,c), because the
+birth cohort is part of that observer's past. This finite-sample result
+supports such a violation at L512 under the stated target.
+
+It does **not** revive the older direction-only claim: its independent
+secondary remains −0.040 +/- .570 and did not reproduce +1.629 +/- .391.
+Adding c changes the observer. The enlarged observer can expose memory
+even when a coarser observed process is Markov; these two statements are
+not contradictory. Nor does one nonzero mean identify the full successor
+law, the minimal state, original U, or a continuum operator.
+
+Next attention goes to two concrete questions, not another feature scan:
+
+- Fix this same negative-sign completion-creation prediction and score it
+  once on an independent prefix block with the exact evaluator. Prefer a
+  correctly replayable independent archive before collecting new samples.
+  If it is another previously inspected archive, say so; do not call it
+  prospective new acquisition. Do not reuse the present block as the second
+  confirmation or pool it to manufacture a stronger number.
+- Derive a continuous-kernel prediction for creation versus survivor
+  selection. The [new label-time identity](completion-pair-label-curvature-20260930.md)
+  gives hazard'=[mean c+2 mean e−Var(c)]/(1-t)^2. Fixed-count data cannot
+  be substituted by t=b/N, and a two-insertion effect cannot be extended
+  to a 735-insertion window by assuming constant curvature. Scaling and
+  survival-selection cancellation are the actual next mechanism questions.
+
+The finite L4 counterexamples remain exact; this is separate larger-size
+evidence, not an asymptotic extension of their proof. These are priorities,
+not locks on parallel research.
+
+## 4. Execution and reproducibility
+
+Eight completed batches were retained locally. After the owner's request
+to prioritize Huawei for medium/large work, local incomplete workers were
+stopped and the remaining six batches ran on TV2N0X, six workers, in
+420.381 seconds including compilation and a 16-old-prefix cost check.
+Actual available budget was 14.5 CPU / 25 GiB; GCC10.3.1, Python3.9.9,
+ARM64. Other machines were not operated. The prior user pause and abandoned
+partial runs are not included in successful-batch runtimes and do not add
+samples. Their completed checkpoints were preserved.
+
+The engine/geometry code was unchanged between local and cloud evaluation.
+Replay source: `8b5eee0d47b6c220c2ba46a5bae08d2d19b392a4`; cloud handoff:
+`0684c3b4eee0a647e643540ecfcdcd2918a70977`. Four small physical pair-set
+controls and a 128-old-prefix identity/cost check ran once before full
+replay. The scorer's small two-cell arithmetic control and one complete
+production scoring pass were performed; no full suite or old census rerun.
+
+[Run and raw rows](../analysis/exact-completion-pairs-20260929/data/run.json) ·
+[Results and covariance](../analysis/exact-completion-pairs-20260929/results/result.json) ·
+[Compact table](../analysis/exact-completion-pairs-20260929/results/RESULT.md) ·
+[Method and reproduction](../analysis/exact-completion-pairs-20260929/README.md).
+The old four-probe report is preserved unchanged.
diff --git a/notes/geometric-source-window-identity-20260930.md b/notes/geometric-source-window-identity-20260930.md
new file mode 100644
index 000000000..373ae9fbe
--- /dev/null
+++ b/notes/geometric-source-window-identity-20260930.md
@@ -0,0 +1,70 @@
+# Finite-window geometric susceptibility from the original uniform source
+
+2026-09-30. New finite response identity; the consumer of the
+[clock-preserving geometric intervention](birth-selection-intervention-20260930.md).
+This changes the mechanism question, not the precision target of the old
+birth-history contrast.
+
+## Identity
+
+Fix a rank-one prefix A at count b. Let m=N-b, c its completing vacancies,
+s=m-c, and d_v the synergy degree of a safe vacancy. Set mean_d=2e/s.
+Apply the previously specified source kick once, then return to the uniform
+source. Let S_h(theta|A) be survival through h insertions, h>=1.
+
+At theta=0 the first-site likelihood score is d_v-mean_d for safe v and
+zero for completing v. Given the unordered set B of the next h sites, all
+orders are equiprobable. Survival depends only on A union B. If it survives,
+every member of B was safe at A; otherwise its contribution is zero.
+Consequently
+
+    S_h'(0|A) = E_uniform [1{rank(A union B)=1}
+                          * (sum_{v in B} d_v/h - 2e/s)].       (1)
+
+This is an exact likelihood derivative, with an exact conditional average
+over the h possible first sites. It does not approximate a small nonzero
+theta by finite differences or hold geometry fixed during continuation.
+Degrees are those at the starting prefix, not degrees recomputed later.
+For s=0 survival and response are zero. Assume 1<=h<=m.
+
+Proof at the subset level: the perturbed probability of B relative to its
+uniform value is (m/h)*sum_{v in B} P_theta(first=v|A). Differentiate and
+multiply by the endpoint survival indicator. The endpoint only sees the
+set, so internal-order averaging leaves the expectation unchanged. For
+h=1 the safe scores sum to zero. For h=2 it reproduces
+
+    S_2'(0|A) = -s Var_safe(d)/[m(m-1)],
+
+but h=735 uses its actual continuation, not that expression extrapolated.
+No assumption about the sign for arbitrary h is made.
+
+## Concrete consumer and decision
+
+The input is the existing square-L512 safe-insertion block: 14x10000 full
+Fisher-Yates permutations, b=155385. Replaying the same seeds also recovers
+their unused continuation. Use **h=735**, the unchanged destination156120
+from the original completion-hazard contract, with no lag or subgroup scan.
+
+Primary: average (1) over the original rank-one risk population at b.
+Accompanying quantities are its ordinary endpoint survival and the exact
+two-step susceptibility. Save their complete aligned 14-batch covariance.
+The relevant question is whether a kick invisible to the occupation clock
+and immediate exit can influence this specified much longer rank window.
+
+- A resolved response supplies a finite-size transmission beyond local
+  curvature, for this explicit source and observer.
+- A small/unresolved response despite a nonzero local susceptibility leaves
+  rapid loss, cancellation and insufficient precision as distinct possibilities;
+  it is not an invitation to search a more significant lag.
+- Neither outcome identifies original U, continuum scaling or the fraction
+  of passive early/late association mediated by this intervention.
+
+These prefixes have already informed the project; the new readout is not
+independent evidence for the old history effect. The other worker's
+independent-archive replication is a separate question and remains separate.
+
+[Contract](../analysis/geometric-source-window-20260930/contract.json) and
+[engine](../analysis/geometric-source-window-20260930/engine.cpp) are fixed
+before this response is computed. The old topology/pair algorithms are
+reused; the only change to their source is an entry-point guard for inclusion.
+Medium-size replay goes to Huawei, not this Mac.
diff --git a/notes/geometric-source-window-readout-20260930.md b/notes/geometric-source-window-readout-20260930.md
new file mode 100644
index 000000000..c188fd312
--- /dev/null
+++ b/notes/geometric-source-window-readout-20260930.md
@@ -0,0 +1,99 @@
+# L512: an invisible immediate source has a resolved735-step rank response
+
+2026-09-30. The [fixed finite-window contract](geometric-source-window-identity-20260930.md)
+has now been evaluated, not left as another tool or production proposal.
+
+**Result.** A single clock-preserving geometric kick has a clearly resolved
+response after the original735-insertion window. It does not disappear
+between local completion creation and this endpoint. The source preserves
+the current exit probability, so this is an actual delayed geometric
+transmission, not a change of the occupation clock.
+
+| Conditional on original rank_b=1 | Estimate | Aligned14-batch SE |
+|---|---:|---:|
+| d/dtheta survival at b+735, theta=0 | **-0.01104203191** | **0.00015766645** |
+| Baseline survival at b+735 | 0.4447553973 | 0.0021409587 |
+| Exact two-step source susceptibility | -0.000059071990 | 0.000000469939 |
+
+The735-step derivative is about70 batch SE from zero. These are derivatives
+per unit theta in exp(theta*degree), not finite-theta probability changes.
+All three coordinates are correlated readouts of the same original block;
+the full3x3 covariance and14 deletion vectors are retained. The two-step
+quantity is a geometric identity, not a second independent experiment.
+
+## What was actually evaluated
+
+- Square NN occupied-site torus, L512, N262144; unchanged b155385 and
+  endpoint156120. Exactly one735-step window, no lag/angle/size/feature scan.
+- The original safe-insertion block's140000 complete Fisher-Yates
+  permutations were replayed. There are55028 rank-one prefixes. All140000
+  first-five-field tuples match the earlier raw prefixes.
+- At each rank-one prefix A, enumerate its exact synergy graph once,
+  then follow the **original uniform continuation** through735 arrivals.
+  The estimator is the exact conditional score identity
+
+      1{rank_{b+735}=1} [sum_{v in next735} degree_A(v)/735 - 2e_A/s_A].
+
+  It averages over which of that unordered arrival set came first. It
+  does not recompute initial degrees later, simulate a finite-theta model,
+  or extrapolate the two-step curvature.
+- Conditional risk denominators are recomputed on each whole-batch
+  deletion. No early/late subgroup was added to this new contract, and no
+  precision or sample extension was triggered by the result.
+
+This is a new mechanism response on **previously analysed prefixes**, not
+independent evidence for their passive birth-history contrast. The separate
+[independent archive reproduction](completion-creation-independent-replication-20260930.md)
+is already recorded and is not pooled with this calculation.
+
+## What now follows, and what still does not
+
+1. **The finite-window channel exists at this large size.** The selected
+   micro-source is invisible to immediate rank and to the occupation clock,
+   but its effect reaches future rank after735 insertions. A pure clock
+   explanation of this intervention is insufficient.
+2. **This is a second-birth actuator.** Since the kick acts only after
+   rank one is reached, the first-birth distribution is unchanged. Thus at
+   the fixed count endpoint, unconditional deltaM=deltaE_top=deltaP2, with
+   conditional survival response multiplied by minus the original rank-one
+   risk probability. It is not an observed change of the first-birth source.
+3. **The result does not yet attribute the natural memory signal.** A strong
+   response to an intervention does not determine how much of the passive
+   early/late association arises at birth, during safe transport, or through
+   survivor selection. That separation is the next mechanism calculation,
+   using the already derived safe-path/weight identity, not more precision
+   on this response.
+4. **No label-time or original-U substitution.** A count endpoint is not a
+   fixed p. The [one-sided source map](one-sided-birth-source-normal-response-20260930.md)
+   proves what survives moving-root projection and supplies an exact L4
+   symmetric positive control; a numerical L512 label-time map still needs
+   its proper count mixture. No continuum exponent, universal amplitude,
+   physical H4 attribution or original norm-4/U identification is claimed.
+
+In particular, this closes “does the supplied source transmit at all beyond
+two steps?” for the fixed L512735-step experiment. Do not automatically
+repeat it at higher precision or start a size ladder. The useful next
+question is source/selection attribution or the appropriate endpoint
+mixture for a specified consumer.
+
+## Execution and artifacts
+
+TgFr7R (`83750ac4eae34bbbbc64b894b854dd8c`) executed14workers in519.145seconds,
+with ARM64/Python3.9.9/GCC10.3.1, live14.5CPU and25GiB limits. A32-old-prefix
+cost/correspondence check carried zero additional scientific weight.
+The compiler was installed in the task's own installroot because the
+restarted base image lacked g++; other workspace directories were untouched.
+
+One support worker reviewed the exact score/source correspondence and one
+specified L4,h3 rational control; the main investigator did not repeat
+that verification. After transfer, one standard-library score in the local
+research Python3.11 environment computed the result and all-row alignment.
+No full suite, old census, fresh stochastic block or numerical theta grid.
+
+[Result and covariance](../analysis/geometric-source-window-20260930/results/result.json),
+[brief report](../analysis/geometric-source-window-20260930/results/RESULT.md),
+[execution](../analysis/geometric-source-window-20260930/execution.json),
+[source review](../analysis/geometric-source-window-20260930/verification.md).
+Raw compressed rows, batch commands, exit statuses and source hashes are
+retained in the same directory. Final resource release is recorded in the
+execution file, not inferred from an ended SSH connection.
diff --git a/notes/one-sided-birth-source-normal-response-20260930.md b/notes/one-sided-birth-source-normal-response-20260930.md
new file mode 100644
index 000000000..a8018de12
--- /dev/null
+++ b/notes/one-sided-birth-source-normal-response-20260930.md
@@ -0,0 +1,146 @@
+# A delayed second-birth source survives the moving-root E_top projection
+
+2026-09-30. Finite author derivation with an explicit square-L4 positive
+control. This supplies a complete **dynamic source -> second birth ->
+M/E_top -> moving-root normal response** map. It is not identification of
+the original norm-4 source or the separately normalized six-coordinate U.
+
+## 1. What is unchanged: first birth and the occupation clock
+
+Apply the [normalized safe-degree kick](birth-selection-intervention-20260930.md)
+only at a fixed insertion count b and only when the current rank is one.
+Before b the source is uniform; after this one kick it is uniform again.
+Total completing probability c/m and total arrival rate remain unchanged.
+
+Couple all sources with the same arrival times and the same prefix through b.
+If a prefix is affected, its first birth has already happened. If its rank
+is zero or two at b, it is not affected at all. Therefore its first-birth
+count J1 is unchanged pathwise, and so is its label time T1 under the common
+clock. The entire occupation-count law K(p) is also unchanged. The second
+birth J2, and hence the joint birth law, can change.
+
+At any later observation p, write P_r(theta,p) for rank probabilities and
+g(p)=partial_theta P_2(0,p). Then exactly
+
+    partial_theta P_0 = 0,
+    partial_theta P_1 = -g,
+    partial_theta P_2 = g,
+    partial_theta M = partial_theta E_top = g,              (1)
+
+where M=P2-P0 and E_top=P2+P0. These are **source derivatives**, not the
+ordinary p derivatives. The latter generally move both birth marginals.
+
+A single-kick response conditional on rank_b=1 must be multiplied by the
+unperturbed rank_b=1 probability before making an unconditional rank claim.
+The current L512735-step calculation uses a fixed count endpoint; it is not
+yet the binomial count mixture that yields a numerical label-time g(p).
+
+## 2. Moving the balance root does not erase a nonzero response
+
+At theta=0 the finite model is ordinary iid occupied-site percolation.
+Let p_* be its interior balance root, M(0,p_*)=0. Differentiating the moved
+root p_*(theta) gives
+
+    p_*'(0) = -g(p_*)/M_p,
+    d/dtheta E_top(theta,p_*(theta)) |0
+        = g [1 - (E_top)_p/M_p]
+        = g * [-2(P0)_p / M_p].                            (2)
+
+All p derivatives are at the baseline root. In a nondegenerate finite
+torus, (P0)_p<0 and (P2)_p>0 for every interior p: each is a nonconstant
+monotone Boolean event, and differentiating the product measure gives a
+sum of pivotal probabilities. At least one pivotal configuration exists;
+every configuration has positive weight for 0<p<1. Thus M_p>0 and the
+factor in (2) is strictly between0and2.
+
+**A nonzero one-sided second-birth source therefore has a nonzero
+moving-root E_top response of the same sign.** It cannot be entirely a
+thermal coordinate shift: a nonzero p shift would change P0, whereas this
+source leaves P0 unchanged. This is an exact two-coordinate statement,
+not a classification of a local CFT field or a result for another normalizer.
+If g(p_*)=0 by cancellation, the conclusion does not assert a nonzero signal.
+
+Equivalently the actual two-parameter response Jacobian satisfies
+
+    det d(M,E_top)/d(p,theta) = -2(P0)_p*g.                 (2a)
+
+When g is nonzero it has rank two: temperature and this second-birth source
+are locally independent response directions. This uses an explicitly
+normalized transition source, not extra derivative-jet coordinates or a
+rank increase created only by relabelling the same scalar source.
+
+## 3. An explicit symmetric finite source with g(p)>0 everywhere inside
+
+Use the already computed square-L4 preparation
+
+    A={0,1,4,5,8,12},  v=x+4y.
+
+At b=6, activate the source precisely when the occupied set belongs to
+A's orbit under translations and the square dihedral group. There are64
+distinct such sets, all rank one, c=0, with the same continuation law by
+graph symmetry. At theta=0 their total prefix probability is
+
+    alpha = 64/binom(16,6) = 8/1001.
+
+This source is invariant under lattice translations and dihedral symmetries;
+it is deliberately constructed and history-dependent, not the physical
+norm-4 Q-source. The previous exact full-lag calculation for A gives negative
+single-kick survival derivatives at h=2,...,6 and zero at the other h.
+It is reused here, without a new lattice or permutation enumeration.
+
+The order statistics of the common label clock are independent of the
+site-choice process. With K(p)~Binomial(16,p), setting q=1-p yields the exact
+unconditional second-birth response
+
+    g(p) = alpha sum_k binom(16,k) p^k q^(16-k) [-S'_{k-6}(0|A)]
+         = (96/35) p^8 q^8 + (32/9) p^9 q^7
+           + (1088/525) p^10 q^6 + (256/385) p^11 q^5
+           + (16/165) p^12 q^4.                            (3)
+
+Only k=8,...,12 contribute; for k<=6 the kick has not acted, and the
+one-insertion rank response is zero. Every displayed coefficient is
+positive, proving g(p)>0 for every 0<p<1, including the balance root.
+Together (2)--(3) give a strict nonzero normal E_top response in the actual
+unconditional finite percolation ensemble at the baseline source.
+
+The old L4 static rank table supplies the baseline polynomial. One ordinary
+floating-point root evaluation, only to show scale, gives
+
+| Quantity | Value |
+|---|---:|
+| baseline p_* | 0.5906721123 |
+| baseline E_top | 0.6454881965 |
+| g(p_*)=delta M=delta E_top | 0.00017057649 |
+| moving-root factor -2(P0)_p/M_p | 0.98801982 |
+| root derivative | -0.00003425820 |
+| normal E_top derivative at moving root | **0.00016853296** |
+
+The sign theorem relies on exact positive coefficients and strict
+monotonicity, not on those decimal digits. Symmetrizing the control changes
+its activation probability, not its local-to-global response law.
+
+## 4. Consequence, without changing the original-source question
+
+This control fills a conceptual gap: a source can be invisible to current
+rank and to the whole occupation clock, then acquire a genuine normal
+topological response through subsequent geometric growth. “The immediate
+global observable is unchanged” is therefore not a general obstruction
+to temporal transmission. Here the source acts on transitions, not merely
+by reweighting already conditioned static configurations.
+
+The source is intentionally selective (activation probability8/1001 at L4).
+Its existence does **not** attribute natural square-lattice anisotropy,
+norm-4 residuals or the old within-K source to this mechanism. The original
+U still requires its actual source, six coordinates, normalizer and
+moving-root forward map. A finite constructed normal response is a positive
+control for that reasoning, not a substitute for those missing columns.
+
+For the ongoing L512 source-window job, first obtain its actual735-step
+transmission. Its single-count result cannot be put into (2) by replacing
+p with k/N; doing so would silently change clocks. The general identities
+(1)--(2) already hold, but a numerical label-time consumer needs the proper
+count mixture. No extra size, p grid or high-precision root task is launched.
+
+[Exact coefficients and illustration](../analysis/one-sided-birth-source-20260930/RESULT.md),
+[result](../analysis/one-sided-birth-source-20260930/result.json),
+[short table consumer](../analysis/one-sided-birth-source-20260930/derive.py).
diff --git a/notes/projected-component-pair-criterion-20260929.md b/notes/projected-component-pair-criterion-20260929.md
new file mode 100644
index 000000000..2abd0573e
--- /dev/null
+++ b/notes/projected-component-pair-criterion-20260929.md
@@ -0,0 +1,237 @@
+# Projected component-pair criterion for exact safe-pair synergy
+
+2026-09-29. Author algorithm and proof for occupied-site torus homology.
+This supplies the geometric enumeration behind the
+[completion-pair identities](completion-pair-synergy-20260929.md).
+Implementation and benchmark results belong to the parent's separate note;
+this note makes no code-performance, empirical-validation, or novelty claim.
+The later [complete same-prefix readout](exact-completion-pairs-readout-20260930.md)
+reports the actual L512 refinement; the original probe result quoted below
+is retained as the motivation, not the latest precision.
+
+## 1. Graph, ambient rank, and transverse potential
+
+Let V=(Z/LZ)^2, N=L^2, L>=3. Use the simple square nearest-neighbour
+graph with steps ±(1,0), ±(0,1) (z=4), or add the two steps
+±(1,1) for the triangular graph (z=6).
+For an oriented edge u->v, let delta(u,v) be its actual lattice step,
+including across a periodic seam; delta(v,u)=-delta(u,v).
+It is not the difference of the two canonical coordinates in [0,L)^2.
+L>=3 ensures that distinct physical edges do not collapse to parallel
+edges or loops. The proof below uses this simple-graph convention.
+
+For occupied A, let H(A) be the rational span of the winding vectors
+of all cycles in its induced graph, and r(A)=dim H(A). A closed walk's
+lifted displacement is Lh, where h in Z^2 is its winding vector.
+Assume r(A)=1 and choose a primitive integer D=(dx,dy) spanning H(A),
+with a fixed sign convention. Define the integer linear functional
+
+    ell(x,y)=dx*y-dy*x,        ell(u->v)=ell(delta(u,v)).          (1)
+
+This is **global ambient rank one**, not an assumption that exactly one
+occupied component wraps. Every occupied component has cycle windings
+in QD; several may wrap in that direction and others may have rank zero.
+At least one old cycle has nonzero winding. Under any rank-one insertion,
+H(A) is contained in the new one-dimensional space, so D persists.
+
+In each occupied component C choose a root with t(root)=0 and define
+t(u) by summing ell along an occupied path from the root to u. Two such
+paths differ by closed walks whose projected displacements are zero:
+ell(Lh)=0 for h in QD. Hence t is path-independent and
+
+    t(u')-t(u)=ell(u->u')       on every occupied edge.           (2)
+
+It is a scalar potential on C, even when C wraps; a full vector-valued
+lift need not be path-independent. Each component has its own additive
+gauge. In an extension write its potential as T(u)=t(u)+g_C.
+
+For any B containing A, r(B)=1 iff the edge values ell admit a scalar
+potential on every component of B. Indeed, such a potential exists iff
+all closed-walk sums vanish, iff H(B) is contained in ker ell=QD.
+The old nonzero winding prevents rank zero. Conversely, any nonzero
+projected closed-walk sum gives a winding independent of D, hence rank
+two, even if that walk lies in a component separate from the old wrap.
+
+## 2. Single-site safety and the contact sign
+
+For a vacancy v and each occupied neighbour u in component C, form
+
+    a_v(C;u)=t(u)-ell(v->u).                                    (3)
+
+An extended potential must satisfy T(u)-T(v)=ell(v->u), hence
+T(v)=a_v(C;u)+g_C. Thus v is immediately completing iff some C gives
+two different values in (3). Inconsistency prevents a potential and
+forces rank two by the criterion above. Conversely, if
+each incident C gives a single value a_v(C), choose T(v) arbitrarily
+and set g_C=T(v)-a_v(C). All contact edges then satisfy the potential
+equations. Components not incident to v retain arbitrary gauges.
+
+Consequently, different values from *different* components do not make
+v completing: their gauges are independent. No contacts, or just one
+contact in each of several components, are safe. Record one a_v(C)
+per distinct incident component for every safe v; repeated contacts
+within a component have already been checked for consistency.
+Let I(v) be that component set, so |I(v)|<=z.
+
+## 3. Exact two-site criterion
+
+Take distinct safe vacancies v,w and write q=T(v)-T(w). Every shared
+component C in I(v) intersection I(w) imposes
+
+    q=q_C:=a_v(C)-a_w(C).                                      (4)
+
+If a physical lattice edge v->w exists, it additionally imposes
+
+    q=q_edge:=-ell(v->w).                                      (5)
+
+In particular, the edge constraint has a minus sign. Let Q(v,w) be the
+set of the values in (4), together with (5) when applicable. Then
+
+    {v,w} is a synergy edge iff |Q(v,w)|>=2.                    (6)
+
+Here a synergy edge means r(A+v+w)=2 although both single insertions
+have rank one. Empty Q or any number of identical constraints is
+consistent; a singleton constraint never causes synergy.
+All comparisons are integer equalities, never congruences modulo L:
+reducing modulo L would erase the very winding differences being tested.
+
+Proof. A potential on A+v+w must satisfy every equation, so conflicting
+values rule it out and force rank two. Conversely, choose q to equal
+their common value, or arbitrarily when Q is empty. Set T(w)=0 and
+T(v)=q. For every component incident to v choose g_C=q-a_v(C); for
+one incident only to w choose g_C=-a_w(C). On a shared component these
+choices agree by (4). Other components retain arbitrary gauges.
+Single-site safety ensures all contacts satisfy their equations, and
+(5) handles the only possible edge between v,w. This constructs the
+required potential and proves rank one. It also covers isolated sites,
+disconnected enlarged graphs, and multiple old wrapping components.
+
+## 4. Discover all nonlocal edges without false candidates
+
+Fix any total ordering of occupied component identifiers. For every
+safe v and every unordered pair C<C' from I(v), emit one record
+
+    (C,C',lambda_v,v),        lambda_v=a_v(C)-a_v(C').           (7)
+
+Group records first by (C,C'), then by the exact integer lambda value.
+For each component pair, enumerate the Cartesian products of every
+two *unequal-lambda* buckets, considering each bucket pair once.
+Every emitted unordered vertex pair {v,w} is a true synergy edge:
+
+    lambda_v-lambda_w
+      =[a_v(C)-a_w(C)]-[a_v(C')-a_w(C')]=q_C-q_C' != 0.         (8)
+
+The vertices need not be physically adjacent. No subsequent rank test
+is needed to filter these emissions. Equal-lambda buckets are not
+crossed with themselves, even if they contain many vertices.
+
+Separately visit each unordered physical edge with two safe endpoints.
+For each shared occupied component compare q_C with -ell(v->w), using
+one consistent orientation. Emit {v,w} once for this visit if any
+comparison disagrees. This too emits only true synergy edges.
+An adjacent safe pair with no shared component passes this check;
+its physical edge alone cannot supply a conflicting constraint.
+
+Completeness follows directly from (6). Any conflict either compares
+two shared-component values, detected by (8), or a shared-component
+value with the physical-edge value, detected by the adjacent scan.
+There cannot be two distinct physical-edge constraints because the
+lattice graph is simple. These exhaust the possibilities.
+
+Deduplicate all emissions by the canonical unordered vertex pair.
+Initialize every safe vertex's degree to zero. For each first occurrence
+of {v,w}, increment e and both endpoint degrees. The result is exactly
+the completion graph G_A, its edge count, and every degree, including
+isolated safe sites. Duplicates are repeated true edges, not false
+candidates. Local contacts discover even arbitrarily separated pairs.
+
+## 5. Gauge invariance and a sign check
+
+Changing t on C to t+b_C sends a_v(C) to a_v(C)+b_C. Single-component
+consistency and each q_C are unchanged. A component-pair key changes
+lambda_v to lambda_v+b_C-b_C', the same translation for every record
+in that pair, so bucket equality and inequality are unchanged.
+Root choices therefore cannot change safety, edges, or degrees.
+Replacing D by -D negates the potentials and all constraint values
+(up to gauges), also preserving consistency and the resulting graph.
+
+The existing [square L4 witness](completion-pair-synergy-L4-20260929.md)
+B={0,2,4,5,8,12}, with v=x+4y, checks both signs by hand. Take D=(0,1),
+so ell=-x. The component C={0,4,5,8,12} has t=0 on x=0 and t(5)=-1;
+the isolated component C'={2} has t(2)=0. Contacts give
+
+    (a_3(C),a_3(C'))=(1,-1),   (a_6(C),a_6(C'))=(-2,0).
+
+Thus lambda_3=2 and lambda_6=-2 detect the nonadjacent edge {3,6}.
+For the adjacent pair 6->7, a_7(C)=1 gives q_C=-3, whereas the physical
+step is (+1,0), so q_edge=+1. Their mismatch detects {6,7}.
+These are algebraic checks of the cited witness, not a new census.
+
+## 6. Output-sensitive cost and its limits
+
+A spanning-forest traversal computes components and t in O(Nz).
+Scanning vacancy contacts, checking repeated-component values, and
+ordering each incidence list costs O(Nz^2). There are at most
+
+    M=sum_(v safe) binom(|I(v)|,2) <= N*binom(z,2)               (9)
+
+records. Grouping them costs O(Nz^2) with expected constant-time exact
+hash dictionaries. Use exact integer word arithmetic and sufficient
+width for potentials, differences, and counts; floating tolerances
+would change the criterion. Bounds here count word operations and
+use expected hash costs, not measured runtime or worst-case hashing.
+
+For a particular true edge, only pairs from I(v) intersection I(w)
+can emit it, at most binom(z,2) times. Each bucket pair visited has a
+nonempty Cartesian product, so its iteration cost is charged to emitted
+true edges; singleton-bucket groups incur only preprocessing cost.
+The adjacent scan visits O(Nz) physical pairs; merging their ordered
+incidence lists costs O(z) per pair, hence O(Nz^2) in total.
+Deduplication costs expected O(1) per emission and stores each edge once.
+Therefore, with e=|E(G_A)|, the total expected bounds are
+
+    time O(Nz^2+z^2*e),        storage O(Nz^2+e).                (10)
+
+Store buckets and stream their products into the deduplication set;
+do not materialize a list of all repeated emissions. For fixed z this
+is O(N+e) time and storage in the stated model. It is not a guarantee
+of worst-case subquadratic work: if e is quadratic, explicit output is
+quadratic. No assertion of a typical e or an implementation speedup
+follows from (10). Comparison-tree dictionaries would add log factors.
+
+## 7. Same-target conditional averaging, not recursive closure
+
+Put m=N-|A|, nu=nu_2(A), and s=m-nu. For s>0, take a uniformly random
+safe site U and define Y=nu_2(A+U)-nu. The existing successor identity
+gives Y=deg_G(U), hence the exact prefix-conditional measurement is
+
+    E[Y|A]=2e/s=2e/(m-nu).                                    (11)
+
+This replaces the safe-probe mean by its conditional expectation for
+the same prefix and target. It removes within-prefix probe randomness
+but leaves variation of e/s among prefixes and birth cohorts. For q
+independent safe probes, in any fixed prefix cohort the variance law is
+
+    Var(Ybar_q)=Var(2e/s)+E[Var(Y|A)/q].                        (12)
+
+Retain the original exact (D,nu) cells, cohort definitions, support,
+weights, and prefix/batch dependence. Empirical weights depend on the
+prefixes, so conditioning on the full prefix sample also preserves the
+weighted estimator's expectation. Apply cell-dependent consumer factors
+inside cells as in the source contract. If s=0 the conditional mean is
+undefined; set neither it nor its missing cohort contribution to zero.
+
+The [latest safe-insertion result](safe-insertion-independent-block-20260929.md)
+reports 140,000 new prefixes, 220,112 probes, and 1,233 nonzero increments;
+its matched increment contrast is +0.00668183 +/- 0.01208706 batch SE,
+unresolved. Sparsity alone does not quantify the removable variance in
+(12), and this derivation supplies no new cohort estimate. Replaying old
+prefix seeds with exact evaluation reuses those configurations: it is
+neither independent validation nor new prospective production.
+
+Exact degrees also give a_j=#{v safe:deg_G(v)=j}, the current full
+safe-successor increment law a_j/s. This does not close the future graph:
+inserting v removes v and its graph neighbours from the safe set, while
+previously noncompleting triples may create new edges among survivors.
+The current degree histogram does not encode that update; even the
+current graph alone is not generally a recursive closure certificate.
diff --git a/notes/safe-insertion-independent-block-20260929.md b/notes/safe-insertion-independent-block-20260929.md
new file mode 100644
index 000000000..b95f78347
--- /dev/null
+++ b/notes/safe-insertion-independent-block-20260929.md
@@ -0,0 +1,151 @@
+# Independent L512 safe-insertion block: unresolved drift, unreplicated direction signal
+
+2026-09-29. A new, fixed 140,000-prefix square-site L512 block, **not** a
+reanalysis of the preceding 140k block. Acquisition source and contract were
+pushed at `55fa265629701e04b879b774ae7d3b97fefa845f` before sampling. The
+scorer was pushed at `f340aa55134213be71d22941fae2ccb7fd068cb7` while the
+job was running and before any production response was read.
+
+**2026-09-30 follow-up:** [exact conditional averaging](exact-completion-pairs-readout-20260930.md)
+of these same prefixes resolves the same target at −.003449±.000772,
+SE15.65x smaller. This note retains the original four-probe readout;
+the refinement is not independent confirmation and leaves the secondary unchanged.
+
+**Outcome.** The prespecified completion-creation contrast is unresolved.
+The prespecified independent replication of the earlier direction-only
+completion-count contrast also does **not** reproduce that signal. This
+lowers the empirical standing of the earlier retrospective direction
+challenge; it does not establish a Markov model or invalidate the exact
+finite L4 counterexamples.
+
+## 1. Actual new acquisition and primary readout
+
+Keep the old insertion counts `a=154646`, `b=155385`, N=262144. Among
+rank-one prefixes at b, early means `J1<=a`, late means `a<J1<=b`.
+For each prefix, four independent with-replacement choices from its safe
+vacancies are inserted in **separate copies**. Their completion-count
+increments Y are averaged within prefix. A rejected immediately completing
+vacancy is resampled; this gives the uniform safe-site law. Probe random
+numbers use a separate stream from the site permutation.
+
+The primary contrast compares prefix-average Y at the same exact primitive
+direction D and integer completion count nu. Cell overlap weights are
+`n_E*n_L/(n_E+n_L)`. The secondary compares current nu at the same D, under
+its own weights, repeating the earlier chosen statistic independently.
+
+| Prespecified quantity, early minus late | Estimate | Aligned batch SE |
+|---|---:|---:|
+| Primary: safe-insertion completion increment, same (D,nu) | +0.00668183 | 0.01208706 |
+| Implied mean synergy-edge contrast, same (D,nu) | +356.3978 | 645.0460 |
+| Implied two-next-insertion survival contrast, same (D,nu) | −6.25403e−8 | 1.13192e−7 |
+| Secondary: current completion count, same D | −0.0398651 | 0.5698779 |
+
+Weighted early/late means of Y are 0.0956368 and 0.0889549. All errors are
+14-whole-batch delete-one SEs, with support and weights recomputed after
+each deletion. The full six-dimensional covariance and all deletion vectors
+are saved. Four probes are **not** counted as four independent prefixes.
+
+The middle two rows are exact-map estimators from the same probes, not
+independent outcomes or observed binary exits. For m=N-b, s=m-nu in each
+cell, the [pair-synergy identity](completion-pair-synergy-20260929.md) gives
+
+    Delta(mean e) = s*Delta(mean Y)/2,
+    Delta(two-step survival) = -s*Delta(mean Y)/[m(m-1)].
+
+Apply s inside each cell before averaging. None of these two-step quantities
+replaces the old 735-insertion survival endpoint. Different weighting targets
+do not define an explained/mediated fraction.
+
+## 2. The independent replication changes the scientific interpretation
+
+The earlier [retrospective within-direction readout](directional-hazard-contrast-20260929.md)
+was **+1.62917 +/- 0.39098** completion sites. The same defined comparison
+on this new block is **−0.03987 +/- 0.56988**. Old and new blocks are not
+pooled. The prior positive point estimate cannot continue to be presented
+as a stable empirical exclusion of direction-only closure.
+
+Both observations remain in the record. The new result neither proves zero
+nor explains why the old selected comparison was larger. In particular,
+there is no basis here to assert a discovered engine error, a physical
+sign reversal, or a universal disappearance of memory. Its proper status
+is **retrospective signal not independently reproduced**.
+
+For the primary, the new block has not resolved a difference in the mean
+creation of completion sites after matching D and nu. Even a precise zero
+of that weighted mean would not establish the entire successor law, exclude
+opposing cell effects, or prove recursive closure. These finite count-clock
+observations do not settle the continuum law.
+
+The exact L4 [weak-Markov counterexamples](completion-pair-synergy-L4-20260929.md)
+remain true. Their existence disproves universal finite-size closure; it
+does not guarantee a detectable history effect at L512 or under this one
+weighted moment. Likewise the earlier L3 closure remains valid at L3.
+
+## 3. What the acquisition actually resolves
+
+Of 140,000 prefixes, 55,028 are rank one at b: 24,201 early and 30,827
+late. No rank-one prefix has all remaining sites immediately completing.
+The primary's 1,190 common cells retain 54,463 prefixes, **98.973%** of
+rank-one risk. Missing overlap is reported, not filled by binning or a
+different conditional definition.
+
+There are 220,112 valid probes but only 1,233 nonzero increments (about
+0.560%); 1,215 prefixes have at least one nonzero probe. Thus the simulator
+has measured many independent prefixes but most continuation measurements
+return zero. This sparsity is a directly observed limitation of this probe
+design; it alone does not quantify a variance decomposition or prove the
+entire SE is continuation noise.
+
+The experiment therefore redirects attention:
+
+- Treat L512 direction-only closure as unresolved, with an unreplicated
+  retrospective challenge. Do not keep using the old point estimate as an
+  accepted constraint on the next mechanism.
+- The next numerical gain should come from a **variance-reduced measurement
+  of the same completion-creation target**, for example analytic conditional
+  averaging or an unbiased, explicitly weighted sampler. An unweighted
+  convenience sample of high-contact sites changes the source and is not a
+  replacement. No such sampler is claimed implemented here.
+- Another full small-size census would not settle the large-size issue;
+  merely extending these sparse uniform probes has less decision value than
+  reducing their noise or deriving a continuum-kernel prediction. Theory on
+  the joint near-critical law remains the parallel primary consumer.
+
+This is attention allocation, not a prohibition on parallel ideas. No extra
+samples, time cuts, directions, feature fits or response-selected subgroup
+analyses were added after seeing the result.
+
+## 4. Execution and reproduction
+
+Only TV2N0X (`4a8d1d443419434889e49148ed0a7ba6`) was operated, in the new
+directory `/workspace/Matching-One-TV2N0X/safe-insertion-20260929`.
+Its 14.5-CPU / 25-GiB actual cgroup budget was measured, and 14 workers used.
+The restarted ARM64 container required GCC installation; GCC 10.3.1 and
+Python 3.9.9 ran the acquisition. No other machine, data directory or old
+production block was altered.
+
+The 32-prefix excluded benchmark forecast 353.99 s of ideal-parallel work;
+actual complete acquisition was **519.843 s** (14 x 10,000). The benchmark
+screen is an estimate, not a promised wall time. The declared sample count
+was retained; there was no response-dependent extension. A 20-site probe
+control on the two known physical L4 witness configurations ran once, not
+a new census. The scorer had a small in-memory arithmetic check, not a
+full test suite; the main agent read it before its one complete data pass.
+
+All raw compressed rows, seeds, source/binary/batch hashes and timestamps
+are in [data/run.json](../analysis/safe-insertion-production-20260929/data/run.json).
+[Result JSON](../analysis/safe-insertion-production-20260929/results/result.json)
+retains full covariance and per-cell sufficient statistics;
+[RESULT.md](../analysis/safe-insertion-production-20260929/results/RESULT.md)
+is the compact numerical report. Results were scored locally in the existing
+research Python 3.11 environment, with no new local dependencies.
+
+```bash
+python3 analysis/safe-insertion-production-20260929/analyze.py \
+  --output-dir /tmp/matching-safe-score-new
+```
+
+The scorer preserves existing outputs; choose an unused output directory.
+Do not rerun the acquisition simply to reproduce the score. The remote
+raw files are retained, while the completed job's machine is returned to
+Ready after confirming no other research processes remain.
diff --git a/notes/safe-path-selection-coupling-20260930.md b/notes/safe-path-selection-coupling-20260930.md
new file mode 100644
index 000000000..351b87cd3
--- /dev/null
+++ b/notes/safe-path-selection-coupling-20260930.md
@@ -0,0 +1,85 @@
+# Directly sample transport versus selection without a path of completion censuses
+
+2026-09-30. Finite sampling identity for the next mechanism calculation.
+The [735-step source transmission](geometric-source-window-readout-20260930.md)
+is now resolved. Its missing interpretation is not more precision: how
+much of passive history memory is already in safe transport, and how much
+is introduced by survivor selection?
+
+The earlier [safe-path formula](birth-selection-intervention-20260930.md)
+uses W=product(1-c_i/(N-i)). Computing c_i by scanning the whole lattice at
+every insertion is unnecessary. The following coupled sampler realizes
+the same distinction using local hypothetical-completion queries.
+
+## Coupled path and survival flag
+
+Start at a specified first-birth rank-one configuration with flag I=1.
+At each subsequent insertion count:
+
+1. Visit the remaining vacancies in a fresh uniform random order until a
+   safe site is encountered. Insert that site in the reference path.
+2. If the **first** vacancy examined would have completed rank two, set
+   I=0 permanently. Continue the reference path even after I becomes0.
+3. If no safe vacancy exists, send the reference path to a cemetery and
+   set I=0. Do not loop forever or silently cap attempts.
+
+Only the first examination updates I; later rejections do not constitute
+additional natural time steps. The inserted site is uniform among current
+safe vacancies. Completing-site rejection is tested before altering the
+geometry. Rejected sites remain vacant and may be examined again next step.
+
+With s safe out of m vacancies, for each safe v,
+
+    P(accepted v) = 1/s,
+    P(first examination safe, accepted v) = 1/m.
+
+Thus the first-examination flag and accepted site's identity are independent
+conditional on the current geometry, with flag probability s/m. Until the
+flag dies, the chosen sites therefore reproduce exactly the **killed
+original uniform process**; the reference path always follows the safe
+process. Conditional on its whole accepted path,
+
+    E[I_end | safe path] = product_i s_i/m_i = W,
+    E[I_end*f(A_end)] = E_safe[W*f(A_end)].                 (1)
+
+There is no product of noisy completion-count estimates. I is a Bernoulli
+realization of the full survival weight, not an approximation to W.
+For a chosen endpoint cell C, condition on the reference endpoint in C:
+
+    E[f | I_end=1,C] - E[f | C]
+       = Cov(f,I_end | C)/E[I_end | C].                    (2)
+
+Equation (2) gives selection; the unweighted safe-path means give
+entrance/transport. Apply the same formula to the two fixed birth cohorts
+and subtract, retaining their separate cell probabilities and normalizers.
+The resulting components are correlated, not separate evidence blocks.
+
+## Why this can be an actual bounded calculation
+
+A vacancy's completion status is decided by its at most four occupied
+neighbours and their lifted union-find potentials; the current engine
+already performs this test inside its completion-count loop. Extracting
+that one-site query removes the whole-lattice scan from each safe step.
+The all-vacancy pair graph is only needed once, at the chosen endpoint,
+if the readout is Y=2e/s. Random-order rejection terminates even at s=0;
+for s>0 its expected number of probes is (m+1)/(s+1).
+
+This is a consumer-specific algorithm, not another generic validation
+framework. The saved original permutation determines each first-birth
+configuration and its count. New safe continuations need a distinct RNG
+stream; they are conditional branches of those same archived prefixes,
+**not new independent preparations**. Direct rank0-to-rank2 births are
+outside a rank-one entrance source and remain separately counted.
+
+The next readout is the already specified endpoint completion creation
+Y and its natural versus selection-neutralized early/late contrast. A
+nonzero unweighted contrast would reject selection-only for that target;
+weighting can add, oppose or dominate it. It would still not uniquely split
+birth geometry from subsequent safe evolution. No new descriptor, cutoff,
+size ladder or forced statistical significance is required.
+
+**Execution update:** the [coupledL512 continuation](safe-transport-selection-readout-20260930.md)
+is now complete. The safe-reference history contrast remains negative
+(-.00253706+/-.00063649), while selection's contribution is unresolved
+(+.00033199+/-.00028738). This is distinct from the earlier735-step source
+response and uses new conditional futures of the same archived entrances.
diff --git a/notes/safe-transport-selection-readout-20260930.md b/notes/safe-transport-selection-readout-20260930.md
new file mode 100644
index 000000000..cdb43e21a
--- /dev/null
+++ b/notes/safe-transport-selection-readout-20260930.md
@@ -0,0 +1,220 @@
+# Safe transport versus survivor selection: one mechanism comparison
+
+2026-09-30. Contract and implementation committed before new continuations
+at `5054c02383242473834045fcbddc7379d83acfb9`, in Draft #838.
+**Completed.** The negative early-minus-late completion-creation contrast
+persists after removing natural survival reweighting. For this specified
+reference comparison, selection-only is disfavoured; entrance/safe transport
+is needed. The selection contribution itself remains unresolved, not zero.
+
+| Same1198 endpoint cells, same weights | Estimate | Aligned14-batch SE |
+|---|---:|---:|
+| Natural-alive contrast | -0.002205065286 | 0.000787091700 |
+| Uniform-safe reference contrast | **-0.002537058794** | **0.000636492674** |
+| Selection = natural minus reference | +0.000331993508 | 0.000287375294 |
+
+The reference contrast is about3.99 batch SE from zero. Selection's point
+estimate opposes it, but its sign is not resolved (about1.16SE); do not call
+that a proved cancellation or a measured mediation percentage. These are
+correlated components of one experiment, not three independent confirmations.
+
+All140000 original J1 values match. There are97666 rank-one entrances
+(56286early,41380late),42329 no-entrance prefixes and5 direct rank-two
+entrances. Every reference entrance reaches b; none has an undefined safe
+endpoint. The natural flag survives on54850 endpoints (24216early,30634late).
+Common support contains54295natural endpoints and94123reference endpoints.
+The strong cohort difference in overall survival is retained; an unresolved
+selection term for this matched Y contrast does **not** make selection
+irrelevant to survival generally.
+
+Weighted natural early/late means are .09610311808/.09830818336; reference
+means are .09540936877/.09794642757. The exact decomposition and singular
+3x3 covariance are saved with all14 deletions and cell means in
+[result.json](../analysis/safe-transport-selection-20260930/results/result.json).
+This is new conditional continuation of old entrances, not a new independent
+prefix block. The outcome supports moving the main question to entrance
+versus safe growth under explicit sources, rather than more precision or
+more descriptions of the same endpoint.
+
+## What changes the explanation
+
+The previous natural-history contrast has two possible origins. Birth
+cohorts can enter/evolve through different safe geometries, or natural
+exits can select different survivors out of otherwise comparable cohorts.
+Matching the endpoint direction and completion count does not match the
+**path of past exit hazards**. Endpoint matching alone cannot separate them.
+
+The [exact rejection/flag coupling](safe-path-selection-coupling-20260930.md)
+now supplies an operational distinction. At count i, with m_i vacancies
+and c_i completing sites, the killed natural transition on rank-one states is
+
+    K_i(A,A+v) = (1-c_i/m_i) P_safe,i(A,A+v),
+    P_safe,i(A,A+v) = 1/(m_i-c_i), for safe v.
+
+Thus a natural safe path has relative weight
+
+    W = product_i (1-c_i/m_i)
+
+under the always-safe reference source. This is a pathwise killing weight,
+not a proposed additional endpoint descriptor. The natural-alive flag I
+has E[I | safe path]=W. For each cohort g and endpoint cell C,
+
+    mu_natural(g,C)-mu_safe(g,C)
+       = Cov_safe(Y,I | g,C)/E_safe[I | g,C].
+
+Subtract early and late means using the same overlap weights throughout:
+
+    Delta_natural = Delta_safe + Delta_selection.
+
+**Selection-only for this specified comparison requires Delta_safe=0.** A
+resolved safe-reference contrast requires information already in entrance
+and/or safe transport. If both terms are appreciable with opposite signs,
+survival selection masks part of that contrast rather than creates it.
+These predictions are fixed before reading the new continuations. No fourth
+descriptor or adaptive time window will be added to explain the outcome.
+
+### What the selection term means physically
+
+Completing vacancies are persistent along a safe rank-one path: once adding
+v would create rank two, enlarging the occupied set without adding v cannot
+make it non-completing. Safe insertion never occupies one of these sites.
+Consequently c_(i+1)=c_i+d_i, with d_i>=0 the chosen safe site's synergy
+degree. The killing weight therefore remembers **when completion sites
+were created**, not just how many exist at the final endpoint.
+
+For two paths with the same entry and terminal counts and equal terminal c,
+if c_i on one path is at least as large at every intermediate count, its W
+is no larger; the inequality is strict when an interior factor differs and
+both weights are positive. Natural survival prefers later completion
+creation under this pointwise comparison. This is an exact ordering, not a
+claim that an arbitrary endpoint Y contrast must have a particular sign.
+Endpoint c matching cannot remove it. No new feature is needed to test the
+aggregate consequence: the coupled flag already samples the full weight.
+
+| Fixed result pattern | Mechanism consequence for this comparison |
+|---|---|
+| Safe-reference contrast unresolved, selection resolved | Selection can account for the observed contrast; not proof of all-history closure |
+| Safe-reference contrast resolved, selection unresolved | Entrance/safe transport is needed; quantify selection as unresolved, not exactly zero |
+| Both resolved, same sign | Entrance/transport and selection reinforce |
+| Both resolved, opposite signs | Selection attenuates or reverses the entrance/transport contrast |
+
+These are interpretation branches, not significance-driven instructions to
+extend the sample. Endpoint conditioning is retained in both laws; "without
+selection" here means without the **natural survival reweighting**, not
+without all conditioning or with an unmodified iid source.
+
+## One endpoint, unchanged history groups
+
+- Source: all 140000 original square-L512 permutations, grouped in their
+  original 14 batches; replay to the **first** birth, not to the old risk set.
+  A prefix that originally completed before b is still an eligible entrance.
+- Entrance: rank-one first births J1<=b; direct rank-zero-to-two transitions
+  and no-birth prefixes are counted separately. Early J1<=154646, late
+  154646<J1<=155385.
+- One distinct seeded uniform-safe continuation per entrance to b=155385.
+  No-safe intermediate states enter a cemetery. The first proposal's
+  completion status updates the natural flag; extra rejection queries do
+  not advance count time.
+- Endpoint C=(primitive direction, integer c), Y=2e/(262144-b-c).
+  States with no safe endpoint have an undefined Y and are counted, not
+  replaced by zero. Only cells containing both natural-survivor cohorts
+  contribute; weight n_E*n_L/(n_E+n_L) is reused for reference means too.
+- One three-component result and its singular full covariance, recomputing
+  every weight and support on 14 aligned whole-batch deletions. New futures
+  are conditional branches of archived entrances, **not new independent
+  preparations or untouched holdout**.
+
+The reference is explicitly modified dynamics. Its surviving branch has the
+natural killed-path law, but the reference contrast is not a unique causal
+split between imported birth geometry and subsequent safe growth. This
+finite count-clock calculation does not identify original U or continuum
+fields. The two earlier original-block contrasts are context, not numbers
+to subtract from this new realization or pool as additional votes.
+
+## If safe transport persists: an identifiable next comparison
+
+Do not try to "hold J1 fixed and swap early/late labels": the cohorts are
+defined by J1, and the birth configuration's cardinality is J1. That is not
+an intervention identifying age separately from birth geometry.
+
+Instead specify two actual source columns: an entrance-law perturbation
+delta alpha_j with zero total mass at each j (preserving the first-birth
+count law), and a row-sum-zero safe-transition perturbation delta P_i
+(preserving safety and the insertion clock). Let P_(j:b) be the reference
+safe propagator, Z=sum_j alpha_j P_(j:b)1_C, and mu its conditional endpoint
+mean. Their distinct response formulas are
+
+    entrance: Z^(-1) sum_j delta alpha_j P_(j:b)[(f-mu)1_C],
+    growth:   Z^(-1) sum_(j<=i<b) alpha_j P_(j:i)
+                        delta P_i P_(i+1:b)[(f-mu)1_C].
+
+The centered terminal function includes the changing endpoint-cell
+normalizer. The sources must be realizable, not arbitrary fitted columns;
+these formulas define the comparison rather than supplying such a source
+automatically. At least two nonredundant readouts are needed to distinguish
+the two response columns; a single scalar contrast cannot identify them.
+These are within-cell formulas. An overlap-weighted multi-cell response
+must either fix its external cell weights in advance or include their
+derivatives; it must not silently hold data-dependent weights constant.
+
+In particular, **Y and two-step survival are a bad two-readout pair** here.
+Within fixed (k,D,c), with m=N-k,s=m-c,
+
+    survival_2 = s(s-1)/(m(m-1)) - s*Y/(m(m-1)).
+
+Their source columns are exactly proportional after the within-cell
+normalizer. More precision cannot restore the missing rank. Pooling
+different c cells with different coefficients can change the apparent
+weighted relation but would not create a new within-cell physical readout.
+
+A same-contract long-window survival readout is a genuine candidate for the
+second row: the already calculated L4 preparations A and C have equal c,e,Y
+but different lag-three source responses. Use a real propagator rather than
+replace it with two-step curvature. Whether entrance and growth columns
+are independent for a specified source pair is the next mechanism question,
+not answered by the current decomposition and not a request for an automatic
+size/precision ladder. The earlier735-step susceptibility cannot be pasted
+in as a column under a different preparation or normalization.
+
+### Static M/E_top also cannot separate two first-birth-preserving sources
+
+There is a second structural rank bound, complementary to the previous
+[one-sided-source positive control](one-sided-birth-source-normal-response-20260930.md).
+Suppose BOTH proposed sources preserve the complete first-birth marginal
+and hence P0(p), and are evaluated at the same p. For column r let
+g_r=partial_r P2. Probability conservation gives
+
+    partial_r (P0,P1,P2) = g_r (0,-1,1),
+    partial_r (M,E_top) = g_r (1,1).
+
+All such source columns are proportional in these static readouts. At the
+moving root their (root position, root E_top) columns remain proportional:
+
+    partial_r (p_star,E_top(p_star))
+       = g_r (-1/M_p, -2 P0_p/M_p).
+
+Thus the genuine rank-two **thermal p plus one geometric source** Jacobian
+already proved is not rank-two identification of **two geometric sources**.
+More precise M/E_top data at one p cannot distinguish the latter. Multiple
+p values or a joint-time/geometry readout may supply a nonproportional row,
+but that must be demonstrated under the same source definition. This bound
+does not apply indiscriminately to the missing original norm-4/U columns,
+whose sources and observables have a different contract.
+
+## Reproduction
+
+See [contract](../analysis/safe-transport-selection-20260930/contract.json),
+[runner](../analysis/safe-transport-selection-20260930/run.py) and
+[scorer](../analysis/safe-transport-selection-20260930/score.py).
+Huawei TgFr7R completed14workers in664.094seconds, ARM64/Python3.9.9/GCC10.3.1;
+actual14.5CPU/25GiB limit. All14 engine exits were zero. The new task directory
+reused only this task's persistent compiler in a distinct compilation subtree.
+The32-old-prefix cost probe predicted430.389seconds and was excluded; the
+full task took longer, with no sample extension. Output hashes and commands
+are in [execution.json](../analysis/safe-transport-selection-20260930/execution.json).
+
+One support worker reviewed source/coupling/covariance and ran one fixedL4
+vacancy-query control plus one exact analytic coupling example; see
+[verification.md](../analysis/safe-transport-selection-20260930/verification.md).
+Main ran the fixed scorer once in research-py311 after retrieving the files.
+No extra precision production, new size or repeated all-repository test suite.
diff --git a/notes/square-birth-transfer-and-iqr-clock-20260929.md b/notes/square-birth-transfer-and-iqr-clock-20260929.md
new file mode 100644
index 000000000..8a3cbf451
--- /dev/null
+++ b/notes/square-birth-transfer-and-iqr-clock-20260929.md
@@ -0,0 +1,324 @@
+# Square 4/8 birth transfer and the population-IQR clock
+
+2026-09-29. Continuation of the
+[two-time density derivation](two-time-pivotal-density-bound-20260929.md),
+for #778 / #773. Author proof using named standard probability inputs;
+not independent certification, an exact exponent calculation, or a novelty claim.
+
+**Outcome.** The preceding argument transfers to square-site percolation on
+its own pivotal clock. It does not have to wait for a proof of the exponent
+5/4. The relevant NN/matching arm conventions are compatible, and the
+required qualitative power bounds and near-critical comparisons belong to
+the general-lattice theory. Every subsequential joint birth limit is locally
+absolutely continuous and has two distinct finite births on a fixed
+nondegenerate torus.
+
+A second argument connects this clock to the **population** IQR of the
+archived insertion births: the width is bounded above and below by constant
+multiples of `N*a_eta`. Hence the IQR normalization cannot by itself manufacture
+or remove a limiting zero-gap atom. No limiting width constant, copula
+uniqueness, finite-sample guarantee or cross-lattice equality is asserted.
+
+## 1. Exact graph and arm conventions
+
+Black occupied sites use the square nearest-neighbour graph `G`. White sites
+use the matching graph `G*`, obtained by adding both diagonals in every
+square face. Only the original square vertices are random Bernoulli sites.
+An auxiliary face centre used to draw white paths is **not** an independent
+site, a second source or an extra occupation variable.
+
+Let `A_(2j)(r,R)` mean alternating black NN and white matching arms in a
+square annulus. Arms are truncated to proper inner-to-outer crossings, with
+no other boundary visits; fixed lattice collars are harmless. Their cyclic
+order is alternating. Set
+
+```text
+pi_4(r,R) = P_pc[A_4(r,R)],
+pi_6(r,R) = P_pc[A_6(r,R)].
+```
+
+The critical parameter is square **site** `pc`, not the square-bond value
+1/2. There is no need to insert a numerical estimate of pc in this proof.
+
+### Compatibility with the planarised-white separator
+
+The stronger-looking face-capacity convention in #823 is compatible with
+these alternating events. Fix the selected black proper crossing arcs and
+cut the annulus along them. Each white arm starts in a different intervening
+strip, by alternation. A white matching edge cannot cross a black NN arc:
+its diagonal lies inside a square face and its endpoints are white.
+Therefore each selected white path stays in its own strip.
+
+An open square-face interior belongs to just one such strip, since black
+NN arcs run on its boundary. Two white arms in different strips cannot use
+the same face interior. Inside each strip, erase loops and replace a white
+diagonal by the two edges through a face centre. The resulting paths cannot
+share face centres or white vertices with paths in another strip. Conversely,
+a face-centre path projects to a white matching path. Thus planarisation
+does not discard the proper alternating crossings used here. Fixed collar
+changes handle contacts with the annulus boundaries; this is not an equality
+of the earlier degenerate L3/L4 thin-annulus census conventions.
+
+This argument uses the alternating **whole collection**. It does not say that
+arbitrary vertex-disjoint white paths, without intervening black arms, are
+geometrically disjoint: crossing white diagonals alone would be a counterexample.
+
+The four-arm event rooted at a pivotal site's neighbours and the annular
+event with fixed microscopic radius have comparable probabilities. Arm
+separation and a fixed finite-energy extension join their endpoints in a
+bounded collar. The collar cost is uniform because p stays in a compact
+subinterval of `(0,1)`. No extra mesh-dependent power is introduced.
+
+## 2. Probability inputs actually used
+
+The primary sources read for this transfer are:
+
+1. Harry Kesten, *Scaling relations for 2D-percolation* (1987),
+   [DOI](https://doi.org/10.1007/BF01205674),
+   [full text](https://math.bme.hu/~balint/oktatas/perkolacio/percolation_papers/kesten_2.pdf).
+   Page 112 explicitly includes square site and matching graphs. Lemmas 4--6
+   supply separation/gluing, Lemma 7 (2.59) supplies the two-scale gain,
+   Lemma 8 compares pivotal probabilities below
+   characteristic length, and (4.5), (4.28), (4.33), Theorem 4 give the
+   characteristic-length relation and the two-sided comparison.
+2. Pierre Nolin, *Near-critical percolation in two dimensions*,
+   [arXiv:0711.4948](https://arxiv.org/abs/0711.4948).
+   The downloaded text's Proposition 16, Theorem 26, Proposition 32 and
+   Lemma 37 give the arm/length formulations; section 8.1 explicitly discusses
+   their square/matching extension and the strict four-/six-arm power bounds.
+   This transfer uses that general-lattice scope, not the triangular exact
+   values. Journal numbering differs.
+3. Van den Berg and Nolin, *On the four-arm exponent for 2D percolation at
+   criticality*, [arXiv:2008.01606](https://arxiv.org/abs/2008.01606),
+   [author full text](https://ir.cwi.nl/pub/31186/31186.pdf).
+   Section 2 fixes the square NN/matching arm convention. The end of section 5
+   proves `pi_4(3n)<=C*n^-1*sqrt(pi_2(n))`; RSW decay of `pi_2` yields the
+   strict power above one used for the label/insertion comparison below.
+
+Here is the precise package consumed, in lattice units. Constants need not
+be known numerically. There are positive `kappa_4,kappa_6,beta` such that
+
+```text
+c (r/R)^(2-kappa_4) <= pi_4(r,R) <= C (r/R)^beta,
+pi_6(r,R) <= C (r/R)^(2+kappa_6),                        (1)
+pi_4(r,R) asymp pi_4(r,s)*pi_4(s,R),                    (2)
+```
+
+uniformly over nondegenerate annuli, with fixed-radius changes absorbed in
+constants. The upper power beta may be decreased so that `beta<2`.
+The statements are uniform inequalities, not assumptions that limiting
+critical exponents exist.
+
+For clarity, the lower bound in (1) is an **annular**, not just a rooted,
+input. It can be recovered from Kesten's (2.59) (printed page 137, checked
+against the rendered formula). For dyadic r=2^j<R=2^k, divide that estimate
+by `|p-pc|` and let p tend to pc. The finite-box probabilities are
+polynomials in p, and the critical square-crossing derivative obeys the
+standard Russo/separation comparison `sigma'_pc(R) asymp R^2*pi_4(1,R)`.
+Thus, for a fixed zeta>0,
+
+```text
+r^2*pi_4(1,r) <= C*(r/R)^zeta*R^2*pi_4(1,R).
+```
+
+Quasi-multiplicativity gives
+`pi_4(r,R)>=c*(r/R)^(2-zeta)`. Dyadic rounding and fixed collars yield
+the stated uniform form (decrease zeta if needed). This explicitly explains
+why a mere claim about a limiting rooted exponent is not being substituted
+for the two-scale estimate consumed by the pair sum.
+
+In addition, below characteristic length the
+corresponding near-critical arm probabilities are comparable to critical
+ones, and
+
+```text
+|p-pc| L(p)^2 pi_4(1,L(p)) asymp 1.                    (3)
+```
+
+Subcritical rectangle crossings decay exponentially in size divided by
+`L(p)`. The matching graph has its own critical density `pc*=1-pc`; (3) and
+the crossing comparisons apply to both members. These are literature inputs,
+not theorems rederived by the finite square census.
+
+**What is not imported:** triangular colour-switching between arbitrary
+polychromatic words, 5/4, 35/12, 4/3, SLE6 convergence, or square conformal
+invariance. We use only the indicated alternating words and matching-pair
+estimates. Colour exchange of the entire graph pair is different from a
+claim that arbitrary colour words on the same non-self-matching graph agree.
+
+## 3. Square joint-birth corollary
+
+Let the mesh eta square tori approximate a fixed flat torus of positive
+systole and finite area, so `N asymp eta^-2`. In rescaled arm notation set
+
+```text
+a_eta = eta^2/pi_4(eta,1),
+lambda_i = (T_i-pc)/a_eta,                i=1,2.
+```
+
+Every rank-changing site forces four arms by the punctured-disk replacement
+argument in the preceding note. A direct `0->2` forces six (or eight) arms by
+the offset/spine argument in #773 and the square separator in
+[#823](https://github.com/LightChainr/Matching-One/blob/3d0176a9f394f7d7228766709731493fd15095c5/analysis/queue-20260919/815/NOTE.md).
+Section 1 identifies the arm events used for their probability bounds.
+
+For each bounded time window choose the local annulus radius below the
+minimum physical characteristic length in that window. Relation (3) and
+the lower four-arm power guarantee that this radius is a positive constant
+independent of eta. This avoids applying near-critical comparison above its
+stated range. Equations (1)--(2) make the two-time pair sum integrable:
+
+```text
+sum_{dyadic eta<=rho<=R} rho^2/pi_4(rho,R) <= C_R.
+```
+
+For the same-site atom, the two qualitative powers give the concrete
+vanishing estimate
+
+```text
+pi_6(eta,R)/pi_4(eta,1) <= C_R eta^(kappa_4+kappa_6).    (4)
+```
+
+This is not the triangular 5/3 rate and does not determine the optimal square
+rate. The earlier density proof now gives, for every finite Lambda,
+
+```text
+f_eta(s,t) <= C_Lambda,          -Lambda<=s<t<=Lambda,
+P(gap<=delta, both births in the window)
+   <= C_Lambda delta + C_Lambda eta^(kappa_4+kappa_6).    (5)
+```
+
+For tightness, apply (3) and the arm powers as in the earlier note to make
+the physical characteristic length tend to zero when `|lambda|` tends to
+infinity. Cover the torus by finitely many embedded charts. An essential
+black cycle at the negative endpoint forces a fixed-size rectangle crossing,
+whose probability tends to zero by subcritical exponential decay.
+
+At the positive endpoint use the **white matching** configuration, not white
+NN: the finite digital Alexander identity gives
+
+```text
+r_black,NN(p) + r_white,matching(p) = 2.
+```
+
+White occupation density is `1-p=pc*-Lambda*a_eta`. Swapping colours and
+the graph pair leaves the unrestricted alternating four-arm event the same,
+up to the already declared microscopic conventions. Its pivotal clock is
+therefore comparable to `a_eta`. Matching subcritical decay controls
+`P(lambda_2>Lambda)`. No false square-site symmetry `p <-> 1-p` on the
+**same** NN graph is invoked.
+
+**Corollary (author proof from the above standard inputs).** The square
+birth pairs are tight on their pivotal clock. Every subsequential limit has
+a locally bounded two-dimensional density and distinct finite coordinates;
+the corresponding rank paths have two separate unit jumps in Skorokhod J1.
+This completes the square input transfer claimed as conditional in the
+earlier note. A unique law, exact clock power or cross-lattice law equality
+does not follow.
+
+## 4. The archive clock is comparable: a separate probability argument
+
+The stored simulation records permutation birth counts `J1,J2`. Label time
+uses independent sorted uniform labels; the uniform ordering permutation is
+independent of those order statistics. In this coupling,
+
+```text
+T_i = U_(J_i),
+E[T_i | J_i=j] = j/(N+1),
+Var(T_i | J_i=j) = j*(N+1-j)/[(N+1)^2*(N+2)]
+                <= 1/[4*(N+2)].                       (6)
+```
+
+The bound holds even though `J_i` depends on the entire permutation.
+Consequently, with `X_i=(J_i/(N+1)-pc)/a_eta`,
+
+```text
+E[(X_i-lambda_i)^2] <= 1/[4*(N+2)*a_eta^2].             (7)
+```
+
+The square bound from van den Berg--Nolin gives
+`pi_4(eta,1)<=C eta^(1+epsilon)` for some epsilon>0. Thus
+`sqrt(N)*a_eta asymp eta/pi_4(eta,1) -> infinity`, and (7) tends to zero.
+For triangular site the standard four-arm bound also gives this property.
+Hence both individual birth times, not only their difference, have the same
+subsequential laws in insertion and label clocks. This step needs a power
+above one; the earlier pair-density integrability needed a power below two.
+
+Let H_eta be the equally weighted **population** mixture of `X1,X2`, and let
+`w_eta=Q_.75(H_eta)-Q_.25(H_eta)`, with lower quantiles. We claim
+
+```text
+0 < c <= w_eta <= C < infinity                         (8)
+```
+
+for all sufficiently small eta.
+
+**Upper bound.** Tightness of the pair and (7) put more than 3/4 of the
+mixture inside a fixed bounded interval; choose the window with each tail
+less than 1/4. Both quartiles are then inside that interval.
+
+**Lower bound.** Fix a window whose pair escape probability is less than a
+small tau. Equation (5)'s joint density implies, uniformly over intervals I
+of length h, for the label mixture,
+
+```text
+P(mixture label coordinate in I)
+   <= tau + 2Lambda*C_Lambda*h + C_Lambda*e_eta.         (9)
+```
+
+Transfer to the insertion mixture by enlarging I by b at each end and adding
+`P(|X_i-lambda_i|>b)`, which vanishes for each fixed b by (7). Choose tau,
+b and then a fixed positive h so that the resulting upper bound is strictly
+less than 1/2. Any interval between the .25 and .75 lower quantiles contains
+at least half the distribution, including in the discrete case. Its length
+cannot be less than h. This proves (8); no unproved convergence or strict
+positivity of a limiting density at a quartile was assumed.
+
+In the simulation's count units, write W_eta for the population IQR of the
+equal mixture `(J1,J2)`. Positive affine scaling of lower quantiles gives
+
+```text
+W_eta = (N+1)*a_eta*w_eta asymp N*a_eta.                 (10)
+```
+
+Therefore the population-normalized gaps
+
+```text
+G_J = (J2-J1)/W_eta,
+G_T = (N+1)*(T2-T1)/W_eta
+```
+
+differ by `o_P(1)` using (7)--(8). Their denominators relative to the pivotal
+clock stay in a compact positive interval. In particular, since soft mass is
+monotone in its resolution,
+
+```text
+lim_{delta->0} limsup_{eta->0} E[(1-G_T/delta)_+] = 0,   (11)
+```
+
+and the same holds for G_J. To justify the latter directly, split on
+`|G_T-G_J|<=delta` and use the small-ball criterion at resolution `2delta`;
+then take eta->0 before delta->0. Do not interchange these limits.
+
+This supplies a population-level justification for the archive's exponent-free
+normalization. It does **not** prove that `w_eta` converges, calculate it,
+replace the empirical W by a known deterministic value, or make square and
+triangular primitive tori have the same modulus.
+
+## 5. What can and cannot now be said about the existing data
+
+- The code's `label_IQR` uses exactly `(N+1)*(T2-T1)/W`, with W estimated from
+  the pooled insertion births. Its target now has a clock-comparison argument.
+- The population proof does not validate a fixed finite sample's empirical
+  quantiles or imply a confidence guarantee for 14 batches. Those remain
+  sampling questions; the existing delete-one calculation re-estimates W.
+- No result file or production sample is changed or regenerated for this note.
+  No exponent is fitted and no larger-size job is launched.
+- The next missing process object is the actual joint limit / copula and its
+  torus dependence, not another scalar direct-jump exponent. The same-site
+  arm suppression, two-time density, tightness and width comparison address
+  different links and should stay separately visible.
+
+The literature input is old mathematics, not a claim that this project has
+newly proved square critical exponents. The contribution here is the explicit
+transfer to the ambient-rank birth process and its stored insertion readout.
diff --git a/notes/stochastic-parity-block-encoding-20260929.md b/notes/stochastic-parity-block-encoding-20260929.md
new file mode 100644
index 000000000..cf2a53921
--- /dev/null
+++ b/notes/stochastic-parity-block-encoding-20260929.md
@@ -0,0 +1,325 @@
+# A distributed stochastic rank-three code, and its single-readout cost
+
+2026-09-29. Bounded sidecar for #809/#833; consumers are the
+[deterministic encoding obstruction and physical product queries](deterministic-block-encoding-obstruction-20260929.md).
+This is an elementary parity-sharing construction with exact proofs, not a
+novelty claim or a new full-machine positive-realization theorem.
+
+**Answer.** Private stochastic encoding does evade the deterministic
+four-state obstruction: for every `k>=2`, two independent uniform parity
+shares give four disjoint physical preparation distributions, every proper
+subset of blocks reveals nothing, and the **full tensor-response rank is
+exactly three**. But the explicit physical product-query family has at most
+`2^(1-k)` total-variation distinction for one logical bit. Even an arbitrary
+single monotone Boolean readout has sharp distinction only
+
+    b_k = binom(k-1, floor((k-1)/2)) / 2^(k-1).
+
+Thus this is a genuine change of encoding hypothesis, not a demonstrated
+reliable redundant memory. The explicit product-query family even admits
+a three-label normalized positive initial-memory model, matching its linear
+rank. Tensor-response rank three is **not** asserted as an upper bound on
+the entire physical future-query language.
+
+## 1. Encoding and its private-randomness contract
+
+Logical states are `(u,v) in {0,1}^2`. Independently sample
+
+    U uniformly on {x in {0,1}^k : xor_i x_i = u},
+    V uniformly on {y in {0,1}^k : xor_i y_i = v}.
+
+Physical block `i` receives corner `(U_i,V_i)` of the original five-column
+preparation. The random samples/encoder seed are not disclosed to the query
+controller. They are not additional measurements. A code distribution
+contains `2^(2k-2)` equiprobable physical preparations. The four supports are
+disjoint and partition all `4^k` preparations; observing the entire sampled
+configuration would identify both logical bits exactly.
+
+Every set of fewer than k physical blocks is uniformly distributed on its
+four-corner configurations, independently of `(u,v)`. Indeed every proper
+coordinate marginal of a uniform parity class is uniform: after fixing r<k
+coordinates there are exactly `2^(k-r-1)` completions of either parity.
+Independence of U and V then gives the block claim. In particular every
+individual block is uninformative for `k>=2`. This is truly distributed
+stochastic sharing, not a random name for the one-informative-block
+deterministic exception.
+
+The distinctions are between four **mixed preparations**. Disjoint support
+does not imply that an allowed observation can identify which support was
+chosen, and does not by itself impose four states on every positive model
+for a restricted observation language.
+
+## 2. Full tensor expectation and exact rank
+
+For disjoint subsets `A,B` of `{1,...,k}`, the admitted tensor monomial is
+
+    q_(A,B)(U,V) = product_(i in A) U_i * product_(i in B) V_i.
+
+Disjointness expresses the local response `(1,U_i,V_i)`: at most one of the
+two coordinates can appear in each block. Write `sigma_u=(-1)^u` and
+`sigma_v=(-1)^v`. Uniform parity gives
+
+    E_u[product_(i in A) U_i]
+      = 2^(-|A|) [1 + 1{A=all} (-1)^k sigma_u].
+
+For proper A this is its uniform marginal. For A=all, the all-ones vector
+has parity k mod 2 and either has probability `2^(1-k)` or is excluded.
+Consequently the encoded response is exactly
+
+    E_(u,v) q_(A,B)
+      = 2^(-|A|-|B|)
+        [1 + 1{A=all} (-1)^k sigma_u]
+        [1 + 1{B=all} (-1)^k sigma_v].                 (2.1)
+
+Since A and B are disjoint, both cannot equal all. No `sigma_u sigma_v`
+term occurs. Every tensor response lies in the three-dimensional span of
+`1,sigma_u,sigma_v`. This is an upper bound for the **declared tensor
+language**, including all its linear combinations.
+
+Conversely, the constant query, the all-U product and the all-V product
+give respectively
+
+    1,
+    2^(-k) [1+(-1)^k sigma_u],
+    2^(-k) [1+(-1)^k sigma_v].                         (2.2)
+
+These three columns are independent, so rank is exactly three for every
+finite k. In row order `(00,10,01,11)` the only relation is
+
+    response_00 - response_10 - response_01 + response_11 = 0.
+
+The unseen joint logical parity requires the microscopic character
+`product_i (-1)^(U_i+V_i)`, which uses both coordinates in every block and
+is absent from this tensor space. This explains how disjoint underlying
+supports coexist with dependent response rows.
+
+## 3. What the actual product queries read
+
+The prior note's section 5 constructs a **single two-new-row rank experiment**
+with deterministic masks for every disjoint A,B, whose final NN occupied
+rank on the `w x 5` torus, `w>=5k`, equals `q_(A,B)` pointwise. Averaging that
+same experiment over this stochastic preparation yields (2.1). No new
+geometric argument or graph enumeration is needed here.
+
+For distinguishing u=0 from u=1 with v fixed, the only informative column
+in this product family is `A=all, B=empty`. Its two Bernoulli output laws
+have total variation
+
+    TV_product = 2^(1-k),
+    optimal equal-prior single-query error = (1-2^(1-k))/2.    (3.1)
+
+All other product queries have identical laws for the two u values.
+Choosing a query randomly independently of the logical bit, even while
+reporting the query label, cannot improve this maximum. The v statement
+is symmetric. Although linear inversion of (2.2) recovers `sigma_u` from
+the exact response expectation, that inversion is not a single Boolean
+measurement or a reliable single-preparation decoder.
+
+### An exact three-label positive model for this single-query family
+
+There is a stronger operational distinction than rank alone. Put
+`a=1{u=k mod 2}`, `b=1{v=k mod 2}`, and `c=2^(1-k)<=1/2`. Encode the
+logical mixed preparation into three latent labels with probabilities
+
+    pi_(u,v) = (1-(a+b)/2, a/2, b/2).
+
+For the all-U query use success probabilities `(0,2c,0)` on these labels;
+for all-V use `(0,0,2c)`. Every other product query has a logical-state
+independent success probability p by (2.1); use `(p,p,p)`. All encoding
+and response entries are nonnegative and normalized as probabilities.
+The resulting binary output law agrees with every encoded product query.
+The latent labels are statistical labels, not asserted physical preparations.
+Thus the minimum common normalized **initial** memory for this family of
+single experiments is exactly three: the rank gives a lower bound of
+three and this explicit construction attains it. Disjoint physical
+supports do not preserve the old positive-versus-linear gap here.
+
+This is not a recursive model for growth, nor a model of multiple
+correlated measurements of the same sampled configuration. It also does
+not treat every bounded linear combination of response expectations as
+an additional allowed Boolean measurement: a signed linear combination
+can take latent response probabilities outside [0,1]. The construction
+serves the actual product-query columns and their randomized choices.
+
+### Fresh independent encodings have an explicit repetition cost
+
+One explicit operational cost: repeat the all-U product on n **fresh,
+independently randomized encodings** of the same logical state. Under one
+hypothesis all outcomes are zero; under the other each is Bernoulli with
+success probability `c=2^(1-k)`. The optimal equal-prior error is exactly
+`(1-c)^n/2`. To reach error eta<1/2 requires and suffices
+
+    n >= ceil(log(2 eta) / log(1-c)).                    (3.2)
+
+For fixed eta this is order `2^(k-1)`. Repeating a noiseless measurement on
+the same sampled preparation is not a new independent encoding and cannot
+be substituted into this calculation.
+
+### The physical full-language boundary remains
+
+The constructed physical product family has rank exactly three after
+encoding. A larger actual future-query family has rank **at least three
+and at most four**; proving that its columns all lie in the local tensor
+space would be an additional upper-bound input. The earlier physical
+product construction supplied inclusion, not that upper bound.
+
+For example, the abstract monotone function
+`f(U,V)=product_i U_i V_i` would have encoded expectation
+
+    2^(-2k)[1+(-1)^k sigma_u][1+(-1)^k sigma_v],
+
+and would restore rank four together with (2.2). It uses both coordinates
+per block, and is **not** asserted to be a realizable future rank query in
+the original geometry. It demonstrates why monotonicity alone cannot
+prove the full-physical-language rank-three upper bound.
+
+## 4. Sharp single monotone readout bound
+
+Let `mu_0,mu_1` be the uniform even/odd laws on `{0,1}^k`. Let
+`f:{0,1}^k -> {0,1}` be coordinatewise nondecreasing, and observe just
+`Y=f(U)`. The following result holds for every k>=1:
+
+    TV(Law_mu0(Y),Law_mu1(Y))
+      = |E_mu0 f - E_mu1 f| <= b_k,
+    b_k = binom(k-1,floor((k-1)/2)) / 2^(k-1).          (4.1)
+
+The constant is attained by a Hamming-weight threshold. Hence the optimal
+equal-prior error among **all abstract monotone Boolean readouts** is
+`(1-b_k)/2`; the decoder may reverse its binary decision as appropriate.
+Attainment in this abstract class is not a claim that the threshold can
+be implemented by the allowed physical rank queries.
+
+### Proof by layer averaging
+
+Let `a_j` be the mean of f over vectors of weight j. Couple a uniform
+j-subset to a uniform `(j+1)`-subset by adding a uniformly chosen missing
+coordinate. Monotonicity implies
+
+    0 <= a_0 <= a_1 <= ... <= a_k <= 1.
+
+Set `d_m=a_m-a_(m-1)>=0`, so `sum_(m=1)^k d_m<=1`. The constant layer term
+cancels, and the elementary alternating binomial tail identity gives
+
+    sum_(j=0)^k (-1)^j binom(k,j) a_j
+      = sum_(m=1)^k d_m sum_(j=m)^k (-1)^j binom(k,j)
+      = sum_(m=1)^k d_m (-1)^m binom(k-1,m-1).         (4.2)
+
+The identity follows directly from Pascal's rule, whose two tails cancel
+except for their first term. The absolute value of (4.2) is at most the
+largest coefficient `binom(k-1,floor((k-1)/2))`. Dividing by `2^(k-1)`
+gives the difference of the two conditional expectations and proves (4.1).
+
+For `f=1{|U|>=m}`, only `d_m=1` is nonzero, and its signed difference is
+`(-1)^m binom(k-1,m-1)/2^(k-1)`. Choose m with m-1 a middle index to attain
+the bound. No regularity, symmetry of f, or enumeration is assumed.
+The same proof works for monotone functions into [0,1] interpreted as
+success probabilities of a randomized binary output.
+
+For k=2,3,4 the sharp TVs are respectively `1/2,1/2,3/8`. By the central
+binomial estimate, `b_k ~ sqrt(2/(pi*k))`; even the best monotone binary
+readout loses all constant advantage as k grows. This polynomial loss
+should not be confused with the stronger exponential loss (3.1) for the
+explicit product family.
+
+### Conditioning on V and on independent query noise
+
+For a fixed logical v, V is sampled independently of the **entire** U
+vector under both u hypotheses. For each fixed V=z, a Boolean readout
+`f(U,z)` monotone in U satisfies (4.1). Averaging proves the same upper
+bound for Y alone. Even revealing z gives
+
+    TV(Law_0(V,Y),Law_1(V,Y))
+      = E_V |E_mu0 f(U,V)-E_mu1 f(U,V)| <= b_k.        (4.3)
+
+The same argument permits independent public query randomization or
+independent future-site randomness, conditional on which the output is
+monotone in U. V need not itself be observed or have a product distribution
+across its coordinates; its fixed-parity law is allowed.
+
+**Important side-information condition:** merely having a marginal law
+independent of u is insufficient. The side variable
+`W=(U_1,...,U_(k-1))` has the same uniform law under both parities, but
+observing W together with the monotone output `Y=U_k` recovers u exactly.
+Conditioning here requires independence from the sampled U, as our V and
+the independent query random seeds have. Revealing the encoder's private
+seed would change the experiment.
+
+### Consequence for a single final physical rank
+
+Fix the appended row masks and V. Adding old U sites cannot decrease the
+occupied NN ambient rank on that fixed final torus. Because the old first
+row is empty, the rank is Boolean (0 or 1). Thus every **single final-rank,
+nonadaptive** query with future-site randomness independent of the old
+sample satisfies (4.1), at any fixed chosen height, even if its response
+does not lie in the declared tensor span. This applies to the original
+deterministic or independently noisy prescribed source experiments.
+
+It does not bound TV of an entire vector of intermediate outputs: several
+monotone coordinates together can reveal parity (the identity vector is
+a simple abstract example). Nor is it a claim about unrestricted feedback
+controllers: fixing their random seed need not leave a monotone final
+function of U when the later commands depend on earlier observations.
+Multiple preparations and their costs are separate contracts as well.
+
+## 5. Hypothetical noisy access to all shares is not a free physical readout
+
+Suppose, under a stronger observation model, all k U bits can be read with
+independent bit errors `E_i~Bernoulli(epsilon)`, `0<=epsilon<=1/2`, and one
+observes `Y=U xor E`. Parity decoding then has exact error
+
+    Pr[xor_i Y_i != u] = [1-(1-2epsilon)^k]/2.         (5.1)
+
+Indeed `E[(-1)^(sum E_i)]=(1-2epsilon)^k`, which is the even-minus-odd
+error probability. More precisely, putting `a=(1-2epsilon)^k`, the full
+noisy-share output law is
+
+    Pr[Y=y | u] = 2^(-k)[1+(-1)^u a (-1)^(|y|)].     (5.2)
+
+Its even/odd TV is exactly a, so parity decoding is Bayes optimal for
+equal priors in this stronger model. For any fixed `epsilon>0` it tends
+to a fair guess as k grows. Noiseless full-share access gives TV=1; it is
+not a monotone Boolean readout.
+
+If only a monotone Boolean function of the noisy shares may be read,
+(5.2) scales every difference in section 4 by a. The sharp TV is therefore
+`a*b_k`, again attained by a weight threshold in the abstract class.
+This is distinct from (5.1), which uses the entire vector and a nonmonotone
+parity decoder.
+
+Independent future instruction flips in a physical query are not, without
+a separate channel construction, independent BSC errors on measured old
+shares. Neither (5.1) nor (5.2) is claimed as that physical channel's law.
+
+## 6. One bounded exact control and interpretation
+
+The [standard-library checker](../analysis/stochastic-parity-block-encoding-20260929/check.py)
+performs one run at k=2,3,4. It enumerates the four finite code supports,
+all `3^k` tensor monomials, and the monotone Boolean functions generated
+recursively from sections `f_0<=f_1`; it does not scan a larger encoder
+space or rerun the previous deterministic census. Integer counts and
+`Fraction` arithmetic check the moments, rank, private marginals, sharp
+TVs, threshold equality, and the separate BSC formulas at epsilon=1/10
+and 1/4. All-k statements rely on the displayed proofs.
+The one run checked 468 tensor expectations and 194 monotone Boolean
+functions, with zero failures; the three computed tensor ranks were all 3.
+The three-label positive construction above is a displayed algebraic proof,
+not an extra computation or an output of this checker.
+
+[result.json](../analysis/stochastic-parity-block-encoding-20260929/result.json)
+records both the computed scope and the missing full physical-language
+upper bound. Run from the repository root:
+
+```sh
+python3 analysis/stochastic-parity-block-encoding-20260929/check.py
+```
+
+The four mixed preparations have exact tensor response dimension three
+at width `w>=5k` and two new query rows; no constant single-readout
+reliability is gained by increasing k. The constructive lesson is narrow
+but useful: **randomness evades the deterministic rank obstruction, while
+a single monotone Boolean readout cannot make this particular code a
+large-k reliable single-shot replacement.** This is not a general
+stochastic-encoder impossibility theorem. It does not settle arbitrary
+future-query rank, adaptive transcript decoding, full-language positive memory,
+or a different geometry. No cloud work, navigation changes, full-machine
+audit, or novelty certification accompanies this sidecar.
diff --git a/notes/two-time-pivotal-density-bound-20260929.md b/notes/two-time-pivotal-density-bound-20260929.md
new file mode 100644
index 000000000..fdb110a2b
--- /dev/null
+++ b/notes/two-time-pivotal-density-bound-20260929.md
@@ -0,0 +1,388 @@
+# Two-time pivotal density: an arm-based route to distinct limiting births
+
+2026-09-29. Consumer: #778 / #773. Author derivation, not an independently
+reviewed theorem or a claim of literature novelty. No new simulation is used.
+
+**Main result.** On a fixed nondegenerate torus, two different rank-changing
+sites have a locally uniformly bounded **joint near-critical time density**,
+provided macro-pivotality forces four alternating arms and the usual
+near-critical arm estimates hold. The proof uses independent labels in
+disjoint spatial annuli, not independence of the two births. Together with
+six-arm suppression of a direct double birth and tightness of the birth pair,
+this excludes coalescence of two distinct insertions in a scaling limit.
+
+The triangular-site corollary below supplies the probability inputs from
+standard near-critical theory. The square-site statement remains explicitly
+input-conditional here: the precise 4/8 arm contract must be carried through,
+but **an exact square four-arm exponent is not required**. The result does not
+identify the limiting copula, prove square/triangular universality, or identify
+the original-U operator.
+
+This advances the earlier [soft-gap criterion](birth-gap-noncoalescence-20260929.md):
+the two-time upper bound is now derived, rather than merely requested.
+
+## 1. Model, clock and explicit hypotheses
+
+Let a fixed flat torus have positive systole and finite area. Approximate it
+by a periodic planar site lattice of mesh `eta`, with `O(eta^-2)` sites and
+the usual `O(rho^2/eta^2)` local packing bound. Work in embedded disks smaller
+than the injectivity radius. The torus shape is fixed, not an arbitrarily
+thinning family.
+
+Give each site an independent uniform label `U_x`. The occupied graph at p
+contains exactly sites with `U_x<=p` and all their lattice edges. Let
+
+```text
+r_eta(p) = rank im[H1(occupied graph) -> H1(torus)],
+T_i      = inf{p:r_eta(p)>=i},                 i=1,2,
+a_eta    = eta^2 / alpha_4(eta,1),
+lambda_i = (T_i-p_c)/a_eta.
+```
+
+Here `alpha_j` is the critical probability of a specified alternating arm
+event, with a fixed microscopic inner convention. Changing that convention
+or the fixed reference radius 1 changes comparison constants. Filling
+contractible occupied faces does not change this ambient image rank.
+
+For each finite `Lambda`, assume:
+
+**A1 — deterministic local importance.** A site that changes ambient rank
+has four alternating arms in its punctured neighbourhood to a fixed radius
+`R=R_Lambda>0`. A direct rank `0->2` update has at least six alternating arms
+to that radius. The radius can be decreased; it does not depend on eta.
+
+**A2 — compact-window arm stability.** For
+`p=p_c+u*a_eta`, `|u|<=Lambda`, four- and six-arm probabilities in embedded
+annuli with outer radius at most R are bounded above by a constant times
+their critical probabilities, uniformly over the inner radius. Conditioning
+the centre site's label does not alter an arm event outside the centre.
+
+**A3 — critical quasi-multiplicativity and integrability.** Up to fixed
+changes of radii, `alpha_4(r,t)` is comparable to
+`alpha_4(r,s)*alpha_4(s,t)`. For some `kappa>0`,
+
+```text
+alpha_4(r,R) >= c_R (r/R)^(2-kappa),     eta<=r<=R.       (1)
+```
+
+It is enough to assume directly that the dyadic sum
+`sum_{rho=R*2^-j, rho>=eta} rho^2/alpha_4(rho,R)` stays bounded.
+No equality with a power law is needed.
+
+**A4 — direct-jump suppression.**
+
+```text
+e_eta(R) := alpha_6(eta,R)/alpha_4(eta,1) -> 0.           (2)
+```
+
+**A5 — birth-pair tightness.** The pairs `(lambda_1,lambda_2)` are tight on
+the finite real plane. Section 5 supplies a sufficient finite-size argument;
+this is not inferred from a finite Monte Carlo window.
+
+## 2. The two-time bound
+
+### Proposition: uniformly bounded off-diagonal density
+
+Under A1--A3, the law of `(lambda_1,lambda_2)` restricted to different birth
+sites has a density `f_eta(s,t)` on `s<t`. For every finite Lambda,
+
+```text
+0 <= f_eta(s,t) <= C_Lambda,
+              -Lambda<=s<t<=Lambda,                    (3)
+```
+
+uniformly for sufficiently small eta. A1--A4 also give
+
+```text
+P(lambda_1=lambda_2 in [-Lambda,Lambda])
+    <= C_Lambda e_eta(R).                               (4)
+```
+
+Consequently, for `0<delta<=1`,
+
+```text
+P(0<=lambda_2-lambda_1<=delta,
+  lambda_1,lambda_2 in [-Lambda,Lambda])
+    <= C_Lambda delta + C_Lambda e_eta(R).               (5)
+```
+
+The constants depend on the window and geometry. This is not a global
+Lambda-independent density or tail bound.
+
+### Proof: fix the two labels before discarding the birth conditions
+
+For distinct sites x,y, condition on
+`U_x=p_c+s*a_eta`, `U_y=p_c+t*a_eta`, with `s<t`. Their joint label density
+in `(s,t)` is exactly `a_eta^2`. All other labels remain independent uniforms.
+The ordered birth-site events partition the different-site event, so
+
+```text
+f_eta(s,t) = a_eta^2 * sum_{x!=y}
+  P(x causes birth 1 and y causes birth 2 | U_x=p(s), U_y=p(t)).   (6)
+```
+
+Conditional probabilities at prescribed label values can be defined by
+integrating over the remaining finite set of labels; (6) holds almost
+everywhere and gives a density representative with the asserted bound.
+
+Let `rho=dist(x,y)`, initially `eta << rho << R`. The first pivotality
+requires four arms around x at time s; the second requires four arms around
+y at time t. Retain only these necessary subevents:
+
+1. four arms around x from microscopic scale to `rho/8`, at s;
+2. four arms around y from microscopic scale to `rho/8`, at t;
+3. four arms around x from `2rho` to R, at s.
+
+Their supports are spatially disjoint and avoid **both conditioned sites**.
+Each is a function of the labels in its own support. They are therefore
+independent even though the first and third use one threshold and the second
+uses another. Arm stability and fixed-radius comparisons bound (6)'s
+conditional probability by
+
+```text
+C_Lambda alpha_4(eta,rho)^2 alpha_4(rho,R)
+    <= C_Lambda alpha_4(eta,R)^2 / alpha_4(rho,R).        (7)
+```
+
+This is not a factorisation of two overlapping global pivotal events, and
+not a Markov approximation of the rank history. No new mixed-time arm theorem
+is smuggled into A2: only single-time arm events on disjoint sets are used.
+
+For `rho` within a fixed factor of eta, omit the two inner events and use the
+outer event starting at a sufficiently large fixed multiple of eta. Its
+bound agrees with (7) up to constants. For `rho>=R/8`, retain disjoint
+four-arm disks of a smaller fixed radius around x and y. Their product gives
+the same required uniform far-pair contribution.
+
+For a close dyadic distance bin, the number of ordered pairs is at most
+`C eta^-4 rho^2`. Substituting (7) in (6) and using
+`alpha_4(eta,R) <= C_R alpha_4(eta,1)` gives
+
+```text
+f_eta(s,t)
+  <= C_Lambda [1 + sum_{dyadic eta<=rho<=R}
+                          rho^2/alpha_4(rho,R)]
+  <= C_Lambda [1 + C_R sum_j (R*2^-j)^kappa]
+  <= C'_Lambda.                                        (8)
+```
+
+The cancellation of `eta^-4` against `a_eta^2 alpha_4(eta,1)^2=eta^4`
+is the useful clock normalization. The spatial singularity is integrable
+because the four-arm power is strictly below two.
+
+A same-site double birth has only one label density. Sum its six-arm bound
+over sites and integrate over the time window:
+
+```text
+O(eta^-2) * O(Lambda*a_eta) * C_Lambda alpha_6(eta,R)
+    <= C'_Lambda e_eta(R).                             (9)
+```
+
+Finally integrate (3) over the triangular band `0<t-s<=delta` inside the
+window, whose area is at most `2Lambda*delta`, and add (9). This proves
+(3)--(5).
+
+## 3. Limiting consequences and the remaining unidentified object
+
+Under A1--A5, every weak subsequential birth-pair limit has a locally bounded
+two-dimensional density and is supported on `lambda_1<lambda_2`. To see the
+density assertion, restrict to a compact square: the off-diagonal measures
+are dominated by `C_Lambda` times area, while the diagonal measure tends to
+zero by (4). Domination passes to the weak limit, using a slightly larger
+square at its boundary. Tightness prevents mass escaping to infinity.
+
+Equivalently, if `G_eta=lambda_2-lambda_1` and
+`tau_eta(Lambda)=P((lambda_1,lambda_2) outside [-Lambda,Lambda]^2)`, then
+
+```text
+E[(1-G_eta/delta)_+]
+   <= tau_eta(Lambda) + C_Lambda delta + C_Lambda e_eta(R).   (10)
+```
+
+Take `limsup_eta->0`, then `delta->0`, then `Lambda->infinity`. The result is
+zero, supplying exactly the earlier noncoalescence criterion. One must not
+exchange those limits or erase the tail term to claim a global O(delta) law.
+
+There is also a path consequence: along a convergent birth-pair subsequence,
+the two-jump rank paths converge in Skorokhod J1 on compact intervals whose
+endpoints are not limiting birth times. Distinct ordered jumps can be matched
+by a small time change. The local density makes every fixed endpoint a
+non-birth almost surely. Thus the resulting subsequential paths have two
+separate unit jumps, not a merged jump of size two.
+
+**Not supplied:** uniqueness of that subsequential law; its copula or density
+formula; a Markov property; a rank-one direction-mark limit; an explicit
+constant for the measured soft mass. Distinct births do not determine their
+dependence. Identification of the torus observable in the continuum process
+is a separate, more informative next theorem.
+
+## 4. Why the local arm hypotheses fit triangular ambient rank
+
+Use regular triangular-site percolation, whose white matching connectivity
+is also triangular NN connectivity. All local disks below are embedded in
+the torus and have lattice polygon boundaries, up to harmless fixed changes
+in radius.
+
+**Four arms for any rank change.** Delete the centre x and examine black
+connections inside a disk D around it. If all black branches incident to x
+which reach the boundary of D are connected to each other within `D\{x}`,
+every traversal through x in a global black cycle can be replaced by a
+black traversal in that one local component. Branches not reaching the
+boundary only add cycles supported in D. The replacement differs from the
+original cycle by cycles in a disk, which have zero ambient homology.
+Hence opening x cannot change ambient rank in this case.
+
+A rank change therefore requires at least two different components in the
+punctured disk, each incident to x and reaching its outer boundary. Select
+two disjoint black representatives. Planar separation in the two intervening
+strips supplies a white path from the microscopic hole to the outer boundary
+in each strip. They are disjoint and alternate with the black paths: four
+arms. This argument concerns local components, which can be connected outside
+D; it does not wrongly require two different global clusters.
+
+**Six arms for a direct double birth.** Before such a birth the ambient
+black rank is zero. Each lifted black component and its nonzero deck
+translates are disjoint. The rank increment from adjoining x is generated
+by differences of deck offsets among its contacts in each base component.
+Two independent differences require either three different lifted copies
+of one component, or two pairs from different components. The attachment
+paths joining the corresponding translated neighbourhoods reach a fixed
+fraction of the systole; their initial branches give respectively three or
+four different lifted black components to that scale. Alternating white
+separators give six or eight arms, and eight implies six at the macroscopic
+annular level. The same offset argument works for six triangular neighbours;
+it does not use the square degree-four restriction.
+
+These are deterministic disk/lift arguments, not an extrapolation from the
+square L3--L5 census. The existing square-NN/white-matching version is recorded
+in #773's [attachment/arm note](https://github.com/LightChainr/Matching-One/blob/9ef22d1cc531c09dedf9b1195ab6aad8ef79668f/notes/rank-jump-two-arm-extraction-20260914.md)
+and #823's [separator proof](https://github.com/LightChainr/Matching-One/blob/3d0176a9f394f7d7228766709731493fd15095c5/analysis/queue-20260919/815/NOTE.md).
+The triangular argument uses ordinary planar white paths; it does not identify
+crossing square diagonals as disjoint white arms.
+
+## 5. Probability inputs and triangular corollary
+
+Two primary papers were read with the arXiv retrieval skill:
+
+- Pierre Nolin, [Near-critical percolation in two dimensions](https://arxiv.org/abs/0711.4948),
+  DOI [10.1214/EJP.v13-565](https://doi.org/10.1214/EJP.v13-565).
+  In the downloaded arXiv text: Theorem 26 (near-critical arm comparison),
+  Theorem 20 (polychromatic exponents), Proposition 32 / (7.13) (finite-size
+  relation), Lemma 37 (uniform subcritical exponential decay), and section 8.1
+  (general-lattice scope). Numbering differs in the journal version.
+- Garban, Pete and Schramm,
+  [The scaling limits of near-critical and dynamical percolation](https://arxiv.org/abs/1305.5526),
+  [published JEMS version](https://ems.press/journals/jems/articles/15407),
+  DOI 10.4171/JEMS/786. Definition 1.2 uses an exponential-clock convention;
+  section 10.2 gives correlation-length control, and section 11.2 discusses
+  near-critical arms. Its quad-crossing path convergence is **not** quoted as
+  an already-proved torus-rank path continuity theorem.
+
+### Local estimates
+
+For a compact time window, choose R smaller than the torus injectivity radius
+and the minimum physical characteristic length at its two endpoints. R can
+depend on Lambda. Arm comparison then gives A2 without extrapolating a
+below-correlation-length theorem to arbitrarily large radii. Standard critical
+quasi-multiplicativity and the four-arm bound give A3. For the triangular
+model, the four-/six-arm exponents yield
+
+```text
+e_eta(R) = eta^(35/12-5/4+o(1)) = eta^(5/3+o(1)).        (11)
+```
+
+Only its vanishing is consumed. This is a **direct-atom error bound**, not a
+newly fitted gap exponent. GPS's weaker six-arm estimate also suffices.
+For their exponential clock, passing to the affine p-clock used here has
+derivative bounded above and below on each fixed window for small eta
+(at p_c=1/2 its limiting time factor is 2). No clock equality is assumed.
+
+### Tightness from finite-size decay, not from mean gap
+
+Here is the torus reduction needed for A5. Write `n=eta^-1`, and let `m` be
+the subcritical characteristic length in lattice units at
+`p=p_c-Lambda*a_eta`. The finite-size relation and quasi-multiplicativity give,
+when `m<=n`,
+
+```text
+1 asymp Lambda * (m/n)^2 / alpha_4(m,n).                 (12)
+```
+
+Usual critical annular bounds
+`c s^(2-kappa) <= alpha_4(sn,n) <= C s^beta`, with
+`0<beta<2`, imply
+
+```text
+c Lambda^(-1/kappa) <= m/n <= C Lambda^(-1/(2-beta)).    (13)
+```
+
+Constants can be enlarged for bounded Lambda. For large Lambda the same
+finite-size relation forces `m<n`; otherwise `m^2 alpha_4(1,m)` is, by the
+four-arm lower bound and quasi-multiplicativity, at least a fixed multiple
+of `n^2 alpha_4(1,n)`, contradicting (12)'s defining relation at large Lambda.
+Thus the physical characteristic length stays positive on each fixed window
+and tends to zero as Lambda grows. Exact 4/3 scaling is unnecessary here.
+
+Cover the fixed torus by finitely many embedded charts of a fixed radius
+smaller than its systole. An essential black cycle must cross one of a finite
+collection of annuli in these charts. Such a crossing forces a crossing of
+one of finitely many rectangles with fixed positive dimensions. Uniform
+subcritical exponential decay therefore makes its probability tend to zero
+as `m/n->0`. There is no union over `eta^-2` individual starting sites.
+It follows that
+
+```text
+lim_{Lambda->infinity} limsup_{eta->0} P(lambda_1<-Lambda)=0.
+```
+
+For triangular site percolation, complementary ambient ranks sum to two and
+the white graph has the same lattice. Applying the same argument to white
+at `1-p` gives the upper tail `P(lambda_2>Lambda)`. The other two tails are
+smaller by ordering. This proves A5.
+
+**Triangular corollary (author proof using the stated standard inputs).**
+On any fixed nondegenerate triangular torus approximation, the near-critical
+birth pairs are tight; every subsequential limit has a locally bounded
+two-dimensional density and almost surely two distinct finite births.
+The rank paths consequently have only unit jumps in those subsequential
+limits. Unique continuum law is not asserted.
+
+### Square-site consequence is conditional, but exact exponents are not the gap
+
+**Follow-up:** [Square birth transfer and the population-IQR clock](square-birth-transfer-and-iqr-clock-20260929.md)
+supplies the convention/standard-input transfer and the nondegenerate population
+width comparison left open in this section and item 3 below. The original
+conditional formulation is retained here to separate the two proof steps.
+
+For square black NN / white matching connectivity, the same theorem consumes
+the actual alternating 4/8 arms. Nolin's general-lattice discussion indicates
+the relevant RSW-based ingredients, including four-arm power below two and
+six-arm power above two. It is therefore too strong to say that this strategy
+must wait for a proved square exponent of 5/4 or square conformal invariance.
+
+This note does not silently identify that general discussion with every
+endpoint and planarised-white convention in #823. A square upgrade should
+state the arm comparisons and quasi-multiplicativity under that convention,
+and use the square/matching complementary finite-size tails. Once those
+inputs are supplied, (6)--(10) require no new two-time argument. It would
+prove noncoalescence on the model's own pivotal clock, **not** equality with
+the triangular clock, amplitudes or limiting copula.
+
+## 6. What changes in the research allocation
+
+1. The triangular positive-control target is now a proof-level density result,
+   not just a request for more samples or a numerical extrapolation.
+2. The next joint-process work is the square 4/8 input transfer and continuum
+   torus-observable identification / uniqueness. This is narrower than
+   reproving the entire near-critical construction.
+3. The measured width-normalized gap uses the mixture IQR of `(J1,J2)`.
+   Equating it with the canonical pivotal clock requires a nondegenerate
+   width comparison; this note does not insert an unmeasured conversion
+   constant. The existing pilot is compatible evidence, not proof of (3).
+4. Do not infer a Markov process, independent births, a universal density,
+   or an original-U mechanism from this anti-concentration result.
+
+No Monte Carlo block, exact torus census, old test suite or cloud session was
+run for this derivation. External literature was retrieved to check the
+actual probability inputs. The arXiv search API did not return promptly;
+known-paper retrieval succeeded, so no completeness-of-search claim is made.
diff --git a/scripts/check_birth_gap_counterexample.py b/scripts/check_birth_gap_counterexample.py
new file mode 100644
index 000000000..d88ac1701
--- /dev/null
+++ b/scripts/check_birth_gap_counterexample.py
@@ -0,0 +1,77 @@
+#!/usr/bin/env python3
+"""One cheap exact check for the birth-gap note; no percolation simulation."""
+
+from fractions import Fraction as F
+from math import comb
+import json
+
+
+def average(pairs, fn):
+    return sum((fn(*pair) for pair in pairs), F(0)) / len(pairs)
+
+
+def beta_cdf(alpha, beta, x):
+    n = alpha + beta - 1
+    return sum((comb(n, k) * x**k * (1 - x)**(n - k)
+                for k in range(alpha, n + 1)), F(0))
+
+
+def beta_soft_direct(n, d, a, delta):
+    """Integrate (1-a*t/delta) times beta density, expanded as a polynomial."""
+    if d == 0:
+        return F(1)
+    x = min(delta / a, F(1))
+    return d * comb(n, d) * sum((
+        (-1)**k * comb(n - d, k)
+        * (x**(d + k) / (d + k)
+           - a / delta * x**(d + k + 1) / (d + k + 1))
+        for k in range(n - d + 1)), F(0))
+
+
+def main():
+    counterexamples = []
+    for n in (2, 4, 16, 64):
+        a = [(F(0), 1 + F(1, n)), (F(1), F(2))]
+        b = [(F(0), F(2)), (F(1), 1 + F(1, n))]
+        assert sorted(s for s, _ in a) == sorted(s for s, _ in b)
+        assert sorted(t for _, t in a) == sorted(t for _, t in b)
+        assert all(s < t for s, t in a + b)
+        endpoints = sorted({x for pair in a + b for x in pair})
+        for t in endpoints:
+            for rank in range(3):
+                assert average(a, lambda s, u: F(int((s <= t) + (u <= t) == rank))) == average(
+                    b, lambda s, u: F(int((s <= t) + (u <= t) == rank)))
+        assert average(a, lambda s, t: t - s) == 1 + F(1, 2*n)
+        assert average(a, lambda s, t: t - s) == average(b, lambda s, t: t - s)
+        delta = F(1, 4)
+        z_a = average(a, lambda s, t: max(F(0), 1 - (t-s)/delta))
+        z_b = average(b, lambda s, t: max(F(0), 1 - (t-s)/delta))
+        assert z_a == 0
+        assert z_b == max(F(0), F(1, 2) * (1 - F(4, n)))
+        counterexamples.append({"n": n, "Z_A_quarter": str(z_a), "Z_B_quarter": str(z_b)})
+
+    n = 9
+    hist = {0: F(1, 5), 1: F(1, 5), 3: F(2, 5), 7: F(1, 5)}
+    overlap = lambda m: sum((h * max(d - m, 0) for d, h in hist.items()), F(0))
+    for d in range(1, n + 1):
+        assert overlap(d-1) - 2*overlap(d) + overlap(d+1) == hist.get(d, 0)
+    for m in range(1, n + 1):
+        assert 1 - (overlap(0)-overlap(m))/m == sum(
+            (h * max(F(0), 1-F(d, m)) for d, h in hist.items()), F(0))
+    assert 1 - (overlap(0)-overlap(1)) == hist[0]
+
+    scale, delta = F(2), F(3, 4)
+    x = min(delta/scale, F(1))
+    soft = F(0)
+    for d, weight in hist.items():
+        closed = F(1) if d == 0 else (
+            beta_cdf(d, n+1-d, x)
+            - scale/delta * F(d, n+1) * beta_cdf(d+1, n+1-d, x))
+        assert closed == beta_soft_direct(n, d, scale, delta)
+        soft += weight * closed
+    print(json.dumps({"status": "passed_exact_fraction_checks", "counterexamples": counterexamples,
+                      "beta_mixture_soft_mass": str(soft), "direct_atom": str(hist[0])}, indent=2))
+
+
+if __name__ == "__main__":
+    main()