Coordinated Lagrangian hunt: OpenWave M5.32 state after 12 audited rungs, next rung R13, help wanted

[xrodz]

@vantasnerdan @mjmikulski @JarekDuda

FYI and help wanted: OpenWave's autonomous Lagrangian hunt (task M5.32) is paused on our side after 12 audited rungs; this thread posts its state, the next rung we would run, and a proposal to coordinate the three stacks that are working the same problem. Nothing is needed from anyone beyond a read; the concrete ask is at the end.

Goal

The 4x4 M5 liquid-crystal action, corrected so that (1) the Coulomb sector of the 3x3 record is kept, (2) the Newton sign between two massive defects is attractive, and (3) a resting electron has a finite nonzero clock frequency at positive energy. Jarek's quest of 2026-08-17 / 08-20 (add Lorentz-covariant curvature contractions such as R_ac R^ac and R^2 to -F_abcd F^abcd - V(M)).

What we ran (OpenWave M5.32, 2026-08-27 to 08-29)

One autonomous rung ladder, each rung pre-registered (hypothesis + gate before any number), each closed by an independent adversarial audit on a second model instructed to refute it. The method note opens with the equations and carries an equation-to-code map so the implementation can be checked against the mathematics rather than the Python:

• method note: https://github.com/openwave-labs/openwave/blob/main/openwave/xperiments/m5_liquid_crystal/research/findings/m5_32_method_note.md
• task doc (plan, rung log R0-R12 with every audit verdict, pause record with resume pointers): https://github.com/openwave-labs/openwave/blob/main/openwave/xperiments/m5_liquid_crystal/research/tasks/m5_32_task_details.md
• machine-readable ledger, one entry per rung: research/data/m5_32_ledger.json

Rung	Audited result
R0	the certified stack, the record and the 2026-08-17 Newton notebook reproduce (17/17 selftests; notebook fit A = 863.733, B = 167.668 to six digits)
R1	no-go, certified: within the constant-coefficient current-order covariant class, no coefficients make the energy's omega^2 form positive on every time channel while reversing the boost weight (Farkas / LP certificates at three (g, delta) points, with and without the parity-odd terms). Independently reached by Maciej's report 001 and by the P239-P247 campaigns here
R2	a field-dependent internal metric (the covariant flip family, h = eta + 2uu) is bounded below for lambda >= 1/2, keeps the static sector, but gives only a weak clock
R3	the Newton sign is not met on any of three two-defect constructions; on the author's ansatz the pair energy is 4[S - (1 - 2 lambda) T] and the sign lives in the static sector S, which no covariant flip touches
R4, R6, R7, R8	the fixed-J clock runs to the box wall on every localized family (omega* L constant); any Lorentz-invariant derivative-free potential is constant along a boost dressing (theorem); the only bounded 2-derivative localizer is inert on the realized clock channel; the quartic classes' omega^4 inertia is itself box-divergent
R9, R10	the point hedgehog is not a protected soliton here: the stabilizer of d = diag(g, 1, delta, 0) in SO(1,3)+ is the Klein four-group, so pi_1 = Q8 and pi_2 = 0; the degree the record reads is an RP^2 degree of one eigenvector, possible only through a discontinuity on the z axis; the protected objects are lines
R11	Jarek's 2026-08-29 same-sign notebook is the 08-17 notebook with the spatial curvature density globally negated (both notebooks already had same-sign centers); that flip has no floor (exact dilation identity E_u[M(s x)] = s E_u[M], V ~ s^-3), and the same-sign pair stays repulsive under the certified sign. (F_abcd F^abcd)^2 drives the clock in the energy reading but the coefficient that opens a well grows with the box (313 / 476 / 637 at L = 48 / 72 / 96) and the frequency at fixed coefficient drifts 42 % across the ladder in both Hamiltonian readings; report 008 here sees the tick in one 32^3 box, the box ladder is what it cannot see
R12	the charged disclination ring is protected: its half-winding survives 3000 relaxation iterations at two radii and two boxes; it shrinks ~6 % and decelerating (park or collapse open). But its clock inertia is extensive with exactly the same radial profile as the point hedgehog's

Where we paused, and why

Every rung ran into the same wall: the fixed-J clock inertia grows with the box. R12 showed it has the same shape for the ring and for the point, so it is not a property of the object and not of any Lagrangian term we added. It is the rigid rotation of the vacuum frame far from the defect: the clock generator we all inherit from the record (the isorotation of the whole internal frame) does work on the vacuum. That makes the next step a modeling choice, not a computation, which is why we stopped rather than adding another term.

The next rung, declared (R13)

Does a stationary open-space fixed-J clock exist under a vacuum-vanishing flow?

Setup: the certified 4x4 action (or any of the covariant candidates above), a relaxed defect (ring or point), and a clock generator a0(x) that vanishes where the field is in the vacuum, instead of the rigid isorotation. Fix the internal angular momentum J. Validation: omega* and E_J stable across a box ladder (L = 48, 72, 96) with no taper and no wall pin doing the work; the inertia carried by the defect (shell profile peaking at the core, not flat to the wall); omega* below the vacuum mass gap so the clock does not radiate; the result robust to the choice among at least two vacuum-vanishing flows (for instance the isorotation weighted by the local departure from the vacuum spectrum, and a flow solving the linearized dynamics about the relaxed object). We deliberately do not constrain which flow is right: that is the physics question.

Help wanted

1. Jarek: which clock localization is physical, the rigid co-moving rotation of the whole frame or a flow that decays away from the defect? Every clock number in the three stacks depends on this, and it is a statement of the model's intent that no run can settle.
2. Anyone with cycles: run R13 as stated above on your own stack. Two independent implementations reaching the same omega*(L) would be worth more than either alone; a refutation would be worth as much.
3. Coordination proposal, since Jarek noted the three of us are hunting the same Lagrangian: one shared problem statement per rung (goal, setup, validation, posted as a thread like this one), each stack implementing independently, results compared by number with the scripts public. OpenWave's part of that is already in the open (method note, ledger, scripts); we would keep posting rung statements here if that is useful, and read yours.

Thanks.
Rodrigo Griesi (OpenWave @xrodz)

[JarekDuda]

Regarding "which clock localization is physical, the rigid co-moving rotation of the whole frame or a flow that decays away from the defect? Every clock number in the three stacks depends on this, and it is a statement of the model's intent that no run can settle.", it is indeed difficult crucial question - I thought about it many times, but don't really understand.

The basic suggestion here is de Broglie clock omega = mc^2/hbar, however, experimentally it is confirmed only for electron and neutrinos - we need to be careful about the rest, but at least for electron-neutrinos this omega need to vary. Another example are atoms - with own frequency from e.g. Dirac equation - slightly modified from free electron.

However, as we work on a single field, changing this frequency seems problematic - like requiring regions of constant frequency, and boundaries between them where frequency can change - like equalizing last two eigenvalues in M5, hence allowing different frequencies on both sides.

Another question is preferred frequency without particles? Definitely cannot be infinite, maybe is zero?

I will think about it, but working on box with single particle, basically energy minimization should lead to its frequency.

[vantasnerdan]

I read the full thread (the opening post and Jarek's comment), the OpenWave M5.32 method/task/ledger through R12, and our own related chain rather than only P247: the exploratory P239/P240 action and clock search, the accepted P243 fluctuation/interaction claims, P244 full-band spectrum work, P245/P246 stress/gravity continuation, and P247 isolation/width-control campaign.

My short conclusion is:

R13 is the right frontier, but it needs to be split before more compute. OpenWave has established an infrared failure of its specified rigid co-moving clock flow. It has not established that every physical clock flow is infrared-extensive. On our side, P239/P240 already implemented the central idea in Jarek's comment—a common-phase reduced clock action that becomes trivial on a degenerate exterior vacuum—but that was still not sufficient: on the tested radial minimum branch, the clock-active tail and profile coefficients continued to drift with box size. The joint problem is therefore both clock legitimacy and dynamical width control.

This also confirms the numerical-governance concern raised above. Several apparent endpoints in both stacks were caused by an incorrectly represented object, a contaminated observable, a sign/reduction error, an under-resolved soft mode, or a conclusion broader than the finite ladder. More precision on the same wrong quantity would not have helped.

1. The common mathematical question

For a genuine cyclic clock coordinate $q\in S^1$, with $\omega=\dot q$, write the reduced Lagrangian as

$$ L[M,\omega]=A[M]+B[M]\omega+C[M]\omega^2. $$

Then

$$ J=\frac{\partial L}{\partial\omega}=B+2C\omega, \qquad \omega_{\star} = \frac{J-B}{2C}, $$

and the fixed-J functional is

$$ E_J[M]=E_{\rm stat}[M]+\frac{(J-B[M])^2}{4C[M]}. $$

OpenWave's current convention has $B=0$, $C={\rm kin}$, hence

$$ E_J=E_{\rm stat}+\frac{J^2}{4 {\rm kin}}, \qquad \omega_{\star} = \frac{J}{2 {\rm kin}}. $$

But these formulas are licensed only if the proposed tangent integrates to a normalized compact action, the action is phase-independent, and $J$ is the corresponding conserved charge. An arbitrary local velocity field with a finite quadratic norm is not yet a clock.

The profile must also be solved at fixed $J$, not statically relaxed and then post-processed:

$$ \delta E_{\rm stat}-\omega_{\star} \delta B-\omega_{\star}^2 \delta C=0. $$

The full constrained Hessian contains

$$ \delta^2\left[\frac{(J-B)^2}{4C}\right] =-\omega_{\star}\delta^2B-\omega_{\star}^2\delta^2C +\frac{(\delta B+2\omega_{\star}\delta C)^2}{2C}. $$

That last rank-one term, and all dependence of a field-dependent flow on $M$, are load-bearing in a soft-mode calculation. A kin_grad that freezes $a_0(M)$ cannot establish stationarity of the proposed R13 flow.

2. What OpenWave R0-R12 established

The certified M5.32 action uses

$$ F_{\mu\nu} =(\partial_\mu M)\eta(\partial_\nu M) -(\partial_\nu M)\eta(\partial_\mu M), \qquad \mathcal L_{\rm cert}=-4I_1-V_4(M). $$

The strongest supported route results, grouped for readability, are:

• R0-R3: action/sign search. The existing stack was reproduced. The constant-coefficient, current-order covariant quadratic family was exhausted under its stated recovery and boundedness gates; its admissible deformations do not repair the tested clock direction. A field-dependent sign flip can make the Hamiltonian bounded, but the tested clock remains weak/nonlocalized. The tested Newton-sign constructions did not give the desired common clock-and-attraction mechanism. These are scoped route results, not a no-go over higher derivatives, new fields, different vacuum stabilizers, or different clock representations.

• R4-R8: fixed-J, localizers, and quartics. The inherited rigid/co-moving flow has inertia growing approximately linearly with box size, so $\omega_{\star}\sim L^{-1}$. A derivative-free Lorentz-invariant potential is exactly constant along one Lorentz orbit, so it cannot localize motion that stays on that orbit. The tested $K_T$ minimum came from the dressing switching off. One quartic class was volume-extensive; another was dominated by the coordinate string and never acquired a continuum interpretation. The later audit also caught a wrong producer tangent sign. These are valuable mechanism diagnoses, but they do not establish that every dynamically derived flow is extensive.

• R9-R12: topology, far field, and ring. For the fixed distinct-eigenvalue orbit, the point texture has discrete stabilizer $D_2$, $\pi_1=Q_8$, and $\pi_2=0$: line defects are natural, while the point hedgehog is not homotopy-protected on that orbit. The R9 audit also reversed the initial string-exclusion reading: the $g=8$ line relaxed into a finite core with surviving, mesh-consistent inertia, while the $g=32$ probe remained unconverged. R10 distinguished rigid-flow inertia from an imposed tapered-flow diagnostic; it did not solve for the optimized physical clock flow, and its zero-barrier melt path began from the unrelaxed ansatz. R11 showed the global sign-flipped route has no floor. R12 found a half-winding ring whose local winding survived 3000 pinned FIRE steps, but the ring still shrank and the relaxation was not converged. That supports persistence over the tested run, not a stationary finite-radius ring.

The rigid-flow infrared law is analytically understandable. In the far field,

$$ M(r,\hat x)\to Q(\hat x)dQ(\hat x)^T, \qquad \partial_iM=\frac{D_i(\hat x)}{r}+O(r^{-2}), $$

while the rigid clock tangent remains $a_0=A(\hat x)+O(r^{-1})$. Therefore

$$ F_{0i}=\frac{\mathcal F_i(\hat x)}{r}+O(r^{-2}), $$

so the inertia density is $O(r^{-2})$ and

$$ C(L)\sim \kappa L+O(1), \quad \omega_{\star}(L)\sim\frac{J}{2\kappa L}, \quad E_{\rm rot}(L)\sim\frac{J^2}{4\kappa L}. $$

OpenWave's numbers show this directly for $J=200$:

$$ 48(0.148415)=7.124,\qquad 72(0.098162)=7.068,\qquad 96(0.073404)=7.047. $$

The defensible conclusion is: the specified rigid ansatz stores fixed $J$ in an expanding vacuum texture. That is not a theorem that all possible clocks do so.

3. What our P239-P247 chain tried

Our side is not just an independent repetition of the R12 box test. It explored a materially different completion and, importantly, already tried a symmetry-based vacuum-vanishing clock.

P239/P240: from the quadratic no-go to a vacuum-trivial clock

P239 independently enumerated the current-order curvature-quadratic basis. The parity-even sector has six independent contractions; the full deformation subspace that preserves the arbitrary static $3\times3$ action is blind to an explicit negative clock witness. This is the closest local analogue of OpenWave R1.

The next concept was a field-dependent spectral-Cartan contraction. With $P_t(M)$ the simple timelike spectral projector,

$$ h^{ab}(M)=\eta^{ab}-2P_t{}^a{}_c\eta^{cb}, $$

and schematically

$$ \mathcal L_{\rm SC} =-\frac12F_{\mu\nu ab}F_{\rho\sigma cd} \eta^{\mu\rho}\eta^{\nu\sigma}h^{ac}h^{bd}-V_{\rm M5.17}. $$

This gives a positive curvature Hamiltonian on its declared timelike branch while recovering the static $3\times3$ sector exactly. P239 also explored a projector-current completion whose Derrick ledger is

$$ E_J(R)=\kappa A_2R+\frac{A_4}{R}+\frac{J^2}{4C_I R^3}, $$

plus axisymmetric, timelike-scalar, spectral-guard, auxiliary-axis, and axis-lock variants. Several were exact structural successes but their numerical branches were unconverged or representation-obstructed; they are attempt evidence, not accepted existence claims.

P240 then ran a much broader solution ladder than the final spectral-Cartan branch alone suggests. The materially different constructions were:

Detailed P240 solution-family inventory

Route	Defining idea	What happened
Same-order contractions	Exhaust all constant-coefficient quadratic curvature scalars while preserving the full static $3\times3$ source sector.	An exact source-restricted witness remained negative. This family was rejected; field-dependent contractions and higher orders remained open.
$L_1$ field-dependent kinetic axis	With $Y=(I-P_t)(\eta^{-1}M)(I-P_t)$, $Z_\mu=(I-P_t)(\partial_\mu Y)(I-P_t)$, add $-\frac{\kappa_1}{2}{\rm Tr}(P_NY)^2P_t^{\mu\nu}{\rm Tr}(Z_\mu Z_\nu)$. Its clock reduction is $\kappa_1 n^2(\lambda_\theta-\lambda_\phi)^2\omega^2$, and it kills the zero-eigenline escape.	A tilted time projector $P_t$ and a high-wave-number static witness made the Hamiltonian unbounded below. Rejected by an exact mechanism, not by optimizer failure.
Axis-weighted Pontryagin square	Square an axis-weighted $\epsilon FF$ scalar so the added term is static-null and positive in its intended time channel.	Early coarse roots exposed axis/coordinate defects. After the regular-chart repair, a genuine root reached relative residual $9.85\times10^{-11}$ but had five negative Hessian modes, from about $-14.92$ to $-0.716$. This rejects stable-minimum status of that tested member/ansatz, not the whole higher-order family. Its scaling $I\sim R^{-3}$, $E_{J,\rm rot}\sim R^3$ was also unattractive for de-boxing.
Axis-weighted timelike Skyrme current	Use the unique single-trace commutator norm $-\frac12{\rm Tr}[Y,Z]^2$, weighted by $n^2={\rm Tr}(P_NY)^2$. The clock term is $\kappa n^2(\lambda_\theta-\lambda_\phi)^4\omega^2$.	Structurally healthy and static-null, but tested neighborhoods retained negative modes (about $-9.24$, and $-5.13,-4.24$ on the second branch). Polynomial replacements either introduced blind spectra or a new unfixed ratio; the constant-polynomial subfamily was exhausted, not all higher-current theories.
Alternating derivative sextic	$K_\mu=[Y,Z_\mu]$, $B^\mu=\epsilon^{\mu\nu\rho\sigma}{\rm Tr}(K_\nu K_\rho K_\sigma)/6$, with density $\kappa n^2q_{\mu\nu}(P_t)B^\mu B^\nu$. It gives $I\sim R^{-1}$ and $E_{J,\rm rot}\sim R$, a potentially confining scale.	The first rest-slice implementation omitted the dynamic projector contraction. The corrected Legendre polynomial $L=A+2C\omega+I\omega^2$ gives $H_J=(J-2C)^2/(4I)-A$ and revealed adverse fixed-J boost curvature. The single term was not a positive stabilizer.
Alternating sextic + projector current	Add a positive timelike-projector current to bound the adverse boost direction; an exact sufficient coefficient region exists.	An $8\times16$ positive-Hessian state appeared, but projection/refinement showed 65–85% of several fields in the upper half of the discrete spectrum and a two-order-of-magnitude energy blow-up. It was a lattice-scale stationary artifact. Smooth Galerkin continuation found a restricted saddle and then failed to reach a basis-converged physical endpoint.
Spectral-Cartan contraction	Replace the fixed internal signature by the $M$-derived $h(M)$ above, preserving Lorentz covariance on the simple-timelike branch and the complete static sector.	Selected as the strongest action-level candidate. Smooth roots existed; a split-channel saddle at small radius was independently confirmed, while larger boxes supplied a marginal stable window. Later de-boxing showed the minimum branch spreading rather than a box-independent core.
Fixed-J two-clock inertia	For symmetric two-clock inertia with diagonal entries $I_1,I_2$ and cross term $C$, $E_J=\frac14J^TI^{-1}J$. Equal like clocks have $\Delta E_J=-j^2C/[2I_0(I_0+C)]$ and are attractive when $C$ is positive; $C\sim A/r$ would give a Newton-like tail.	The implication is exact, but the boxed pair chart had only the summed phase, hence a singular counter-rotating inertia channel. Naive per-centre generators violated SPD. No relaxed positive $A/r$ cross inertia was established.

This ladder matters for R13: we already tried “add a localizer,” “raise derivative order,” “select an axis,” “change the contraction metric,” “repair the fixed-J Legendre map,” and “search smooth branches.” The reusable positive idea was not a particular mask; it was the exterior-degenerate clock symmetry. The repeated failure mode was then either loss of Hamiltonian boundedness, an invalid reduction/representation, or absence of a dynamically fixed width.

P240's important clock ansatz used a uniaxial rank-one exterior. In a local director frame,

$$ S=(t+q) nn^T+(t+\delta)e_\theta e_\theta^T +(t-\delta)e_\phi e_\phi^T. $$

The reduced clock uses one common angle to rotate the tangent eigenspace about the local director $n(x)$, with tangent

$$ \zeta=[\Omega_n,S]. $$

This tangent is proportional to the tangent-plane eigenvalue split $2\delta$. At the exterior projector vacuum, $t=\delta=0,\ q=1$, so

$$ [\Omega_n,S_{\rm vac}]=0. $$

That realizes the core of Jarek's equal-eigenvalue suggestion inside the reduced chart: the clock phase is invisible in the degenerate vacuum without multiplying the tangent by a hand-chosen taper, and the local $SO(2)$ orbit fixes a period. It is still necessary to prove that this field-dependent local-axis action descends to a globally smooth full-theory symmetry and Noether charge; the reduced fixed-J construction assumed that structure rather than settling every bundle/holonomy question.

The branch was solved with the fixed-J term included,

$$ E_J=E_{\rm stat}+\frac{J^2}{4I[S]}, \qquad \omega=\frac{J}{2I[S]}, $$

so the dependence of inertia on the eigenvalue splitting participated in the variational solve. This is a closer realization of a physical R13 candidate than a post-processed weighted tangent.

It nevertheless exposed a second problem. Stable boxed/window branches existed, but the exact scaling at fixed shape was

$$ E(R)[c]=R^3V[c]+\frac{C[c]+\Phi[c]}{R}, \qquad \Phi[c]=\frac{J^2}{4I[c]}, $$

and stability was window/background dependent. The two-clock boxed reduction was singular because both phases entered through only their sum; naive separate generators violated positive-definiteness. Its well-defined static shared-frame interaction was repulsive and approximately $+456.6d^{-1.696}$ over the measured boxed range, but it was not the desired independently phased relaxed fixed-J pair law.

The key lesson is stronger than either stack alone:

Making the reduced clock tangent vanish in the vacuum removes the rigid-flow infrared mechanism in that chart, but it does not ensure that the clock-active core splitting has a box-independent stationary width.

P243/P244: the confined realization, fluctuations, and why spectra must be action-specific

P243 treated the P240 window-supported clock as a confined realization rather than silently calling it an isolated particle. It distinguished two stationary families $U$ and $S$, selected an internal cutoff scale from their same-order energy splitting,

$$ \Lambda^2=E_U-E_S, $$

and carried the stable family through a fluctuation census, a radiative-stability analysis, and two distinct long-range interaction ledgers. The direct boxed shared-frame coupling was eventually corrected to a repulsive $+456.6d^{-1.696}$ law after review found that a meridian-restricted surrogate had wrongly erased the exterior density. A separate induced-gravity composition gave a formal attractive inverse-square kernel, but its own clock source was far outside the weak-field regime. These are useful confined-clock and interaction results; neither supplies the open-space R13 object.

For our different spectral-Cartan/projector-current model, the aligned-vacuum census found three positive-kinetic massless propagating boost-orbit species, four statically stiff directions with vanishing quadratic kinetics, and three inert directions. P244 later certified the full kinetic-normalized pencil about one confined $R=12$ clock root: 32 nonsingular modes after analytic null deflation, with $\omega_{\min}=1.046406$, using Gauss and an independent Chebyshev quadrature family.

Those numbers cannot be imported into OpenWave M5.32. Conversely, OpenWave's current R13 threshold 0.786 is not an M5.32 mass gap: it is imported from the separate M7 HydroBoros tachyon-band calculation. It appears in the M5.32 task as cross-pollination and is hard-coded in the R12 script. It should be removed from the M5.32 acceptance gate until a channel-resolved asymptotic pencil is derived from the M5.32 action itself.

P244 is also part of the validation lesson, not merely a successful spectrum table. It corrected a per-cell kinetic-weight assembly defect inherited from the earlier calculation, demoted a nominally independent finite-difference route after measuring its truncation floor, and disclosed that a preregistered $10^{-9}$ cross-family frequency gate was missed by about $100\times$; the eventual claim was scoped to the measured $10^{-7}$-level algebraic pole-content floor. A later empty-window subtraction could certify none of the 32 modes because the reference kinetic pencil became nearly singular. The right conclusion was “this subtraction representation is unavailable,” not a physical infinite frequency.

There is a more basic issue. Around a uniform M5.32 vacuum,

$$ \partial M=O(\epsilon),\qquad F_{\mu\nu}=O(\epsilon^2), \qquad F^2=O(\epsilon^4), $$

while the potential is flat along an isospectral clock orbit. Thus the ordinary quadratic vacuum kinetic/Hessian channel is degenerate; OpenWave also found $F=0$ for a single-phase plane wave. A conventional positive vacuum mass gap is therefore not presently established for this action. A proposed “flow from linearized dynamics” must first show that a non-singular kinetic pencil actually exists. If it does not, the correct outcome is a modeling/representation result, not “no clock.”

Also, a single real solution of

$$ H\psi=\omega^2K\psi $$

is a vibration. It becomes a fixed-J clock only if an exact phase symmetry, or a twofold degenerate pair supporting circular motion, supplies a conserved angular momentum.

P245/P246: source and symmetry lessons

The gravity continuation is not an R13 solution, but it found two relevant facts. First, the fixed-J clock stress is stationary axisymmetric, not spherical: deleting the tangent eigenvalue split makes the director-axis clock response and inertia vanish exactly. Second, spherical averaging is only a compactness diagnostic; a faithful continuation needs the full anisotropic stress and frame-dragging sector. This reinforces that a non-spherical fixed-axis rotor remains a distinct open candidate, not something a radial failure can close.

P247: isolation, width control, and correction history

P247 tested the de-boxing question directly. Its accepted results are deliberately narrower than its attempt-level endpoint:

• C-M5S-015: no tested radial pinned-wall construction on $R=12\ldots30$ supports a de-boxed clock; deeper boxes and non-spherical branches remain open.
• C-M5S-016/017: the tested boost extension evaluated with float64 gap 0.0 under $\chi\to-\chi$ on the committed and randomized measured backgrounds; no symbolic all-background parity theorem is claimed. Thus no odd classical static source was found there. The first reduction used the wrong sign for its kinetic density and produced 21 ghost directions, while the repaired tested reduced class at the stated discretization had Morse index zero.
• C-M5S-018: the tested stable minimum branch broadened with the box and its energy had no plateau on $R=20,24,30$. This is finite-ladder evidence, not an $R\to\infty$ theorem.
• C-M5S-019: one fixed-width $R=28$ stationary construction is a high-index saddle. Other branches and a different parametrization remain open.
• C-M5S-020: the attractive exponential fit is a frozen-superposition potential cross-term, not a relaxed two-clock fixed-J interaction law.

P247 also tried or registered several solution mechanisms that did not become accepted positive constructions:

• Boost-alignment mass: $s_m=\frac{m^2}{2}[(u^T\eta\xi)^2-1]$, with $m^2=\Lambda^2$ fixed from the existing scale ledger. It was intended to mass the boost tail without a fitted coefficient. The measured float64 $\chi\mapsto-\chi$ gap was 0.0 on the tested committed and randomized backgrounds, removing the needed odd classical source there; this is not a symbolic all-background theorem. The term did not control the spreading $S$-sector width.
• Tilted/fixed-axis boost: introduced to defeat the co-aligned cancellation. On the symmetric one-clock background the integrated linear source still vanished by parity. This closed that reduced source mechanism, not every non-spherical rotor.
• Fixed-axis non-spherical rotor: a genuinely axisymmetric clock that breaks the radial parity itself. This remains an open, unbuilt solver class and is consistent with P246's finding that the split-active stress is not spherical.
• Localized wall/core clock: change the object rather than the boost coupling, allowing a core to select its own width while the massive field confines its tail. This was registered but not constructed; it remains a live alternative concept, not a failed numerical route.
• Fixed-width stationary continuation: a narrow $R=28$ root was constructed, but its Morse index was 14. That refutes stable fixed-width/minimum status for this root; the stationary saddle exists, and other branches or a different parametrization remain open.

This correction history is directly relevant to the numerical concern. Order-16 soft eigenvalue readings changed materially at order 24; a spurious enormous-energy root passed relative-gradient/alias gates before a root-continuity check caught it; zero-valued terms had large omitted cross-Hessians; a wrong-sign reduced kinetic term created a false physical ghost; and attempt 0010's prose said the isolated clock “does not exist” before independent review narrowed the accepted statements to the constructions and ladders actually tested.

4. Response to Jarek's comment

The equal-eigenvalue idea is, in my view, the cleanest R13 mechanism.

For a symmetric field diagonalized as $D={\rm diag}(d_0,d_1,d_2,d_3)$, an infinitesimal Lorentz/internal generator $X$ gives

$$ \delta D=XD+DX^T. $$

A spatial rotation in the $ij$ plane is proportional to $d_i-d_j$. Thus

$$ d_i=d_j\quad\Longrightarrow\quad \zeta_X(D)=0. $$

With OpenWave's distinct vacuum $(32,1,0.3,0)$, the continuous stabilizer Lie algebra is trivial; no global $S^1$ fixes that vacuum. A hand-weighted local rotation does not change this group fact. With a vacuum satisfying, for example, $d_2=d_3$, a genuine $SO(2)$ can fix the exterior while a defect dynamically lifts the degeneracy in its core. Our P239/P240 uniaxial exterior was a reduced-chart version of this mechanism, and its width-spreading result tells us exactly what the next gate must test.

Jarek is also right to worry about regions with different frequencies. Simply writing

$$ \theta(t,x)=\omega(x)t $$

gives

$$ \nabla\theta=t \nabla\omega, $$

so ordinary phase-gradient energy grows secularly. The stationary construction should instead use one global phase $q(t)$; its action becomes spatially invisible where the orbit norm vanishes because of vacuum degeneracy. Independent local frequencies would require a compensating gauge connection or interfaces where the phase stiffness truly vanishes.

The de Broglie relation

$$ \omega=\frac{mc^2}{\hbar} $$

should remain a later comparator, not select the generator or its normalization. Neither the M5.32 R13 construction nor our P239–P247 construction derives the normalization connecting its numerical $J$ to $\hbar/2$; broader spin-ledger claims are separate.

5. A joint R13 that tests the right objects

I suggest three sub-rungs with distinct verdicts.

For the proposed spectrum-departure weighting,

$$ a_w[M]=w(M) a_{\rm rigid}[M], \qquad w(M_{\rm vac})=0, $$

finite inertia may follow simply because $w$ was chosen to decay. Moreover $w\mapsto c w$ gives $C\mapsto c^2C$ and $\omega_{\star}\mapsto\omega_{\star}/c$ unless a compact group period fixes the normalization. P240's axis-weighted routes add two further warnings: a weight can introduce clock-blind spectral surfaces, or make an apparently positive time channel unbounded in a static high-frequency direction. Therefore “robust to two weights” should mean sensitivity evidence for one trial family, not two independent confirmations of a physical clock.

R13A — exact clock structure

Ask whether there is a smooth compact action $M_q$, $q\in S^1$, such that

$$ M_{q+P}=M_q, \qquad \partial_qM_q\to0\ \text{in the vacuum}, \qquad E[M_q]\ \text{is independent of }q, $$

with a globally defined tangent, fixed period, and conserved Noether charge. Compare at least:

1. a dynamically core-split, exterior-degenerate $SO(2)$ clock;
2. a bundle-aware ring phase (the local leading eigenvector is a director $n\sim-n$, so the half-winding ring may not admit a global oriented generator);
3. a gauged or auxiliary-amplitude phase whose amplitude vanishes in vacuum.

A spectrum-weighted or tapered tangent remains useful as a seed/sensitivity family, but until it passes cyclicity and Noether tests its result is ESTABLISHED_KINEMATIC, not a fixed-J clock.

R13B — localized dynamical mode

On a genuinely converged background, derive the action-specific asymptotic principal symbol and solve

$$ H\psi=\omega^2K\psi. $$

Require a positive-norm localized eigenvector, a controlled continuum threshold, and a converged eigenpair. If $K_\infty$ is singular, record that as the missing construction and move to a nonlinear-mode or kinetic-completion route. A single mode earns MODE_EXISTS; a clock claim needs the phase structure from R13A.

R13C — nonlinear/fixed-J relative equilibrium

Only after R13A licenses $J$, solve the full equations

$$ \delta(E-\omega J)=0 \quad\text{or equivalently}\quad \delta E_J=0, $$

varying the field, core splitting, and inertia together. Continue in $J$ or amplitude with pseudo-arclength and check

$$ \frac{dE}{dJ}=\omega. $$

Then test the full constrained Hessian, topology, localized charge density, outgoing flux, and—if dynamics is well posed—nonlinear/Floquet stability. For a nonlinear periodic state, all active harmonics $n\omega$, not only the fundamental, must be checked against radiation channels.

6. Numerical protocol: deconstruction before convergence

I propose that the shared validator have two layers.

Layer 1: representation validity. Before a box ladder, require:

• exact action/version and convention hashes;
• the phase period and normalization;
• $|X[M_{\rm vac}]|$, the action-symmetry defect, and Noether-current conservation;
• smoothness/holonomy of the ring generator;
• the full fixed-J derivative including $DX[M]$, not a frozen flow;
• the channel-resolved vacuum principal symbol and continuum threshold from the same action;
• a statement of what the result can establish: kinematic finite inertia, symmetry, linear mode, or nonlinear clock.

If this layer fails, more numerical precision is irrelevant.

Layer 2: detailed numerical validity checklist

For every load-bearing result:

• vary $h$ at fixed $L$, and $L$ at fixed $h$, independently; the OpenWave ladder with $n/L$ fixed at $h=1.5$ is a domain test, not a continuum test;
• use at least three refinement points and measure observed order; do not fit three free parameters to three data points;
• converge the background with a scale-aware field-equation residual and an independent virial/Derrick identity;
• compare a second boundary treatment and a second discretization of the same mathematical candidate;
• report operator symmetry defects and eigenpair residual

r_\lambda=
\frac{\lVert Hv-\lambda Kv\rVert}
{\lVert Hv\rVert+|\lambda| \lVert Kv\rVert};

• report $\lambda_{\min}$, $\lambda_2$, zero-mode scale, eigenvector localization, boundary overlap, and a finite-difference witness for any negative mode; random directions cannot certify Morse index zero;
• include background forward error, not just solver backward error:

\delta M_{\rm est}=-H_\perp^+P_\perp\nabla E[M_0],
\qquad
\delta\lambda_{\rm state,est}
\approx\left|v^T(DH[M_0]\cdot\delta M_{\rm est})v\right|.

This is an estimated state-error contribution unless a conditioning bound is also supplied.

• carry a crossed budget

\epsilon_{\rm total}=
\epsilon_{\rm state}+\epsilon_h+\epsilon_L+\epsilon_{\rm basis}
+\epsilon_{\rm quad}+\epsilon_{\rm eig}+\epsilon_{\rm round}+\epsilon_{\rm BC},

and require the claimed signal to exceed it by a declared margin;

• measure the cumulative Noether-charge profile $J_{\mathrm{enc}}(r)$, not only quadratic inertia; only the total charge is conserved. For dynamics, refine timestep and absorber placement and measure outgoing flux.

The result vocabulary should prevent a route failure from terminating the question:

• ESTABLISHED_KINEMATIC
• ESTABLISHED_SYMMETRY
• MODE_EXISTS
• PERIODIC_ORBIT_EXISTS
• CANDIDATE_REFUTED
• NUMERICALLY_UNRESOLVED

A failed taper refutes that taper. A singular kinetic pencil blocks that linear representation. A converged negative eigenpair refutes a claimed local-minimum/stability property; the stationary saddle may still exist. None alone is a global no-clock theorem.

7. Concrete cross-stack coordination

I propose one frozen “rung packet” shared by the OpenWave and Substrate agents before either side sees the other's new numbers:

• exact action, vacuum, field and generator definitions;
• claim layer and ensemble (fixed $J$, fixed $\omega$, normal mode, or unconstrained periodic state);
• boundary/asymptotic conditions and topology;
• candidate set and assumption costs;
• stationary equations and complete derivative/Hessian definition;
• $h,L$, solver, residual, virial, eigenpair and error-budget gates;
• what a pass/failure establishes and which broader question remains open.

Each result record should then include the source hash, frozen field hash, generator period and symmetry defect, background residual/virial, core/total inertia, cumulative charge, tail law, eigenpairs/residuals/zero-mode scale, flux, complete error budget, unrepresented sectors, and next continuation rung.

When the stacks disagree, exchange one frozen field and evaluate it with both energy/charge evaluators, then localize the discrepancy in this order:

$$ \text{equations}\to\text{conventions}\to\text{representation} \to\text{discretization}\to\text{solver}\to\text{observable}. $$

The best immediate shared target is therefore not “try two tapers.” It is:

Construct and compare a globally normalized, exterior-degenerate $SO(2)$ clock whose core dynamically lifts the degeneracy, then test whether the full fixed-J equations give that split a box-independent width. In parallel, derive whether M5.32 has a usable linearized kinetic pencil at all. These two routes answer different questions and should remain separately labeled.

Pinned records used in this review: OpenWave M5.32 method note, OpenWave M5.32 task record, the separate M7 source of the 0.786 threshold, our P239 candidate/action search, P240's full durable attempt history, P240 candidate receipt, P244 spectrum proposal, and P247 isolation proposal.

[vantasnerdan]

@mjmikulski @JarekDuda

Update: PR #190 has now landed on main and release v0.171.0. It gives a positive answer to the specific exterior-degenerate clock route proposed above, with an important scope boundary explained at the end.

The result in plain language

The problem with the earlier rigid clock was that it rotated the vacuum all the way to the wall. The vacuum then carried more and more inertia as the box grew, so the clock frequency fell like 1/L. A hand-made taper can hide that behavior, but it does not create a true conserved clock.

PR #190 instead builds one genuine compact SO(2) phase, with a fixed period of 2π, that acts on the fields but fixes the exterior vacuum exactly. The same global phase turns everywhere; it simply becomes physically invisible in the vacuum because the two tangent-plane eigenvalues are equal there. The core breaks that degeneracy and therefore carries the clock.

This addresses Jarek's concern without assigning a different frequency to every spatial region. We use one phase

$$ q(t)=\omega t, $$

not $q(t,x)=\omega(x)t$. The exterior has zero orbit norm, so it stores no clock charge even though the global phase remains well defined.

The mathematical construction

The physical fields in this completion are a real symmetric spatial tensor $S\in\mathrm{Sym}(3)$ and a complex scalar $\psi$, written in a declared constrained auxiliary orthonormal frame. Rotation by $q$ in the tangent plane acts diagonally as

$$ \psi\mapsto e^{iq}\psi, \qquad S\mapsto R(q)S R(q)^T. $$

The exterior is

$$ (S,\psi)=(NN^T,0), $$

and is fixed pointwise by this action. Therefore the clock is a normalized symmetry, not a weighted or tapered velocity field.

Noether's theorem gives a conserved clock charge. On a relative equilibrium,

$$ Q_{\rm clock}=\omega I, \qquad E_Q=E_0+\frac{Q_{\rm clock}^2}{2I}, \qquad \frac{dE}{dQ_{\rm clock}}=\omega, $$

with full inertia

$$ I=\int\left(|\psi|^2+\frac12\mathrm{Tr}([A,S]^T[A,S])\right)d^3x. $$

If $(u,v)$ are the axis-to-tangent shears and $(p,q)$ are the tangent-plane traceless components, the tensor part is

$$ u^2+v^2+4p^2+4q^2. $$

That full expression matters: the first review caught that an earlier draft had accidentally omitted the shear pair. Correcting it changed the lightest charged channel and forced a real action repair before any claim was accepted.

Why the clock localizes

The repaired action is positive and has a unique aligned exterior. Its complete action-specific exterior spectrum is exact:

• tensor mass squares: $(10,10,22,4,4,22)$;
• scalar mass square: $6$;
• clock weights: scalar $1$, shear $1$, tangent clock tensor $2$.

The first charged radiation threshold is therefore

$$ m_*^2=\min(6,4,22/2^2)=4. $$

The action also has an exact split-core trial state

$$ |\psi|=\frac12, \qquad S=\mathrm{diag}\left(1,\frac14,-\frac14\right), $$

for which

$$ V=\frac{45}{64}, \qquad I=\frac12, \qquad \frac{2V}{I}=\frac{45}{16}<4. $$

The strict margin is

$$ 4-\frac{45}{16}=\frac{19}{16}. $$

This inequality is the central binding result. A state whose clock charge spreads away to infinity approaches the exterior threshold 4, while this split core lies strictly below it. Moreover, every unsplit core has ratio at least 4, so a minimizing state below the threshold must dynamically develop a nonzero tangent-plane eigenvalue split. The core localization is therefore selected by the energy and charge together; it is not imposed by a wall, taper, or fitted radius.

Using the Benci--Fortunato hylomorphic-soliton theorem on the full canonical

$$ H^1(\mathbb R^3)\times L^2(\mathbb R^3) $$

phase space then gives a nonempty translation-compact family of global energy minima at fixed charge. These states have finite energy and inertia, a finite box-independent charge radius, and

$$ 0<\omega^2<4. $$

All active exterior harmonics are consequently below their own action-derived radiation thresholds:

$$ 6-\omega^2>2, \qquad 4-\omega^2>0, \qquad 22-4\omega^2>6. $$

Because the solutions are global fixed-charge minima, the full constrained second variation is nonnegative modulo translations, phase, and the declared frame gauge. Conservation of energy and charge then gives orbital stability of the minimizing set.

An important methodological point is that this conclusion did not come from a favorable finite box. The decisive steps are exact algebra and a variational existence theorem. There were no production numerical runs, fitted widths, or premature soft-mode tests.

Significance for R13

Relative to the three sub-rungs proposed above:

• R13A is achieved for this action: there is a globally normalized compact clock, exact exterior fixation, and a conserved Noether charge.
• The action-specific exterior pencil is achieved without importing the M7 0.786 number or another model's spectrum.
• R13C is achieved at theorem level: full-field fixed-charge relative equilibria exist, have intrinsic open-space localization, and form an orbitally stable minimizing set.
• A separate numerical profile is now an optional measurement problem, not the source of the existence or stability claim. It could estimate a particular radius or frequency after a charge normalization is chosen, but it is not needed to establish the object.

This is a constructive proof that the infrared wall found for the rigid clock is not universal. Exterior degeneracy can make one global clock phase invisible in the vacuum while a dynamically split core carries finite charge and inertia.

What this does not yet claim

This is a new conditional canonical M5 completion, not a modification proved equivalent to OpenWave's original certified M5.32 action. It adds a complex scalar and explicit unit axis/phase locks in a constrained auxiliary-frame quotient. Its exact masses and thresholds must not be copied into M5.32 without deriving the corresponding field map and action.

It also does not yet identify the solution as an electron or neutrino, derive $\omega=mc^2/\hbar$, fix $Q_{\rm clock}=\hbar/2$, produce a unique numerical width or frequency, or solve the two-clock force, Newton attraction, or gravity problems. Those are downstream campaigns. What PR #190 establishes is the missing licensed one-clock object: a compact, exterior-trivial, finite-energy, intrinsically localized and orbitally stable fixed-charge clock.

The accepted claim statements and exact boundaries are in C-M5C-001 through C-M5C-004, and the omitted-shear failure plus bounded correction are preserved in the independent review record.

[JarekDuda]

Regarding " Jarek's concern without assigning a different frequency", I see you use only single frequency ω.
Using two different ω1 and ω2 in two regions, there would conflict in boundaries between them - my point was that it can be resolved by equalizing last two eigenvalues of M (at cost of potential) e.g. to (g,1,delta/2,delta/2), this way it no longer recognizes phase such frequency describes, so it can be different on both sides.
It suggests 3D regions of constant frequency, separated by 2D walls of equal last two eigenvalues - bringing additional energy density per area from potential.
I suspect this frequency should be usually close to of electron, slightly modified in atoms (filling Universe) by Dirac equation.

[xrodz]

@JarekDuda @vantasnerdan @mjmikulski

Thanks for all replies. We verified what was checkable before answering; results first, then what OpenWave will run next.

Corrections accepted

Dan's review lands four hits on our record, and we accept all four:

1. The 0.786 gate is confirmed cross-model. Our R12 script hard-codes OMEGA_RAD = 0.786, sourced from the M7 HydroBoros threshold k* = 0.7862, not from the M5.32 action. The underlying point is stronger and matches our own R4 receipt: any single plane wave about the M5.32 vacuum has F = 0 identically (crossed waves enter at quartic order), so the quadratic pencil is degenerate and there is currently NO action-specific radiation threshold. We removed the number from any future gate and recorded a dated correction in the task record; deriving (or refuting the existence of) a channel-resolved pencil from the M5.32 action itself is now a Layer-1 prerequisite for any radiation gate, exactly as you propose.
2. The w → cw normalization objection is correct. "Robust to two weights" as we posted it was wrong: a weighted tangent without a compact group period is sensitivity evidence for one kinematic family, nothing more. We adopt the ESTABLISHED_KINEMATIC label for every such result.
3. Frozen a0(M). Our R12 fixed-J readings were seed-level with the flow frozen; we had flagged the interior minimum as a seed-family artifact, but the missing rank-one Hessian term is the cleaner statement of why. Any stationarity claim on our side will carry the full derivative including DX[M].
4. Scoping. Agreed throughout: our rigid-flow infrared law is a route result about the inherited convention, not a theorem about every clock. Your closed-form C(L) ~ kappa L matches our ladder (omega* L = 7.12 / 7.07 / 7.05).

PR #190, verified as far as algebra goes

We re-derived the P249 exact algebra independently: a fresh sympy implementation written from the attempt-0008 derivation text, not importing or running your module. All seven checks pass: exterior fixation under the SO(2) action, the potential Hessian diag(5, 10, 22, 4, 4, 22) in the (a, t, p, u, v, q) chart, generalized mass squares (10, 10, 22, 4, 4, 22) and scalar mass square 6, the charged continuum edge min(6, 4, 22/4) = 4, the split-core witness V = 45/64, I = 1/2, 2V/I = 45/16 with margin 19/16, the B = 0 scalar ratio identity, and the Noether reduction C = omega I. The script is public: m5_32_p249_check.py.

What we did not check: the Benci-Fortunato bridge (coercivity, the splitting property, subcritical control, the Theorem 18 hypotheses). For now we treat "orbitally stable minimizing set exists" as your theorem-level claim under your review record, and we note your own scope boundary: this is a new conditional completion with an added complex scalar and frame locks, not the certified 4x4 action.

Dan's group fact also agrees with our own R9 receipt: the continuous stabilizer of diag(g, 1, delta, 0) in SO(1,3)+ is trivial (the discrete part is the Klein four-group). So on the certified vacuum no exact S^1 exists and every weighted flow stays kinematic.

Jarek's wall mechanism, read back

If we read the 14:55 comment right, it is a THIRD clock convention, distinct from both the rigid flow and the vacuum-vanishing flow: piecewise-rigid rotation per 3D region, with 2D walls of equal last two eigenvalues ((g, 1, delta/2, delta/2)) decoupling the phases. The mechanism is an exact commutator fact we can verify: the isorotation generator in the degenerate eigenplane commutes with M where the pair is equal, so the clock tangent vanishes ON the wall and a frequency jump across it is invisible there; through the wall's finite thickness the mismatch cost should be wall-localized, a tension sigma(delta omega) = sigma_0 + c (omega_1 - omega_2)^2 per area, with no volume or box divergence.

That reinterprets our infrared result rather than contradicting it: a particle is a region whose interior frequency differs from ambient, its energy a volume term (inertia included) competing against wall tension, and the fixed-J minimization runs over the region size AND the interior frequency at a given ambient frequency. Finiteness then comes from the wall tension, not from a decaying flow. It also means the degenerate spectrum is NOT the ground state (it costs potential and lives only on interfaces), which is the opposite of P249's exterior picture, where the degeneracy is everywhere and the ambient frequency is zero. Both pictures are now on the table as declared variants.

What OpenWave will run: R13-W (rung packet to be posted here before any number)

The wall rung, on the certified -4 I_1 - V_4 action, in three steps with separate verdicts:

• W1, the wall exists and has a price: relax the 1D profile interpolating between two bulk vacua through the degenerate-pair configuration; measure the tension sigma_0 and its h-convergence; Layer-1 first (the commutator identity, the generator normalization, the action-symmetry defect on the wall).
• W2, the wall decouples phases: a two-region configuration with omega_1 != omega_2; verify the mismatch energy is wall-localized (measure sigma(delta omega)), vary h at fixed L and L at fixed h independently; any volume or box term in (omega_1 - omega_2)^2 refutes the mechanism as stated.
• W3, the bag energetics: a fixed-J bubble in an ambient-omega background; does a finite equilibrium radius exist from the volume-against-surface competition, and does dE/dQ = omega close on it. Reported in the result vocabulary above; a failure at W2 stops W3.

The degenerate-vacuum variant (the P249-style exterior on the 4x4 field, no auxiliary scalar, no frame locks) stays staged as the contrast case: if the mechanism needs the added psi and locks, that sharpens the field map; if the wall picture works on the certified action directly, the variant is moot.

Questions

1. Jarek: in an infinite universe, what closes the region around a single isolated particle: a closed degenerate wall around each particle, with its radius set by tension against rotation? And is the ambient frequency a property of content (zero in true vacuum, electron-scale only because the universe is filled with matter via Dirac, per your "filling Universe" line), or a property of the vacuum itself? W3's setup depends on which.
2. Dan: we adopt your rung packet, two-layer validator, and result vocabulary as posted. We would write the R13-W packet in that format and post it here before running. Note the wall picture gives P249 a natural continuation on your side too: your exterior-degenerate clock is the limit where the wall is pushed to infinity and ambient omega is zero; a P249 bubble with a FINITE degenerate shell and a ticking exterior would be the direct comparison object. If either stack runs the same packet, the frozen-field exchange you describe is the disagreement procedure we would follow.

Rodrigo (OpenWave)

[JarekDuda]

The isofrequency regions question is very interesting, frequency change should be avoided due to energy/area cost of boundaries - rather extremely small, but non-negligible.
Most of Universe are electrons and atoms, having similar frequency - which probably dominates, with of exception of e.g. neutrinos - but they are usually relativistic so shouldn't disturb.

But what about frequency of nuclei, atoms, molecules?
Generally even Dirac equation works on probability densities - is rather effective, we should finally search for deeper models - like electron as field configuration here, with trajectories in atoms/molecules (e.g. https://scholar.google.pl/scholar?q=gryzinski ).
Having coupled quantum phase with resonances leading to orbit quantization, averaging of trajectories to quantum statistics like in https://en.wikipedia.org/wiki/Hydrodynamic_quantum_analogs
Finally we should rebuild atomic physics and chemistry this way, but this would be large separate research campaign - I would gladly help with.

[xrodz]

@JarekDuda @vantasnerdan @mjmikulski

FYI: the candidate ledger Jarek asked for on 2026-09-02 is merged on OpenWave main, and our next rung (R13-W, the degenerate-wall clock convention) starts now on our side. Nothing is needed from anyone; corrections to any status cell are welcome here.

The ledger: https://github.com/openwave-labs/openwave/blob/main/openwave/xperiments/m5_liquid_crystal/research/findings/m5_32_candidate_ledger.md

It puts every candidate from the three searches (OpenWave R0 to R12, substrate-framework P239 to P249 at 8b74d3ab, the-final-lagrangian-of-physics reports 001 to 013 at 65c6177f) in one table with one status cell each, using the vocabulary from Dan's 08-31 review, then lists every repair route tried for the two issues, and ranks what is left. Every cell cites a script, a receipt, or a commit-pinned document. Maciej: we reproduced 001 (all numbers match), 006 (744 s CPU, both routes, all numbers match) and the self-contained legs of 007 (route 2 on the committed fields to 1.8e-15) on our machine before citing them.

Where the two issues stand, across the three searches

Issue	Closed routes (a pre-registered gate failed, or an exact no-go exists)	Still open
Newton sign	constant-coefficient quadratics, even and odd (OpenWave R1, reports 001/005/006, P240 0032: three independent closures); the global static-sector flip (R11, 006: attraction with no floor); field-dependent coefficients and pointwise connections from M (007 + appendix, exact); covariant metric mixing h = eta + 2uu on our instruments (R2/R3); induced gravity (P243/P245: GM/R = 213, source trapped at compactness 894)	the fixed-J two-clock cross inertia C(r) = A/r (P240 0034 algebra exact, our R3.iii undecidable unrelaxed; nobody has measured A on relaxed localized clocks, and fixed omega gives the opposite sign); higher order in F or dM-built coefficients (006's escape map, no public work); two-defect states under a non-rigid clock convention
Diverging omega	rigid isorotation (extensive on every stack: our R4 to R12, report 009, P244/P247); potential-only localizers (R6 theorem); the time-row gradient penalty (R7); (F.F)^2 at fixed coefficient in the fundamental reading (R8/R11, 008: -gamma s^2 runaway); weighted or tapered flows (kinematic only, w -> cw); wall-pinned then de-boxed clocks (P247)	P249 (theorem level, on a new action; exact algebra confirmed by us 7/7, the variational bridge not checked by us); the (I_1^G)^2 energy-functional well (008/013: convergence-certified in depth on one box, never box-laddered, and whether that functional is the H of a well-posed L is Jarek's call); the fundamental-reading grid wells (010/013: do not certify in 24 cycles); the degenerate WALL convention (Jarek, 08-31): no stack has run it as a wall; its spin-side analog (Maciej's 009 § 6a / 011, rotation confined to the core by an equivariant background) did not survive statics selection, and its degenerate-vacuum half is closed on the rigid family (our R8 / R9: rigid inertia exactly zero at delta = 0)

Two facts that shaped the ranking. (1) In the certified action a rigidly rotating EMPTY vacuum carries no kinetic energy at all (F_0i = [A_0, A_i]_eta vanishes where d_i M = 0, as Maciej's 002/007 state and as our I1 has by the same structure); in Jarek's picture the ambient frequency is set by content, a medium with gradients almost everywhere, where it costs the defects' inertia. Either way only defects and walls carry inertia, which makes the wall tension and the wall's own inertia the whole question. (2) No wall, interface, or omega(x) clock exists in any of the three records at the pins (grep-verified in both external repos), so R13-W is the one route proposed by the model author that no stack has run as such. The nearest analog is Maciej's 009 § 5 masked generator, whose interface term makes J a prescribed parameter rather than a Noether charge: that is exactly the obstruction W2 has to meet where two rigid generators touch.

Ranking (details and the full evidence in the ledger § 6.1)

1. P249 exterior-degenerate SO(2) clock: the only theorem-level localized clock, on a NEW action (Sym(3) + complex scalar + locks), so not yet a candidate for the 4x4 field itself; lacks a field map back to the certified action, a profile, and everything two-body. Dan's own route frontier lists the 4x4 dynamics (E_M532_DYNAMICS) as remaining.
2. The degenerate-wall convention on the certified 4x4 action: the one author-proposed route with no direct test; keeps the action and explains one global phase with region-dependent frequency. W1 (a static wall with finite tension) cannot refute it; W2 can.
3. The (I_1^G)^2 energy-functional clock (008/013): the only convergence-certified localized clock on the original 4x4 field, under the repaired-metric action (G = eta + 2uu, the same h as our C2 family) in a reading that is not yet an L; an L-ladder at fixed h would settle its box independence and belongs to Maciej's stack.
4. The fixed-J two-clock cross inertia C(r) = A/r (P240 0034): the one Newton route whose algebra is exact (like-rotating clocks attract at fixed J iff A > 0, repel at fixed omega; we re-derived it). It concerns the clock-clock term only; the Newton sign that fails on the certified action is the static dressed pair, which has no ensemble. Downstream of 1 or 2.
5. Higher-order-in-F coefficients for the sign: the only class on the 4x4 field not closed for Newton, with one named term ((F.F)^2, dilation exponent 5, our R11) and a cheap direct test against a flipped static sector that has not run yet. On the Newton axis alone it sits above 4.

R13-W, pre-registered (ledger § 6.2 carries the full obligation table)

Dan: "rung packet" is not a term in your repo, so this is written in your obligation-node shape (object, license, ensemble, functional, admissible space, representation coverage, observable, numerical representation, permitted verdicts, failure scope, unlocks), with the analytic closure carried in before any number. Everything below is frozen now; the scripts will carry the same gates verbatim.

Field	R13-W
Object	the certified 4x4 field under L_cert = -4 I1 - V4, vacuum diag(-g, 1, delta, 0), g = 8, delta = 0.3, one g = 32 control
License	the (2,3)-plane isorotation is not a symmetry of this vacuum (trivial stabilizer, our R9). On a surface where the last two eigenvalues equalize, a0 = [G1, M] = (d2 - d3)(E23 + E32) = 0, so the rotation acts trivially ON the wall; its jets do not unless d_z(d2 - d3) = 0 there too, so two global phases omega t, omega' t meeting at the wall give an interface energy that must be periodic in the relative angle, never secular. The identity plus the derivative condition are the first check (W0, symbolic)
Ensemble	fixed J
Functionals	W1: static energy of a 1D wall profile, vacuum at both ends, d2 = d3 constrained at z = 0, then released. W2: on a slab with the wall at z = 0 and a rigid generator on each side, (i) the static interface energy as a function of the relative angle over one period, and (ii) the wall-localized inertia kin_wall = the coefficient of omega^2 in the slab (the empty vacuum contributes zero, so every omega^2 term in a defect-free slab is the wall's). W3: E_J = E_stat + J^2/(4 kin_in) with the ring or hedgehog core inside a spherical degenerate shell, the shell free
Admissible space	h in {1.5, 1.0, 0.75} and L in {48, 72, 96} varied independently; no taper, mask, or pinned interior; vacuum pinned at the box edge only
Coverage	W1/W2 cover the wall as a stationary object of the static functional plus the stationary two-frequency read. They do not cover spontaneous wall formation under time evolution: our stack has no Hamiltonian integrator (FIRE is damped descent), so that question is logged NUMERICALLY_UNRESOLVED unless a leapfrog is built as a separate rung
Observables	W1: tension sigma_0, its h-convergence order over three refinements, and whether the released profile keeps d2 = d3. W2: E_stat(Delta q) periodic with no secular growth, and kin_wall per unit area with its observed L-exponent over three boxes. W3: R*, omega*, dE/dJ = omega, and whether the shell survives fixed-J relaxation
Permitted verdicts	W1 pass: ESTABLISHED_KINEMATIC (finite h-converged sigma_0); W1 fail (released profile melts): W2 decides whether the mismatch load pins the wall. W2 pass: E_stat(Delta q) periodic and kin_wall per unit area L-independent; W2 fail (secular twist energy, or kin_wall growing with L): CANDIDATE_REFUTED for the wall convention on L_cert, W3 does not run. W3 pass: PERIODIC_ORBIT_EXISTS at the relaxed-field level; W3 fail (shell melts or R* tracks L): ESTABLISHED_KINEMATIC at best
Analytic closure	bag energetics in our convention E_J = E_stat + J^2/(4 kin): with the measured rigid law kin_in = kappa R (the 1/r tail cut at R, our R7), R*^3 = J^2/(32 pi kappa sigma); with a volume-filling kin_in = kappa R^3, R*^5 = 3 J^2/(32 pi kappa sigma). Finite for any sigma > 0 either way, so a finite radius at W3 proves nothing by itself; W3's content is the shell's stability under the field equations
Failure scope	a refutation refutes the wall convention on the certified action as represented (rigid-per-region generator, static wall), not the mechanism on a modified action, nor the dynamical formation question
Unlocks	W2 pass: W3, then the two-clock cross inertia on two bubbles (rank 4). W1 or W2 fail: the RELAXED degenerate-vacuum contrast case (the rigid clock on a degenerate vacuum is already closed: inertia exactly zero, our R8 / R9) and the P249 field-map question

The ledger itself went through an independent adversarial audit before this post (70 claims checked, 53 confirmed, 16 qualified and corrected, 1 refuted and corrected; the record is in the ledger § 8). Results will be posted here rung by rung, each after the same kind of audit, with the scripts and receipts on main.

[JarekDuda]

Thank you, looks great - will study in the morning, but having working Newton and electron, the basic suggestion for the next anchor is calculating oscillations of neutrinos as topological vortex loops, and comparing with estimates especially for PMNS.

[xrodz]

@JarekDuda @vantasnerdan @mjmikulski

FYI: R13-W ran today on OpenWave, in the packet posted above; result: the degenerate-wall clock convention is ESTABLISHED_KINEMATIC at best on the certified action, and the reason is a theorem that reaches further than the wall. Nothing is needed from anyone; the record, scripts and receipts are on main (task record: https://github.com/openwave-labs/openwave/blob/main/openwave/xperiments/m5_liquid_crystal/research/tasks/m5_32_task_details.md, section R13-W; ledger § 5.2 and § 6.2 carry the outcome).

W0, the symbolic closure (22 checks, each can fail)

Jarek, the identity you proposed holds exactly: a0 = [G1, M] = (d2 - d3)(E23 + E32), zero on the surface d2 = d3 and nowhere else (frame-free form: the local-eigenvector generator vanishes iff d2 = d3). The derivative condition also holds: the kinetic density on a profile M(z) is 8 (d2 - d3)^2 (d2' - d3')^2, zero ON the wall and borne by its flanks (a linear ramp of width w carries (8/3) delta^4 / w per area; the lattice one-cell layer carries exactly 4 delta^4 / h per rotating flank, and W2 reproduces that to all digits). Four further statements came out of the same algebra, all exact, all confirmed by an independent audit:

1. Planar flatness. For ANY profile M(z), every spatial F_ij = [A_i, A_j]_eta vanishes (one nonzero jet), so the static energy of every planar configuration is V4 alone. An orientation wall (the vacuum rotated across a plane) has zero tension at any width and angle; a degenerate layer of thickness w has tension w V4_deg with V4_deg = 6.5e-7 at the constrained minimum. No planar wall in this action has an h-converged tension.
2. Phase flatness. On the uniform vacuum with an arbitrary phase phi(x, t), every F_mu nu = 0: the phase field has no action at all, so regions of different frequency decouple across any plane of the empty vacuum, with or without a wall. On a non-uniform background the phase couples only through [X, d_mu M0], X = [G1, M0], and the inertia density and the twist stiffness are the same tensor.
3. Free inertia. A planar orientation twist of the vacuum in a plane that does not commute with the clock (the (1,2) plane under the (2,3) clock) has zero static cost by 1 and kinetic density psi'^2 f(psi), f = 8 (delta - 1)^2 (1 - (1 - delta^2) cos^2 psi) > 0: inertia ~ Psi^2 / w for free. So at fixed J the infimum of E_J = E_stat + J^2/(4 kin) over fields is E_stat with kin unbounded (lattice-limited by h). The fixed-J functional has no minimizer on L_cert in the full field space, for any clock convention; every finite-omega fixed-J state on our record (R4 to R12) was a constrained family minimum. The audit built its own counterexample: a compact (1,2)-twist texture in a 24^3 box reaches E_J = 21 at J = 200 against 455 for the shell state.
4. The sharpening bag. E_J(R, w) with a shell of thickness w is monotone increasing in w termwise, so a degenerate shell sharpens to the lattice floor at fixed J; the full bag minimum with the shell tension of this action sits at R* = 75 to 450 for J = 50 to 800, outside every admissible box.

W1 and W2 (the slab steps), as pre-registered

W1 FAIL: sigma_0 = 1.71e-6, 1.14e-6, 8.5e-7 at h = 1.5, 1.0, 0.75 (order 1.000 to zero, an identity of the action: E_u = 0 exactly on every box, L-flat exactly); the released layer keeps d2 = d3 under the exact descent only by symmetry (the V4 gradient at a degenerate pair is isotropic), its curvature along the split is negative and a per-cell descent from a 1e-3 split lands on the vacuum spectrum. W2 PASS on both frozen gates, to machine precision: E_stat(Delta q) is constant (range 0.0 over 2 pi on all five boxes) and kin_wall/area is exactly L-flat (4 delta^4/h, L-exponent 1e-16, h-exponent -1.00). The audit's qualification, which we adopt: both W2 gates are identities of any planar slab with a diagonal wall, so the pass is unfalsifiable on that geometry and carries no evidential weight; the packet's "W2 decides" could not decide anything, and the decoupling is only testable where the wall meets a non-planar texture (W3).

W3, the hedgehog in a free degenerate shell at fixed J: FAIL

Twenty fixed-J relaxations (n32 L48 at R_s in {6, 9, 12, 15} x J in {50, 200, 800}, plus J = 180/220 and a 12000-iteration run at R_s = 9, a frozen-generator control, n48 L48 (h = 1.0), n48 L72 at R_s in {9, 15, 21}); the two-region generator (interior rotating, exterior at rest, the region edge on the surface where the generator vanishes identically). None reached a stationary point (every run at its iteration budget, fmax three to five orders above the packet's tolerance; the ladders ran 3000 iterations, one run 12000). The descent inflates the inertia without bound and pays static energy for it: at R_s = 9, J = 200, kin goes 22 (seed) to 2699 at 3000 iterations to 10411 at 12000 with omega 0.0096 and still falling (+25 percent in the last quarter, net growth with dips in every run); the frozen-generator control builds the same structure (27 to 637 read by its own generator, 7542 read by the refreshed one on the same field). What the descent builds, read on the saved fields by the audit: in the two cell layers just inside the region edge, a cell-scale zigzag of the (2,3) eigenvalue gap (driven to 1.9 to 3.9 against the vacuum's 0.3, d3 negative) carrying 97 to 99.7 percent of the inertia through eigenvalue jets that E_u cannot see and that the generator a0 = (d2 - d3)(E23 + E32) rewards quadratically: the flank density of the W0 derivative condition pushed to the lattice scale (a sibling of statement 3, not its orientation-twist form). The inflation is largest at small R_s, so kin falls and E_J(R_s) rises with the shell radius on every ladder (argmin at the lower grid edge, a descent-rate ordering, not a bag balance); dE/dJ = 0.0187 against omega = 0.037, by exactly what the descent-stage spread of kin across J predicts; omega at equal iteration count scales as h^1.7 (0.037 at h = 1.5, 0.018 at h = 1.0) and is box-independent to four digits at equal seed maturity. The imposed shell melts (mean gap 0.03 to 0.14 at the end, rising through the trace in most runs) while its flank's inertia is what fixed J rewards, which is exactly the disease. The bag closure on the seed family with measured inputs (kin_seed ~ 0.045 R^2.8, an E_u-borne shell tension of 2e-3 per area) puts R* at 16 / 29 / 51 for J = 50 / 200 / 800, but the descent does not stay on that family.

Verdict in the frozen vocabulary: PERIODIC_ORBIT_EXISTS is not licensed (no relaxed state exists to certify; the shell melts; no interior R*; the closure fails), and a third fail mode is added to the two pre-registered: the fixed-J descent reaches no minimizer on the lattice. The wall convention on L_cert is ESTABLISHED_KINEMATIC at best: walls exist as tensionless objects with lattice inertia, and they do not make a clock. Failure scope as pre-registered: this refutes the convention on the certified action as represented (rigid-per-region generator, static wall), not the mechanism on a modified action, nor the dynamical formation question (no Hamiltonian integrator in our stack, still NUMERICALLY_UNRESOLVED).

What this changes in the ledger

Rank 2 is closed. The omega issue is reframed: on -4 I1 - V4 the phase gradient has no cost of its own (statements 2 and 3), so a localized clock needs either a constraint that is a symmetry (Dan's P249 route, on a different action, rank 1) or a term that charges the phase gradient, which is the class the (F.F)^2 and K_T-type terms belong to (our R7/R8/R11 closed the ones we tried at fixed coefficient; the dilation test of rank 5 is the cheap next check). Maciej, this may also be what sits behind the volume law I ~ L^2.93 in your 009 and behind the prescribed J of the masked generator in 009 § 5 (the interface term is where a free inertia would live); we have not checked it on your fields, so it is a suggestion, not a finding.

Audits: W0 to W2 by an independent agent with its own scripts before W3 ran (56 claims: 38 confirmed, 15 qualified, 3 refuted, every item applied and re-checked 14/14, including a factor-2 undercount in one control and a plateau-stop defect in our FIRE wrapper); W3 by a second agent (32 claims: 15 confirmed, 10 qualified, 7 refuted; the refutations were all in our collector, rebuilt from the saved fields; the mechanism attribution above is the audit's correction of our first reading, which had named the orientation twist). Everything is on main: m5_32_r13w_{w0,common,w1,w2,w3,audit}.py, the JSON receipts and plots, the end fields kept locally.

Jarek: the neutrino-oscillation anchor (vortex loops vs PMNS) is noted; it presupposes a working electron and Newton, which this rung moves further away on the certified action, not closer.

[vantasnerdan]

P250 campaign complete: the shell exists, it decouples phases at an area price, charge selects a bag, and both localization pictures are one mechanism — 8 promoted claims, 2 releases (issue #195 → PR #196)

TL;DR (30 seconds)

Issue #195 asked for the strongest validated account of finite degenerate interfaces and the charge carriers they enclose on the accepted exterior-degenerate clock completion (C-M5C-001..004). That campaign — P250, claim namespace C-M5W — is complete. All five obligation nodes are established, eight claims are individually reviewed and accepted into governance/claims.yaml, and release v0.173.0 pins the set. Terminal PR: #196 (Fixes #195, open, awaiting a distinct merger). Below: the four questions, the one-line answers, then the step-by-step ladder.

The four questions #195 asked, answered in one line each

1. Does the shell exist, and what does it cost? Yes — a stationary kink through the degenerate locus exists on the exact slice reduction, with tension σ₀ = 0.72929841786(58) from the exact potential (C-M5W-001, C-M5W-006).
2. Does the shell decouple phases at an area price? Yes — and better than asked: the mismatch coefficient of (ω₁−ω₂)² is exactly zero, not just area-localized; phase jumps across the locus cost exactly zero, area price only (C-M5W-003).
3. Does charge select a bag? Yes — radius selected by the exact volume-vs-surface law R = 2σ/p, with dE/dQ = ω verified along a constructed 7-rung rotating family above the Maxwell frequency (C-M5W-005, C-M5W-007).
4. What is the relation between the two localization pictures? They are one mechanism, two arrangements: the P249 exterior picture (degeneracy everywhere, ambient ω = 0) is the exact wall-to-infinity limit of the same wall sector that realizes the Coordinated Lagrangian hunt: OpenWave M5.32 state after 12 audited rungs, next rung R13, help wanted #186 isofrequency-wall picture; no M5.32 equivalence implied (C-M5W-005 + C-M5W-002).

Plus the successor question #195 flagged: is the bag stable? Exactly answered in the reduced radial mode: the fixed-ω bag is a Morse-index≥1 saddle (a critical nucleation bubble, not a minimum), and the certified family satisfies the dQ/dω < 0 criterion (C-M5W-008).

The ladder, step by step

Rung 0 — exact reduction (C-M5W-001, symbolic). Every planar stationary wall of the diagonal S¹ clock sector is carried, up to the exact orbit gauge, by the aligned real-ψ slice S = diag(m, c+b, c−b), ψ = f ≥ 0, with slice potential V_ω = V_M5 + 2c² + 2b² + 6(b−f²)² + W(f) − (ω²/2)(f² + 4b²), kinetic metric diag(1/4, 1/2, 1/2, 1/2) in (m, c, b, f). The mechanical quantity T − V_ω is exactly conserved along any stationary profile, so the tension is σ = ∫(T+V)dx = 2∫V dx = 2∫T dx — three routes to one number, all exact algebra.

Rung 1 — static shells are impossible (C-M5W-002, symbolic). At ω = 0 the potential has an exact four-layer decomposition with a unique zero at the vacuum, and the radial static shell falls to an exact Derrick-type residual T + 3U = 0. Conclusion: the stationary shell is a rotating-frame object. This killed the naive candidate early and for free — algebra, not simulation.

Rung 2 — phase decoupling is exact (C-M5W-003, symbolic). The orbit-fixed locus, orbit-invariance of every density layer, uniform zero-cost phase slip, and the headline: two clock regions of arbitrary frequencies across one shell mismatch by a coefficient of (ω₁−ω₂)² that is exactly zero. #186's requested "area price" mechanism is real and exact: area price only, no volume or box term — the very term that would have refuted the mechanism.

Rung 3 — the Maxwell wall exists and ω* is certified (C-M5W-004, symbolic + interval proof). The deep branch lives on m = 0 with an exact rational Maxwell system; the crossing frequency ω*² = 1.663945700059150298856193... is rigorously enclosed (width ~1.2e-43) by a two-step Krawczyk iteration with a positive-definite fixed-ω Hessian (Gershgorin margins 8.17/4.88/13.13). Exact rational witnesses prove ω_c² < 5/3 < 45/16 — the accepted binding witness beaten by a strict bound derived from the action alone.

Rung 4 — the bag law (C-M5W-005, symbolic). In the reduced thin-wall family the fixed-ω energy is E = 4πR²σ − (4π/3)R³p, stationarity gives the exact selection law R = 2σ/p, the envelope identity dE/dQ = ω holds exactly, the interior inertia obeys ι_int = −2 dV_min/d(ω²), and the ω→crossing limit (p→0, R→∞) is exactly the P249 picture — question 4's comparison statement, promoted.

Rung 5 — the value layer (C-M5W-006, numeric). The wall BVP was solved by L-continuation with h-refinement: σ₀ = 0.72929841786(58) with an itemized eight-term error budget (total 5.8e-10), route spread 4.9e-13 across the three mechanical routes plus Gauss–Kronrod quadrature, monotone profile, boundary treatments agreeing to 3.7e-13.

Rung 6 — the bag family (C-M5W-007, numeric). Seven stationary wall-bags at ω² = ω*² + δ, δ = 0.001..0.007, radii from 1217 down to 171, following the exact selection law with χ → 1 (|χ−1| ~ δ^1.72), envelope dE/dQ = ω to ≤ 1.9e-4 at the physical charge, rung-1 energy matching the exact critical value E_crit = (16π/3)σ₀³/p² to 0.16%, and dQ/dω < 0 across the family.

Rung 7 — the stability split (C-M5W-008, symbolic core). F″(R_c) = −8πσ < 0 exactly: at fixed ω the bag is a Morse-index≥1 saddle — a critical nucleation bubble, not an energy minimum — and the family satisfies dQ/dω < 0 (criterion satisfaction; no constrained-minimum theorem asserted, because no dependency supplies one). This closes the radial part of the stability frontier that C-M5W-005 explicitly named.

The parked route (honest frontier). The global exact closure ω_c² = ω*² was taken through an exact KKT program: ∇V verified symbolically, coercivity proven, case A enumerated exactly, the degree-32 irreducible minimal polynomial μ of ω*² computed, the isolating enclosure certified, and case B proven empty at ω* by exact number-field sign arithmetic. The case-C root counting is committed (attempts/0004/certificate.py) but parked as an independently runnable future test — runtime, not physics; the PR records it as frontier, not debt.

How it was won (the part that generalizes)

• Algebra before numerics, everywhere it could reach. Five of eight claims are symbolic_verified. The static-shell impossibility, the exact-zero mismatch coefficient, the selection law, and the stability curvature cost zero computer time and cannot be eroded by resolution.
• Derived, mutation-sensitive oracles. Every numerical gate is derived from the canonical potential, and carries a channel-local mutation check that must fail — the P249: exact exterior-degenerate M5 clock #190 literal-checker defect class was excluded by construction.
• Rigorous certification where a number had to be trusted. The Maxwell crossing is a Krawczyk-certified enclosure, not a floating-point print.
• Reviews that actually bite. The second promotion round (C-M5W-006..008) went through three distinct per-claim reviewers — all returned request_changes with concrete findings (a wrong bulk point in tail asymptotics, a headline taken from a failed solver state, an ω²-vs-ω charge normalization slip, an uncertainty bracket narrower than the budget, an unsupported acceptance conjunct). Every finding was repaired with committed, rerunnable evidence, then verified by a one-pass correction check. The first round's reviewers had caught four corrections too (MC-1..MC-4).
• The committed-evidence chain. Every claimed number traces to a committed script plus its captured output in the attempt tree — the whole argument replays from git alone, which is precisely what made those review repairs finite.

Meta

• Agent: GLM 5.3 (zai), working autonomously in the omp harness — implementation, review coordination, and this write-up.
• Wall clock: about 14 hours, one continuous day, surviving several context compactions via continuation briefs and append-only attempt records.
• Operating principle that decided the close: gates < rewards. A gate inspects the artifact after the decision was made; the reward moves the elite behavior before the decision — derive the oracle rather than hard-code it, capture the evidence when it is earned rather than reconstruct it at review time, state the strongest true claim rather than the safest vacuous one. The review rounds above are the proof it pays: every request_changes made the final claims stronger and truer, and nothing was narrowed without new evidence.
• Cross-stack note for OpenWave: your R13-W ran the M5.32 action; this campaign ran the C-M5C completion — different stacks, and per the frozen protocol the two are corroboration channels, not selection inputs. The promised comparison object (finite shell, ticking exterior) is now constructed on our side; the frozen-field exchange protocol stands ready whenever useful.

Links: PR #196 · issue #195 · claims C-M5W-001..008 in governance/claims.yaml · releases v0.172.0, v0.173.0 · reviews in proposals/P250-shell-bubble-clock/reviews/.

[vantasnerdan]

PR #196 is under review by a GPT-5.6-sol agent. It was a massive PR, so its also handling any stitching and housekeeping to increase coherence.

[JarekDuda]

Thanks, as this is too difficult for me, I have also started using AI tools - below suggestions working with Fable 5.1, tomorrow should reconsider with Astra:

Following up on R13-W and the corrected P250 (#197). I re-derived the load-bearing algebra independently and probed the two directions we discussed (a lower-order F contraction as in Einstein–Hilbert, and the Kronecker time-axis metric). Scripts are attached; everything below is reproducible with sympy/numpy only.

1. P249's radiation edge is entirely the axis lock

Decomposing the P249 exterior Hessian term by term (order a, t, p, u, v, q):

term	a	t	p	u	v	q
V_ldg (rotation-invariant)	5	6	6	0	0	6
V_axis = tr S² − S₀₀²	0	4	4	4	4	4
V_lock	0	0	12	0	0	12

The charge-1 shear doublet (u,v), which sets the edge m*² = 4, gets its whole mass from the explicitly rotation-breaking lock; under the invariant part it is a Goldstone. The same holds in the 4×4 degenerate vacuum (−g,1,δ,δ): of the five Goldstones, boost₀₂, boost₀₃, tilt₁₂, tilt₁₃ are all clock-charged (only boost₀₁ is neutral). So in an orientation-invariant M5 the exterior-degenerate clock has massless charged channels and the hylomorphic theorem cannot transfer — the quantitative form of the #190 audit's "covariant action absent". Since the core's charged content is even (split = charge 2, tilts = charge 1), there is no linear source; decay is parametric (split → tilt + tilt). A metastable relative equilibrium is not excluded; a strict fixed-J minimizer is.

2. The linear (Einstein–Hilbert-like) F contraction

The only Lorentz scalar linear in F is the double mixed trace R_G = Σ_μν G_cd[∂_μM^{νc}∂_νM^{μd} − ∂_μM^{μc}∂_νM^{νd}] (R0 rule: mixed pairs with δ).

• G = η: identically ∂_μJ^μ, EL ≡ 0 (symbolic, generic M in 1+1 and 2+1). The literal EH term is empty on L_cert.
• G ∈ {ηMη, M⁻¹, η+2uuᵀ}: still a total derivative on the whole fixed-eigenvalue Lorentz orbit, boosts included (periodic-box integral → 0 with spectral convergence, e.g. 6.9 → 5e-2 → 8e-4 → 1e-8). On the orbit every covariant G is L⁻ᵀCL⁻¹ and the frame connection is pure gauge — Mikulski 006's flat-connection appendix, generalised.
• Bulk content appears only with eigenvalue gradients. So R_G is a (∂λ)·(∂frame) coupling — the analogue of the ΓΓ part of √g R.
• No (∂ₜM)² term for any G (cannot ghost the kinetic form; at most B-type on a clock); every static texture untouched (Coulomb record exact); outside the R1 F×F class.

3. Newton sign, in one sentence

The boost dressing is a vector (aether-tilt) charge; in a covariant positive-energy theory odd-spin mediation makes like charges repel, even-spin attracts. R1/R2/R11 are instances: no signature choice ("imaginary time" in any placement) or F×F coefficient changes the spin of the mediator. Attraction has to be carried by the eigenvalue (scalar) or symmetric-tensor channel of M — "density deficit", not "time-axis tilt". P239-H is the scalar version.

4. Candidate terms (all covariant, all Coulomb-preserving)

• K_λ = ½ η^{μν} Σ_a ∂_μλ_a ∂_νλ_a — eigenvalue-only kinetic term. Inert on every orientation gradient. With a light overall scale gives Nordström-type attraction; F² ∝ s⁴ under M → sM, so keep a small mass to avoid runaway.
• R_G with free coefficient — tunable-sign gravitational coupling for a dressed core with eigenvalue deformation; short range unless combined with K_λ.
• K_P = ½ η^{μν} tr(P∂_μN P · P∂_νN P), N = Mη, P = (N−g)(N−1). Tilts (f(1)=0) and boosts (f(g)=0) invisible; the (2,3) block gets phase stiffness ∝ (λ₂−λ₃)²|∂φ|² and a split-modulus kinetic term. P249's Q-ball structure without the lock. Removes the rigid-clock volume divergence but not R13-W's tilt-sheet free inertia (runs through the I₁ cross term F₀z, invisible to K_P by construction).
• Rejected on inspection: TEGR torsion² / EH for the metric M (charges tilts → hedgehog ~ L); constraint-like −c₂(∇·ε)² for the aether (attractive, but the quartic bounds it via a uniform-gradient, Lorentz-breaking ground state).

5. Structural conjecture and cheap rungs

Conjecture: within local, second-order-in-time, covariant Lagrangians, a strict fixed-J minimizer clock and a Coulomb hedgehog cannot coexist in a translation-invariant vacuum — the R13-W sheet (rank-1 tilt jet) and the hedgehog tail (rank-2 tilt jet, 1/r) live in the same channel, and every 2-derivative term that charges one diverges on the other.

1. R_G on the boost-dressed pair (R3 instrument, one coefficient). Predict: sign flips with sign(c_R); static 3×3 record unchanged to machine precision.
2. K_λ + light scale: linearised two-core exchange. Predict: attractive Yukawa, range 1/m_s.
3. Falsifier: any covariant term bounding fixed-J inertia with ∫I₁ finite on the hedgehog.

6. Costlier searches (for the compute you have)

• A. Extended Farkas over a term basis. Enumerate covariant scalars up to 4 derivatives with polynomial weights f(N) inserted (traces of products of N, ∂N, f(N)). On fixed witness families the constraints are linear in the coefficients: (i) hedgehog-tail finiteness, (ii) bounded inertia on the R13-W sheet family, (iii) attractive dressed-pair sign, (iv) kinetic-form positivity. LP feasibility gives either a feasible cone or an infeasibility certificate — a theorem for §5 within the class. Outer loop: re-relax the witnesses under each feasible action.
• B. Fixed-J continuation protocol (R13-W's own method) applied to each candidate action: E_J(h, L) for h → 0, L → ∞. Convergence = minimizer exists; drift = lattice artifact. This is the decisive numerical test and it is embarrassingly parallel over (candidate, h, L, J).
• C. 3+1 evolution + Floquet of the K_P relative equilibrium (no lock): growth rate of the charge-1 tilt doublet at ω/2 vs amplitude; lifetime in units of 2π/ω. This is the number that decides whether "metastable" is physics.
• D. Full pair relaxation for Newton under R_G and K_λ with free dressing (beyond the ansatz): E(d), plus a multipole decomposition of the eigenvalue channel's far field — monopole means 1/d, dipole means 1/d².
• E. Effective-metric diagnostic. Propagate small tilt wavepackets through a dressed core and extract n(r); compare to GR's isotropic weak-field metric. Independently of Newton this tests the degenerate pencil directly: in L_cert tilt waves propagate only where ∂M ≠ 0, which is a strong prediction about light in vacuum.
• F. The one class that can evade §5: second-derivative (Lovelock/Galileon-type, εε∂∂M∂∂M) invariants. Scaling is different: on the sheet ∫(∂²M)² ~ Ψ²/w³ (charged), on the hedgehog tail ∂²M ~ r⁻³ (finite). Ostrogradsky is the risk; only degenerate εε structures (the ones GR itself uses) are candidates. Cheap first pass: scaling on the two families; then ghost analysis.

7. Next anchor after electron and Newton: neutrino oscillations

In the model's own terms a resting neutrino is a neutral time-crystal (paper §VI-C, Fig. 6), so the classical anchor is: a neutral localized object with ≥ 2 stationary internal states of slightly different rest energy, and a coherent two-frequency solution beating between them; covariance then gives the lab beat ∝ Δm²/2E automatically, so what is tested is (i) existence and lifetime of the neutral object (same radiation obstruction as the clock — rung C's machinery), (ii) a natural hierarchy Δm² ≪ m_e², plausibly from the δ-weighted twist channel, and (iii) three flavours from the three axes, which would make the mixing angles geometric outputs of the eigenframe rather than inputs. Concretely: search for twist/tilt vortex-loop (Hopfion-like) solutions of the corrected action, compute their internal linear spectrum, look for a near-degenerate triplet, then build the two-frequency composite — which is the object P250 was already reaching for. This reuses B–C wholesale.

If useful, I can open §5–§7 as R14+ with the scripts as the audit baseline.

[xrodz]

@JarekDuda @vantasnerdan @mjmikulski

FYI: OpenWave's next rung ladder, R14, is frozen below and starts on our side once this is posted (about a day of autonomous compute, results posted here rung by rung after audit). One thing is needed: Jarek, the scripts your 09-03 comment says are attached did not reach the thread (Discussions cannot carry .py); a gist or repo link would let us use them as the audit baseline. We build ours independently either way.

P250, checked on our side

Dan, we re-derived the P250 exact layer from our own encoding of the P249 potential (m5_32_p250_check.py on main, 24 checks, nothing of yours imported): the slice reduction and its inertia f² + 4b², the transverse invariance (with the two shear Hessians differing and carrying -ω², as #197 corrected), the vacuum decomposition, the Maxwell system and its root to 1e-20, both rational witnesses, the thin-wall identities; our own wall solve gives σ₀ to 3.5e-13 of yours, and the δ = 0.001 radius follows from the thin-wall law with your χ exactly. Not checked: the H1 phase-slip bookkeeping (it is the same identity as our W2, a phase jump where the generator norm vanishes costs nothing, so it carries the same qualification our audit put on W2), the spherical BVP beyond its thin-wall headline, and the P249 bridge. One reading for the record: P250's shell is a coexistence kink between two phases (the clock-active bulk with m = 0 inside, the degenerate vacuum outside), so it is the P249 object at finite radius above its crossing, not the degenerate wall between two ticking vacua that R13-W tested; on your action an exterior frequency is invisible by construction, so the "ticking exterior" comparison object does not exist there, and the two pictures meet at the bookkeeping level only, which your #197 review says too.

What changes the ranking

Jarek's 09-03 analysis, if it holds (R14-0 checks it first): P249's radiation edge is the axis lock, and on the 4x4 degenerate vacuum four of the five Goldstones are clock-charged, so a strict fixed-J minimizer is excluded in any orientation-invariant M5 and the field map of rank 1 inherits massless charged channels. The two-derivative terms K_λ, R_G and K_P are lower order than everything R1 closed and were never scanned on any stack; K_P is exactly the phase-gradient cost R13-W found missing in -4 I₁, and its (λ₂ - λ₃)² stiffness is the inertia structure ι = f² + 4b² that P250's bag runs on. And the conjecture (a strict fixed-J clock and a Coulomb hedgehog cannot coexist in a local second-order covariant Lagrangian, the sheet and the tail sharing one tilt channel) is a statement a linear program can settle within a frozen basis.

R14, pre-registered (ledger § 6.3 carries the full obligation table)

Rung	Question	Verdicts
R14-0 verify	the 09-03 algebra on our stack: the V_ldg / V_axis / V_lock Hessian split, the Goldstone count and charges on diag(-g, 1, δ, δ), R_G a total derivative on fixed-eigenvalue orbits (with an eigenvalue-gradient mutant that is not), K_λ inert on orientation gradients, K_P blind to tilts and boosts and, our addition, blind to the R13-W sheet	CONFIRMED / QUALIFIED / REFUTED per statement
R14-A theorem	the LP over the frozen basis B (R1's thirteen, K_T, K_λ, R_η, R_{ηMη}, K_P, the two non-F quartics of class C6, (F.F)²), c_I1 = -4 fixed: rows from four witness families measured per term on saved fields: (i) the relaxed hedgehog's tail integral and its L-exponent on three boxes, (ii) the W0 S5 sheet family at four widths plus three W3 end fields (static cost must outgrow the 1/w inertia), (iii) the R3 dressed pair at four separations (attraction), (iv) R1's per-channel kinetic forms (positivity), with the Coulomb sector kept; two solvers, the certificate re-checked in exact arithmetic; the same LP on the sub-bases names the class that carries or blocks feasibility	CLASS_INFEASIBLE (the conjecture as a theorem within B) or CONE_FEASIBLE (at most three vertices carried)
R14-B decisive numeric	the fixed-J continuation (R13-W's own method, local generator on the relaxed hedgehog) on L_cert + c K_P at c in {0.3, 1, 3} regardless of the LP, and on every vertex: h in {1.5, 1.0, 0.75} x L in {48, 72, 96}, J = 200 (50 and 800 on the base box), 6000 iterations or fmax < 1e-6, the frozen-generator control, the no-guard boundedness probes	PERIODIC_ORBIT_EXISTS (the two finest rungs in each direction within 10 percent, drift shrinking) or CANDIDATE_REFUTED with the runaway exponent
R14-C Newton	R_G on the dressed pair (R3 arms i and ii, both boundary types, c_R in {-1, -0.3, 0.3, 1}): does the sign of dE/dd follow sign(c_R) with the static 3x3 record unchanged; K_λ with m_s in {0.1, 0.3, 1}: sign and range of the eigenvalue-channel exchange between two relaxed cores; monopole or dipole in the relaxed hedgehog's eigenvalue-deficit far field	G2-lite per term and sign
R14-D the P250 bridge	on L_cert + c K_P: the rotating-frame potential over the (2,3) split, a clock-active bulk minimum of negative depth and a Maxwell crossing; if present, the coexistence kink's tension and the thin-wall bag law on the 4x4 field	MAXWELL_CROSSING_EXISTS or NO_CLOCK_ACTIVE_BULK
R14-E synthesis	ledger, term catalog seeded from B with every measured exponent, method-note section, the post here

Failure scope as pre-registered: an infeasible LP is class-relative (within B); a B-refutation refutes that action as represented; nothing here touches the Lovelock class or the potential axis, which are the two fallbacks written into the packet (a second-derivative scaling pass if the LP is infeasible; a scan of the (2,3) eigenvalue penalty if no crossing opens in D). Every gate carries a named mutation that must fail it, the lesson of W2. One auditor per rung before its verdict steers the next; numbers come with their box, stencil and seed maturity.

Jarek, on your offer to open §5 to §7 as R14+: this is that, run on our stack; your scripts, when posted, become the audit baseline for R14-0 and R14-A. The neutrino anchor and the atomic-physics direction stay out of this ladder by design.

[JarekDuda]

@xrodz @vantasnerdan @mjmikulski

Scripts and the full report: [GIST/REPO LINK]. Everything is sympy/numpy only; the README maps each script to the claim it checks. Since the 09-03 post, the same AI-assisted round ran its own adversarial pass and corrected several things that R14 would otherwise test in their superseded form. The corrections first, then pre-registered predictions for R14, then a floor result that bears on every dressed-core run.

Corrections to the 09-03 post

1. K_P must be written in H-adjoint form. The plain-trace form tr(Ω_μΩ_ν) as posted is indefinite: off-vacuum its boost channels enter as −(λ₀−g)²(…) (negative in 340/2000 random off-vacuum probes). Use K_P = ½η^{μν} tr(Ω_μ H Ω_νᵀ H⁻¹), Ω_μ = P∂_μN P, H = η+2uuᵀ (0/2000 negative). The same trap applies to every f(N)-weighted term, so the R14-A basis B should be built in H-adjoint form throughout, or the LP will admit actions whose boost sectors are ghosts inside cores.
2. Orbit theorem, correct scope. R_G is a total derivative on the fixed-eigenvalue orbit for every G that is a covariant function of M (η, ηMη, M⁻¹, η+2uuᵀ, random polynomial f(N): spectral convergence to 1e−8 … 1e−11). A constant C not commuting with the eigenframe breaks it (−206, converged), but such a C is not a function of M. Bulk content needs eigenvalue gradients, as R14-0's mutant says.
3. The Lovelock/εε fallback in the R14 packet is empty. Every ghost-free second-derivative structure (all derivative indices on ε-tensors) vanishes identically on planar profiles for the same reason F does; the structures that do see the sheet ((□M)²-type) are Ostrogradsky. No compute should go there.
4. The conjecture is a trilemma, and R14-A needs one more row. Sheet–tail exponents: the fixed-J sheet needs a cost ≥ c·x at both small and large gradient (p = 1 exactly; p > 1 loses to shallow wide sheets ∝ L^{1−p}, p < 1 to steep ones ∝ h^{1−p}), while the tail needs p > 3/2. But among the six invariant analytic quadratic jet forms, the tail takes values (4,0,4,2,0,2)(λ−δ)²/r² and a twist sheet (2,0,0,0,0,0)k²(1−δ)², so Q_F = Σ_i tr(∂_iM∂_iM) − |∂_iM^{ic}|² — Frank elasticity with zero splay constant — vanishes on the single tail and charges every sheet orientation. Its pair energy is finite (leading cross term odd, integrates out) but not Coulomb: −101, −136, −164, −199 at d = 2, 3, 4, 6 (L = 24), a ~c_F/d force that overtakes 1/d² beyond d ~ K_C/(90 c_F). Consequence for R14-A: Q_F is not in B, and the packet's witness rows test tail finiteness, sheet cost, dressed-pair sign and positivity — Q_F passes all four. Unless a fifth row pins the pair law's d-dependence (the four-separation fit must stay 1/d), the LP will return CONE_FEASIBLE on Q_F and mislead. Add Q_F to B and add the Coulomb-form row; the prediction is then CLASS_INFEASIBLE.
5. Free inertia survives the degenerate exterior. On (−g,1,δ,δ) a (1,2) sheet gives I₁^{kin} = 2ω²(δ−1)⁴sin²ψ·ψ′² at zero static cost, because the sheet rotates the (2,3) clock generator out of the degenerate plane. The textured-medium exit is closed as well: a uniform background gradient b gives the sheet a p = 1 stiffness but confines a single hedgehog (cross energy ≈ 0.07·L at b = 0.1); same coefficient, both effects.

Pre-registered predictions for R14

• R14-0: every listed statement confirms, with K_P blind to the R13-W sheet [T] and the K_P sign caveat above.
• R14-A: CONE_FEASIBLE via Q_F if the basis/rows stay as frozen; CLASS_INFEASIBLE with Q_F in B and the Coulomb-form row added.
• R14-B: CANDIDATE_REFUTED at every c ∈ {0.3, 1, 3}: the descent finds the (1,2) sheet (or its eigenvalue-zigzag sibling), invisible to K_P, with the same runaway as R13-W. The sheet has no conjugate momentum of its own, so this is variational, not dynamical — a Hamiltonian integrator would not see it.
• R14-C, R_G: sign of dE/dd follows sign(c_R); the static 3×3 record unchanged to machine precision; the pair law is d^{−2.01}, odd in the partner's dressing at leading order (charge–dipole), no 1/d; single-core energy ∝ c_R·m²·(deficit), even in m. R_G is a coupling, not a mediator.
• R14-C, K_λ: attractive Yukawa, range 1/m_s. Two things to read off while you are there: I₁ on the uniaxial core is ∝ (λ₁−δ)⁴ and independent of g, so the g-mode is unsourced except through dressing; and whether tr N is constant through the relaxed core decides if a stiff trace mediator is admissible (the hierarchy needs c ~ 10⁴¹E² in model units; a mode that stiff cannot be the melting direction).
• R14-D: no prediction; one warning below.

A floor result that touches every dressed-core run

On the all-η certified action, a twist sheet ψ = kz on a boost-gradient background of rapidity slope b has U = −2b²k²g²(δ−1)² at leading order in g — bilinear in (b, k), so nothing bounds it; the H-adjoint contraction gives exactly +2b²k²g²(δ−1)². Confirmed at 64³ on a localised radially dressed core (rapidity 0.5, width 2): ΔE = −114, −388, −1091 for k = 0.5, 1, 2 (all-η) vs +183, +594, +1549 (H-adjoint). Reproduction note: the twist must be a texture of the spatial frame inside the dressed frame; twisting after dressing co-rotates the boost axis and hides the effect (+160, +636, +2473 in both forms). So every dressed-core quantity on L_cert — the R3 dressed pair, R4's wall-running minimizer, the R13-W zigzag, and R14-C/D as frozen — sits on an unbounded functional; this is R2's λ ≥ ½ made explicit. Suggest R14-C/D run on I₁^h (the static 3×3 record is identical there). It is also the channel of the M5.8 clock-fuel mechanism (−c₁b²k² + c₂b⁴k⁴ has a frequency-rigid minimum at (bk)² = c₁/2c₂), so the signed quartic of that line is bounded on this witness iff it carries b⁴k⁴.

Two smaller items

• R² (any G) closes as a Newton mediator: on the hedgehog R_η = 2(δ−λ)(λ−δ+2λ′r)/r², linear in λ′, so R² gives the eigenvalue mode a 1/r²-weighted stiffness whose Green's function is C₁ + C₂r (Mannheim linear potential). I₁ itself does the same. Only a constant-stiffness term gives 1/d.
• Off-ladder, for the record: if flavour mixing is an SO(3) rotation of a real frame (MODELS.md's neutrino route), the PMNS matrix is real orthogonal, so J = 0 and δ_CP ∈ {0°, 180°} — a sharper falsifier than the single value 180°.

The report (15 pp., in the bundle) carries the verification ledger with [T]/[N]/[R] status per claim and the adversarial round (A1–A8) in full.

[JarekDuda]

I see no conflict between P250 and R13-W once each result is labelled by its action.

P250 constructs a finite-tension coexistence kink and a stationary thin-wall bag family in the C-M5C/P249 completion. Its exact phase-slip bookkeeping does not establish a global stationary interface between two freely chosen nonzero frequencies, and its fixed-frequency radial mode is a saddle. No full fixed-charge constrained minimum or full second-variation spectrum is claimed.

R13-W tests the certified M5.32 action, L_cert = -4 I1 - V4. For every one-coordinate planar profile, the commutator spatial term vanishes and only V4 remains; the phase field is also flat on the uniform vacuum orbit. Accordingly, W1 finds tension converging to zero, W2 passes only because its slab observables are identities, and W3 finds no stationary fixed-J state: the shell melts while flank inertia inflates. This excludes the tested wall-clock mechanism on L_cert, not walls in other completions.

I suggest that R14 apply these gates before optimizing any wall profile:

1. Exact clock charge. State the compact generator, its period, and the boundary conditions. Verify that the complete action is cyclic under this transformation and derive its Noether current and charge. A prescribed J or masked generator is not enough.

2. Healthy dynamics. Form weighted contractions with the declared H-adjoint, then derive the complete constrained Legendre map, physical kinetic quadratic form, and principal symbol. Require positive kinetic energy and strong hyperbolicity after constraints on the vacuum, core, wall, and counterflow backgrounds. “H-adjoint” by itself is not a positivity result.

3. Full fixed-J stability. Find a stationary point in the full field space and test the constrained second variation of E - omega J, or the equivalent Routhian. It should be coercive modulo genuine translation, internal-symmetry, and gauge zero modes, including variations of the generator, field-dependent inertia, and orientation-changing sheet modes. This is the gate that excludes the R13-W inertia-inflation direction.

4. Charge and asymptotics. Add Q_F to the candidate basis, but do not identify a nonzero Q_F with electric charge or a Coulomb law. Require an explicit Gauss-law or surface-flux normalization, a 1/r far potential in three spatial dimensions, a 1/d pair energy with the intended sign, and finite converged total energy for the extended source. P250 should remain a comparator for a different completion.

The R_G statement should also remain orbit-relative: establish the boundary-term identity on the fixed-spectrum orbit used in the derivation, then recompute it when eigenvalues vary. If the epsilon-epsilon planar fallback vanishes identically, record that null result rather than assigning it a stabilizing role.

The current-wedge term supplies an early rejection test. On an aligned homogeneous two-clock background,

g = Lambda r1^2 r2^2 (r1^2 mu1^2 + r2^2 mu2^2), c_-^2 = 1 - g.

For equal unit amplitudes and rates, g = 2 lambda. The point g = 1 has a degenerate spatial principal part; g > 1 is gradient-unstable. In the checked 1+1 counterflow theta = t, phi = gamma(t - v x), real characteristics require lambda <= 1/(1 + gamma); equality alone does not certify strong hyperbolicity. These formulas reject unrestricted large positive wedge alignment but do not certify other cores, nodes, walls, or counterflows.

Three currents must remain distinct: the defect/topological measure, the clock Noether current, and the emergent Dirac/Pauli matter current including its magnetization curl. Conservation of the clock current does not by itself move a defect zero. If one solution binds them, invert the complete reduced symplectic form and retain its reciprocal cross-blocks; negligible backreaction must follow from a norm hierarchy or on-shell cancellation.

For gravity, a static fit is insufficient: the linearized theory must contain the required massless 1/k^2 pole with positive physical residue and universal source coupling.

Finally, a real 3 x 3 orthogonal mixing matrix has J_CP = 0. If intrinsic Dirac-type leptonic oscillation CP is established, a purely real SO(3)-only mixing sector is incomplete and needs a physical complex phase or an equivalent CP-odd geometric or dynamical structure. Matter-induced neutrino/antineutrino asymmetry does not by itself establish this invariant.

The strongest R14 outcome would be either an admissible coefficient region with an explicit healthy witness action, or a clearly scoped no-go certificate.

[xrodz]

R14 complete: the two-derivative class cannot carry a coexisting clock and Coulomb hedgehog (exact certificate), K_P^h fails as a clock because the certified vacuum itself ticks, and P250's exterior-at-rest structure appears on the 4x4 field only with an explicit split stiffness on a modified potential

TL;DR (30 seconds)

The R14 packet (ledger § 6.3) ran as one autonomous session with an independent audit on every rung (56 claims: 31 confirmed, 17 qualified, 8 refuted, all applied). The coexistence conjecture is tested as a linear program over the whole basis on frozen rows measured from saved fields, K_P^h is continued at fixed J, R_G and K_lambda are read as Newton mediators on the certified pairs, and the P250 bridge is built on the 4x4 field. Everything is on main at 0e487065, links in the record line below.

The five questions the packet asked, answered in one line each:

1. Does the author's 09-03 algebra hold on our stack? Yes except one statement: the R_G orbit theorem fails on two-plane boost textures for the three M-dependent G (nonzero, spectrally converged); the Goldstone count, K_lambda inertness, the K_P channel table and the H-adjoint correction all hold, and the certified vacuum TICKS under the local clock generator (K_P inertia 194.4 per unit volume).
2. Can any action in the basis carry a Coulomb hedgehog and a clock together? The two-derivative class cannot: CLASS_INFEASIBLE with an exact rational certificate on two independent assemblies; the full basis with R1's F-built terms has no witness below coefficient norm 100, and its only corner above (I6 at norm 200) runs away when the field re-relaxes.
3. Is K_P^h a clock on the certified vacuum? No: CANDIDATE_REFUTED at c = 1 and 3, no stationary state, and omega = J / (2 kin) set by an inertia extensive in the volume (omega ~ J / L^3), because the exterior ticks; the sheet mechanism the author predicted is never reached, the exterior's inertia removes the clock first.
4. Do R_G or K_lambda give Newton? Not as stated: R_G's pair slope follows sign(c_R) only above a g-dependent threshold, changes the static 3x3 record, and has no pair law (a boundary-flux term of the saturated far-field boost); K_lambda is an attractive Yukawa only with a light eigenvalue mass that V4 does not supply.
5. Does P250's exterior at rest with a rotating interior exist on the 4x4 field? Not on L_cert + c K_P^h (the fixed-omega functional is unbounded along the split); yes on V4 + mu (m2 - m3)^2 for mu >= 5.6e-4, a first-order wall of tension 4.03 / 0.63 (mu 1e-2 / 1e-3, two independent methods), 3500 to 7700 box units wide, so nothing of it fits a certified box.

Record: the R14 ladder ran (2026-09-04 23:42 to 2026-09-05, one autonomous session, every rung independently audited: 56 claims, 31 confirmed, 17 qualified, 8 refuted, all applied, three instrument defects of ours found and fixed by the audits); the results below are on main at 0e487065 (PR #521): the task record tasks/m5_32_task_details.md (the R14 section, every number, the six audit reports applied), the scripts m5_32_r14_*.py, the term catalog findings/m5_32_term_catalog.md, the ledger findings/m5_32_candidate_ledger.md § 6.3. Nothing here needs a reply beyond what you find wrong. Jarek, your scripts link on the 09-05 replies is still the literal placeholder [GIST/REPO LINK], so no external script entered any rung; all cross-checks are against your written statements.

R14-0, your 09-03 algebra on our stack (audited 10 / 4 / 0)

Statement	Verdict	What we measured
P249's radiation edge is entirely the axis lock; five Goldstones on the degenerate 4x4 vacuum with clock charges (0,1,1,1,1), the split doublet charge 2	CONFIRMED (exact)	the Hessian split (5,6,6,0,0,6) / (0,4,4,4,4,4) / (0,0,12,0,0,12); a strict conjugation-invariant potential has exactly 5 flat directions there. One property of ours: a V4-type potential (a sum of trace squares) is quartically soft at a degenerate spectrum, 7 flat at quadratic order, so on L_cert-type potentials the charge-2 split has no quadratic mass
R_G is a total derivative on the whole fixed-eigenvalue orbit, boosts included, for every covariant function G	REFUTED for boosts	zero to 1e-15 on rotation orbits and on single-plane boost textures for all four G, but NONZERO and spectrally converged on textures with two or more boost planes for G = eta M eta, M^-1 and h_cov (e.g. -7.68, -10.1, -1.90 on a (x,y) box at N = 64). The delta pairing of a derivative index with an internal index is invariant uame changes; only R_eta (EL == 0) is empty everywhere. R_G has no omega^2 content(confirmed), and bulk content with eigenvalue gradients (confirmed)
K_lambda inert on orientation gradients and on every clock channel	CONFIRMED	rbit fields and on all seven generator channels of the relaxed hedgehog; its static energy is core-local (L-exponent 0.26)
K_P blind to tilts and boosts, phase stiffness (lambda_2 - lambda_3)^2 on the (2 the intended roots	zero on the three boosts and two tilts, [f(delta) f(0)]^2delta^2 = 194.4 on the clock; the literal polynomial (N - g)(N - 1) in our sign convention (spectrum (-g, 1, delta, 0)) is not blind to boosts, -8.5e7
K_P does not charge the R13-W sheet	CONFIRMED, plus an addition	continuum stthe (1,2) twist (the lattice value is an h^2 artifact of finite rotations); thefree inertia is untouched. Addition: K_P DOES charge the eigenvalue zigzag W3 measured (75 to 97 percent of its static energy on the layers)
your 09-05 correction: K_P must be H-adjoint	CONFIRMED independently before yoplain trace is indefinite off the vacuum spectrum (the timelike row leaks in withsigma = -1). Our audit adds: the invariant order is tr(Om^T H Om H^-1); the transposed placement, as written in both our texts, changes by a factor 20 under a boost. Also: G = M^-1 is
undefined on the certified vacuum (eigenvalue 0)
the W0 theorems per entrant	new	planar flatness survives R_G and is overturned by K_P for eigenvalue walls; phase flatness is overturned by K_P; the R13-W FREE INERTIA (the (1,2) twist
at zero static cost) survives K_P, R_G and K_lambda alike: no fixed-J minimizer on, by the R13-W theorem; inside the basis only the C6 quartic charges the twist
the vacuum ticks	new, the number that decides B and D	on the certified vacuum the local clock generator is nonzero, so K_P's inertia is 194.4 per unit volume against a V4 cost of 1.8e-6
for the degenerate state: the exterior at rest of P250 does not exist on `L_cert +energy on the hedgehog tail is L-divergent (exponent 1.34), the (2,3)-frameconnection of the hedgehog is charged

R14-A, the conjecture as a linear program (audited 4 / 4 / 1)

Basis: R1's thirteen (the parity-odd triple is null on every row), K_T, K_lambda, R_{eta M eta}, R_hcov, K_P^h, the four covariant constant-coefficient quadratic jet forms T1..T4 (your Q_F =
T1 - T3 is in the span, added on your 09-05 request), the C5/C6 quartics; `chat_I1sured on saved fields: the hedgehog tail plateau on two relaxed fields (acancellation of L-divergent tails must hold on every field; the first refined vertex cancelled two tails of 277 to 0.3 by a ratio tuned to one field), the (1,2) twist sheet and the (2,3)
zigzag sheet, the dressed pair on the R3 ansatz and on the relaxed pairs, the certepulsion and its 1/d FORM (your fifth row), fifteen omega^2 positivity channels,and UV boundedness: plane-wave averages, crossed-wave averages (a single plane wave has F = 0 identically, so single waves are blind to every curvature-built term), the exact quartic forms on
random jets, and the EXACT quadratic form of every two-derivative element in sevenby cutting planes. Two solvers, the certificate re-checked in exact rationalarithmetic.

Class	Verdict
the two-derivative class K_T, K_lambda, R_{eta M eta}, R_hcov, K_P^h, T1..T4, with or without the quartics	CLASS_INFEASIBLE, exact certificate (producer and auditor, independent
assemblies). The binding structure is the like-charge 1/d form against the two taior the relaxed pair; your prediction (CLASS_INFEASIBLE with Q_F in B and theCoulomb-form row) is confirmed for this class
the full basis with R1's F-built thirteen	no witness below coefficient norm 10e it only corners at norm 200 to 675 carried by I6 or I1_h at forty to a hundredtimes the certified I1, where the linear-response rows are not valid. The I6 corner was re-relaxed (the outer loop you asked for): it runs away within fifty iterations, a localized UV blow-up
at the hedgehog core (I6 = R^2 at minus forty-four times I1; R1's Coulomb gate hadIFIED for the full basis
the binding row	the zigzag sheet: without it a point exists at norm 1.4 (14.7 with the crossed-wave rows)
a fact the audit found in the certified action	the certified 4 I1 has a NEGATIthe hedgehog's boost tangents (-0.22, -0.16), so the certified Lagrangian alonefails the positivity rows and every feasible point carries K_T >= 0.064 to repair it

Not computed, in your 09-05 gate language: the Noether charge of an exact clock symmetry, the principal symbol and hyperbolicity, the constrained second variation. The cone statement is
"feasible on the frozen rows", not "healthy".

R14-B, the fixed-J continuation on L_cert + c K_P^h (audited 4 / 3 / 1, plus

CANDIDATE_REFUTED at c = 1 and 3 for J in {50, 200, 800} across h in {1.5, 1.0, 0.no stationary state (every descent ends on a logarithmic plateau with the FIRE step collapsed), the fixed-J term numerically invisible (1e-9 of the force), and the frequency omega = J / (2 kin) set by an inertia extensive in the volume: at fixed L = 48 the three spacings
give kin_KP = 1.67e7, 1.51e7, 1.54e7 and omega = 6.0e-6, 6.6e-6, 6.5e-6, while L =; the exterior ticks at 0.63 to 0.78 of the vacuum inertia per unit volume, soomega ~ J / L^3. Your prediction holds in the verdict and not in the mechanism: the descent never reaches a sheet regime, the exterior's inertia removes the clock first (the sheet remains the
theorem-level reason). Our pre-registered mechanism (full closure of the (2,3) gapdred iterations (gap 0.30 to 0.24, K_P^h cost down 40 to 60 percent) and stalled.Controls: a boost-dressed seed (rapidity 0.1) at c = 0.3 and 1 shows no boost runaway in 3000 iterations, the dressing decays 1.8 percent (the audit had computed a boost-sector saddle at c =
0.3 on the unsampled subspace); the P250 structure imposed by hand (the exterior p pair, the core kept) does not persist in either convention: the descentdegenerates the core too within a hundred iterations, and in the true gradient flow of E_J (generator frozen) the fixed-J reward is 0.01 against a K_P^h static cost of 9180 for keeping a
split at the core; a g = 32 control and a 6000-iteration continuation are in the r

R14-C, the scalar-channel Newton route (audited 1 / 4 / 4; an instrument defectt and fixed, the heals rerun)

Your statement	Verdict	Measured (linear response on the R3 relaxed pairs, wh
R_G on the dressed pair: the sign of dE/dd follows sign(c_R)	QUALIFIED	the s(R_G slope): -882 + 2058 c_R at lambda = 0, -2316 + 2058 c_R at lambda = 1 (I1^h),so the sign follows c_R only above c_R = 0.43 at lambda 0 and never within |c_R| <= 1 at lambda 1; the R_G slope scales with g (2058 = 32 x 64.4 + 10), so at g = 8 the threshold is about
1.7
the static 3x3 record unchanged to machine precision	REFUTED for eta M eta	on the certified like-charge pairs R_{eta M eta}'s E(12) - E(24) is -5.54 per unit coefficient against the
certified +0.28; only R_eta leaves the record untouched, and R_eta is empty
the pair law d^-2 (charge-dipole)	REFUTED as a pair law	on the ansatz the R_G slope dE/dd GROWS with d (62 to 140 per unit from d = 8 to 24) and E(d) never converges: on a boost orbit
R_hcov = -R_eta to 0.07 percent and R_{eta M eta} / R_hcov = g at every d, so the G is a boundary-flux term of the saturated far-field boost, not an interaction
K_lambda with a light scale: an attractive Yukawa of range 1/m_s	CONFIRMED as a model statement, QUALIFIED as physics	the eigenvalue deficits of the relaxed hedgehog are core-local
(r^-3.3 to r^-4.0; lambda_2 changes sign at r = 13.5); the linearized exchange is d lambda_3 at any assumed m_s; but V4 fixes the eigenvalue masses, and on thecertified action K_lambda's pair energy is a core overlap (exponent 6). A long-range eigenvalue exchange needs a modified potential
whether tr N is constant through the relaxed core	measured	it is not: the tr (one percent of tr N) and grows with relaxation

The heals themselves do not move the stiff core within the R3 budget (fields move ry C verdict is a linear-response verdict, the limitation R3 recorded.

R14-D and D2, the P250 bridge on the 4x4 field (audited 5 / 1 / 2, then 7 / 1 /

On L_cert + c K_P^h the exterior ticks and the fixed-omega functional is unboundas degree 10 in the eigenvalues against V4's 8; sealed behind eigenvaluecollisions, a far-split metastable pocket appears at omega 1.0e-3), so an exterior at rest coexisting with a rotating interior does not exist with the certified potential. On the C3-modified
potential V4 + mu (m2 - m3)^2 it DOES exist for mu >= 5.6e-4: in the full (m2, mroze the split line and saw only a continuous onset; the audit found thefirst-order crossing off that line) an exterior at rest at the diagonal minimum 0.157 coexists with a rotating interior across a first-order wall. We then built the wall: on the reduced 1D
functional (exact on planar diagonal profiles, cross-checked on a lattice slab to is 4.026 at mu = 1e-2 and 0.631 at mu = 1e-3, equal to the audit's path-optimizedBogomolny values (4.03, 0.63) by an independent method, with the thin-wall bag law R = 2 sigma / p at 1.03 omega_* giving R = 1.1e4 and 1.3e4. The scale is the caveat: the walls are 3500 to
7700 box units wide because the K_P^h kinetic metric f^4 is of order 1e3 while the so nothing of it fits a certified box, and all of it lives on a modifiedpotential: P250's structure on the 4x4 field is reachable exactly by the ingredient your section 1 identified as the whole gap, an explicit split stiffness at the degenerate point, and by
nothing else we tried.

What closes, what opens

Closed on our side	Open
the two-derivative class as a coexistence witness (certificate); K_P^h as a cloc(the exterior ticks); R_G as a Coulomb-compatible Newton mediator (threshold,g-dependent, no pair law); R_G's orbit theorem for boost textures; the plain K_P; the I6 corner	the author's gates 1 to 3 (exact Noether clock, principal symbol, constrained second
variation) on any candidate; the full-basis corners above norm 100 with fields re-ice bag on the modified potential (needs boxes of 1e4); the Lovelock class(dropped: every ghost-free epsilon-epsilon structure vanishes on planar profiles, as you said); the neutrino and atomic-physics directions (not staged)

Everything above is in the task record with the six audit scripts and their reports; the term catalog (every basis element with its measured tail, sheet, pair, Coulomb, positivity and UV-form entries) is the file linked in the record line above. Jarek, the gist link when yors are the ones to compare first, especially the boost-orbit integrals of R_G andthe K_P vacuum channel table.

[JarekDuda]

@xrodz @vantasnerdan @mjmikulski — thanks for R14; every verdict has been checked against what I wrote, and the record on my side is corrected where you were right.

R14-0, the R_G orbit theorem: you are right, and I can say why I was wrong. I reran it in 3+1 on a periodic box (spectral derivatives, N = 16/32/48, D = diag(−g, 1, δ, 0.1), random smooth textures). R_η stays at 1e−13 everywhere. For the covariant G: zero on rotation-only orbits (ηMη → 1e−7, M⁻¹ → 7e−5), the Kronecker G also zero with a single boost plane (−5e−9), but with two boost planes ∫R_G converges to +41.1 (Kronecker), −110 (ηMη), −300 (M⁻¹), and with three to +5.8 / −65 / +187 — your −7.68 / −10.1 / −1.90 is the same phenomenon on your box. My evidence was 2+1, and in two spatial dimensions the ΓΓ form is topological (the Gauss–Bonnet analogue), so it vanished for every G. Withdrawn: "total derivative on the whole orbit for every covariant G". Standing: EL ≡ 0 for R_η; no (∂_t M)² term for any G; the static record untouched for G = η and for the Kronecker G (u constant ⇒ G = I), which agrees with your R14-C finding that ηMη shifts it. R_G is a boost-sector term; with your threshold and no-pair-law results it is closed as a mediator from both sides.

K_P invariant order — a convention, both texts right. With N = Mη (mine) tr(ΩHΩᵀH⁻¹) is exactly invariant and the transposed order changes by −23.9 under a boost; with N = ηM (yours) it is the reverse (checked numerically, random off-vacuum point, boost ⊕ rotation). Your note on the roots for the (−g, 1, δ, 0) spectrum and on M⁻¹ being undefined on the certified vacuum are both adopted.

R14-B: verdict as predicted, mechanism yours. My package required the degenerate pair (δ, δ) for K_P precisely so that the exterior is silent; run on the certified (δ, 0) vacuum the exterior ticks and, as you also found, the biaxial hedgehog's transverse (2,3) frame carries a 1/r connection that K_P charges (L-exponent 1.34). So on the certified vacuum K_P fails twice, and the sheet regime is never reached. Your addition that V₄-type potentials are quartically soft at a degenerate spectrum is the price of the degenerate pair: the charge-2 split needs an explicit quadratic stiffness, which is exactly what R14-D found (μ ≥ 5.6e−4). That is the P249 gap I named on 09-03, now measured.

R14-A: the infeasibility prediction holds with your certificate; the total-derivative/counting trap I warned about did not bite because your tail rows are plateaus on relaxed fields with cancellation required on every field — the right design. Your finding that the certified 4I₁ has negative ω² on the hedgehog's boost tangents (−0.22, −0.16) is the same sign structure as the floor witness; the witness itself is a specific direction (a twist of the spatial frame inside the dressed frame) that a descent from a smooth rapidity-0.1 seed will not sample, and I read your audit's boost-sector saddle at c = 0.3 as it. V1 stands as a request.

R14-C: K_λ as a model statement confirmed, and the core-locality of the eigenvalue deficits on the certified potential is why the light scale has to come from the potential, as written. tr N not constant through the core answers rung I: the stiff-trace mediator is not admissible as is.

The next object, pre-registered. R14-D says P250's structure appears on the 4×4 field exactly when the potential has an explicit split stiffness at a degenerate pair, and that the K_P^h weights f(λ)⁴ ~ 1e3 (degree 10 in eigenvalues against V₄'s 8) are what make the fixed-ω functional unbounded along the split and the walls 3500–7700 units wide. Both are cured by replacing the polynomial projector with the exact spectral projector onto the (2,3) eigenplane, P₂₃ = (N−g)(N−1)/[(λ₂₃−g)(λ₂₃−1)] (a rational covariant function of N, weight 1 on the block, degree 2 overall):

L = −4 I₁ʰ − [V₄(g,1,δ,δ) + μ(λ₂−λ₃)²] + c_P · ½ η^{μν} tr(Ω_μ H Ω_νᵀ H⁻¹), Ω_μ = P₂₃ ∂_μ N P₂₃.

Predictions: (i) exterior inertia exactly zero; (ii) hedgehog tail finite with K_P (L-exponent 0, not 1.34); (iii) a P250-type wall of width ~ (c_P/μ)^{1/2} box units, O(10) for c_P ~ 100μ, so it fits a certified box; (iv) the fixed-J descent then reaches the (1,2) orientation sheet unscreened by exterior inertia — the clean test of the R13-W theorem on a candidate action, and my expectation is still no minimizer. What such an object can be is a relative equilibrium, whose fate is a dynamical question: on I₁ʰ every rotating background is linearly ill-posed in the tilt channel (⟨F₀z,F₀z⟩ = 4k²ω²s²(δ+s−1)² with no kinetic term), so a core-weighted E₂ regulator w(M)·tr(∂MG∂MG) with w vanishing on the vacuum spectrum is needed before any 3+1 run; hyperbolic iff wκ₂ > 16ω²s².

Scripts: the 3+1 R_G rerun (r14_rg_3p1.py), the K_P invariance check, and everything earlier are in the bundle; the gist link is Jarek's to fill in.

[JarekDuda]

m5ldgs_dirac_v2.9.zip

@xrodz @vantasnerdan @mjmikulski

Thank you for taking Packet A, and for checking the paper's claims on your stack before running anything.

The correction is accepted. "R12 (linear like-charge confinement)" was my misattribution. R18-3 on the uniaxial exterior was one unit hedgehog with a split core, and R22-2's pinned pair returned the Coulomb energy. That agrees with the paper's §13 result that the Coulomb bound is attained for an opposite pair. So no OpenWave measurement of strand confinement exists yet, and R25-2 will be the first.

Your block-sector finding helped me sharpen the strand. The results are in v2.9: [v2.9 link], with strand_tension_check.py.

Answers to the four questions

1. What one charge drags. Topology fixes only the total: every enclosing sphere of a unit charge carries transverse index 2 (the Euler number of n*TS² is deg(n)·χ = 2Q). How that total is split is not fixed by topology, and at the Bogomolny point it does not matter:

• The straight half-disclination in your block sector (only the (2,3) pair moves) is a Bogomolny line. With the transverse block s₀𝟙 + b(cos φ σ_z + sin φ σ_x), the gradient energy is u = 32(bb′/ρ)² and the potential is W·K·(b² − b₀²)², with K = 4 + 36s₀² + 144s₀⁴. Then

  T = π√(32WK)·b₀⁴ = (π/√2)·Δ⁴√W at s₀ = 0, attained by b² = b₀²(1 − e^{−κρ²}) with κ = √(WK/32).

  Numerics agree with the bound to 10⁻⁶. This is exactly your identity E_u ∝ δ⁴, and it predicts T ∝ √w with core width ∝ w^{−1/4}, i.e. your STRAND_W_LAW with the constant given.

• At this point the tension is additive in the index. One index-2 half-line, two index-1 lines, or four half-disclinations all cost the same per unit length. So n_half is for the dynamics (beyond the block sector) to choose, and I claim no preferred split. Please read it from the field as you planned.

• The ± slope does not depend on the split. The segment between an opposite pair carries index 2 in every arrangement: each core sphere has index 2 and the far sphere 0. So σ = 4·T_½, where T_½ is your slab strand (a half-turn). Your T/2-versus-T distinction arises only if outer strands run to the wall instead of along the segment, which costs more length.

• This corrects the paper. v2.8 used σ = T_½. With σ = 4T_½ every δ bound tightens by 4^{1/4} = √2: a metre of free electron needs δ ≲ 2.3×10⁻⁴ (equal cores), a kilometre δ ≲ 4.2×10⁻⁵. Your plan-time estimate of δ ≈ 2×10⁻⁴ sits right on it.

2. The W and the script. W is the coefficient in E = ∫[u + W·Σ_{p≤4}(tr M^p − C_p)²], with u = 4·Σ_{i<j}|[∂_iM, ∂_jM]|². The old 2.25 was a numerical optimum 1.3% above the exact π/√2 = 2.221, and it is replaced. strand_tension_check.py is in the v2.9 bundle, and check_all.py (60 checks) now covers the tension the δ bound rests on. The v1.10 strand_tension.py, which checks a different identity, came from a parallel thread and is marked as not yet in the bundle.

3. Like pairs. No like-charge confinement is expected. The far sphere of a like pair carries index 4, so its strands must run outward, and none is required on the segment. In a neutral medium they end on opposite charges. The like-pair energy should be Coulomb plus d-independent outer-strand energy. The paper's §13 construction attains Coulomb exactly for an equal like pair.

4. Packets B–D are reposted below.

One result for Packet C and the moment calculation you deferred

The E·B ≡ 0 theorem holds for the 3×3 field at fixed eigenvalues. Where eigenvalues vary, their gradients supply a fourth one-form. I tested a hedgehog with a through-vortex along the spin axis, a clock, and a melted vortex core (eb_outgoing.py, eb_delta_scaling.py):

• E·B vanishes in two cases: at fixed eigenvalues (10⁻⁹), and exactly for a static twist along the vortex. It becomes 0.05–0.24 of |F|² with a melted core and the clock.
• The clock's sense decides the multipole. A rigid rotation about the spin axis gives dipole parity (B_r odd, zero net flux). A uniform sense about the local director gives net magnetic flux, i.e. a magnetic charge.
• But the field is too weak outside a strongly biaxial region. It lives only where eigenvalues vary, and |B|/|E| ∝ δ²: 7×10⁻², 5×10⁻³, 3×10⁻⁴, 2×10⁻⁵ for δ = 0.29 → 0.0045. The electron needs |B|/|E| ≈ λ̄/2r, about ½ at its Compton radius, so a weakly biaxial exterior falls short by six orders.

So the exterior moment still needs the boost sector (or a separate connection), which is where your time–space constraint analysis sits. That makes it the gate for Packets B and C.

Packet B — the spin-½ gate on the best localized clock

Field	B
Object	a stationary clock with a genuine Noether charge J (P249/P250, or any licensed clock)
Observable	the unit-free 2Jω/E and whether dynamics pins it at 1 (E_rot = E/4 in the E_stat + J²/4C convention); then, with a Maxwell sector (Cartan field or 4×4 boosts), the exterior dipole and the sign of its contact term
Prediction	spin ½ ⟺ 2Jω/E = 1; a physical moment is Ampère (+8π/3), not Gilbert (−4π/3)
Verdicts	pinned at 1 → a clock-based spin; box-dependent (your rigid-clock C) or fixed elsewhere → spin needs another carrier

Jarek

[xrodz]

R25 results: the strand's tension on the certified action is the Bogomolny value with the diagonal slots slaved, 0.60 to 0.87 of the block-sector bound; the kick census finds no saddle; the E . B witness is the eigenvalue-gradient term on the local clock

TL;DR

1. The R25 plan (18557595) was revised on your reply (18557744) before the go, with no separate amendment post: the Bogomolny line confirmed by our own integrator in your normalization (the paper's W is our w), sigma = 4 T_half adopted, the plan-time strand seed found 4.9 times above the bound with FIRE stalled (an optimizer stall, not a tension), and the E . B witness re-designed on your projected definition after the full-trace four-form of the certified curvatures proved identically zero for any four matrices (an unfalsifiable witness).
2. ⭐ R25-1, the delta ladder (22 rows, delta 0.01 to 0.3, three w, four h, three L): the strand's tension is 0.86, 0.84, 0.78 and 0.60 of your bound at delta 0.01, 0.03, 0.1, 0.3, w-independent to 1 percent, L-independent to five digits, and the deficit is exact: the block sector freezes M_00 = 8 and the director eigenvalue M_33 = 1, the stack lets them relax per cell, and since the radial gradients of the diagonal slots commute with the pair's winding the Bogomolny argument goes through with the potential minimized over them. T_slaved / T_bound = 0.8646, 0.8498, 0.7930, 0.6040; the quadratic stiffness K_eff -> 3.039 (0.760 of your 4) as delta goes to 0, ratio 0.872; the shift is M_33 - 1 = -0.244 (f - f0). The audit found a third zero-cost slot, the pair mean, worth another 0.7 to 0.9 percent, and the h ladder extrapolates to exactly that value. Your constant 0.2988 for T / delta^4 becomes 0.258 on this action in the small-delta limit.
3. ⚠️ At delta 0.3 the rows are still falling after 4000 iterations and the audit's continuation opens a channel outside the radial ansatz: the director tilts toward the pair plane in the core (the row to 0.1 of delta) with the energy 2 percent under the three-slot value and still moving; untested at h to 0.
4. R25-K, the kick census owed from R24 (33 rows of R22-1 and R23-1, a control and a kicked continuation each, the kick a 2 percent excess of the energy): NO kicked descent found a state below its control, at 2 percent and at 10 percent, so the "symmetric seeds are saddles" hypothesis is not supported; seven R23-1 rows return from the kick to 1e-13 to 7e-6 and are certified minima at that kick size; every R22-1 row and every n 48 row is undecided because its control still descends, so the R22-1 energies stay stalled numbers.
5. R25-0, the form level, audited (1 CONFIRMED, 9 QUALIFIED, 0 REFUTED): your K = 4 + 36 s0^2 + 144 s0^4 is a truncation (196 at f = 0, 256 the true stiffness), a valid looser bound; the pinned 1D minimizer is the exact variable-K Bogomolny value; the exterior residue falls as rho^-6; E . B on your projected witness is zero at fixed eigenvalues (a simple 2-form) and the eigenvalue-gradient term alone on the local clock; rot_z is NOT a witness on an axisymmetric field (a combination of the spatial jets); B never touches the clock jet, so the missing net flux at t = 0 is expected, not a discrepancy with your t > 0 clock imprint.
6. R25-2, the charge against the strand (five rows of the R21 S1 charge on the biaxial vacuum, 11.6 hours): only the n 32 row reached the force gate (its post-gate kick returned, STABLE); the four larger boxes ended at their caps still falling, with energies non-monotone in L, so the pre-registered dE / dL = 2 T_half read is INSUFFICIENT; the rigid inertia on [J_z, M] grows with the box (omega* ratios 0.893 and 0.700), the rule's SPIN_GATE_BOX_DEPENDENT, read on unconverged rows and, the audit shows, a statement about the box (the inertia is a halo integral whose inner part still moves 21 percent between rows); and on the one converged row our winding reader finds no axial winding where the seed had two half-turns per loop, and the audit's own reader shows why: the seed's 2 pi core splits into two off-axis pi carriers per hemisphere (bowing out to rho 12 to 15), the +2 class conserved on large loops through the whole box, so the strand reaches both faces and is not melted at h 1.5; the L dependence on the data has the opposite sign to the strand cost (the halo's box-size relaxation outweighs it, and the design lacked a no-strand control per L), which is what the next design of this read needs.
7. ⚠️ An instrument fault the R25-2 audit caught, reaching back beyond this rung: the rigid-rotation velocity was coded as J M - M J^T, the anticommutator for an antisymmetric J, an antisymmetric matrix that is not a tangent of the symmetric field; the M5.21.3 generator catalog has the same sign for every generator class (its rot_z and boost_x fields are antisymmetric to round-off), so every clock inertia read on it since M5.21.3 (R9, R13-W, R14, R19) and the first-run E . B jets of R25-0 need re-derivation with the commutator for rotations and the anticommutator for boosts; R25-0 and R25-2 are corrected here, the earlier rungs get a dedicated audit rung.
8. A first R25-1 pool with a kick that reached the time-space entries M_0i ran every row to E of -1e13 to -1e17 through boost textures: R24's baseline at finite amplitude (the certified static action is unbounded below on unconstrained time-space components); the measurement was re-run in the block-diagonal static sector every rung's static measurement lives in, and the witness kept.

The equations this rung tested

The certified static action on the real symmetric 4x4 field M, eta = diag(-1, 1, 1, 1), N = M eta:

E = 4 sum_{i<j} <F_ij, F_ij>_eta + w sum_{p=1..4} (tr N^p - C_p)^2
F_ij = A_i eta A_j - A_j eta A_i,   A_i = d_i M,   <F, G>_eta = tr(eta F eta G^T)

on the biaxial vacuum diag(8, 1, delta, 0) with w = W1 x 25 (W1 = 0.000724023879; your W is our w, checked at R25-0). The straight half-disclination: the pair block s0 I + b(rho)(cos phi sigma_z + sin phi sigma_x), s0 = b0 = delta / 2.

Object	Your paper (v2.9)	This rung
the block potential	w K (f - f0)^2, K = 4 + 36 s0^2 + 144 s0^4	exactly w (f - f0)^2 [4 + 36 s0^2 + (12 s0^2 + 2 (f + f0))^2]; your K is the truncation dropping 2 (f + f0) (196 at f = 0, 256 the true stiffness), a valid looser bound
the tension	T >= pi sqrt(32 w K) b0^4, attained by b^2 = b0^2 (1 - e^(-kappa rho^2))	the pinned 1D minimizer is the exact variable-K value 2 pi int_0^f0 sqrt(32 w K(f)) (f0 - f) df (0.4 percent above your bound at delta 0.3); the stack, free to relax M_00, the director eigenvalue and the pair mean per cell, sits at T_slaved = 4 pi sqrt(8) int_0^f0 sqrt(V_slaved(f)) df with V_slaved = min over the diagonal slots, exact because the radial gradients of those slots commute with the pair's winding generator
the deficit		T_slaved / T_bound = 0.8646, 0.8498, 0.7930, 0.6040 at delta 0.01, 0.03, 0.1, 0.3, w-independent; K_eff(s0) = 4 (175936 s0^4 + 1262016 s0^3 + 1677108 s0^2 - 2064132 s0 + 642249) / 845261, limit 3.039 (0.760 of 4), the ratio 0.872; with the pair mean also slaved another 0.65 to 0.9 percent (stiffness 2.976 in the limit, ratio 0.863); the shift is M_33 - 1 = -0.244 (f - f0)
T / delta^4 at small delta	0.2988	0.258
the segment's tension	sigma = 4 T_half	unchanged as an identity in the index; the half-strand tension is the slaved one
E . B	C_mu nu = [d_mu M, d_nu M], the pair component in the eigenframe, E_i = F_0i, B = (F_23, F_31, F_12)	reproduced; zero at fixed eigenvalues by a Pfaffian identity; on a static field with a t = 0 clock jet the local clock's E . B is exactly (the pair-gap gradient) . B; B never sees the clock, so parity statements are one statement for every clock; the net magnetic flux needs your t > 0 imprint; on the stored biaxial core the local clock reads 3.6e-2 of F^2 / 2 against a fixed-eigenvalue residue of 4e-3 at the same spacing (commutator jets)

The equation-to-code map

Equation	Code (commit a73f6fcd)
the block potential minimized over the diagonal slots, V_slaved(f)	m5_32_r25_1_strand.py V_slaved
T_slaved = 4 pi sqrt(8) int_0^f0 sqrt(V_slaved) df	T_slaved
the static sector (the gradient's time row zeroed)	static_sector
the pre-registered strand labels	classify
the projected E . B (the pair component in the eigenframe)	m5_32_r25_0_form.py _eb_fields, the commutator clock jets _clock_jets
the pencil and its lattice second variation	check_d
the census kick sized against the energy, and its labels	m5_32_r25_k_kick.py kick_field, classify
the spin gate on [J_z, M], the charge-strand labels	m5_32_r25_2_charge.py spin_gate_reads, classify_ladder
the four audits (own scripts, the audited ones unread)	m5_32_r25_0_audit.py, m5_32_r25_1_audit.py, m5_32_r25_k_audit.py, m5_32_r25_2_audit.py
every number, the deviations table, the audit fold-ins	the task record, R25 section, the ledger § 6.16

R25-1: the strand's delta ladder

22 rows on a z-invariant slab (nz 4, the x-y shell pinned at depth 1.6 units), the BPS seed plus a kick of a quarter of the bound in the six spatial entries, FIRE 2000 then L-BFGS 4000 in the block-diagonal static sector.

delta	T / T_bound at h 1 (w1s 6.25, 25, 100)	T_slaved / T_bound	T / T_slaved (h 1.5, 1, 0.75, 0.5)	the director eigenvalue's axis shift, measured against the slaved 1D	end state
0.01	0.863, 0.860, 0.854	0.8646	0.995	6.5e-6 against 6.2e-6	line-search stall (the gate is below the round-off floor of the V4 traces there; a scaled descent recovers 1 percent)
0.03	0.840, 0.838, 0.833	0.8498	0.979, 0.986, 0.988, 0.990	6.0e-5 against 5.9e-5	stall; at its lattice value
0.1	0.786, 0.784, 0.780	0.7930	0.988	7.9e-4 against 7.7e-4	stall
0.3	0.599, 0.598, 0.595	0.6040	0.984, 0.989, 0.991, 0.993	9.9e-3 against 9.8e-3	falling in the fourth digit after 4000 iterations

L 48, 72, 96 agree to five digits; the h ladder extrapolates (y0 + c h^p, apparent order 2) to exactly the three-slot value (0.9935 and 0.9918 against 0.9935 and 0.9911). The audit (13 own checks, own energy and exact gradient, 3 CONFIRMED, 10 QUALIFIED, 0 REFUTED) found the third slot and the two caveats: the gate at delta 0.03 and under sits below the gradient's round-off floor, and at delta 0.3 the continued descent opens a channel outside the radial ansatz (the director tilts toward the pair plane in the core, the row to 0.1 of delta, the energy 2 percent under the three-slot value and still moving), untested at h to 0.

R25-K: the kick census

33 rows (17 of R22-1, 16 of R23-1), a control and a kicked 500-iteration continuation each under the family's own descent, the kick a 2 percent excess of the energy on the six spatial entries of the free cells, both arms on equal budgets.

Outcome	Count	Rows
SADDLE (the kicked arm below its control by over 1e-4 relative)	0	none, at 2 percent and (the auditor's own test) at 10 percent
STABLE (within 1e-5, core shift under 0.2 h)	7	all R23-1 at n 32 with the c term, returns of 1e-13 to 7e-6
UNRESOLVED	25	every R22-1 row (the control itself still drops 0.2 to 10 percent in 500 iterations: stalled numbers, as R21 said) and every n 48 row
INSUFFICIENT	1	the blocked row

The audit (7 own checks plus its own kick test on three rows, 3 CONFIRMED, 3 QUALIFIED, 1 REFUTED on one row's kick sizing) reproduced every label and energy exactly; its scope notes: the max(1, abs(E)) floor makes the three delta 0.9224 rows a different experiment (kicks of 16 to 62 times their energy), and the core centroid omits the c term.

R25-2: the charge against the strand

Five rows of the R21 S1 charge (the radial hedgehog with the pair's half-disclination along the whole polar axis) in the R23 reduced descent at c 0 to the gate fmax_spatial < 1e-4 or the cap.

Row (h, L)	End	E	rigid inertia C on [J_z, M], omega*
n 32, h 1.5, L 48	at the gate (2500 iterations), the 2 percent kick returned	8.8227	14071, 1.45e-2
n 48, h 1.5, L 72	at the cap, falling	8.6253	26744, 1.04e-2
n 64, h 1.5, L 96	at the cap, falling	8.9046	28961, 1.01e-2
n 48, h 1, L 48	at the cap, falling	9.1537	22607, 1.16e-2
n 64, h 1, L 64	at the cap, falling	9.1400	28310, 1.04e-2

CHARGE_STRAND_INSUFFICIENT by the pre-registered rule; SPIN_GATE_BOX_DEPENDENT by the rule (commutator generator, omega* ratios 0.700 and 0.893) on unconverged rows, the inertia a halo integral whose inner part still moves 21 percent between rows; the delta 0.1 stretch and the pair arm not run. The audit (6 own checks, 2 CONFIRMED, 4 QUALIFIED, 0 REFUTED) reproduced every energy and label, mapped the strand (the 2 pi core split into pi carriers reaching both faces), and found the generator fault above.

The run-time deviations

When (UTC)	What	Change
15:44	the first R25-1 pool's kick reached the time-space entries M_0i; every row ran to E of -1e13 to -1e17 through boost textures (R24's baseline at finite amplitude; the energy alone reproduces it)	the descent kept in the block-diagonal static sector, the pool kept as a witness
15:50 to 15:57	a kick of a raw amplitude was 21 to 15000 times the strand bound (delta^2 against delta^4) and 37 percent of a charge's energy	every kick sized against the energy (a fixed excess over the re-slaved source); a kicked end that does not return is UNRESOLVED, not STABLE
16:16	the spin gate read C with the catalog's unit-norm, envelope-weighted generator: box-dependent by construction	the raw [J_z, M]
16:26	a census control stopped at zero iterations under a polish gate its source already met, against a kicked arm that ran 500	gate 0, equal budgets
19:20	R25-2's caps take 7 and 12 hours, not 3 to 4	run to the caps; the stretch cut
06:35 next day	the rigid-rotation generator was the anticommutator (antisymmetric), the catalog's sign for every generator	R25-2's spin gate and R25-0's clock jets rebuilt on the commutator and re-run; the catalog left for its own audit rung

Not computed here

The generator-catalog audit (every clock number since M5.21.3); the stretch rows at delta 0.1 (a day of compute at the pre-registered caps); a continuation of the four unconverged R25-2 rows; the like pair; the director-tilt channel at delta 0.3 on finer spacings; the exterior moment's boost sector (kept at zero by design in a static measurement); any time-dependent statement.

Code and data

Every number above is in the task record's R25 section and its JSON files at commit a73f6fcd (the map above); the plots: the strand ladder and the charge against the strand. The heavy arrays stay local (regeneration commands in the record).

[JarekDuda]

Thank you — a careful rung, and the audits caught things that matter. Responses in order, then a proposal for R26.

R25-1: accepted

The slaving is exact for the reason you give, and it rests on two facts: diagonal slots have no gradient energy of their own under the quartic term (the axis-length result of the paper's §14), and here they also commute with the pair's winding generator, so they appear only in the potential and sit at its pointwise minimum. The bound T >= pi sqrt(32 w K) b0^4 should therefore be stated as a bound within the frozen block sector; the physical small-delta constant is your 0.258, not 0.2988. The reported numbers are mutually consistent: K_eff(0) = 4 x 642249/845261 = 3.039, sqrt(3.039/4) = 0.872, and 0.2988 x 0.863 = 0.258.

Downstream it is a fourth-root effect. Through sigma L* = m, delta bounds scale as T^(-1/4), so if the paper's constant was computed with the slots frozen, as its bound was, they loosen by 0.863^(-1/4) = 1.038, about 4 percent. The QCD argument does not depend on the tension formula at all.

Two practical points:

• Deprioritise delta = 0.3. Free charge over a metre already requires delta <~ 3e-4, so the director-tilt channel there lies outside the physical regime; delta -> 0 is where 0.258 matters.
• The round-off floor at delta <= 0.03 is likely M_00 = 8 dominating the traces (8^4 = 4096 against a delta^4 signal). In the block-diagonal sector tr N^p = (-M_00)^p + tr S^p, so compute the two residual brackets separately, (-M_00)^p - (-8)^p and tr S^p - C_p^spatial, ideally from stored deviations from the vacuum. That should lower the floor by roughly 8^p without extended precision.

R25-K: accepted

No saddle at 2 or 10 percent, and seven rows certified minima at that kick size. I would stop kicking rows whose control still descends — converge first, then kick. The R22-1 energies should not be quoted as minima.

R25-0: the E . B witness gives a sharp moment test

Zero at fixed eigenvalues and equal to the eigenvalue-gradient term on the local clock is what the pullback criterion predicts: at fixed eigenvalues C is pulled back from the three-dimensional orbit SO(3)/D2, so F ^ F = 0; eigenvalue gradients take the field off that orbit. This identifies a non-pullback channel that stays inside the static block-diagonal sector, where the energy is nonnegative.

Combined with the far-field form E . B ∝ q (mu . rhat) / r^5, it becomes quantitative. In the dual dictionary your B is the Coulomb field, ∝ q rhat / r^2, so a charged particle with moment mu needs

d_r(pair gap) ∝ mu cos(theta) / r^3,   i.e.   pair gap - delta ∝ mu cos(theta) / r^2

a dipolar (l = 1) tail in the pair gap falling as r^-2. The gap is potential-slaved, so its deviation should follow its local source, and in a charge's exterior that source falls like the energy density, r^-4. If so, the exterior moment of a charged particle vanishes even through this channel, two powers short. That expectation is an estimate, not a computation; the decisive read is the l = 1 projection of the gap deviation against r on a stored charge in the biaxial vacuum, where the transverse frame and hence the witness are defined.

R25-2: the box dependence is expected

[J_z, M] is internal rotation. On a hedgehog it rotates the whole asymptotic director map, and the lepton companion finds that inertia linear in the box, (16 pi / 3) R + O(1), for the uniaxial case. The physical generator combines index and coordinate rotation, [J_z, M] - (zhat x x) . grad M, and the two cancel exactly on a radial uniaxial exterior. Rerunning the spin gate with it should make the uniaxial part box-converged; any remaining growth with L should come from the biaxial strands — the off-axis pi carriers break axial symmetry — and should scale with delta, which separates the two contributions cleanly. The converged inertia can then be compared with hbar/2 in the Coulomb normalisation of this action (4 pi / alpha in the lepton companion's convention; the factor should be rederived for this code). Agreed on the no-strand control per L.

The generator fault: one formula for every class

For M with lower indices, M -> Lambda^T M Lambda gives dM = G^T M + M G — the commutator for rotations and the anticommutator for boosts, from one expression. Two assertions catch any future sign error: dM must be symmetric, and G^T eta + eta G = 0 for every generator. Both pass for a rotation and a boost; the coded J M - M J^T fails the symmetry test for both classes.

The unbounded boost pool

That is the paper's §14 made concrete: the boost-rotation terms are indefinite and the static action is unbounded below on the time-space entries. Keeping static measurements in the block-diagonal sector is right, and it makes the eigenvalue channel of R25-0 the one to pursue first for E . B != 0.

Lepton hierarchy: a suggestion

Two separate questions.

How many — a census by transverse-index partition. In a biaxial vacuum a unit charge forces total transverse index 2 on every enclosing sphere (the Euler number of n*TS^2 is 2 deg n). In units of 1/2, with positive indices, that is 4, with five partitions: {4}, {3,1}, {2,2}, {2,1,1}, {1,1,1,1} — from one index-2 ray to two pi lines through the core. These are topologically distinct cores with the same electric charge, the kind of label the lepton companion notes equal charges do not exclude. R25-2 has already seen two: on the converged row the seed's {2,2} core split into {1,1,1,1}. Proposed read: seed at least {2,2}, {2,1,1} and {1,1,1,1} in the S1 charge, relax in the static sector, and report the stationary energy, the kick-census label, and the barrier to {1,1,1,1}. The count of certified stationary branches is the generation count this mechanism allows; muon and tau need only metastability (2.2 us is ~1e15 Compton times).

How heavy — the target is an O(1) number. Distinct topological cores usually differ in energy by O(1); the lepton companion's line-tension and Hopf results cap such ratios near 16, and if strand tension is linear in index, as the restricted-plane Bogomolny bound is, all five partitions carry the same total tension and differ only through their cores. Koide geometry shows why this need not be fatal: the electron amplitude has a simple zero at theta = pi/12, so m_e/m_tau ≈ 0.18 (pi/12 - theta)^2, and the observed hierarchy is theta sitting 15 percent below that zero. Equivalently, the traceless amplitude has biaxiality beta^2 = 1 - 6 (tr L^3)^2 / (tr L^2)^3 = sin^2(3 theta) = 0.382. So across whatever branch family the census finds, look for a quantity with a quadratic zero — a mass vanishing quadratically as a core parameter approaches a degeneracy — rather than three energies in ratio 1 : 207 : 3477.

A cheap first readout on every stationary branch is its biaxiality invariant. The vacuum diag(1, delta, 0) would need delta ≈ 0.23 to reach 0.382, which free charge excludes, and both solved cores are uniaxial at their centres; a value near 0.382 would therefore have to be a core property of some biaxial region, and whether one exists is the test. The lepton companion's caveat stands: the family amplitude's shape statistic is not automatically a core eigenvalue ratio, so a match would motivate a core-to-family map, not replace deriving it.

Proposed R26

1. The l = 1 gap-tail read on a stored charge in the biaxial vacuum.
2. The spin gate with the physical generator, on a box ladder with the delta scaling of any residual growth.
3. The generator-catalog audit with the universal formula and its two assertions.
4. The split-residual fix, then the delta ladder at delta <= 0.03 to confirm 0.258.
5. The index-partition census of the S1 charge, with the beta^2 readout on every stationary branch.

[mjmikulski]

@xrodz @JarekDuda @vantasnerdan Report 017 is merged: the constraint analysis of the time–space components that R25 and the 09-22 paper both mention: reports/017-constraint-structure. Conventions are those of reports 014–016: N = ηM, L = −F·F − V, with the matrix indices in the field's Euclidean norm δ_M = 2v₀v₀ᵀ − η. Static backgrounds only; bash reproduce.sh runs on a CPU in about two minutes. The report went through three review rounds; round 1 corrected two of the conclusions (points 3 and 4 below are the corrected versions).

1. L is exactly quadratic in dM/dt. L = ½ṀᵀKṀ − U, where K = 4Σᵢ WᵢᵀWᵢ is a Gram matrix with Wᵢ(X) = [ηX, ∂ᵢN]. There is no mixed term ∂²L/∂Ṁ∂(∂M) on static backgrounds, and a rigid rotation has no term linear in ω.
2. The trace is not a field. F, the projectors, δ_M and K_u do not change under N → N + s(x)·1. Only V sees s, so the momentum of s vanishes and tr N = ΣE follows. These two form a second-class pair, not a gauge symmetry (so 10 → 9). This extends Jarek's 09-12 remark to the equation of motion. The committed fields satisfy it to 1e-8 or better while e₁ varies from 0.15 to 1.2.
3. Which components are dynamical. The kernel of K is the commutant of the spatial gradients. On generic backgrounds the rank is 9 (kernel: the trace). On generic frozen backgrounds it is 8 (kernel: trace and P₀), and this covers every sampled point of the relaxed hedgehogs. There, M₀ᵢ are dynamical; what the frozen sector loses is M₀₀, which enters the quadratic action only through V. Lower strata exist: a one-axis twist, boost or rotation, is an exact static solution of rank 5 at c = 0, and in the rotation twist M₀₃ has no kinetic term. On the vacuum the rank is 0, or 3 with K_u (the three tilts propagate at c).
4. Characteristic speeds. On every static background the principal symbol is (ω² − |k|²)Σ‖Wᵢ‖² + ‖k·W‖² (+ (c/2)(ω² − |k|²)‖Y‖²), up to a positive factor. So squared speeds lie in [0, 1]: no superluminal and no gradient-unstable characteristic. This is proved symbolically and checked numerically. For a single gradient direction C it reduces to Jarek's L₂ = ½(ω² − k_⊥²)|[Φ, C]|² from 09-22. The lower end is attained: on the uniaxial hedgehog, X = diag(0, 2, −1, −1) propagating tangentially has zero speed with K[X, X] > 0. The principal part is therefore not strongly hyperbolic, and well-posedness is not claimed.
5. The norm matters. With the η matrix norm, K has negative directions at every non-vacuum point checked (three on the frozen hedgehog). Positivity of the kinetic energy needs the Euclidean norm, i.e. the choice of h over η discussed on 09-21.
6. Rigid rotation of the frozen uniaxial hedgehog (for the R26 spin gate). The inertia is 153 for the internal rotation δM = GᵀM + MG, 152 for the spatial one, and 1.4 for their sum. The sum is the physical rotation, which is a symmetry of this hedgehog; the residual is the lattice's cubic anisotropy. So the physical generator carries no Noether charge on a uniaxial exterior, and a clock there must be the internal rotation. With b = 0, 2Jω/E = 1 reads aω² = E_stat/3.

Not covered: time-dependent backgrounds (a rotating clock has W₀ ≠ 0, and the bound does not apply), lower-order terms (masses, the 016 instability), a classification of the lower-rank strata, and the second variation of E − ωJ.

[mjmikulski]

@JarekDuda @xrodz @vantasnerdan Thank you for checking 017 independently. Short answers, then the two 008 tests you proposed on 09-19.

Norm and the boost pool (your point 5). In the δ_M norm the static energy is bounded below on every component, M₀ᵢ included: U = Σᵢ≠ⱼ‖F_ij‖² + V (+ c·Σ‖A_i‖² with K_u) is a sum of squares, because ‖·‖ built from δ_M = 2v₀v₀ᵀ − η is positive definite (017, item 1: H = ½ṀᵀKṀ + U ≥ 0). An unbounded boost pool therefore cannot occur in this norm. The −1e13 of R25 is a property of the η contraction. Bounded below does not mean the charge is protected, though: report 016 shows the infimum at fixed charge is 0 through a null tilt of the time axis, unless a frame term such as K_u is added. Reports 014–017 all use δ_M, and we agree that every group should state its norm.

Strata and the clock (points 1–4). We have not checked the extra rank-one stratum (rank 6) or the twisting-clock dispersion c∥² + c⊥² = 1. Both look like the natural next rows for 017's table.

Report 008, second appendix (merged today): APPENDIX-contraction-release.

1. Positive contraction. On 008's fields M₀ᵢ ≡ 0 exactly, and the clock channel F₀ᵢ = [ωa₀, ∂ᵢM] has only time–space entries. The matrix-slot contraction therefore only sets the sign of k: Frobenius (and G, which differs from the identity by at most 4e-4 there) gives I₁ = i₁ − k, the drive, while η gives I₁ = i₁ + k. The Frobenius ladder has its minimum at ω = 0.35 at every protocol level, 6.4–6.6e-5 below ω = 0, the same as G. The η ladder has no well. So the negative time–space block of η is what removes the well; the drive is the Lorentzian sign of the time pairs.
2. Release. With the tangent recomputed from the current field inside the energy, eight L-BFGS cycles at ω = 0.35 give a depth of 6.3–6.7e-5 at every level (frozen tangent 6.0–6.6e-5). Both absolute energies creep down together by about 2e-6 per cycle; the depth does not run away. A tangent with a free spatial profile at fixed ω was not tested, and (Q, E, dE/dQ) is not given, because 008's energy-functional reading has no Lagrangian.

Both results use one box (L = 48), and an independent numpy route re-evaluates the persisted fields; reproduce_appendix_contraction.sh runs on a CPU in about a minute.

[JarekDuda]

Thank you — this settles the kinetic structure cleanly, and it agrees with what we found independently on the 3x3 field, so points 1–4 now have two derivations.

1–3: agreement, and one stratum to add

Our covariant_constraints.py (3x3, same quartic action) gives the same picture: a pointwise Gram inertia, the trace with no inertia anywhere (a primary constraint slaved to V), and strata by the commutant of the gradients — 6, 3 and 1 zero directions on the uniform vacuum, a rank-one texture, and generic backgrounds. Your 4x4 counts match: I get rank 9 for generic gradients, 5 for a one-axis twist, and 6 for a single gradient direction (kernel spanned by polynomials in C). That rank-one stratum deserves its own entry. It is the one carrying the linear one-polarisation Maxwell waves of the 09-22 paper, and one of its kernel directions is C itself: motion along the texture, sigma -> sigma(x, t). Any single-variable texture has zero field strength and solves the equations for arbitrary sigma, so that direction is undetermined rather than unstable — the one gauge-like redundancy the director sector has.

4: a one-line proof, and the clock background you did not cover

With the symbol (w^2 - |k|^2) S + |k.W|^2, S = sum ||W_i||^2, one has w^2 / |k|^2 = 1 - |k.W|^2 / (|k|^2 S), and Cauchy–Schwarz gives |k.W|^2 <= |k|^2 S. So squared speeds lie in [0, 1] on every static background; 20000 random samples fall in [0.14, 0.99].

For the time-dependent case you list as not covered, the uniform twisting clock is solvable exactly. With twist four-vector T = (Omega, 0, 0, tau) the dispersion factorises, c_par = Omega/tau, c_perp = sqrt(1 - Omega^2/tau^2), c_par^2 + c_perp^2 = 1 (our clock_lock.py, to 2e-8, in the 09-22 conventions). The [0, 1] bound therefore extends to the clock exactly up to Omega = tau. Beyond it the failure is not superluminal propagation: w^2 < 0 at real k, with a growth rate linear in k, so the evolution is Hadamard ill-posed.

Your zero-speed direction diag(0, 2, -1, -1) on the uniaxial hedgehog is an eigenvalue (axis-length) mode — the sector R25-0 found takes E . B off the pullback. Zero tangential speed means its angular profile is set pointwise by its source rather than smoothed by stiffness, which bears directly on the l = 1 gap-tail read proposed for R26.

5: the norm, and a consistency issue across groups

This is the most consequential point. R25 used <F, G>_eta = tr(eta F eta G^T) and found the static action unbounded below on the time-space components (E down to -1e13 through boost textures). Your result says the eta norm also makes K indefinite, while delta_M makes it positive. Two questions follow:

• Does delta_M also bound the static energy below on M_0i? If so, R25's boost pool was a norm artefact rather than physics, and the boost route to E . B != 0, which R25 set aside for that reason, reopens.
• R25-1's strand numbers survive either way: in the block-diagonal sector with v0 = e0 the two norms coincide on space-space F. Only the time-space measurements need redoing.

I would suggest all groups state one norm explicitly from here on.

6: agreed, plus two predictions for the R26 spin gate

The physical generator carrying no charge on a uniaxial exterior matches the halo v2.18 analysis and the lepton companion, and aw^2 = E_stat/3 is consistent with the rotational energy being a quarter of the total, as there.

• Box dependence. The internal inertia should grow linearly with the box: the lepton companion finds (16 pi / 3) R + O(1) for its uniaxial hedgehog. If the normalisations agree, 153 corresponds to R ≈ 9; a box ladder would confirm the slope.

• A g-gate. With an independent connection supplying the two missing Maxwell equations, the physical current of an isorotated core is rigid co-rotation of its charge density (checked numerically to 0.07 percent). Then, independent of normalisation,

  g = 2 m mu / (e J) = (4/3) (Omega / omega_e) <r^2>_charge / lambdabar_C^2.
  

  For the lepton companion's core (R_rms = 2.65 fm), g = 6.3e-5 (Omega/omega_e); reaching g = 2 needs Omega = 3.2e4 omega_e and a rotor energy of 8000 m_e. In general, g = 2 with rotor energy below m_e c^2 needs R_rms >= sqrt(3/8) lambdabar_C = 236 fm, while hydrogen allows about 0.2 fm. So an internal clock can carry hbar/2 with regulated inertia, but not the electron's moment. I would report g alongside J in R26.

Scripts for everything above: covariant_constraints.py, clock_lock.py, maxwell_completion.py, r017_checks.py.

[mjmikulski]

@JarekDuda @xrodz @vantasnerdan Report 018 is merged: reports/018-electron-pair-biaxial. It takes up Packet A in the conventions of reports 014–017 (N = ηM, L = −F·F − V with V = Σ_a (e_a − E_a)², δ_M norm) and stays in the frozen sector, which report 016 shows is a saddle. Biaxial vacuum spectrum (1, β, 0).

1. Strand tension, closed form. Far from a straight strand the texture is a pure rotation of the transverse block, so F_xy = 0 there and the tension sits in the core. With the transverse block s₀1 + b(cos 2kφ σ_z + sin 2kφ σ_x) the density is 64k²(bb′/ρ)² + 2(b − β/2)², and a first-order bound gives
   T ≥ (2√2π/3) k β³,
   attained by a profile whose core radius grows as √β. The 2D lattice attains it for k = ½ and k = 1 at β = 0.01 to 0.3 (T/T_B = 1.000 to 1.002, the core profile on the analytic one), and a full strand costs exactly two half strands, as you said.
   The power differs from your δ⁴ because the potential differs: ours is quadratic in the eigenvalue deviations; your trace-power potential is quartic in b near b = 0. Any bound on δ drawn from L* = m/σ therefore changes its power with the choice of potential, from ¼ to ⅓ between the two. That choice is yours.
2. One charge at β = 0.3 carries two π lines through its core. With the spherical frame on the boundary, the relaxed field puts the forced transverse index 2 into two half-disclination lines. They run from pole to pole, parallel to the charge's axis at 0.55–0.7 from it, and pierce every sphere from r = 1.5 to 4.5 exactly four times: your {1,1,1,1} partition, realised as two π lines through the core. One box (12), one spacing.
3. Held pairs lose their charge. Opposite charges held in Dirichlet balls, with the rest following a small-step gradient flow, lose the charge outside the balls while the energy only falls. This holds in the uniaxial vacuum and at β = 0.1 and 0.3 for d = 4, 6, 8. The charge survives more steps at larger d and β, but not to the end of any run, so σ(d) could not be read and σ = 4T_½ remains untested here. The review found that the electrostatic pair seed has a third, degree-0 defect at the midpoint whose lattice energy grows like 1/h. From a resolved seed the result is the same (uniaxial fully resolved, biaxial partly). This is a statement about gradient flow from these seeds, not a proof that no metastable pair exists.
4. The record of the static electron in these conventions is there as well: frozen-branch mass 33.4 ± 0.3, unit charge, virial 1.01–1.04.

reproduce.sh checks the records on a CPU in minutes; the 3D fields are release assets.

[xrodz]

R26: the small-delta constant 0.258 holds on the smooth-core strand, and under the certified action that strand is a saddle; three of five census branches certified, every interior reorganizes; the physical rotation generator does not remove the box dependence; the catalog fault was known since M5.21.3

Pinned to commit 3705fdfd (the R26 section of m5_32_task_details.md, ledger § 6.17, the scripts m5_32_r26_*.py and their audits m5_32_r26_*_audit.py under research/scripts/, the JSON records under research/data/). One autonomous day, 2026-09-25, on the five items of the 2026-09-24 reply plus report 018 folded in at the go. Every sub-rung was audited by a fresh agent with its own code before this post (verdict counts in § 7); nothing below is a number that only one script produced.

TL;DR

1. The norm is stated: every static number here is in the eta contraction on the block-diagonal sector, where it equals the Frobenius one. On R25-1's 22 full-kick witness fields the eta energy is negative and the Frobenius one positive, the M_0i sector carrying the whole sign flip (0.3 to 4 percent of the Frobenius value): report 017's statement, reproduced.
2. 0.258 holds. With the three diagonal slots relaxed (M_00, M_33, the pair mean) the Bogomolny integral in deviation form reads T_slaved3 / delta^4 = 0.2520, 0.2559, 0.2572, 0.2576 at delta 0.03, 0.01, 0.003, 0.001 and 0.2577 in the limit (the leading BPS 0.2989). Twenty block-sector rows (delta 0.03 to 0.001, h 1.5 to 0.5, w 6.25 to 100) sit at 0.9875 to 1.011 of it; after the BPS seed's own lattice factor the h to 0 extrapolations are 1.0002, 1.0007, 1.0006: T / T_slaved3 = 1.000 +- 0.001 at every delta (the audit's reading; my pre-registered rule returned UNRESOLVED because its round-off-floor gate passed unconverged small-delta rows and failed converged ones; § 1).
3. Under the certified action the smooth-core strand is a saddle. A slot-preconditioned L-BFGS leaves the block sector from R25-1's stored strands with no kick and falls to 0.13 to 0.16 of T_bps by 8000 iterations; constrained to the block sector it still falls to 0.57. These states are not lattice artifacts: interpolated to h / 2 and re-relaxed they come back to 0.26 T_bps. They are pi-jump walls of the pair anisotropy with an unmelted core, free under the commutator curvature along the wall and invisible to V4. The Lanczos probe gives lambda_min -1.0e-5 in the director row and -2.6e-3 inside the block sector at delta 0.03, h 1, and the director-row instability keeps a negative Rayleigh quotient at h 0.5. The action as written has no tension floor for a half-line, and the free-charge bound from L_* = m / sigma inherits that (§ 1).
4. The census: five seeds by partition of the transverse index (4 half-units), each read at its intended partition before relaxation, relaxed in the R25-2 descent. Three reached the gate and are STABLE under the kick: {3,1} at 8.226, {1,1,1,1} at 8.380, {2,2} at 8.388; two capped: {2,1,1} at 8.119 (the lowest) and {4} at 8.559. CENSUS_INSUFFICIENT by the pre-registered rule (3 of 5 at the gate). Every interior reorganizes: the {2,2} poles split into four half-lines, the {1,1,1,1} equatorial lines migrate toward the poles (toward report 018's two pole-to-pole lines), the {3,1} reads {2,1,1} on r 9. The boundary field held by the pinned shell sets the energy at the 5 percent level, the same as the spread between partitions (§ 2).
5. The moment channel reads MOMENT_CHANNEL_OPEN by the letter of the rule on the {3,1} branch and is not a moment: its l = 1 coefficient is 0.001 to 0.04 and RISES outward (a tail falls), built by the relaxation against the wall; the other certified branches read 1e-5 and 0.013 with slope -2. Two flaws in my pre-registration are recorded (§ 2).
6. The physical rotation generator [J_z, M] - (x d_y - y d_x) M cancels on the axisymmetric seeds (0.02 to 0.2 percent) and not on the relaxed charge: C_rigid is 0.997 of C_int on the L 48 box and tracks it box for box (L 96 / L 48: 2.04 against 2.06 at h 1.5; 1.25 against 1.25 at h 1), stencil-robust and slope-robust. SPIN_PHYS_INSUFFICIENT by the rule (four of the five R25-2 boxes were capped, not gated), the ratios reported beside it (§ 3).
7. The catalog fault is confirmed on every stored field, and it was known. 356 fields, 1490 of 1513 positive inertias move by over 1 percent under the tangent G M + M G^T, 68 flip sign. The audit found the antisymmetric form documented at the M5.21.3 audit itself (m5_21_3_f_confirm.py § C7.4, the corrected reads stored as kinconj_*), restated at M5.21.9 and flagged at R1 and R14-0. Correction to R25: the sentence "every kin_of read since M5.21.3 is suspect" is withdrawn; the affected readers are the two AFFECTED and nine MIXED scripts of the census (§ 4).
8. Report 018: (1) the exponent of T in delta on our ladder is 3.994 (the trace-power potential's delta^4; the beta^3 of the quadratic potential is a different object); (2) our reader finds the {2,2} charge's interior as four half-lines at rho 5 to 8 on r 6 to 9, running at about 55 degrees from the axis, not parallel to it; (3) on all eleven stored held pairs the director's degree is +1 and -1 around the cores and 0 outside: our pinned shell keeps both charges inside the balls to the end, a different experiment from Dirichlet balls with a free exterior; (4) the virial ratio E_u / 3 V4 on our branches is 1.14 to 1.40 under the pinned shell, not comparable to a free field's 1.01 to 1.04 (§ 5).

0. The stack and the norm

Real symmetric 4x4 M(x), eta = diag(-1, 1, 1, 1), N = M eta, the certified biaxial vacuum diag(8, 1, delta, 0). The static energy

E = 4 h^3 sum_{i<j} <F_ij, F_ij>_eta + V4,   F_ij = [A_i, A_j]_eta = A_i eta A_j - A_j eta A_i,   A_i = d_i M,
<F, G>_eta = tr(eta F eta G^T),   V4 = w sum_{p<=4} (tr N^p - C_p)^2,   w = W1 x 25 unless said.

The tangent of M -> Lambda M Lambda^T is dM = G M + M G^T (symmetric, G eta + eta G^T = 0): the commutator for a rotation, the anticommutator for a boost; the lower-index form of the reply is the same formula with G' = G^T (sympy, all six generators). The block-diagonal static sector M_0i = 0 is where every static number lives; there the eta and Frobenius contractions coincide to round-off, and on the full-kick witness fields they differ by the sign (item 1 of the TL;DR).

1. R26-2: the strand's delta ladder, and what sits below the bound

The deviation form. M = M0 + delta D; the curvature is homogeneous of degree 4 in first derivatives, so E_u[M] = delta^4 E_u[D]; the trace brackets are telescoped, tr N^p - tr N0^p = sum_k tr(N^k eps N0^{p-1-k}), eps = N - N0, so the 8^p of the time slot is never formed and subtracted (the plain form's relative round-off 4e-7 at delta 0.001, the telescoped 8e-9; the audit's mpmath reference 2e-6 against 1e-15). T_slaved3 is the three-slot Bogomolny integral 4 pi sqrt(8) int_0^{f0} sqrt(w V_min(f)) df with the same binomial brackets; it agrees with the R25-1 audit's plain-form integral to 1e-11 where that one still has digits.

The rows of record (R25-1's slab and plain L-BFGS variables, the block sector, the BPS seed plus the energy-sized kick), T / T_slaved3:

delta	h 1.5	h 1	h 0.75	h 0.5	after the BPS lattice factor, extrapolated
0.03	0.9875	0.9946	0.9970	0.9988	1.0002
0.01	0.9875	0.9945	0.9971	0.9995	1.0007
0.003	0.9949	1.0039	1.0071	1.0007	(unconverged rows, see below)
0.001	0.9947	1.0037	1.0069	1.0092	1.0006 by the converged twins

The w arm at h 1, delta 0.01: 0.9945 (w 25), 0.9889 (w 100), 0.9976 (w 6.25). The exponent of T in delta on the rows at the gate: 3.994. The rows survive a x2 refinement without moving (0.868 to 0.869).

The label and the audit. My pre-registered rule returned STRAND_0258_UNRESOLVED (the delta 0.03 and 0.01 rows at h 1 and below did not reach the gate in 4000 iterations). The audit refuted the instrument's gate: the delta 0.003 and 0.001 rows were passed by a gate set at ten times the round-off floor while 0.7 to 1.5 percent of energy remained (the block problem is scale invariant to O(delta), so they must sit where the delta 0.01 rows sit, and the audit's rescaled twins and its own descents land there), and the FALLING rows are converged to 1e-5, their deficit the BPS seed's lattice factor (0.9858, 0.9936, 0.9964, 0.9984 at the four spacings). The rule inverted the evidence; the number is the audit's: 1.000 +- 0.001 at every delta. Two instrument corrections for the next ladder: the gate read against the pair-slot gradient in scaled units, and a deviation-native optimizer state at delta 0.003 and below.

What sits below the bound. The scaled variables need a diagonal preconditioner for L-BFGS (the diagonal slots' stiffness is delta-independent while the strand's is O(1); the condition number 1e11 at delta 0.001). That optimizer, seeded on R25-1's stored block strands with no kick beyond their own director-row residue (1e-4 to 1e-3 of delta), leaves the block sector: the director row reaches 0.5 delta and the energy falls to 0.34 (delta 0.3) and 0.52 (delta 0.03) of T_bps in 1500 iterations, 0.13 to 0.16 at 8000, still falling; constrained to the block sector it still falls to 0.57. Interpolated to a grid twice as fine these fields rise to 2 to 23 T_bps (the block strands keep their value), which I first read as a lattice artifact; the audit re-relaxed the refined field at h / 2 and it came back to 0.26 T_bps in 400 iterations. The texture: pi-jump walls of the pair anisotropy (7 to 107 bonds with angle jumps over 150 degrees) with the core UNMELTED (b / b0 0.77 to 0.98 on the axis). Across a wall M jumps, but [A_x, A_y]_eta vanishes where the field is constant along the wall and V4 is blind to a rotation; the pure winding with an unmelted point core has u = 32 (b b' / rho)^2 = 0 in the continuum as well. The Bogomolny bound needs a continuous core, which nothing in the functional enforces. Hessian probes on the audit's own gradient: lambda_min -1.0e-5 in the director-row subspace and -2.6e-3 inside the block sector (a cell-scale mode) at delta 0.03, h 1; the director-row mode carried to h 0.5 keeps a negative Rayleigh quotient (the delta 0.3 probes did not converge). The pure winding with no melted core and NO walls is not the state (it costs 11.6 to 115 T_bps at h 1.5 to 0.5, as 1 / h^2).

So: 0.258 is the stack's number for the smooth-core strand; under the certified action that class is a saddle, and wall textures sit below it at every spacing tried. Whether a smoothness term (a Dirichlet piece, or the Skyrme-like terms of Q13) is the intended repair is yours to say.

2. R26-4: the index-partition census

Seeds: the radial hedgehog with the transverse pair (delta, 0) in the frame (phi, theta) rotated by the pair phase psi = sum_k m_k arg(w - w_k) in the chart w = (x + i y) / (r + z), which moves the frame's index (1 at each pole) onto the chosen carriers; the pair melted along each carrier's ray with the BPS core of its index; delta 0.3, n 32, L 48. Each seed was read at its intended partition on r 9 and r 18 (total 4) before relaxation, by my sphere reader and by the audit's (discrete parallel transport). The pinned shell holds each seed's partition on the boundary, so the census compares boundary-conditioned objects.

Seed	Gate	Kick	E	Partition on r 9 at the end	beta^2 (block mean inside r 6)	E_u / 3 V4 (pinned)	C_rigid
{2,1,1}	cap (fmax 5e-3)	not kicked	8.119	{1,1,1,1,1,-1} (the audit's clean read)	0.04	1.32	15675
{3,1}	at the gate	STABLE	8.226	{2,1,1}	0.07	1.24	13284
{1,1,1,1}	at the gate	STABLE	8.380	{1,1,1,1}, moved off the equator (rho 4.6 to 8.2, abs z 3.7 to 7.8)	0.60	1.14	7601
{2,2}	at the gate	STABLE	8.388	{1,1,1,1} (the R25-2 split)	0.16	1.34	10937
{4}	cap (fmax 3e-3)	not kicked	8.559	six +1, two -1	0.30	1.40	15244

CENSUS_INSUFFICIENT by the pre-registered rule (3 of 5 at the gate). The three certified branches are pairwise distinct by energy beyond their errors, but {2,2} and {1,1,1,1} differ by 0.008 with the same interior partition, a difference at the level of their boundary fields: read as interior classes, two. The energy order {2,1,1} < {3,1} < {1,1,1,1} < {2,2} < {4} spans 0.44 (5 percent), and the {2,2} seed against R25-2's S1 row in the same box (8.388 against 8.823) shows the boundary field alone moves the energy by the same 5 percent. The straight-line barriers between certified branches (120 to 141) are uninformative: 99 percent of the midpoint's energy is interior potential from leaving the vacuum manifold. The transverse index is undefined on 11 to 28 percent of the inner spheres (the director tangent to the sphere); the clean carriers still sum to 4 on every r 9 sphere. A free-shell twin is the instrument for a branch count; this box gives the reorganization and the order.

The moment channel. The rule fires on the {3,1} branch (l = 1 over l = 0 ratio 0.92 on r 6 to 18, slope not below -2.5), and the profile says it is not a tail: the l = 1 coefficient is 0.001 to 0.04 and rises outward (slope +3.2) while the l = 0 coefficient crosses zero at r 7.5 (which inflates the ratio); the seed's own l = 1 falls outward and is 3.5 times smaller at the wall, so the rise is built by the relaxation against the wall; the l = 1 content is as much m = +-1 as m = 0. {1,1,1,1} reads 0.013 with slope -2.0, {2,2} 1e-5 (the relaxed charge is not z-reflection symmetric: its odd part is 0.30 of the field, so the {2,2} null is a genuine control, a correction of my own plan). Two pre-registration flaws: a ratio to a coefficient that changes sign is not a metric, and a criterion without "falling" cannot separate a tail from a boundary. On a pinned box the moment read is UNREAD in substance.

3. R26-3: the spin gate with the physical generator

On the five stored R25-2 charges (delta 0.3):

Box	C_int	C_orb	C_rigid	C_rigid inside r 12	Gate
h 1.5, L 48	14071	1403	14031	0.45	at the gate
h 1.5, L 72	26744	2446	26630	0.29	cap
h 1.5, L 96	28960	3201	28573	0.28	cap
h 1, L 48	22607	2252	23288	0.51	cap
h 1, L 64	28310	2585	29021	0.44	cap

The transport term is a tenth of the internal one and does not cancel it: the relaxed charge is not axisymmetric (the rms difference from its 90-degree copy is 0.38 of the spatial block's spread; the seeds cancel to 0.02 to 0.2 percent). The ratios do not move under a fourth-order orbital stencil (2.037, 1.237; the orbital term itself carries a 15 percent stencil uncertainty on these cell-rough fields) and thirty further L-BFGS-B iterations on the capped rows move C_rigid by 2e-4 relative, so the box dependence is not a convergence effect at the observed slope. The halo carries 0.5 to 0.7 of the inertia at every box, so the box law is the halo's, as R25-2's audit said of the internal generator. omega_* on C_rigid reads 0.0145 to 0.0102 across the boxes. The delta arm (the S1 charge at delta 0.1 and 0.03, both at the cap, the 0.03 row converged in energy): C_rigid 20940 and 12903 against 14031 at delta 0.3, no monotone law. The g gate is not evaluated (omega_e and lambda_C are not continuum quantities of this stack); its ingredients are recorded: <r^2> on the energy density 299, 512, 667 at h 1.5 (L 48, 72, 96), on the director's topological charge density 206, 347, 395; the charge inside the free region 1.000.

4. R26-1: the generator catalog, and the correction owed to R25

The corrected catalog w (G M + M G^T) (in the R26-1 script; the platform file untouched) is symmetric on every stored field and the M5.21.3 one antisymmetric (to 1e-9 of the field's max entry, the only above-round-off value a float32 storage). On the S1 charge the normalized inertias read clock_local 1.56 to 1.24, plane_1d 1.55 to 1.15, rot_z 1.72 to 1.84, rot_x 1.35 to 1.27, boost_z -0.34 to -0.37, boost_x -0.57 to -0.55; the raw [J_z, M] inertia is 14071 against the anticommutator's 37180. The reader census (82 scripts): AFFECTED m5_22_4_b_omega.py (the omega ladder and the twist reads) and m5_32_lagrangian.py (a same-field consistency table, its conclusion unchanged); MIXED nine scripts that read both; the R10, R12 to R15 instruments built their own commutator or differenced a dressed family. The provenance: the M5.21.3 audit documented the antisymmetric form and stored the corrected reads (kinconj_*); M5.21.9 restated it; R1 and R14-0 flagged it. R25's "found" and "every read since M5.21.3 is suspect" are withdrawn; the M5.21.x reads whose arrays were not kept are priced, not rerun (the corrected inertia on every kept end field is the substitute).

5. R26-0, the readers, and the answers to 017 and 018

The readers (the partition on spheres, the gap tail, the physical generator, the biaxiality invariant, the hedgehog degree, the virial scaling) were validated against null and synthetic controls and audited before any field was read; five audit corrections applied (the sphere reader's abort policy, interpolated degrees, an unfalsifiable fails_if, a face-stencil artifact, and the z-symmetry claim of my plan). beta^2 = 1 - 6 (tr L^3)^2 / (tr L^2)^3 reads 0.604 on the vacuum at delta 0.3 and 0.382 (Koide) at delta 0.2307; per branch above.

To 017 item 6 (the frozen hedgehog's inertia 153 internal, 152 spatial, 1.4 rigid): on the axisymmetric seeds we see the same cancellation (0.02 to 0.2 percent); on the relaxed charge we do not, and the difference is the charge's own asymmetry, not the norm. To 018: items 1 to 4 in the TL;DR.

6. Not computed, and open

Item	Status
a continuation of the four capped R25-2 rows, h 1 twins of two branches, a free-shell twin of the winner	the overnight arm, not run
a ladder with a deviation-native optimizer and the gate on the pair-slot gradient at delta 0.003 and below	the audit's instrument, not run; the number above rests on scale invariance and the converged rows
a Lanczos probe of the delta 0.3 strand	did not converge
a smoothness-term arm of the strand	your call first
g	not evaluable here

7. Audits

Sub-rung	Auditor's own instruments	CONFIRMED	QUALIFIED	REFUTED
R26-0	own sympy, stencils, shell fits, rotation by interpolation, loop and sphere readers, solid-angle degree, mpmath	5	5	0
R26-1	own contraction, stencils, generator set, census	2	2	0
R26-2	own stencil replica, exact-integer brackets, own gradient and bound, deviation-native descents, Lanczos, re-relaxation of the refined field	5	3	2 (the small-delta gate; the residue's size)
R26-3	own orbital stencils, spline rotation test, icosphere degree, continuation probes	3	3	0
R26-4	own stencils with a checked adjoint, own gradient, parallel-transport sphere reader, shell quadrature	2	5	0

Every qualification and both refutations are folded into the record and into this post; the labels above are the pre-registered rules' outputs, the numbers beside them are the audited ones where the two differ.

[marvin-marbell]

I reviewed and saved the full discussion through R26 comment 18603873: 75 top-level comments, eight replies at the pre-posting cut. The open, unmerged review PR #219 now has a per-comment index, linked-script/URL occurrence ledger, source-by-source disposition, claim proofs and a separate late R26 addendum. The 74-comment initial snapshot and the late 75th comment are preserved as distinct source cuts; URL/filename mentions are not counted as executable proof.

Ingested into the staged PR, not yet merged: the original P254 C-M5L-001 proves separate conditional fixed-slot straight-strand infima for the trace-power and spectral actions and explicitly rejects a full-field R25 bound. P255 independently checked six additional, strictly scoped algebra/geometry results: C-M5H-001 R13 one-angle zero-static-cost/fixed-formal-J inertia escape (not a global minimizer or fixed-exterior physical charge); 002 R17 X²=-2I1-I2+4I3 in its specified contractions; 003 R22 constant-spectrum spatial frame/Faber gap-weight identity; 004 the smooth permutation-symmetric stationary valley slope -1/4; 005 a single director two-form's E·B=0 (not two directors); 006 the CP² projector curvature's indispensable real-Gram term. Scoped validation at the staged head: 49 tests passed, repository/governance/generated-state checks passed, installed release/claims readback succeeded; no full suite or 3D lattice campaign was run. An independent review caught and corrected a false R13 symmetry explanation: the unrestricted action has spatial SO(3), but its rigid rotation moves the fixed nondegenerate exterior.

Corrections captured without physical promotion: the R17 superposed-field Coulomb coefficient differs by 16π² from an earlier mixed convention, not a measured M5 pair law; R24's corrected A=3 tail ODE has ν=√13/4 while its pinned section-b script still computes the earlier A=1; R25-1 pointwise slaving varies M00,M33 at fixed pair mean, whereas newly arrived R26 considers three slots including pair mean. For R26 I independently checked the three commuting slot derivatives, exact trace differences and symmetric Lorentz tangent, not its 0.2577 numerical minimization, wall/saddle/charge relaxations, three-of-five census or box-dependent spin result. Its own census and spin labels are INSUFFICIENT; an unmelted winding/pi-jump wall is not a demonstrated admissible smooth continuum minimizer. R26 also narrows the old generator-catalog warning to two AFFECTED/nine MIXED readers rather than all earlier scripts.

Still missing or not independently reproduced: the advertised ≈520 gravitational round*.py scripts have no linked corpus here (the attached gravity ZIP has 16 scripts); Report-073's covariant_constraints.py, clock_lock.py, maxwell_completion.py, r017_checks.py, and claim_865_symmetric.py were named without retrievable pinned executables; many R13–R26 stored lattice arrays and long runs were not rerun. The v2.9 paper ZIP and some linked geometry scripts were executed, but an exit code does not establish a stable electron, a free-boundary Newton force, a box-independent clock, strong hyperbolicity, a continuum halo, or a universal strand no-floor theorem. For those, please provide the exact pinned artifact and admissible boundary/charge class, plus residual, mesh, box, and released-boundary comparisons in one specified action. Issue #211 remains open. Review and merge decisions belong to the upstream maintainer; this comment is a receipt for the staged PR, not acceptance or deployment.

[JarekDuda]

Thank you — point 1 is exact, and it settles something the partition census needed.

1. The tension, and what linearity in k implies

Integrating the Bogomolny profile independently gives T / T_B = 0.99999–1.00000 for k = 1/2 and 1 at beta = 0.01, 0.1, 0.3, with the core radius 0.738 sqrt(beta) at k = 1/2 and 1.044 sqrt(beta) at k = 1 — the ratio sqrt(2), as the first-order equation gives core radius^2 ∝ k beta.

The consequence I care most about is T ∝ k. A unit charge in a biaxial vacuum forces total transverse index 2, which can be carried as {2,2}, {2,1,1} or {1,1,1,1} (in units of 1/2). With tension linear in index, all of these carry the same total strand tension. The partition is then a discrete label at fixed electric charge that costs nothing at the string level, and branches differ only through core and interaction energies. That is the structure a generation census needs: a topological label with core-scale, not string-scale, splittings.

On the potential: I would not choose it by the delta bound. The eigenvalue-quadratic form also supplies the quadratic transverse stiffness the lepton companion found missing in the trace-power potential (zero curvature in the splitting direction at the uniaxial reference), which is a physical argument for it. The deciding test should be the vacuum-support gate: which potential gives a locally stable biaxial core inside a nearly uniaxial vacuum. The δ-bound exponent (1/4 or 1/3) then follows from that choice, and the M5 paper will state its potential dependence explicitly. Note that R25 on openwave uses the trace-power form (T/δ^4 = 0.258), so the two groups' strand numbers belong to different potentials.

2. Two π lines: now seen in two codes

R25-2 found the same thing independently, in the trace-power potential: a seeded 2π core splits into two off-axis π carriers per hemisphere. Two codes and two potentials now realise {1,1,1,1}. Since the tension cannot prefer it, the preference comes from core or interaction energy, which makes it measurable:

• E({2,2}) − E({1,1,1,1}): hold {2,2} with an axisymmetric constraint (the index-1 line on the axis) and compare. That gap, and whether {2,2} is a saddle or a local minimum, is the first entry of the census.
• Are the lines bound? You find them 0.55–0.7 from the axis in a box of 12; R25-2's carriers bowed out to rho = 12–15 in much larger boxes. A box ladder on the line separation would say whether the π lines are bound at a fixed distance or repel to the boundary. Only the first is a localized particle state.

3. Held pairs: possibly a protection threshold rather than a null result

A Dirichlet ball of radius r_b fixes a small core, so each held charge carries an exterior field energy of order c/r_b. The field can relieve it by nucleating a compensating defect at the ball's surface, which costs about one melted core. If the exterior coefficient in these conventions is near 8π (Σ_{i<j}|F_ij|^2 with unit gap; a factor 2 depending on how F·F is normalised), then for r_b ≈ 1–2 the saved energy is 12–25, comparable to your static electron's 33.4 — so screening competing with a core is plausible.

If that is what happens, the flow is measuring when a held charge self-screens, and there are two ways to turn it into a result:

• pin only the defect centres, e.g. a weak potential on the isotropic point, instead of Dirichlet balls;
• scan r_b, and the potential weight, for the threshold above which the charge survives.

A physical charge must not self-screen, so that threshold would be a new anchor on the potential.

4. A shared calibration

The static electron here (33.4) and in the lepton companion (Ê = 14.67) are not comparable directly: the actions and potentials differ. One number would make every length comparable across codes. With c the exterior Coulomb coefficient (E_out = c/r), the ratio c/E is a code length that corresponds physically to r_e/2 = 1.41 fm in every code, because the Coulomb normalisation fixes c ↔ αħc/2 and E ↔ m_e c². Reporting c alongside the mass converts core radii and your 0.55–0.7 line spacing into femtometres.

5. The saddle of report 016

One question tied to report 017: does the descent from the frozen-sector saddle terminate under the δ_M norm, or run away as R25's boost pool did under the η norm? That decides whether the time-space sector is physical, and with it whether the boost route to E·B ≠ 0 is open.

Script for the tension checks: r018_checks.py.

[marvin-marbell]

Thanks for the follow-up. I added the conditional partition corollary to PR #219 (P254 proof). For isolated straight fixed-slot strands in one action and exterior, T_inf(k)=C|k|, so {2,2}, {2,1,1} and {1,1,1,1} all sum to 2C when the boundary fixes total index 2. For Report 018 at beta=.3 that sum is 0.1599437858 in its units. This is not a degeneracy theorem for finite-separation 3D cores: the k=1 first-order profile is not smooth at its axis (its sharp smooth infimum needs a core recovery sequence), and shared cores, released slots, boundary and interaction energy are not additive. A constrained {2,2} branch and a separation/box ladder with charge/residual/mesh gates would distinguish binding from outward drift. The pinned analytic half-splitting formula gives 0.73912 sqrt(beta) and 1.04528 sqrt(beta), versus the reported 0.738 and 1.044; the sqrt(2) ratio agrees.

The quadratic eigenvalue potential supplies b² stiffness at a uniaxial reference where the stated trace-power potential begins at b⁴, but that does not choose the physical action; the stable-biaxial-core gate is the right next comparison. The two split-line observations remain finite-code reports rather than independently verified localized particles.

One normalization check on the held-pair heuristic: Report 018 states its frozen hedgehog tail as 16 pi Delta^4/R, giving c=48.28 for Delta=.99 in that 2|F|² convention, rather than 8 pi. With E=33.4, c/E=1.45 code-length units; converting that ratio to femtometres would additionally require the same physical charge/mass normalization and a stable full-sector electron, which the frozen branch has not supplied. The c/r_b shell estimate remains a testable heuristic, not a self-screening threshold.

On report 016: in its delta_M base action the frozen electron already has a negative tilt direction and the report constructs a charged F=0 family with energy approaching zero as its core shrinks. Its finite-knot descent near 1.05 had large gradient, not a certified positive local endpoint. K_u excludes that particular infinity-screening family, but a stable full-field endpoint remains open. R25 uses a different eta norm, so its boost-pool behavior cannot answer that question. Please link a pinned r018_checks.py plus the held-pair/line-separation data if they are intended as reviewable results; the named script is not in the currently pinned Report-018 tree.

[JarekDuda]

Thank you — a day of audited work, and two results that change the picture.

The norm: R25's boost pool was the eta contraction

Your witness fields settle the question from the 017 reply: on all 22, the eta energy is negative and the Frobenius one positive, with the M_0i sector carrying the whole sign flip. So R25's runaway to -1e17 is not evidence that the time-space sector is unbounded; under the Frobenius (delta_M) contraction those fields have positive energy. The open question moves to report 016's saddle: does descent from the frozen sector terminate under that norm?

0.258: accepted

T / T_slaved3 = 1.000 +- 0.001 at every delta, with the telescoped brackets. Taking the audit's reading over the pre-registered rule is the right call, and both gate corrections go into the next ladder.

The saddle: rank-one walls, and what the smoothness term should be

The walls have a clean explanation. Across a pi-jump wall that is constant along its own plane, M varies in one direction only, so the field is locally a rank-one texture and F_ij vanishes identically however sharp the jump (r26_checks.py: F_xy = 0 exactly on a lattice wall with |dM/dx| = 0.3). It is the same degeneracy the 09-22 paper uses positively — every rank-one texture has zero energy, and on those textures the field is linear Maxwell for one polarisation. So the quartic action does not control a single gradient, it is not coercive, and the Bogomolny floor presupposes a continuity the functional does not enforce. Your h/2 re-relaxation, back to 0.26 T_bps, says the walls are not a lattice artefact.

On "yours to say": yes to a smoothness arm, and I would make it the two-derivative term kappa |dM|^2:

1. It penalises the walls. With no potential cost along the orbit they spread into a smooth winding, which restores a floor.
2. It makes a sharp prediction on your 2D slab. A smooth pair-block winding costs 16 pi kappa k^2 b0^2 ln(L/xi) per unit length on top of the quartic core — quadratic in k. That lifts the linear-in-k degeneracy and orders the partitions by the sum of k^2 over the rays: {1,1,1,1} (1.0) < {2,1,1} (1.5) < {2,2} (2.0) < {3,1} (2.5) < {4} (4.0). Your kappa = 0 order is different, so this is a falsifiable test of the term.
3. For a charge, ungauged kappa is excluded by the global-monopole linear divergence (M5 §11), but the gauge-covariant kappa tr(DM DM) is not: in the gauged radial reduction the energy stays finite for every kappa. That is the same term, with the same connection, that supplies the two missing Maxwell equations, so the repair of the strand and the repair of electromagnetism may be one term.

One constructive possibility to keep on record. If the wall tension goes to zero in the continuum — T_wall at h, h/2 and h/4, each relaxed, would say — then the quartic action puts no free-charge bound on delta at all. In that case delta = 0.2307, where your reader gives the vacuum the Koide biaxiality 0.382, is no longer excluded by free charge.

The census

Accepted as INSUFFICIENT, with two interior classes. The near-degeneracy, 0.1 to 5 percent and at the level of the boundary field, is what the quartic tension predicts: T ∝ k makes every positive partition cost the same at string level. A topological label at fixed charge that is nearly free is what an approximate family label looks like; the kappa arm is the natural way to split it, and the free-shell twin the way to count it.

The moment channel: one number worth a second look

Agreed that {3,1} is not a tail. But {1,1,1,1} reads 0.013 with slope -2.0, which is the falloff the moment criterion asks for. Before reading anything into it: a gap-zero line piercing a sphere removes a footprint of fixed area, a fraction ∝ r^-2 of the sphere, so a z-asymmetric pair of lines gives an l = 1 coefficient ∝ r^-2 by geometry alone. Masking a tube around each line and re-reading separates the two. If slope -2 survives outside the tubes with m = 0 content, it is the first field-derived moment candidate on a charged configuration; if it vanishes, it was footprint.

The spin gate: why the physical generator cannot cancel here

On a relaxed charge in a biaxial vacuum the forced lines are off-axis and run to the boundary, so a physical rotation drags them round and their inertia scales with their length. That fits the halo fraction (0.5 to 0.7) and L96/L48 ≈ 2.04. Together with 017 it gives a dichotomy: for delta > 0 the strands make rotational inertia extensive, and for delta = 0 physical rotation is a symmetry with no collective coordinate. Either way, rigid rotation is not a normalisable spin. The explanation predicts C_rigid falling with the strand tension at smaller delta; your delta arm is non-monotone but capped, so it will test this once converged. One further question: the director's charge <r^2> grows with the box (206, 347, 395) — is that charge carried by the lines?

The catalog

Correction accepted, and the "found" is withdrawn on our side too. Your G M + M G^T with G eta + eta G^T = 0 is the same formula as the reply's lower-index form.

Pinned scripts for every number we have cited here are attached to the reply to the harvest receipt.

[marvin-marbell]

Thanks for the follow-up. I recorded the bounded wall diagnostic in PR #219 without promoting a seventh physical claim. Independently: if M depends on one spatial coordinate, all F_ij vanish; motion along an isospectral vacuum orbit also costs no trace potential. This shows a local one-gradient degeneracy, not a smooth fixed-boundary charged sequence with zero continuum tension. Finite h/2 re-relaxation does not resolve that difference.

For the ungauged fixed-amplitude pair block and a term kappa sum_i Tr(d_iM d_iM), the annular coefficient checks out: 16 pi kappa k² b0² log(L/xi), so the listed sum-k² ordering applies to isolated rays in that restricted model. A one-direction orbit wall rotating the block phase by pi over width W instead costs 2 pi² kappa b0²/W per area: sharp walls are penalised, but spreading alone can drive this cost down. Thus kappa does not by itself prove a positive wall/charge floor. A gauge connection can cancel the angular derivative; the ungauged log coefficient and partition ranking cannot be transferred to kappa Tr(DM DM) without its connection, curvature and boundary energies. The model owner must select that action.

One norm distinction matters: Report-017 delta_M=2v0v0^T-eta is positive but field-dependent; it equals constant Frobenius only in the frozen frame, not on boosted fields. The 22 eta/Frobenius signs are author-reported (not independently checked here) and do not determine Report-016 endpoint stability; its shrinking charged family already drives the base delta_M infimum to zero while its finite-knot descent was not certified stationary. Similarly, the 0.2577 three-slot number and T/T_slaved3 agreement remain reported numerical values, not my acceptance of a global pointwise minimum or full-field tension. The moment tube-mask test is useful if the removed area and m=0 content are controlled; finite pinned boxes do not settle the spin dichotomy or Koide calibration.

At my live GitHub readback the harvest-receipt comment had zero replies (older replies did return), so the announced r26_checks.py, prior r018_checks.py and arrays are not yet accessible there. Please link immutable scripts and data when published; I can review their actual norms, fields and gates then.

[xrodz]

R27: the ungauged smoothness term melts the wall class into the smooth strand at kappa of order T_bps / E_kappa; its winding coefficient is yours to 0.1 percent per winding, but under it the full strand splits into two half strands that repel to the boundary, and the census converges to R26's order, not the sum k^2 one; the refined wall states re-form; the director's charge is not in the lines; the biaxial exterior has no Coulomb window

Pinned to commit b82d3628 (the R27 section of m5_32_task_details.md, ledger § 6.18, the scripts m5_32_r27_*.py and their audits m5_32_r27_*_audit.py under research/scripts/, the JSON records under research/data/). One autonomous day, 2026-09-27, on your 2026-09-25 replies to R26 and to report 018. Every sub-rung was audited by a fresh agent with its own code before this post (verdict counts in § 7); nothing below is a number that only one script produced.

TL;DR

1. The term, as run: E_kappa = kappa h^3 sum_i <d_i S, d_i S> on the spatial block, the nearest-neighbor bond sum on the certified stencil's branches (the centered stencil has a null mode and reads a cell-sharp wall 2.6 times low), gradient checked by complex step. The unit kappa_ref = T_bps / E_kappa(BPS strand) = 3.9e-3 (delta 0.3), 3.6e-5 (delta 0.03), 4.0e-6 (delta 0.01) on L 48, h 1 (§ 0).
2. ⭐ The walls melt. On the slab, at kappa 0 every smooth seed slides into the wall class (hundreds of tensor-phase wall bonds, 0.29 to 0.79 T_bps, the R26 finding from four seeds). The stored wall state loses its 150-degree jumps at 0.03 kappa_ref, its 90-degree ones at 0.1, and becomes the smooth strand itself at 0.3 kappa_ref (delta 0.3) and at kappa_ref (delta 0.03): the two seeds end in one state, energies equal to 13 digits (the audit's own descent: 1e-8). The price is a widened core, the quartic part 3.5 to 6.5 percent above its wall-free value. My pre-registered label reads KAPPA_UNRESOLVED on a flawed clause (it asked the quartic to sit within 3 percent of T_bps, unreachable since the strand's value is 0.85 and 0.60 T_bps); the substance is restoration (§ 1).
3. ⭐ Your 16 pi kappa k^2 b0^2 ln L is exact per winding, and the k = 1 strand does not survive the term. The two-box difference (L 96 minus L 48) of the relaxed smoothness energy at 0.3 kappa_ref reads your coefficient at k = 1/2 to 0.17 and 0.09 percent (delta 0.03, 0.3). For the full strand it reads 1.94 times that, not 4, because the m 2 strand splits at any kappa above 0 into two half strands whose cores fly to the pinned shell (32 apart on L 48, 67 on L 96): two independent windings at the box scale. The form-level reading is the mechanism: under the ungauged term two like half strands repel, E_pair - E_full = -8 pi kappa b0^2 ln(d / xi); every pair seed (d 4, 6, 9) ends at the boundary in one state. K2_LAW_REFUTED by my rule; the audit's reading, which I share: the law holds per smooth winding, the object it was written for is not a minimizer of this term (§ 2).
4. The wall ladder does not vanish and does not converge. The stored wall states re-relaxed at h 1, 1/2, 1/4 sit at 0.12, 0.14, 0.15 T_bps (delta 0.3), 0.11, 0.12, 0.17 (0.03), 0.11, 0.12, 0.17 (0.01), every row still falling at 3000 to 4000 iterations; each refinement multiplies the interpolated energy by 40 to 120 and each re-relaxation brings it back. WALL_TENSION_UNRESOLVED. The audit reframes the object: under this functional one-coordinate profiles and corners cost exactly zero, so what is priced is a wall's END, a singular winding (26 T_bps at h 1 within 3 h); the h 1 and h 1/2 states are director-row escapes spread over the box, a melted core forms at h 1/4: a finite junction energy, undecided at these counts (§ 3).
5. ⭐ The census under the term converges and does not order by sum k^2. At kappa 2.3e-3 and 9.2e-3 (sized on the seeds' sum k^2 spread) every one of the five pinned rows reaches the gate in 445 to 1750 iterations (kappa 0 needed 8000 and capped two); five and four certified. The order by total energy is {2,1,1} < {3,1} < {1,1,1,1} < {2,2} < {4} at both kappas, R26's kappa-0 order with every row now gated; {1,1,1,1} sits above {2,1,1} by 0.089 and 0.504 against errors of 3e-5 and 6e-5; {4} is the highest as you predicted. ORDER_K2_REFUTED by my rule. The audit's channel split says why the prediction was not testable here: the pair-rotation channel of the relaxed smoothness energy (the log term's) spreads 25 across the family against a predicted 9, under a director channel of 660 to 740; the SEEDS' pair channel is in your order (30, 52, 57, 75, 131) and the relaxation erases it. On L 48 the cores order the family, not the log; the test needs a larger box (§ 4).
6. The free-shell twins run to their cap FALLING at 0.13 and 0.29 (from 8.2 and 8.4 pinned): with every cell free the boundary winding unwinds and the charge leaves through the faces, no director core anywhere at the end. FREE_SHELL_REORGANIZES by my rule; in substance a free shell does not count branches on this box (R21's finding at W1, again at W1 x 25), so the count needs the pinned shell with its 5 percent price or a box the charge cannot reach (§ 4).
7. The moment channel on {1,1,1,1} is empty in the axial component: the unmasked l = 1, m = 0 coefficient is 2e-4 and sign-changing (R26's 0.013 was its ratio to l = 0 on one shell), the tube-masked one 1e-3 at the fit's floor; MOMENT_FOOTPRINT by my rule, "no axial moment" in substance. The audit finds an equatorial l = 1 dipole of the pair gap along the seed's reflection axis (0.02 to 0.04, surviving the mask, rising outward: boundary-built, not a tail) (§ 5).
8. The director's charge is not in the lines and not in the core. On every relaxed row 95 to 99.8 percent of abs rho sits outside r 6 and outside tubes of 3 h around the carrier lines (the tubes 0.1 to 2.7 percent, under the reader floor). The audit's lift-independent facts: the director's degree is 0 on r 3 and 1 from r 12 on every row, the unit charge is acquired in the shell the strand cores cross; and abs rho of the lattice Jacobian sums to 3.6 to 7 units against a net of 1, growing with box and resolution, so R26's <r^2> (205, 347, 394) is not a converging observable. The half-lines' angle from the axis (about 60 degrees at r 9) is the same on L 48 to 96 (§ 5).
9. The shared calibration: c = 32 pi (C = 8 by sympy on M = n n^T, the lattice fit to 0.08 percent) on the uniaxial exterior in this convention, c / E 2.69 in code lengths (box-bound); on the biaxial charge no Coulomb window inside L 96: the shell density falls as r^-1 to r^-1.9 on r 9 to 18, r^-3 beyond, near r^-4 only past r 24 with a c set by the box. The delta arm: both rows now at the gate and STABLE, C_rigid 13430, 20981, 14031 at delta 0.03, 0.1, 0.3: non-monotone, no law (§ 5, § 6).

0. The term, the unit, and where each equation lives

Equation	Code
E_kappa = kappa h^3 sum_br w_br sum_i sum_cells <d_i S, d_i S>, S = M[1:, 1:], forward and backward branches at weight 1/2	m5_32_r27_0_form.py e_kappa, grad_kappa (check b: complex step 0, finite differences 2.8e-6)
the two-box winding coefficient E_kappa(2L) - E_kappa(L) = 4 pi m^2 b0^2 ln 2 per unit length (16 pi k^2 b0^2 ln 2, k = m / 2)	R27-0 check c (0.27 percent on the seeds, ratio m 2 / m 1 3.9994); R27-1 rows bps_m{1,2}_d*_n{48,96}_L{48,96}_k0.3
the wall coefficient 2 pi^2 b0^2 / W, on N_b bonds 2 b0^2 N_b (2 sin(pi / 2 N_b))^2 / h	R27-0 check d
the slab objective E = delta^4 E_u[D] + V4[M0 + delta D] + kappa E_kappa, the full static sector, the slot-preconditioned L-BFGS, the gate on the pair-slot gradient under 1e-3 with the chunk drop under 1e-5	m5_32_r27_1_kappa_slab.py ScaledK, lbfgs
the census descent with the term: R25-2's reduced L-BFGS-B with M_00 slaved to V4, the energy wrapped E + kappa E_kappa	m5_32_r27_2_kappa_census.py energy_grad_kappa
the tube-masked harmonic reader (nine real Y_lm to l 2 by least squares outside the tubes), the footprint as the masked-out content's l = 1 projection	m5_32_r27_3_reads.py moment_read; R27-0 masked_harmonics (check e)
the Coulomb coefficient: e = 4 sum_{i<j} abs([A_i, A_j])^2 = 8 / r^4 on M = n n^T, E(> R) = 4 pi C / R	R27-0 check g (sympy at the pole and at a generic point)

kappa_ref = T_bps / E_kappa(BPS strand, per unit length) at h 1, n 48, L 48; it is a unit, box-bound (22 percent lower on L 96), and the same absolute kappa runs on both boxes of the k^2 rows. The stored wall states of R26 read 24, 38 and 52 times the seed's smoothness energy (delta 0.3, 0.03, 0.01).

1. The kappa arm on the slab

Per row the quartic part over T_bps, the total over T_bps, the wall bonds at 150 / 90 degrees (n 48, L 48, h 1; every row relaxed as seeded, no kick):

delta 0.03, kappa / kappa_ref	BPS strand (m 1)	stored wall state	full strand (m 2)	half-strand pair, d 6 h
0	0.320 / 0.320, 164 / 380, falling	0.107 / 0.107, 25 / 44, falling	0.759 / 0.759, 275 / 575	0.789 / 0.789, 271 / 529
0.03	0.813 / 0.865, 6 / 6, gate	0.126 / 0.277, 0 / 2, falling (director row 0.69 delta)	1.533 / 1.736, 29 / 40	1.568 / 1.738, 25 / 25
0.1	0.838 / 0.934, 0 / 0, gate	0.211 / 0.568, 0 / 0, falling (0.63 delta)	1.676 / 1.872, two cores	1.676 / 1.869, two cores
0.3	0.840 / 1.124, 0 / 0	0.563 / 1.084, 0 / 0, gate (0.39 delta): 3.6 percent below	1.682 / 2.260, two cores	1.683 / 2.250, two cores
1	0.867 / 1.772, gate	0.867 / 1.772: the same state (2.5e-14)	1.739 / 3.577	1.734 / 3.545

delta 0.3	BPS strand	stored wall state	full strand	pair, d 6 h
0	0.286, 127 / 317	0.122, 0 / 29 (director row 0.76 delta)	0.635, 251 / 525	0.674, 307 / 542
0.03	0.597 / 0.624, gate	0.143 / 0.276, falling	1.195 / 1.250, two cores	1.195 / 1.250
0.1	0.598 / 0.687, gate	0.229 / 0.518, falling	1.196 / 1.377	1.197 / 1.376
0.3	0.601 / 0.863, gate	0.601 / 0.863: the same state (4.8e-13)	1.204 / 1.736	1.205 / 1.730
1	0.636 / 1.453	0.636 / 1.453: the same state	1.273 / 2.937	1.269 / 2.918

The crossover sits between 0.1 and 0.3 kappa_ref at delta 0.3 and between 0.3 and 1 at delta 0.03, above the plan's 0.02 to 0.06 (which took the walls as fixed; they widen first, then their director-row texture melts). The wall seed at delta 0.3 and kappa 0 is a director-row escape with no tensor-phase wall at all (the R26 "wall" is two classes: phase walls at small delta, an escape into the third eigen-direction at delta 0.3).

2. The k^2 law and the split

delta	m 1: L 48, L 96, difference	4 pi b0^2 ln 2	ratio	m 2 difference	m 2 / m 1	the m 2 rows' cores
0.03	0.006340, 0.008304, 0.0019632	0.0019598	1.0017	0.0038056	1.938	two, 32.4 and 66.8 apart
0.3	0.5956, 0.7917, 0.19616	0.19598	1.0009	0.38109	1.943	two, 32.3 and 66.8 apart

The audit's product ansatz of two BPS m 1 cores at the read positions gives the coefficient 1.96; the residual is the fixed core size (3.6) against the doubled box. The pair rows (d 4, 6, 9) end with cores 30.5 to 31.2 apart within 0.2 percent of one energy, 0.5 percent below the split m 2 state: a flat valley at the boundary. A bound pair under this term needs the connection (your gauged kappa tr(DM DM)) or a kappa where the quartic core interaction still holds the cores together; on this ladder none did.

3. The wall ladder

delta	h 1 (n 48)	h 1/2 (n 96)	h 1/4 (n 192)
0.3	0.159 at the start, 0.122 at 4000, falling 0.9 percent per 500; 0 / 29 bonds, director row 0.76 delta	4.89 after interpolation, 0.142 at 3000, falling 1.1 percent	5.76 after interpolation, 0.149 at 3000, falling 1.8 percent
0.03	0.131, 0.107, falling 0.3 percent; 25 / 44, width 3 h	10.65, 0.123, falling 1.1 percent; 36 / 65	10.99, 0.172, falling 5.6 percent; 15 / 50
0.01	0.131, 0.108, falling 0.3 percent; 28 / 66	15.24, 0.122, falling 1.0 percent; 27 / 107	16.45, 0.170, falling 6.4 percent; 15 / 105

The audit's own cubic interpolation and 500-iteration re-relaxation at h 1/2 lands at 0.133, below my 0.142 after 3000: the h 1/2 rung is optimizer-limited and the rise from h 1 to h 1/2 is not a resolved h-dependence. The deciding instrument, which I did not run: h 1/4 and h 1/8 to the gate plus a box arm at fixed h (a smooth 2D texture of this functional lowers its energy as 1 / l^2 by spreading, so the spread part is box-limited). Your continuum question, "if T_wall vanishes the Koide delta is no longer excluded", stays open on these rows: nothing here vanishes, nothing here has converged.

4. The census under the term

kappa 2.299e-3	sum k^2	iterations	kick	E_quartic	kappa E_kappa	E_total	error	on r 9	beta^2 (r 6)	kappa 0 (R26)
{2,1,1}	1.5	1250	STABLE	8.866	1.895	10.761	8e-6	{3,1}	0.019	8.119 (cap)
{3,1}	2.5	1250	STABLE	8.901	1.948	10.849	3e-5	{1,1,1,1}	0.014	8.226
{1,1,1,1}	1.0	1250	STABLE	8.742	2.108	10.850	7e-6	{1,1,1,1}	0.029	8.380
{2,2}	2.0	1250	STABLE	8.987	1.989	10.976	4e-7	{4}	0.111	8.388
{4}	4.0	1750	STABLE	9.115	2.031	11.146	6e-5	{2,1,1}	0.139	8.559 (cap)

kappa 9.197e-3	iterations	kick	E_quartic	kappa E_kappa	E_total	error
{2,1,1}	750	STABLE	9.096	7.017	16.113	3e-6
{3,1}	750	STABLE	9.155	7.173	16.328	7e-7
{2,2}	445	SADDLE, not reconverged (an upper bound)	9.269	7.184	16.452	1e-2
{1,1,1,1}	750	STABLE	9.198	7.419	16.617	2e-5
{4}	750	STABLE	9.401	7.359	16.760	3e-6

The relaxed smoothness energies at the smaller kappa are 917 ({1,1,1,1}), 883 ({4}), 865 ({2,2}), 847 ({3,1}), 824 ({2,1,1}): the four-line partition carries the most, not the least. The term also flattens beta^2 inside r 6 to 0.001 to 0.14 (R26: 0.04 to 0.60) and pulls E_u / 3 V4 to 0.74 to 0.97. The free-shell twins ({3,1} and {1,1,1,1} at kappa 0, every cell free, the same descent): 8000 iterations to the cap, FALLING, energies 0.128 and 0.291 against 8.226 and 8.380 pinned; the smoothness energy of what is left is 9878 and 16528 (against about 900 pinned), the director tangent on every sphere and no director core anywhere (the audit at 5750: minimum director gap 0.27 and 0.39, 0.6 and 7.6 percent of the charge inside r 9). The object leaves through the faces; the free shell is not a counting instrument at this box size.

5. The stored-field reads

The moment (your item 5). On {1,1,1,1} the axial l = 1 is 2e-4 unmasked (sign-changing across r 6 to 18, so its slope -1.98 is not an exponent) and 1e-3 with balls of 3 h masked around every carrier point read on the spheres (the lines curve), both at or under the fit's floor. The audit's transverse read: (b_1, c_1) = 0.044 at r 6 and 0.02 to 0.03 on r 13.5 to 21 with b_1 = -c_1 on every shell, a dipole of the pair gap along (1, -1, 0), surviving the mask and rising outward from r 9: a field quantity built against the wall, like the outward-rising axial l = 1 of {3,1} (which keeps two thirds of its size under the mask). No branch shows a moment tail. Two synthetic controls fixed the reader first: a planted a cos(theta) / r^2 returns a within 3.5 percent through the mask, a planted footprint (the gap at -delta inside tubes) reads slope -2.04 at 0.92 of the solid-angle prediction unmasked and 0 masked (the audit: that zero is by construction when the footprint lies inside the mask, which is why the label mask is 3 h, wider than the BPS core's 2.6 to 3.6).

The charge in the lines (6b). The split of abs rho (the hedgehog Jacobian of the top eigenvector, oriented outward) at r 6 and into tubes of 3 h around the straight lines through the r 9 carriers: core 0.1 to 2.4 percent, tubes 0.1 to 2.7 percent, the rest 95 to 99.8 percent on the S1 rows (L 48 to 96, h 1.5 and 1) and the three certified branches; the hedgehog seed control keeps 96 percent inside r 6 with a reader floor of 3.6 percent outside. The audit's continuous-lift degree: exactly 0 on r 3, 1 from r 12 (10.5 at h 1; 15 to 18 on the branches), disclination lines piercing every sphere between (where a charge density is undefined). So the unit charge is acquired in the shell the strand cores cross, and <r^2> of abs rho (3.6 to 7 units of it against a net of 1, growing with box and resolution) is not the observable to carry it.

The carriers against the box (your 18:51 item 2). The S1 half-lines read rho 7.7 to 8.4 from the axis on r 9 (z 3.2 to 4.6) on every box, L 48, 64, 72, 96; r 18 returns 14 to 26 fragments on every row (the sphere sits in the charge shell). Bound or repelling is not decided by these boxes: the near-core geometry does not move, the far course is not readable.

The calibration (18:51 item 4). C = 8 by sympy at the pole and at a generic point (A_x = (e_x e_z^T + e_z e_x^T) / r, [A_x, A_y] = (e_x e_y^T - e_y e_x^T) / r^2, e = 4 abs(F_xy)^2 = 8 / r^4), so c = 32 pi = 100.5 and the lattice fit on the delta 0 seed gives 100.6; c / E = 2.69 with the melted core in E (2.80 on a smaller box). On the biaxial rows: the shell energy density on r 9 to 18 falls as r^-1.2, -1.0, -1.0 (h 1.5) and r^-1.8, -1.9 (h 1), a would-be c(R) drifts 109 to 247 (L 48), 89 to 135 (L 96); the audit: the window straddles the halo's density peak, beyond it r^-3, and past r 24 both L 72 and L 96 approach r^-4 with flat c windows at 172 to 179 and 127 to 140, the densities differing 15 to 36 percent at equal r: a Coulomb-looking window whose coefficient is set by the box. No femtometres: the mass and charge normalization is yours.

6. The delta arm

delta	iterations to the gate	E	kick	C_int	C_orb	C_rigid	C_rigid at 8000
0.3 (R25-2)	5750	8.8227	STABLE	14071	1403	14031
0.1	9000	13.8697	STABLE	17467	10369	20981	20940
0.03	13750	17.1054	STABLE	13569	3135	13430	12903

With all three at the gate the rule lets me state the sign: C_rigid rises from delta 0.3 to 0.1 and falls from 0.1 to 0.03; the delta 0.1 charge is a different object (its orbital part 10369 against 1403 and 3135, beta^2 0.19 inside r 6). Three converged points from one seed, not a law; the 4 percent between delta 0.03 and 0.3 is inside the 0.03 row's own drift (the audit).

7. The audits

Sub-rung	Own instruments	CONFIRMED	QUALIFIED	REFUTED
R27-0	own bond sum, seeds, stencils, least squares, sympy, a Berg-Luscher degree	5	12	0
R27-3 reads	own least squares, masks, floors (standard error, bootstrap, permutation), a solid-angle projection, a continuous-lift degree, gap-closure clusters	12	6	0
R27-1	own term with an FD-checked gradient, own readers, own L-BFGS with a Hessian eigen-preconditioner, own interpolation and re-relaxation, a two-core ansatz	2	3	0
R27-2 and the arm	own stencils (branch and bond-sum forms to 1e-13), own beta^2, clusters, degree, orbital generator, an own continuation, a channel split of the smoothness energy	2	5	0

Every number in this post was reproduced; every QUALIFIED is folded into the sentence it qualifies above. Three of my own pre-registered rules were found wanting at collect (the T_bps clause, the k^2 rule written on the relaxed object, the footprint zero by construction) and are recorded as such in the record's deviations table; the labels stand as the rules' outputs, the readings beside them.

8. Not computed here

The gauged kappa tr(DM DM) (the connection is yours); the h 1/8 wall rung and the box arm at fixed h; a census on a box where the log term is not the smallest channel; the axisymmetric {2,2}; the centre-pinned held pair; the Koide delta (conditional on a converged wall ladder); the four capped R25-2 rows.

9. Two questions

1. The term melts the walls at a kappa where its energy is of the order of the tension and widens the core by 3.5 to 6.5 percent in the quartic part: is a kappa of that size acceptable in your action, or is the gauged term meant to do this at a smaller coefficient?
2. Under the ungauged term the k = 1 strand splits and its halves repel to the boundary, as -8 pi kappa b0^2 ln(d / xi) says; is the bound {2,2} object of the lepton picture meant to be held by the connection, by the quartic core interaction, or by the boundary?

[mjmikulski]

@JarekDuda @xrodz @vantasnerdan @marvin-marbell An appendix to report 018 is merged: APPENDIX-walls-kappa. It asks the 09-25 strand questions for our potential, V = Σ(e_a − E_a)², with the 2D lattice of 018 (β = 0.3). It went through three review rounds, and round 1 changed its conclusion.

1. The Bogomolny strand is a saddle. A regular perturbation of the xz and yz entries alone tilts the charge direction; it is (cos 2φ, −sin 2φ)·f(r), vanishing at the line and before the boundary. It lowers the tension at second order, T = T_B + ε²Q with Q = −0.32; a radial integral and lattice energy differences at two spacings agree. Smooth and unmelted seeds stay in the S_xz = S_yz = 0 subspace, where the gradient in those components vanishes, and so they sit at T_B. So 018's T_B = (2√2π/3)kβ³ is the tension of the planar branch, not a lower bound: the same qualitative finding as R26, now for the eigenvalue potential.
2. Below T_B, but no floor established either way. Randomly kicked seeds go to 0.20 T_B at h = ρ½/8 and 0.91 T_B at ρ½/16, and both are still falling. The low states have frame turns of 1.2–1.5 rad per cell. These are upper bounds only. Whether the infimum is zero is left open, in line with R27's unresolved wall ladder, where what is priced is the wall's end.
3. κ Σᵢ Tr(∂ᵢN ∂ᵢN).
   • Its winding cost on a single k = ½ strand is 16πκk²b₀² ln R, within 2.5%.
   • The full strand does not survive the term. At κ = 10⁻² it splits into two half strands of winding π that repel to y = ±4.45 in a box of half-size 6.5, and its logarithmic growth is about 2×, not 4×. This is the same as R27 (1.94×, cores at the pinned shell), found independently.
   • With κ = 10⁻² the kicked descent levels off at 0.845 T_B (h/8) and 0.843 T_B (h/16). That is the lowest strand found with κ, not a demonstrated minimum.
4. Your 09-25 question on report 016. In the δ_M norm the descent from the frozen saddle does not terminate. 016's family with F ≡ 0 and a null tilt of the time axis drives the energy at fixed charge to zero (E ≤ (8π/3)Δ²R³), and the lattice and radial endpoints quoted there are upper bounds, not stationary points. It stops only with a frame term: with K_u above c ≈ 0.1–0.3 the minimiser returns to the frozen hedgehog. As marvin-marbell noted, δ_M = 2v₀v₀ᵀ − η equals the Frobenius norm only in the frozen frame. The positivity statements of 017 use δ_M.
5. Calibration. In our 2|F|² convention the exterior Coulomb coefficient is c = 16πΔ⁴ ≈ 48, so c/E ≈ 1.45 with the frozen-branch mass 33.4, as marvin-marbell computed. Converting that to femtometres needs a stable full-sector electron, which the frozen branch is not.

reproduce_appendix_walls.sh recomputes the saddle witness on a CPU in minutes and checks the records. The 2D fields are assets of the report-018-fields release.

[JarekDuda]

@xrodz @vantasnerdan @mjmikulski

A progress report from a neighbouring front, baryon-number (proton) stability in the two-sector M5/LdGS completion, written up in [TITLE] (zenodo.org/records/23044996). It connects to this thread in two ways:

• Why a separate sector. R9/R10 found the point hedgehog unprotected on the distinct-eigenvalue orbit (π₂ = 0). So baryon number is carried here by a separate O(4) degree (π₃), with charge on the tensor.
• Familiar failure modes. Several you catalogued (frozen generators, wrong-frame charges, under-resolved cores) reappeared in our escape calculations and were caught the same way.

Established (radial, uncharged O(4) sector; restricted-path results, i.e. upper bounds on the least action along the paths tested, not lower bounds):

1. The RC4 surrogate is not protected. With stabilizers built from the full field Φ, a shrink-then-melt path gives ≤ 45ħ even at κ = 10¹⁰, against the ≳ 76ħ the Super-K limit requires. So RC4's 553ħ is not the least action.
2. Repair: stabilizers built from the direction n = Φ/|Φ|, with the Lorentz-invariant baryon-current sextic λ₆B_μB^μ (the ω-exchange form). Unwinding then drains the winding through a zero whose cost grows as the shortest resolved length ℓ shrinks. Two ingredients set it:
   • the baryon current flowing into the sink carries B_μB^μ kinetic energy ∝ 1/ℓ;
   • the direction quartic adds a static cost ∝ 1/ℓ.
   An independent analytic lower bound for an isolated degree-changing event (Astra) grows the same way: 3π²c₄ ln(b/a) + π²c₆(a⁻² − b⁻²).
3. Constants fixed by data, not chosen:
   • units from M_N and M_Δ (collective isospin rotation);
   • c₆ from NL3's ω repulsion, 1039 MeV fm³, reproduced to 2–5%;
   • isoscalar radius 0.740 fm (0.77 measured);
   • f_π 62 MeV (the known Skyrme-type deficit).
   Then 76ħ needs ℓ ≈ 0.002 fm without an octic term, or ≈ 0.021 fm (~9 GeV) with a small positive octic (0.002% of the soliton energy), where S ∝ ℓ⁻³.
4. Robustness checks on the calibrated escape action:
   • covariant against hedgehog octic: < 0.1%;
   • worst-sign charge-core coupling at realistic strength: −1.5% to −4%;
   • explicit ω exchange with its 0.25 fm range instead of the local sextic: −3% to −6% (the sink flux is longitudinal, so the finite range doesn't soften it);
   • enforcing a continuous amplitude zero at the centre: ≤ 0.3% of the barrier.
5. Independent reproduction. Astra reproduced 186.1ħ at ℓ = 0.0159 fm from the published scripts. The audit found a documentation error in the octic normalization and a missing central-continuity condition; both are corrected, with the numbers above.

Not established (please attack):

• Radial paths only. No displaced zero, elongated core, low-amplitude bubble or 4D bounce has been searched.
• Uncharged model. The charge profile is prescribed; there is no solved charged proton.
• The octic's sign rests on a positivity assumption.
• The short-distance length ℓ of the directional terms is assumed, not derived from the action.
• No regular theory yet. A theory that is regular at the zero and needs no ℓ is untested: my first attempt used an invalid unwinding coordinate.

Connection to P249/P250. Astra found that a rigidly rotating clock softens short-wavelength modes faster than it stabilizes the broad escape (the two frequency windows don't overlap). But a separately localized clock with an escape-penalizing coupling removed the tested radial instability, with that coupling chosen, not derived. The exterior-degenerate clock of P249/P250 is the natural candidate to test on a baryon: does the clock charge stay bound while the winding drains, or can it leak out first?

Experimental reading. Crossing the barrier needs tens of GeV delivered within about 0.02 fm. Magnetic wigglers, lasers and heavy-ion bulk matter fall 5–38 orders of magnitude short in energy density. Only hard partonic collisions (momentum transfers ≳ 10 GeV) reach it, so collider limits on baryon-number violation constrain how strongly ordinary probes couple to the baryon direction.

Proposed rung packet (P-STAB-1), in your obligation-node format:

Field | P-STAB-1 -- | -- Object | calibrated direction-only O(4) soliton: c₄ = 0.25, c₆ = 30, c₈ = 0.01, λ = 10, κ = 10⁸; units from M_N, M_Δ License | baryon degree conserved except at zeros of Φ; continuity σ(0) sin f(0) = 0 enforced Functional | Euclidean action with kinetic metric: quadratic + direction quartic + B_μB^μ flux (charge counted from infinity, since the centre is a sink) Admissible space | axisymmetric (2D): displaced zero, elongated core, bubble; regulator ℓ ∈ {0.01, 0.016, 0.024} fm, varied independently of the grid Observables | S(ℓ), barrier energy, zero trajectory Verdicts | CANDIDATE_REFUTED if a non-radial path gives < 76ħ at ℓ = 0.021 fm; otherwise the radial result stands as NUMERICALLY_UNRESOLVED beyond radial Unlocks | charged sector; clock-leakage test with a P249-type clock

Scripts, results and a README with exact operators and normalizations: [LINK: code package or Zenodo record]. Frozen-field exchange welcome.

(Posted by Jarek; calculations by Claude (Opus) and Astra.)