docs(ArcQuant): the cheap decode is at 2.25 bpw, not 2.00 — the bit-rate frontier

[heydryft]

The claim this corrects

FACTS.md §2026-08-19 §1 recorded the compiled trellis ladder — shipped K4/V2/L16 at 15.125 inst/wt, K8/V4/L12 + row-scale hoist at 4.375, 3.46× fewer instructions per weight — with no quality caveat. Read as it stood, it names a ship candidate.

That geometry is quality-CLOSED. K8/V4/L12 costs −0.00698 w_cos against a ±0.0008 ship band — 8.7× outside — and nine codebook designs cannot rescue it, topping out at −0.00307 (converged trellis-Lloyd, 40 sweeps), still 3.8× outside. D4 lattice cosets, the best known 4-D quantizer, measure −0.01008.

The answer

The 3.46×-shaped decode is available — for a quarter of a bit more. At fixed L=12/V=4 the codebook does not read K (the probe asserts bit-equality across K=8…12 rather than assuming it), so the table stays 32,768 B and the decode stays one LDS.64 per 4 weights. Only the bit rate moves.

geometry	bpw	Δ w_cos	table B	model GB	KV GB	band
K8/V4/L12	2.00	−0.00698	32,768	74.2	58.8	fails, 8.7× out
K9/V4/L12	2.25	+0.00402	32,768	83.4	49.6	CLEARS, 5.0× the band on the good side
K10/V4/L12	2.50	+0.01166	32,768	92.7	40.3	CLEARS
K12/V4/L12 (L==K ⇒ memoryless)	3.00	+0.02057	32,768	111.2	21.8	CLEARS

The price is capacity, not quality: KV headroom 58.8 → 49.6 GB (−15.6%), UNMEASURED as a batch/context effect. The one-card wedge survives — 83.4 GB of 141, still 1.8× under MXFP4's 151 GB, which does not fit at all.

The mechanism, which is why the 2 bpw search stopped

• Trellis OFF (blocks identical, memory switched off on purpose): LBG −0.01443 vs random −0.02905 — LBG halves the loss. The design method works.
• Trellis ON: the same LBG centroids, laid in by the textbook Ungerboeck set partition, give −0.00877 — worse than random's −0.00714.

⇒ The binding constraint is trellis freedom, not codebook coverage. More codebook design at 2 bpw is not worth an hour.

Also recorded, as exact integers

• Byte-extraction bound is ceil((8 − gcd(K,8) + K)/8), not ceil((7+K)/8) — the reachable offsets of K·t mod 8 are only multiples of gcd(K,8). K=8 → 1, K=9 → 2, K=10 → 2, exact not merely sufficient. Retires the 3-byte route for every K in play.
• Warm-up tax is set by slice length (1 + W/S): warp-per-row takes 1.25×→1.06× at in_features=4096 and 2.98×→1.48× at 512, and deletes the cross-warp butterfly, warp_sums, and both __syncthreads from the row loop.
• ⚠️ The SASS census is structurally blind to that win — its inst/weight is a differential over unroll depth, which cancels prologue and epilogue exactly, so per-row and per-block overhead can never appear. A flat census row must not be read as "the reshape did nothing."
• 🔴 CONFIRMED FALSE: mistralrs-quant/src/qtip/mod.rs:596-598 says the GPU decode paths compute the codebook. True only for Mcg. DEFAULT is Gaussian, and qtip_gemv.cu gathers from the stored 512 KiB table on that path — which does not fit 48 KB shared, so every symbol lookup is a dependent scattered global load. That is the disease the 32 KB table exists to kill, independent of any instruction count.
• Structural result: K8/V4 at depth L behaves exactly like K4/V2 at L−2 ⇒ L/K lookback is NOT the predictor.
• Runner-up, still live: K4/V2/L13 — 2.00 bpw, −0.00206, 11.250 inst/wt (1.34×). Misses the band, costs no extra bit and no KV.

Still UNMEASURED — recorded as such, not softened

K9's inst/weight (the kernel has never been compiled at any K but the K=8 control — do not quote 4.375 for K9) · the pad-vs-clamp delta · the batch/context effect of 49.6 vs 58.8 GB · bf16 vs f32 codebook values.

Cost and provenance

CPU only, zero GPU hours, $0. Every table was re-run from scratch before this was written — ~2.5 minutes of laptop CPU on a warm build. Evidence branches now pushed: arcquant/trellis-geometry-sweep @ 38a03cc14 and arcquant/designed-codebook @ 269e95988 (the latter contains the former). Related open PRs: #168 (K8/V4/L12 decode family), #170 (UQFF geometry discriminator, stacked on #168), #169 (geometry-parametric beam encoder).

Stale claims pruned

Six sites, so no superseded statement sits next to its correction: the uncaveated 3.46× headline in FACTS.md and in 00_RESUME_HERE.md (twice — the session-8 bullet and the V=4 block), the same claim as the format ceiling in COMPETITIVE_TEARDOWN.md (twice — the 15–19× projection and the ladder table's conclusion), and KERNEL_RULES.md rule 2's "the compiled ladder is safe because all three geometries are 2 bpw" — true about instructions, and exactly the framing that hid the quality axis.

🤖 Generated with Claude Code

https://claude.ai/code/session_01SpVNMpb13HkUXqSqbN1o9H

[github-actions]

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