[Lyy-iiis]

The two-full Laurent border has a favorable formal invoice but no admitted recursive precision contract. This contribution supplies a source-bound inverse-interface construction and identifies the actual paid-site cost of one independent-error policy. It claims no new κ and leaves the selected frontier unchanged.

For all eight owned-center macros in each retained orientation, recompute the same deterministic center on unchanged capture fields and negate its fanout. This avoids reversing an erasure and adds no declared Gaussian call. The complete typed inverse recipes preserve the paid profile, conditional on ordinary persistent children providing their actual inverse at the same rank charge and restoring private workspace. Forward COPY/fanout and backward gather/fanout are checked separately.

The complementary census covers all 6,596 actual paid sites. After splitting, 3,608 rank-one jobs alone have main-output error proportional to η/ε. Under the explicitly stated independent-error, linear-precision tariff, their extra cost rejects every concave moment. This is not a lower bound on correlated, adaptive or different implementations.

Validation:

• Isolated optimized standard-library replay passed locally on CPython 3.14.5; metadata/source pins passed.
• Both inverse recipes: 16 macros, eight full 632-role round trips and ten primary rejection controls.
• Paid census: exact rational witnesses, four rejection controls and direct finite-fault checks.
• Separate implementations: all-site exact recount, 36 rational fault trajectories, 96 local inverse round trips, eight full-word round trips, five further controls and all moving-frame events.
• The registered Python 3.12 package replay and unchanged-input check passed in GitHub Actions run 39900321912 at exact head 039b6ea51ab7a7efb38fc6851f345839683251b0. No selected cold replay, full physical/Clifford suite, Lean or outer theorem is claimed.

Remaining obligations: actual paired recursive caller, Gaussian/frame compatibility and fresh independent D factors, finite-grid precision/storage/routing cost, and BIT admission. Invertibility alone does not discharge these.

Scientific base: cef4b52f2a59009865bf8eabde1be3f159063703; refreshed main b91956876e5b4122fd89179f1b57557ff8378f15 leaves relevant inputs, priorities and selection unchanged. Canonical Discussion. Prior two-full border.

Credit Dillon/#4623, Cao Yifan/#4610, Jacob Alan Sussman and the retained native/Clifford/compiler/precision lineage. Prepared by Yiyang (@Lyy-iiis) with substantial OpenAI Codex and separate agent implementation/review. Apache-2.0; no independent human or formal-kernel acceptance.

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[jwr404]

Independent review of the paid-site precision census, at head 039b6ea51ab7a7efb38fc6851f345839683251b0

Scope. This covers paid-sensitivity/PROOF.md: the census and its concluding moment failure. I did not review the inverse-interface part.

1. Recount from the pinned word. I wrote my own replay of research/kappa-round3-warp-20261022/ACTUAL-WORD.json.gz (compressed SHA256 7679daa4…, identical at base). It shares no code with the package's checker or its independent recount.

Following §"Why one exact address suffices", the word is replayed on δ₀ as an exact rational scalar circuit: add, copy, scale and clear are applied, while warps and paid transitions act as the identity.

• Prefix. Forward rows over the chosen inputs. In the lower call (source-forward) x_i = −δ₀. In the upper call (source-backward) x_i = y_i = −δ₀.
• Suffix. Reverse-mode sensitivities: copy a=b adds a's column into b and zeroes a, and clear zeroes. The observations are main X_i − Y_i in the lower call and main X_i in the upper call.
• Witness rule. A site is witnessed when both the prefix row and the suffix column at its register are nonzero, so the first-variation outer product is nonzero.
• Splits. 4=3+1, 5=3+2, 8=3+3+2.

Quantity	Package	Independent
Paid sites (2 × (3,290 climbs + 8 centers))	6,596	6,596
Baseline split paid	{1:4656, 2:920, 3:1386}	identical
Main-visible witnessed split	3608 H1 + 700 H2 + 886 H3	identical (from 4,898 sites)

I did not recompute the ≥ 1/8 coefficient bound. My check is nonzero-ness only.

2. Arithmetic of the conclusion. Target mass 224 + 98 + 752 + 5868 + 3808 = 10,750. Paid mass 4656 + 1840 + 4158 = 10,654, and +3608 = 14,262. Since every target rank is ≥ 4 and every paid rank ≤ 3, for t ∈ [0,1] we have k^t ≤ k·4^{t−1} and r^t ≥ r·3^{t−1}, so T_t/P_t ≤ (4/3)^{t−1}·5375/7131 < 1. ✓

The precision step is also correct. With both fault signs, |2η·s·p/ε| ≤ 2τ and a coefficient ≥ 1/8 give |η| ≤ 8τε ≤ 4τ², hence w ≥ 2k − 2.

3. Scope. These points are already stated in the PR, and none of them is a defect.

• The failure is conditional on the stated tariff: independent per-site worst-case error, base w = k + o(k), and leading cost linear in w. Under a different cost law, or with correlated or recomputed errors (the inverse-interface route), the conclusion does not follow.
• The census relies on the inherited fact that every warp fixes δ₀ with phase exactly 1 (bw_gauss.py::monomial, G0 = 0, q_M(0) = 0). My recount relies on it too, so the census is not independent of that premise.

This is an author-independent check. It is not maintainer acceptance, and κ is unchanged.

Reviewed by James Reed with substantial Anthropic Claude (Claude Code) assistance.