[JosieElliston]

Result

This is an independent audit of #6129 (Nick Petalas, head f496734450c3). I found no error. The sharp nearest-even bound for J normalized Walsh stages, h(J/3+(1-(-1/2)^J)/9), is correct. So are its odd-output refinement h a_(J-1) and its all-grid application to #6119 (CubeJerry, head bdfe5a10d488), which gives 3h/4, or 3/16 at h=1/4. No new κ, and no change to the selected frontier. The card is uniform-transfer-audit, with its scope changed to an audit of this PR.

Package: research/nearest-even-walsh-error-audit/ (base 721c03822).

Proof review

I checked every step of #6129's PROOF.md:

• the sign absorption by oddness;
• the parity recurrence (3) and its non-circular d_j induction;
• the witnesses (4), whose signed errors are all positive;
• the closed form and the comparison with J/2;
• the period lemma 2^(J+1);
• the Audit deferred-row rounding: sharp error bounds and clean-return controls #6119 echo argument. There a=4 is exact on the grid, and the lower inverse leaves {0, ±h/2}, which rounds to 0.

All of them hold. README.md has the table.

Two clarifications. Neither is an error:

1. The Walsh law needs parity tie-breaking, not ties-to-even itself. Ties-to-odd gives the same a_J. The Audit deferred-row rounding: sharp error bounds and clean-return controls #6119 application does need ties-to-even: under ties-to-odd, the echo temporary is left at ±h on odd k.
2. Nearest rounding alone does not give the saving. Floor rounding gives exactly J/2, and ties-away-from-zero already reaches J/2 at J=2.

One more observation: 2^J a_J = ((3J+1)2^J-(-1)^J)/9 is the same sequence as the sharp switch tariff in research/dyadic-rotation-switch-tariff, with the same slope recurrence.

Evidence

python3.12 -I -W error -B research/nearest-even-walsh-error-audit/verify.py uses only the stdlib and runs in about 50 s. It imports nothing from #6129 or #6119. scripts/research_package.py --package ... reports PASS_REGISTERED_RESEARCH_PACKAGE.

• A. Residue-tracking tree sets. These are the complete sets of (stored value mod 2^(J+1-k), exact error) from every leaf residue mod 2^(J+1). They use only R(s+2t)=R(s)+2t and the period lemma, not Prove sharp nearest-even Walsh-stage error (no new kappa) #6129's parity-only shift step. The engine covers every sign pattern for J<=4 and the all-plus pattern up to J=7. The even-output maximum is a_J and the odd-output maximum is a_(J-1) throughout; at J=7 the maximum is 313/128.
• B. Literal 8-point network. I checked every input in [-3,0]^8 in three stage orders, over all outputs. The maximum is exactly 9/8.
• C. Audit deferred-row rounding: sharp error bounds and clean-return controls #6119 surrogate, rebuilt from its definition. I checked every input in each spectator fiber mod 8h (8,192 inputs), plus 300 random period shifts. The sup is exactly 3h/4, and Prove sharp nearest-even Walsh-stage error (no new kappa) #6129's witness attains it. Local only: the rebuild agrees exactly with Audit deferred-row rounding: sharp error bounds and clean-return controls #6119's own forward(rounding=True) on 3,000 random grid inputs.
• D. Controls for the tie rules listed above.

Engine A is complete given the period lemma, which is proved in writing and only sampled in code. The all-J result rests on #6129's written proof, which I found correct.

A read-only OpenAI Codex adversarial review of the proof, the verifier and this README returned no finding in any of the three areas. Its points: engine A's last combination runs at modulus 4, where the residue reduction is still valid; engine C's indexing matches the surrogate. This is AI review only. Hosted CI was not monitored.

Not covered and open next questions

This audit does not cover #6119's arbitrary-defect theorem, bit costs, range guards, reversible history, or any PR6030 child or supplier.

Open next questions:

• Which retained rounded compilers rely on a ties-to-even half-tie cancellation like Audit deferred-row rounding: sharp error bounds and clean-return controls #6119's m=1/2 lower inverse?
• Does a_J survive one intervening grid shear? A first step is to add that stage to engine A.

Credit

The theorem, proof and witnesses are Nick Petalas's (#6129). The surrogate is CubeJerry's (#6119, with OpenAI Codex assistance). The deferred-row context comes from Vlado Vojdanovski (PR1673), and #6129 credits the wider lineage. This PR adds only the audit and its engines.

🤖 Generated with Claude Code