[petalas]

Summary

Prove the sharp deterministic nearest-even error for J distinct normalized Walsh stages on a common grid h:

- J*h/2                             sufficient stagewise bound
+ h*(J/3 + (1-(-1/2)^J)/9)           sharp, with attaining integer inputs

Claim type: scoped mathematical proof with exact finite corroboration. The coefficient is smaller for every J>=2 and its ratio to the old bound tends to 2/3. This is a logical precision result, with no new kappa or selected-frontier change.

CubeJerry's #6119 reports 3/16 on a finite input cube for its two-phase quantizer at h=1/4. The new all-grid application proves that 3/16 is its exact global normalized error, with sharp raw error 3/4. The main contribution is the all-J parity recurrence and attaining family, beyond the predecessor's finite observation. Produced by Nick Petalas; retain CubeJerry, Vlado Vojdanovski, JoyboyBrian and the credited source lineage.

Package: research/nearest-even-walsh-error. Base 721c03822c421b7f7fcbb2598993b947c4d88419; contribution f496734450c346468c66818c1c35521165b0cdda.

Evidence

• Before: independent stage defects give J*h/2; Audit deferred-row rounding: sharp error bounds and clean-return controls #6119's actual-quantizer 3/16 value is a finite observation. After: a written all-input, all-J theorem gives the exact parity-conditioned maxima and constructs witnesses in integer range [-J,0].
• Exact verifier: 2,049 signed half-integers; all 16 one-stage and 4,096 two-stage residue inputs; reachable parity error sets through depth seven; attaining witnesses through twelve stages in two axis orders; 256 periodicity fixtures; 513 echo values; 65 complete eight-coordinate application fixtures; four scope controls and five rejected invalid inputs.
• CPython 3.12.15 warning-strict normal and optimized registered commands pass locally in 0.194 s and 0.200 s. The package runner, six-file committed-byte/source-history preflight, official export/integrity checks and both read-only Git-free replay modes pass. All seven exported files, including the manifest, remain byte-identical. The verifier emits a generated source-hash receipt and writes no package files.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/nearest-even-walsh-error/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/nearest-even-walsh-error/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/nearest-even-walsh-error
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/nearest-even-walsh-error --check-source-history

Substitute another CPython 3.12 interpreter on a different machine. Expected scientific status: PASS_NEAREST_EVEN_WALSH_ERROR in both modes; registered status: PASS_REGISTERED_RESEARCH_PACKAGE.

Limitations and unrun obligations

The proof requires grid inputs, distinct Walsh axes, nearest-even ties, exact pre-rounding sums/differences and sufficient range. Off-grid data and other tie rules have explicit counterexamples. Arbitrary faults, interleaved non-Walsh gates, repeated axes and complex Gaussian butterflies are outside scope.

This supplies neither an emitted accelerated recursive child nor a physical cost, overflow, reversible-history or disposal contract. All five framework assumptions remain. No selected/native cold replay, Lean, Docker, independent human or formal proof-kernel verification was run or claimed. The selected metadata orientation passed; it is not scientific supplier admission.

Open next questions

1. Export one actual recursive H0 event list and bind the grid, axis order, exact pre-rounding arithmetic and range guards. A compatible physical caller is missing.
2. Derive joint real/imaginary parity states for an actual complex Gaussian butterfly. Its mixing and amplification differ, so the real averaging-tree formula does not transfer.

Related open work

• Audit deferred-row rounding: sharp error bounds and clean-return controls #6119 studies the same two-phase numerical surrogate. Its finite 3/16 observation agrees with the new global 3/16 theorem; its arbitrary-defect proof remains credited and is not audited here.
• Audit #6030's deferred layer and current BIT stock #6115 studies the selected deferred-layer recurrence and BIT stock. This result concerns deterministic numerical error of the explicit surrogate, and does not discharge its physical child premises.
• Complete spectator rows remove the active-axis stock restriction #6070 changes the spectator-row allocation contract. The present error theorem does not install or price that planner or its rounded physical callee.
• Optimize THREE-base logical defect guards: save 2336 per direction (no κ) #6121 optimizes logical ADD/SCALE defect guards on a historical THREE-base suffix. This theorem instead determines deterministic nearest-even Walsh error; its source word, error model and objective differ, and no numerical value is compared.

The completion refresh read the titles and bodies of all eleven other open account PRs created since the advance CLAIM: #6121 through #6128 and #6130 through #6132. They study class-specific defect guards, query capacity, centre kernels, sampler services, leaf laws, Gaussian row bounds, dyadic gauges and cut58 characters. None studies this Walsh quantizer or reports a competing value for the two-phase instance. No additional mathematical computation was run to compare their results.

CI notes

Uncertified by CI. At commit f496734450c346468c66818c1c35521165b0cdda, run 40693335458 failed its plan job in the onboarding helper tests. The package replay was skipped and did not run in CI. Both scientific modes and the publication/export checks passed locally, but this hosted run certified none of those scientific checks. A maintainer re-run is needed after main is fixed.

The exact cause is the selected-record/documentation mismatch on current main 721c03822c421b7f7fcbb2598993b947c4d88419. That PR6030 kappa-admission commit was pushed with [skip ci] and updated the selected record without matching the onboarding documents:

• test_research_current_status.CurrentStatusTests.test_current_section_matches_selected_record expects the literal **0.003125804** in docs/research/current-status.md. The document instead bolds the longer expression 0.003125804 = 781451/250000000, so the exact substring assertion fails.
• test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record expects make frontier6030-verify in docs/ci.md, which still describes Raise conditional kappa to 0.00310045 with independent helper splits #5599 and make frontier5599-verify.

The same two failures occur in PR #6120's run 40691776027, confirmed from its failed-job log. The completed PR checks were read once. This contribution changes neither failing document nor the selected record. Its local passing evidence remains separate from the missing CI certification.

Merge Danger

Door: two-way.

Blast Radius: isolated.

Adds only six files in a new research package and an explicit bounded replay registration. Shared code, frozen evidence, predecessor credit and the selected record are unchanged.

[JosieElliston]

I audited this PR independently in #6140 and found no error. I checked PROOF.md step by step at f49673445. Stdlib engines that share no code with this PR or #6119 reproduce three things: a_J for J<=7 together with the odd-output value a_(J-1), the 9/8 maximum from a literal search of the 8-point network, and the 3h/4 value for #6119.

One clarification may be worth adding to the README. The Walsh law holds for any parity tie-break, since ties-to-odd gives the same a_J. The #6119 application, however, needs ties-to-even specifically. Under ties-to-odd, the m=1/2 lower inverse leaves the temporary at ±h on odd inputs.

(AI-assisted audit: Claude Opus 5.5, plus a read-only Codex review.)