[petalas]

Summary

A four-layer quadratic improves exact Walsh-Hadamard entry agreement over every parity interpolant on three consecutive overlap weights, at the same exact rational rank. Claim type: scoped written proof with exact finite corroboration.

n = 16, exact rank = 137 in both constructions
best consecutive triple:  2,633,821,596 agreeing entries
new four-layer choice:    2,713,419,648 agreeing entries
gain:                        79,598,052 of 4,294,967,296 entries

For every n=4m with integer m>=4, use P_m(t)=(-1)^m (2(t-m)^2-5)/3, where t is the integer overlap. Its equality set is m-2,m-1,m+1,m+2. The proof gives exact rank 1+n+binom(n,2), the unique best consecutive triple, and a positive rational gain formula valid throughout that range. This compares two explicit polynomial families, not all low-rank matrices.

This contribution was produced by Nick Petalas. It takes the equal-degree polynomial-comparison question in Jacob Alan Sussman / jacobalansussman's merged #5817, Open next question 1. Credit Alman and Williams for classical consecutive-layer interpolation and monomial-rank background. The new step is the four-layer construction and its exact comparison; no predecessor result is audited. The original decoder card is explicitly changed to this low-rank side result. No new kappa or selected-frontier change.

Evidence

Before, the degree-two consecutive-layer family has best agreement 2,633,821,596 at n=16. After, the explicit rational factorization gives 2,713,419,648 at the same rank 137. research/four-layer-quadratic-parity/PROOF.md supplies the quantified proof; verify.py supplies a read-only exact check.

Local CPython 3.12.15 normal and optimized checks pass 125 dimensions n=16,20,...,512; ten exact count cases including two dense enumerations; 10,920 factorization entries; eight feature-minor cases; and 18 mutation, boundary and scope controls. The n=16 count uses exact layer multiplicities and an independent enumeration by row weight, not a dense replay of four billion entries.

From the worktree root:

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

Any CPython 3.12 executable can replace that interpreter path. No external packages are required. Expected markers are PASS_FOUR_LAYER_QUADRATIC_PARITY, PASS_REGISTERED_RESEARCH_PACKAGE, and PASS_RESEARCH_COMMIT_BYTES.

Both registered commands also ran directly with identical generated package-hash receipts, taking 0.086 seconds each. The complete registered invocation took 0.402 seconds. These are verification durations. Metadata/pins and committed-byte/source-history checks pass. The official committed export passes integrity and declared-input coverage; both warning-strict modes pass in a read-only Git-free export with PATH=/nonexistent, reproduce the same receipts, and leave all seven exported files unchanged.

Limitations

The claim is over the rationals, reals and complexes. It is not a global quadratic or rank-137 optimum. At n=40, a rank-one constant matrix agrees on more entries than this four-layer construction; the verifier explicitly checks that limitation. No per-row error control, operator norm guarantee, numerical stability, physical native realization, correction tariff or complete recursive invoice is supplied. No improved asymptotic non-rigidity exponent or multiplication theorem follows. The all-dimension statement rests on the written proof, not finite testing alone.

Independent human and formal proof-kernel review are unclaimed. Lean and the selected supplier's cold replay were not run; no selected framework assumption is discharged. No local verification obligation was deferred solely to CI.

Open next questions

• What is the best quadratic exact agreement, including constants? Classify the rational level sets at 1 and -1, then optimize their binomial mass. The present result does not supply a global extremal classification.
• Can a higher-degree nonconsecutive equality set improve a complete Walsh-Hadamard construction after row corrections and coefficient preparation? Start with one explicit factorization and a complete bill. The physical consumer remains missing.

Related open work

• #5982 proves spatial cut ranks and one-head travel floors for exact Walsh transforms. This contribution compares entry agreement of low-rank surrogate matrices; no shared numerical invariant is compared.
• #5945 lowers a signed Hadamard connector with dirty scratch. Its exact query-call tariff differs from the rational matrix agreement count here; no same-instance number is compared.
• #956 supplies an exact heterogeneous Gaussian/Walsh provider under its own RAM contract. The present polynomial surrogate provides no physical provider or tariff.

The merged #5817 is the question source. Its separate #5917 follow-up concerns acyclic Boolean-table tensor degree, not this polynomial model. Bounded main, merged/open PR and discussion searches found no matching four-layer comparison; those searches do not prove historical novelty.

The completion refresh inspected this account's eleven other open PRs created since the CLAIM at 2026-10-27T21:04:27Z: #5997 through #5999 and #6001 through #6008. None computes quadratic Walsh-Hadamard entry agreement or a competing same-instance number. No additional computation was run to compare them.

• #6002 gives exact composite-axis chirp factorizations and phase-repair counts. Its rank and invertibility statements concern a different matrix family, rather than Walsh-Hadamard agreement; no numerical agreement or disagreement is asserted.

CI notes

Both hosted checks passed on exact head dcd429d1ca59f0a529d99be4f01dfdb511dfd2e2 in run 40584553082: plan in 4m36s and replay (research/four-layer-quadratic-parity) in 1m20s. The check list was read once after completion; run metadata confirms the same head and successful conclusion. The replay runs both registered normal and optimized commands. All local checks listed above also passed. These automated checks certify the scoped package replay, not independent human or formal proof-kernel acceptance.

Merge danger

Door: two-way. Blast radius: isolated research. Six new package files; shared code, frozen predecessors and the selected frontier are unchanged.