[petalas]

Summary

A Fourier integrality argument strengthens the matched P112 commuting-phase obstruction of Jose Manuel / PR2070.

one matched bank:       at least 140 phase events, previously 139
four independent banks: at least 556 phase events, previously 555

Claim type: scoped mathematical proof with exact finite corroboration. The new step classifies odd nonzero-polar multisets at the half-weight character threshold. Nonnegative Fourier coefficients force a subspace indicator, so k+1 must be a power of two. This excludes both cited average-bound equality cases.

The contribution was produced by Nick Petalas. PR2070's causal inequality, common Pauli-trivial sector, P112 spans and defect-rank bounds are cited premises, without scientific replay or audit. Credit Jose Manuel / jmbarrancoml, eumemic / PR288, Ali Romman / PR468/1862/1969, Rohan Arun / PR1931, Douglas Colkitt and the retained source lineage. Finite-group Fourier inversion and positivity are classical; no global priority for the abstract lemma is claimed.

Advance CLAIM. The six-file package includes the written proof, exact source pins and a standalone checker. Canonical main is pinned at 7c49fe3, priority revision 8, selected PR6030. Scope and overlap limits are recorded in TASK.json.

Evidence

• Before: the cited averaged argument gives 139 and 555. After: the new written proof excludes equality at both counts, giving 140 and 556. No attaining circuit is claimed.
• The new checker exhausts 19,727 multisets in its explicitly bounded domains, with exactly 70 rigid cases. Direct character evaluation, a separate butterfly transform, inversion, multiset counts and Gaussian-binomial counts agree. Thirteen invalid domains are rejected; five semantic control groups pass.
• Warning-strict normal and optimized CPython 3.12.15 direct checks pass in 0.22s and 0.23s locally on macOS arm64, with identical generated receipts. Both registered commands pass and remain far below their 120-second limits. These are check durations, not performance claims.
• Committed bytes, executable modes and declared source ancestry pass for all six files at head 18785eb. Official export/integrity checks pass. Both direct modes also pass from a read-only Git-free copy with an empty executable search path and receipts identical to the checkout.

From the repository root:

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

Expected markers include PASS_MATCHED_PHASE_INTEGRALITY, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Another CPython 3.12 path may be substituted. Receipts are generated on stdout and bind all package files; no hand-written receipt is committed.

Limitations

The theorem retains the cited fixed role-linear, nonzero commuting-phase and common-character hypotheses. It covers one direct matched bank and four independent direct matched banks. It does not strengthen the normalized nonuniform coherent-cover bound. Noncommuting payload adapters, nonlinear or adaptive programs, changed endpoints and new multirole child interfaces remain outside scope.

Finite cases corroborate the new proof; they do not enumerate 139-event or 555-event circuits or verify the predecessor premises. Native realization, physical tariff, routing, ownership/erasure, restoration implementation, precision/tape, stock, compiler and complete signed supplier/outer invoices remain unprovided. No new kappa, supplier admission or selected-frontier change is claimed. All five framework assumptions remain. Selected scientific cold replay, Lean, Docker, independent human review and formal proof-kernel review are unrun or unclaimed.

Open next questions

• Can the 140-event or 556-event lower bound be attained? Start with a polar multiset satisfying the necessary character inequalities, then prove a causal role-linear realization. No such construction or complete physical bill is supplied.
• Can a priced noncommuting payload adapter improve the complete matched endpoint? Specify its tensor-valid action between two phase events. The predecessor already poses this opening; an implementation and price remain missing.

Related open work

The final refresh checked all seven other open contributions created since the advance CLAIM. None studies this matched P112 commuting-phase event count. Exact open-PR searches for the matched-query endpoint and PR2070 also returned no competing open submission. The following adjacent work studies different quantities; no additional computation compares its numbers.

• PR6443 determines correction rank for a unitary reciprocal Clifford encoder, with a different endpoint and encoder contract.
• PR6585 determines isolation capacity under shared phase charts; it does not count matched-query phase events.

CI notes

Both hosted checks passed at exact head 18785ebf02a5976a8c16a747d0095ca4766ca0ae in run 41624919523: plan in 4m26s and replay (research/matched-phase-integrality) in 1m30s. The completed checks were read once; separate run metadata confirms completed success and the matching head.

Hosted replay certifies execution of the registered normal/optimized finite checks and source metadata at this commit. It does not establish independent acceptance of the all-dimension written proof or discharge the inherited physical, supplier or all-size obligations. All local checks listed above also passed. Independent human and formal proof-kernel review remain unclaimed.

Merge Danger

Door: two-way. Blast radius: isolated. Six new files in one opt-in research package. Shared code, predecessor sources and the selected frontier are unchanged.