[petalas]

Summary

Scoped written algebraic proof with exact finite corroboration, produced by Nick Petalas. For any finite dyadic helper-cut pairing K with positive odd Smith factors h_1 | ... | h_r:

max { t : X K Y = I_t, aX and bY dyadic } = #{i : h_i divides ab}
least common odd denominator for t channels = product_p p^ceil(v_p(h_t)/2)

The budgets a,b are prescribed positive odd integers and constrain denominator divisibility. Every feasible common denominator is a multiple of the displayed minimum. Factors (1,27,675) have exact thresholds 1,9,45. The proof constructs attaining maps and excludes larger identity blocks by a prime-local primitive-kernel argument, including rectangular maps and prime powers.

This extends Ali Romman's #5615 dyadic-only interface, which is the case a=b=1. Credit its #5599/#5602 helper splitting, #5609 pairing, #5611 completion, Nick Petalas #5115 row preparation and retained native/controller/compiler lineage. Smith normal form, localization and valuation arithmetic are classical. No literature-wide priority for these methods, predecessor audit, new kappa or selected-frontier change is claimed.

Written proof · Package README · Advance claim.

Evidence

Before, the dyadic unit criterion did not quantify the finite two-sided odd-division requirement. The new proof gives its exact capacity and denominator minimum, with explicit boundary maps.

Local CPython 3.12.15, standard library only, passes:

• 65 non-diagonal chart profiles and 2,340 budget cases, with 1,404 prime-obstruction certificates; all supplied-chart identities and boundary coefficients are exact.
• 72 independent matrix-factorization systems modulo 9, enumerating 159,608 right factors and 12,928,248 left rows. A separate determinant-only false-positive control rejects.
• 30 complete scalar prime-power systems at 3 and 5, enumerating 85,470 factor pairs.
• All 128 odd factors through 255, with 8,256 common-denominator candidates; inclusive bit/dimension/budget positives and 23 rejecting controls.
• Warning-strict direct normal and optimized receipts agree exactly. Their local times are 6.97 and 7.13 seconds; both registered commands pass in 14.42 seconds total, below their separate 120-second caps. These timings establish timeout headroom, not a speedup.
• Source compilation, function-size inspection, metadata/pins, whitespace, exact committed bytes/modes and source-history checks pass. The official export and input-coverage checks pass for nine source files. An optimized read-only, Git-free exported replay with an empty PATH gives the identical computed hash-bound receipt and preserves every exported file's bytes and modes. No separate linter/type-checker was installed.

From the worktree root, with the exact interpreter used locally:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/odd-budget-helper-capacity/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -W error -B research/odd-budget-helper-capacity/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/odd-budget-helper-capacity
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/odd-budget-helper-capacity --check-source-history

Any CPython 3.12 interpreter may replace that path. Expected markers are PASS_ODD_BUDGET_HELPER_CAPACITY, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The verifier generates its receipt from computations and source hashes; no receipt is hand-written. The predecessor proof is an authenticated credited input. Its scientific code and census are not imported or replayed.

Limitations and unrun obligations

The theorem is unrestricted in finite dimension. The executable accepts supplied Smith charts with at most six rows/columns, 256-bit dyadic input entries and odd factors/budgets below 2^20. It does not compute Smith charts. SAFETY.md records runtime bounds and managed-runtime deviations.

No actual selected prefix/suffix modules, physical carriers, common charts, legal productive-target completions, arbitrary-input restoration or implementation of odd division is supplied. Basis synthesis, coefficient magnitude, precision, storage, movement and the complete paid/target and outer invoice remain unpriced. This is an algebraic boundary theorem, not a physical split or a complete multiplication theorem. The five selected assumptions retain their status. Selected cold replay, native execution, Lean and Docker were not run. Independent human and formal proof-kernel review are unclaimed.

Open next questions

1. At one named source cut with a failed dyadic pairing, extract its full saturated modules and Smith chart, then determine whether its exact scalar service can implement the minimum odd denominators. Price both boundary maps, their carriers and restoration before claiming productive gain.
2. Determine whether a minimum-denominator attainer also fits a stated norm/growth budget, beginning with factors (1,27,675). Exact divisibility alone does not control magnitude.

Related open work

• Propose conditional κ 0.003645372367832 from two funded cuts #7244 changes actual P104 helper targets with two funded cuts and reports a conditional kappa. This abstract theorem changes the allowed scalar denominators and computes none of that word's invoice or claimed kappa.

The targeted live searches found no other open PR on prescribed odd-denominator helper capacity. Searches covered current main README formulas, merged Smith/denominator work, open PR bodies, recent account publications and live claims. The cached card search covered 558 open PRs and 7,226 discussion records; focused live queries supplemented it. Arbitrary historical diffs and unpublished work are not exhaustive coverage.

The closeout own-open-PR check read the nine other contributions created since the CLAIM: #7261, #7262, #7263, #7264, #7265, #7266, #7267, #7268 and #7270. They study contraction weights, symplectic completion, Stein norms, terminal rank, Boolean insertion cost, a P120 column space, convolution precision, residue control and algebra automorphisms. None computes this prescribed odd-denominator helper-capacity quantity. No common numerical result is compared or audited.

CI notes

CI did not certify this package. At unchanged head 7de4bd6bd4a6f7fae59b90c941ae5a3b9b3e652d, run 43030268355 failed in plan, Onboarding helper tests. I read the completed checks once and inspected the actual job log. It reports exactly one failure among 760 tests, with 24 skipped:

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record

The AssertionError says that `make frontier7000-verify` was not found in the CI verification document. The package replay job was skipped because plan failed.

The mismatch is already present at canonical base e54523517d0f03e58352e647c403a5af3e525a57: certificates/selected-result.json names make frontier7000-verify, while docs/ci-verification.md describes #6030 and make frontier6030-verify. I inspected both base files and confirmed that both files and the failing test are byte-identical at this PR head. This failure occurs before the package replay and supplies no hosted evidence about the package.

I also inspected all five comparison plan-job logs below. Each shows the same sole failing test, missing-command assertion, 760 tests, one failure and 24 skips:

• Run 42905094314, job 130140894651.
• Run 42903313790, job 130134720799.
• Run 43029427643, job 130552352656.
• Run 43029993808, job 130554266928.
• Run 43030135179, job 130554753063.

The local evidence listed above remains the package's execution evidence. No fix, retrigger commit, push or workflow rerun was made for this repository-level failure.

Merge Danger

Door: two-way. Only eight files under research/odd-budget-helper-capacity are added.

Blast Radius: package. Shared code, pinned predecessor evidence and selected-frontier records are unchanged.