[petalas]

Summary

Prove the exact optimal-second-callee count for the unchanged ordinary wrapper when P=diag(J_2(lambda_1),J_2(lambda_2)) and both eigenvalues satisfy lambda_i^2=1, over every finite field.

one resonant J2 block:       1 optimum       [credited #5556]
two coupled resonant blocks: q^2 optima      [odd characteristic, new]
                            1 optimum       [characteristic two]

Claim type: scoped written proof with exact finite corroboration. The complete parameterization is Q=P+U D S V, S^2=0, with U=[e0 e2], V=[e1^T;e3^T] and D=diag(2lambda_1,2lambda_2). All odd-characteristic optima have determinant one; exactly q^2-1 have rank-one perturbations. The optimum error rank is two, giving agreement on q^6 of q^8 inputs. This does not claim exact interchange in odd characteristic, a native cost advantage or new kappa.

Only six files in research/coupled-resonant-jordan-pair/ are added. The several-block question and fixed-wrapper reduction come from Nicholas Stone's #5556. Nick Petalas's #5228 supplies the general equality characterization, and #6406 treats the different regular J3 first map. Jerry Chen / CubeJerry's #2950 supplies the wrapper optimum and attainer. Predecessor results are cited, not re-audited. Classical block elimination and finite-field counting retain their usual attribution. This contribution was produced by Nick Petalas. Apache-2.0.

Evidence

Before, each resonant block was individually rigid and the several-block count was open. After, the proof forces every optimal perturbation into its image/kernel support and reduces equality to a two-dimensional square-zero parameter. Its determinant filter accepts every parameter, even with mixed signs.

Local CPython 3.12.15 warning-strict normal and optimized verification passes, with identical generated package-hash receipts. The finite checks enumerate 27 field/sign contexts over orders 2,3,4,5,7,8,9,11,13; all 215,848 supported perturbations; all 1,819 accepted wrappers; and every binary four-by-four second map, including 20,160 invertible maps and exactly one optimum. Literal six-operation execution checks 175,832 original input basis columns. Fifteen rejection controls and one coupled positive control pass. Extension fields use exact polynomial coordinates.

Each registered command was timed locally, at 5.39 and 5.46 seconds. The complete registered launcher passed in 11.17 seconds. These are verification durations, not performance-improvement claims. AST compilation, whitespace, metadata/source pins, committed bytes, source-history ancestry and executable modes pass. No registered check is deferred to CI alone.

From the repository root, standard library only:

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

Expected markers are PASS_COUPLED_RESONANT_JORDAN_PAIR, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Another CPython 3.12 path may replace the prefix. Direct verification writes no files and imports no predecessor executable. SOURCES.json pins predecessor prose and source context as historical provenance.

Limitations and unrun obligations

The formal-interface-lemma card is explicitly changed to an algebraic counting problem. This proves no selected assumption, larger block family, native implementation, physical cost advantage or complete multiplication theorem. Restoration, movement, precision, stock, compiler and complete supplier/outer costs remain unpriced. No predecessor scientific suite, selected cold replay, Lean or Docker run was performed. Independent human and formal proof-kernel review are unclaimed. The selected #7000 frontier and frozen predecessors are unchanged.

Open next questions

• Count a pair with at least one eigenvalue squared different from one. Start with distinct regular eigenvalues over F5 and derive its quotient and determinant filter. The present singular-support reduction does not classify it.
• Price an explicit coupled map in a physically specified ordinary caller, including the actual correctness domain and restoration costs. Optimal agreement alone gives no implementation saving.

Related open work

No open PR studying this coupled resonant-pair count or optimal-second-callee quantity was found in the refreshed title/body and targeted Jordan searches. Before DONE, this account's open PRs created since the CLAIM at 2026-11-11T10:03:09Z were checked again: #7191, #7192, #7193, #7194, #7195, #7197, #7198 and #7199, besides this PR. They concern polynomial gap insertions, rounded inverse injectivity, address-twist residues, three-shear COPY floors, cyclotomic precision, compact switches and query-cache stock. None studies this coupled Jordan-pair quantity. References to this PR in their related-work sections do not change their mathematical object. No additional scientific computations or numerical agreement/disagreement claims were made. The relevant #5556, #5228 and #6406 are merged predecessors.

CI notes

CI did not certify this package. At unchanged head 29881951454fdbc1bae3d4142db749d5d726cf14, run 43001498725, plan job 130458470178, failed in Onboarding helper tests. I read the checks once and confirmed the cause directly in the failed 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. Direct inspection of canonical base e54523517d0f03e58352e647c403a5af3e525a57 confirms that certificates/selected-result.json names make frontier7000-verify, while the current section of docs/ci-verification.md still describes #6030 and make frontier6030-verify. The selected record, CI document and failing test are unchanged by this PR; the complete diff contains only the six package files.

The dependent package replay job 130460779558 was skipped because plan failed. No hosted mathematical replay ran. The evidence for this package remains the local normal/optimized, registered, literal-wrapper, metadata/pin, committed-byte/source-history and mode checks reported above.

I also read the failed-step logs for every run below and confirmed the same exact sole failing test, missing-command AssertionError, 760-test total, one failure and 24 skips:

• Other contributor: run 42905094314.
• Other contributor: run 42903313790.
• Prove the sharp rank-two bound for five address-twist contributions (no κ) #7193: run 43000937833.
• Prove the paired-root floor for three-shear cycles off tsu = ±1 (no κ) #7194: run 43001222481.
• Sharpen the composite cyclotomic precision exponent (no κ) #7195: run 43001319803.
• Prove exact minimum persistent stock for query streams (no κ) #7198: run 43001765438.

No fix, retrigger commit, push or job rerun was made for this repository-level failure. Maintainers will repair main. Independent human and formal proof-kernel acceptance of the written theorem remain unclaimed.

Source and coordination

Base and refreshed canonical main are e54523517d0f03e58352e647c403a5af3e525a57, priority revision 8. Advance CLAIM. The delivered scope matches that claim. Current main README searches, merged keyword searches, live open work, recent account work, handoffs and reservations were checked before publication. Unpublished work and arbitrary historical diffs remain coverage limits.

Merge danger

Door: two-way. Blast radius: isolated. One new opt-in research package; no shared code, selected record or predecessor credit changes.