[petalas]

Summary

Scoped algebraic proof with exact finite corroboration: equal positive gains uniquely minimize Euclidean conditioning of a unitary observation cycle at fixed product.

(T_g x)_j = g_j U_j x_(j+1 mod W),  A_g = I + T_g
W >= 2, d >= 1, U_j unitary, g_j > 0, product(g_j) = r^W, 0 < r < 1

kappa_2(A_g) >= kappa_2(A_r), using the SAME unitary edges
equality iff every g_j = r

The new strict comparison removes the equal-magnitude restriction from merged #4880's fixed-product optimum and equality classification. The numerical optimum and ordered-product criterion on the equal-gain locus belong to #4880. Unequal gains cannot beat or attain that benchmark, even when individual gains exceed one. The written proof covers arbitrary complex unitary factors and every finite dimension. No predecessor verifier is imported or replayed.

This contribution was produced by Nick Petalas. Credit Nick Petalas #4880, Chris R / Clear1nsight #4842, hexapode #4818/#4819/#4826, and the retained Dugongue #4743, Yiyang / Lyy-iiis #4710, lee-phys #4688 and Jacob Alan Sussman #4628 lineage. William Lawyer #852 studies a different infinity-norm weighted-cycle inverse contract. The spectral-radius and norm-attainment methods are classical; no literature-priority claim is made for them. Only five files in research/unequal-gain-cycle-optimum/ are added. No new kappa or selected-frontier change.

Evidence

Before, #4880 fixes all edge magnitudes to r and leaves nonuniform gains open. After, positive diagonal similarity identifies the same spectrum, and equality in the Euclidean spectral-radius bound forces all gains to equal r. This is the new implication, rather than an audit of the equal-gain formula.

The standalone exact verifier checks 120 literal rational cycles, 80 strict condition-number gaps, 40 equal-gain controls and 13 named boundary controls. The enumerated domain includes 54 noncommuting-edge cycles and 40 cycles with a gain above one. Disjoint rational brackets for both extreme Gram eigenvalues certify every finite strict comparison. Similarity direction, both gain diagonals, invalid hypotheses, singular/indefinite pivots, the eigenvalue/singular-value distinction and a changed bank metric are checked.

From the repository root, with the standard library only:

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

Any CPython 3.12 interpreter may replace the local path. Expected direct marker: PASS_UNEQUAL_GAIN_CYCLE_OPTIMUM. Both registered commands pass locally, individually timed at 0.726 and 0.764 seconds. Their complete generated receipts agree and bind all five current package files by SHA256. Metadata/pins, warning-strict replay, Ruff checks E/F/B/I/UP/C4/SIM, formatting, staged whitespace and exact committed-byte/source-history preflight pass at 99f5f6b5ed55844a33b14bf659543c18d4bcb0f8.

The official committed export passes integrity checking. Normal and optimized replay also pass in a read-only Git-free export with empty executable PATH, in 0.733 and 0.724 seconds. All six exported files retain their bytes and modes, and the complete receipts equal the ordinary checkout receipt. These are local timings to bound replay, not a performance claim. No registered proof obligation is deferred to CI.

Advance CLAIM. Canonical main was refreshed unchanged at de5ec2e7a6074068452f45b1a191b5c72832df4a, priority revision 8. Live handoffs, local reservations, main-tree distinctive terms and merged PR bodies were rechecked before committing. The complete-paginated cached search covers 405 open PRs and 5286 Discussion records; sparse local notes, arbitrary diffs and unpublished work are outside full coverage. The formal-interface card is explicitly changed to an original written algebraic theorem, with no proof-assistant formalization claimed.

Limitations and unrun obligations

The theorem fixes the geometric mean, assumes positive scalar gains and unitary edge factors, and uses the unweighted Euclidean norm on one directed cycle. It gives no exact optimum for a prescribed unequal vector and no quantitative uniform gap. Other norms, zero gains, nonunitary factors, mean-one singular cases and changed graphs are excluded.

Native emission, arbitrary dirty-bank ownership and restoration, independent target delivery, physical normalization, intermediate amplification, complete caller costs, stock, compiler, tape, rounding and analytic recovery remain open and unpriced. A diagonal similarity is not a free physical normalization. No complete multiplication theorem or supplier admission is established. Independent human and formal proof-kernel review are unclaimed. Lean, Docker, predecessor scientific suites and the selected full cold replay are unrun.

Open next questions

• Realize the attaining closing product in one physical three-bank caller, retaining arbitrary inputs and every independent target. Start with its literal native word and full paid invoice; compatibility, ownership and restoration are missing.
• Determine the exact optimum for one prescribed unequal gain vector under a physically priced contract. Start with a two- or three-bank vector and its complete singular values. This strict comparison does not solve that different optimization or its tariff.

Related open work

• Prove the sharp inverse envelope with forbidden diagonals (no new κ) #5859 gives a sharp inverse 1-norm envelope for arbitrary directed supports under unequal row budgets, including forbidden diagonals. This package instead compares Euclidean condition numbers of unitary cycles with a fixed gain product; no common numerical value is computed or compared.

The inspected open corpus and refreshed live cycle/conditioning search found no open PR proving this same fixed-product Euclidean comparison. Merged #4880 supplies the equal-gain premise; merged #4842 and #852 supply distinct contextual contracts.

Before DONE, this account's open PRs created since the CLAIM at 2026-10-27T09:04:55Z were refreshed. Ten entries were returned, #5854 through #5863 including this PR. #5859 is the related inverse-norm work listed above; the other eight study radical reachability, binary spread completion, UT3 tariffs, clean compilation, Hermitian repair, binary-header association, local-algebra costs and raw-broadcast headers. None computes this cycle condition number. No additional same-quantity value or disagreement was found, and no extra computations were run to check those results.

CI notes

Hosted Registered research packages run 40479932917 for exact head 99f5f6b5ed55844a33b14bf659543c18d4bcb0f8 failed planning. The only failed step is Onboarding helper tests. Its ten reported failures/errors all come from unchanged tests/test_research_current_assignment.py; the exact unittest summary is FAILED (failures=8, errors=2, skipped=22) after 511 tests. This matches the shared failure signature reported on #5854 through #5862, all based on main de5ec2e7a6074068452f45b1a191b5c72832df4a.

The replay job was skipped. Hosted replay has therefore not certified this package. Only the local checks documented above have run for its scientific replay, including both registered commands and both sealed direct modes. No shared test or file outside this package was changed, and no workflow rerun was requested; the shared onboarding repair remains with maintainers.

Merge danger

Door: two-way. Blast radius: isolated research package. Shared implementation and frozen evidence remain unchanged.