[petalas]

Summary

Scoped written proof with exact finite corroboration. In Jeremy Diviney's #836 finite-group formal register model, the sharp paid increment satisfies |Delta Phi| <= c (K+1/K)/2, where K is the spectral condition number of the faithful regular prefix. A two-role C2 family attains equality for every K>=1.

For a complete rank-nine endpoint with eight unit-price calls:

- Necessary paid-prefix spectral condition number: K >= 9/8
+ Necessary paid-prefix spectral condition number: K >= (9+sqrt(17))/8 > 1.64

This strengthens a search obstruction; it does not construct an eight-call endpoint or improve kappa. The proof and exact checker were produced by Nick Petalas. Credit Jeremy Diviney #836 for the signed potential, transport and endpoint laws; Nir Ailon for the retained forward/inverse-potential context; eumemic #288, Swapnil Jain, littleBro #680/#705 and the retained register/frame lineage. The Kantorovich inequality is classical. The new contribution is its sharp paid-increment application and attaining formal family, without a general literature-priority claim.

Evidence

Before: #836 bounds a local coefficient-row/inverse-column product by K and gives the necessary 9/8 spectral threshold. After: the Kantorovich bound replaces K by (K+1/K)/2, and the explicit family proves local sharpness. The predecessor is cited and source-bound; no predecessor executable, example or count was replayed.

At head ed57e9575550cd0203d02c0b025f796b552c7d9c, local CPython 3.12.15 normal and optimized checks pass:

• All 80 distinct rational equality parameters v=p/q, 1<=q<=p<=16, with full regular inverses, separate Fourier-block inverses, exact spectral bases and literal paid increments.
• 64 prescribed-spectrum matrix fixtures in dimensions 1..8 and all 288 row inequalities; nine invalid-input/inverse controls and four semantic comparisons.
• Three disposable optimized mutations reject a loose envelope, an omitted paid call and corrupted cited-source bytes.
• Registered replay, exact committed bytes and source-history preflight, official export integrity and both normal/optimized read-only Git-free export replays pass. All eight exported files, including the export manifest, retain their hashes.

Each registered command took about 0.44 seconds in one local timing, below its 60-second cap. These are replay-budget observations, not runtime speedup claims. The direct verifier emits generated SHA256-bound receipts and writes no files.

Exact reproduction from the repository root:

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

Other machines may substitute their CPython 3.12 executable. Standard library only. Expected markers are PASS_SHARP_RANK_CONDITIONING, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Canonical base and final upstream refresh: 19fdb086ea96af9a9f87936a2b888e10af4ed810, priority revision 8. The original bit-new-register-model card is explicitly changed to this complex conditioning side result. Advance CLAIM. The selected source-binding positive control passed; no selected scientific replay is claimed.

Limitations and unrun obligations

The theorem covers finite groups, complex coefficients, arbitrary scalar mixing and full invertible independent-input words. Local sharpness is proved for every real K>=1; the executable fixtures use real rational parameters. It does not prove complete-endpoint attainability, an infinite-group result, a clean-ancilla result or a bound for a nonfaithful concrete frame representation. Physical/native realization, a complete paid supplier invoice, compiler, stock, precision/tape and analytic recovery remain separate obligations. No selected cold replay, Lean, Docker, independent human acceptance or formal proof-kernel review is claimed. All five selected framework assumptions retain their status.

Related open work

The completion refresh read the titles and bodies of this account's eleven other open PRs created since the advance CLAIM at 2026-10-27T15:17:13Z: #5944, #5945, #5946, #5947, #5948, #5949, #5950, #5951, #5953, #5954 and #5956. None computes #836's signed-rank potential increment or its faithful-regular paid-prefix conditioning threshold. No additional computations were run to check other contributions, and no shared computed value was identified for comparison.

• Prove sharp conditioned matching-layer interpolation (no new κ) #5940 concerns norm interpolation inside conditioned matching layers; it does not compute this potential increment or threshold.
• Prove a sharp ten-unit floor for the nonlinear six-frame seed #5944 supplies a nonlinear six-frame rank-cut obstruction; its charged circuit edges differ from the complex formal-word conditioning here.
• Prove a fourth dirty role keeps the S3 xyx double at six calls (no new κ) #5947 computes a four-role S3 call minimum over F2; this proof permits complex scalar mixers and does not compute that binary minimum.
• Prove the unique optimal Gaussian chirp origin (no new κ) #5949 minimizes a Gaussian chirp's maximum absolute row sum over its origin; this proof bounds Euclidean conditioning of a faithful regular register prefix. The matrices, norms and objectives differ.

The broader published search found no other open submission on the exact quantity. Published searches exclude unpublished work and do not prove novelty.

Open next questions

Can a complete rank-nine endpoint with eight unit-price calls attain the necessary conditioning floor? Start with one fixed formal word and track every signed gain together with the exact full endpoint equations. The local equality family does not supply that endpoint, which remains the blocker to an attainment claim.

CI notes

Both hosted checks passed at exact head ed57e9575550cd0203d02c0b025f796b552c7d9c in run 40530945671: plan in 3m49s and replay (research/sharp-rank-conditioning) in 1m25s. The completed checks were read once. Run metadata confirms completed status, success and the same head as the PR.

Hosted replay executes the registered normal and optimized package commands. It certifies those bounded automated checks; independent human and formal proof-kernel acceptance of the written theorem remain unclaimed. All local scientific, source-history, committed-byte and export checks listed above also passed. No CI failure remains.

Merge Danger

Door: two-way. One isolated research package.

Blast Radius: local. No shared implementation, predecessor package or selected-frontier file changes.