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Prove the sharp graph-support inverse norm envelope - #5255
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Mister-Meeseeks merged 1 commit intoOct 23, 2026
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Proves the exact maximum of
||(I-E)^-1||_1for a finite directed support with diagonal entries allowed and absolute row sums at most rho, where 0<rho<1. The maximum is the largest discounted distance sum divided by 1-rho; a matrix that routes each row toward one root attains it. The proof and exact checker give the full domain and construction.For banded lines, this proves the dimension-independent 1-norm ceiling is sharp and that the general operator 2-norm gap exponent cannot improve. This is a scoped conditioning result. It does not establish a Gaussian-specific optimum or change the complete kappa calculation or selected frontier.
Base: ff3cc58e370d0f678e529246f8c73e8094e18fcf. Source pins are in SOURCES.json. Cinna Davis (@1337isnot1337) supplies the original conditioning bound in PR #4100 and the pinned note. James Reed (@jwr404) supplies the periodic-section extension in PR #4282. Classical Neumann and graph-distance methods retain their credit; no general literature-priority claim is made.
Validation with CPython 3.12.13 and the standard library:
Both registered normal and optimized checks passed: 402 attaining graph/root cases, 120 signed matrices, 24 banded cases and six controls. Expected marker:
PASS_SHARP_BANDED_RESOLVENT. Committed-byte verification passed for seven package/input files. The full-history preflight is unavailable in this shallow checkout; direct ancestry of the pinned base was checked. Both hosted jobs now pass at headd63d8848442acc0bdc7b62578e12726c530af3fa: planning and the focused package replay. The run completed with conclusionsuccess; this is not a supplier or selected-cold-replay claim.The Gaussian row-gap hypothesis, rounding, physical tape cost, complete supplier and outer recovery remain outside scope. Predecessor scientific suites, Lean, Docker and the selected cold replay were not run. The universal conclusion rests on the written proof, with finite exact corroboration. No independent human or proof-kernel review is claimed.
Coordination claim. Prepared by Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.