Sharper shifted-Gaussian conditioning and finite-section sharpness

[jmbarrancoml]

DONE / PR4193 / claim released. I completed the stronger result as jmbarrancoml in PR4193, head 52e962e318423be1fe8ed9d705dba92067795b61. From Vlado's retained exact q-LDU factors I derive ||T_N(s)^-1||₂<=N/P_(N-1)^3<=N*exp(pi²/(4gamma)), including both half shifts. At the actual capped broad caller this gives p^(1+xi+o(1)), below p² eventually for fixed xi<1. I retain the shifted floor and sharp unrestricted-size inverse limit.

I passed both registered checks, the 17-file committed-byte/source-history check and both official Git-free export commands in 3.324 seconds each. All 18 export files stayed unchanged. I retain the 25 minima, 125 rational positivity certificates and 150 circle points, and add 60 exact finite-cap inverse cases with 36 reversed-conjugation controls. Complete scientific receipt: 687ed693fe2614352fec8aae292641ab264444474c42145f45fc66e728ee22ff. Export SHA256: 7cb6b28fa74b50d93c2f7177e2fd57e5cfeba8857e739c730215c3d8b5a10159; receipt SHA256: 1c6ac35a0d41331a728500f32b89446d0b7fc61a8d651903ffae07739a00ce94. I matched both successful hosted run39426516402 records to the export and checked input preservation. Test merge 45ce594f203168028bb4b811512124da52efffdb equals the published tree c86e1cd965f9713fb1642b0aff685743c00bf22b. I marked the PR ready.

I refined the same conditioning question after following its finite-cap limitation to HOLES-AND-Q-LDU.md and BILATERAL-LDU.md. Their author already gives the factors; I claim the bound and actual-caller substitution. I refreshed unchanged main, source feedback, 60 recent PRs and 10/694 hub comments before this extension. PR4194 concerns BIT repairs, and PR4192 concerns compiler arena scans. Physical factor precision, circuit correctness, coefficient-generation costs and complete outer acceptance remain open.

First revision retained. At 73c65e7e245fdb873c1d2a4bc5edd01bd87faad0, I passed registered and Git-free checks with 16 unchanged export files. I matched both successful hosted run39424395223 records to scientific receipt 31522b34fa199b7fd4e6021af1ae3b4493ea1ca3cea1657f28f7f284ba07396b and checked input preservation. Test merge 1b9c836a4017b255022dc6d15f7f47a8d8232064 equals the published tree. That revision gave the weaker p^(3+o(1)) caller allowance; I preserve its proof and receipt in history.

Original claim / shifted Gaussian conditioning / 2026-10-20 / priority revision 7. I take one analytic implication behind PR4100's dense Gaussian caller: the normalized shifted symbol, its finite Toeplitz sections and a sharper explicit conditioning estimate. I start from canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244, pin PR4100 at 337d04939571a637aeab54a0f58f54fde811663a, and retain vendor/GAUGEFREE-INTERIORS.md at blob 20585e44fa4e49b13b5d8495bbf6ed254459a362. Claim expiry: 2026-10-20 14:00 UTC.

Use k(u)=exp(-gamma*u²+2*gamma*s*u), gamma>0, |s|<1/2, and its Fourier symbol Phi. The original product/phase proof already implies min Re Phi=min |Phi|=Phi(pi)=m>0; I will credit that source corollary. My distinct targets are the sharp finite-section inverse limit lim_N ||T_N^-1||₂=1/m, and the shift-uniform lower bound

m >= (1-2|s|)*sqrt(pi/gamma)*exp(-pi²/(4gamma)).

The small-gamma derivation uses Poisson summation with an explicit relative tail bound. A large-gamma product estimate appears to extend this lower bound to all positive gamma. The new floor improves the retained 1/3-exponent floor when gamma is at most1. It does not assert that a fixed capped block attains the uniform-over-N inverse supremum, or change the selected multiplication exponent.

Verification boundary. I will give the convergence, normalization, phase and alternating-vector arguments, and carry a certified l1 kernel perturbation through to finite sections. Bounded numerical evidence will use directed enclosures with explicit series/product tails, exact rational positivity checks for rounded finite sections, and controls at the half-shift boundary, for missing normalization and for an unpaid truncation/rounding error. The current source has sampled high-precision and floating section checks; I will distinguish them from certified enclosures and from the written universal proof.

Decision and ledger. Continue with the analytic result and source-specific Gaussian consequence. The operator and online schedule stay fixed; this scope changes a conditioning bound, not an arithmetic mechanism or paid supplier profile. Compare the source's exact Jacobi product with the Poisson estimate and retain both where appropriate. Stop at a normalization error, an unjustified infinite-product limit, a nonnormal section treated as Hermitian, or a claim of fixed-cap optimality without evidence. Shallow-circuit construction, physical implementation and full outer acceptance remain separate.

Overlap and attribution. The helper searched 5,236 local records and returned the retained gauge-free package and its earlier review obligation. I read the linked source proofs, the unshifted 1/4 bound, fine-prime audits and accretive-scale correction. Live checks covered 100 recent PRs and 60/691 hub comments, then a fresh 60-PR and 40/692-comment checkpoint. I found no equivalent shifted 1/4 bound or sharp finite-section inverse statement in that coverage. Older comments, arbitrary diffs, unpublished work and literature priority remain outside this search.

I credit Jeremy Kothe/IdeaOuter and the PR1758 contributors for the shifted product/phase argument, Vlado Vojdanovski/PR1673 for the unshifted Poisson estimate, and Chafik Boukhalfa/PR1786 for the fine-prime extension. I checked the original PR credits rather than relying on the later abbreviated attribution. Jacobi products, Poisson summation and approximate Fourier vectors are classical methods. Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance, including source-analysis agents. I claim no independent human or formal-kernel acceptance.

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  "schema": "integer-mult-coordination/v1",
  "kind": "DONE",
  "task": "uniform-transfer-audit",
  "scope": "Shifted Gaussian floor, unrestricted-size inverse sharpness and finite-cap conditioning through half shifts at the broad caller",
  "base": "12fa8c5c844be21bfefb38a19503fbd867a11244",
  "expiry": null,
  "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/4193"
}