[JoyboyBrian]

Result

PR4435 leaves a rigorous upper bound open for the broad Gaussian inverse at a phase-cell boundary. This proves an explicit finite-cell bound using PR4193's Toeplitz inverse entry theorem and the retained exact diagonal gauges:

|K^-1[i,j]| <= A_N exp(-gamma theta (i-j)^2),
||K^-1||_2 <= A_N min(N, 1+sqrt(pi/(gamma theta))),
A_N = alpha exp(pi/(4 alpha^2 rho^2)) / P_(N-1)^3,
gamma = pi/(alpha^2 rho), rho=1+theta,
P_m = product_(r=1)^m (1-exp(-2 gamma r)).

The theorem covers every finite contiguous index interval with beta_k=theta*k-rho*c in the closed phase interval [-1/2,1/2], for alpha,theta>0. Keeping the inverse gauges together yields Gaussian off-diagonal decay; separately maximizing their norms discards that information. The theta dependence is rigorously theta^(-1/2) for fixed alpha, with the full alpha-dependent prefactor retained.

At the original alpha=3, theta=1/400, N=400 fixture, the certified norm upper bound is below 8080.184, while the old displayed ceiling is above 713520770.18. The latter is a lower bound on the old ceiling, not on the actual norm. No SVD value or formal full-line boundary condition is used to prove the new bound.

Evidence and scope

• Written finite q-LDU dependency, exact coupled-gauge identity, both index-order inequalities and Schur bound.
• Exact retained source-exponent AST checked on 2,395 pairs; all 160,000 phase inequalities in the original cell.
• 36 directed enclosures of actual finite inverse entries, and 12 independent rational eliminations with 255 entry checks.
• Closed endpoints, shifted origins, inadmissible phases, wrong inverse-gauge sign, missing normalization and missing-pivot controls.
• Registered normal/optimized Python 3.12 checks, committed bytes/source history, official 16-file export integrity and actual Git-free normal/optimized replay pass. Source corruption rejects, exact restoration succeeds and all exported bytes remain unchanged.

This is a written scoped analytic theorem with finite supporting evidence. It does not prove PR4435's sharp asymptotic constant, a matching lower bound, a growing-alpha parameter-family corollary, a new kappa, or a complete physical multiplier. Multiple phase cells, holes, coefficient preparation, rounded arithmetic, circuit prefixes, native calls and outer costs remain separate obligations. The source AST check is expression-level exact refinement, not execution of the original floating-point inversion program. Decimal correctly-rounded exp/sqrt with outward widening is an explicit arithmetic dependency.

Credit Ryanson Jonathan's PR4408/PR4435, Jose Manuel/jmbarrancoml's PR4193, Vlado Vojdanovski and the retained PR1673 Gaussian/q-LDU lineage, Cinna Davis's caller, classical finite q-binomial/Gaussian identities and all source notices. Prepared by JoyboyBrian with substantial OpenAI Codex assistance. No independent human or proof-kernel acceptance is claimed; selected frontier and frozen sources are unchanged.

Open next questions: prove a matching lower estimate/sharp constant, then bind an actual complete Gaussian caller and all preparation/physical costs before changing a numerical guard.

Canonical claim: #1166 (comment)

Submission checkpoint: canonical main81e9bd69e4a5b00ba9c713a1620e36d576d5522e and governing guidance unchanged. Complete09:25–09:26UTC coverage:320openPRs,4233Discussionrecords,6824localHEADnotes. Claims are advisory; unpublished work is outside that coverage.