Source, priority and significance assessment

Henry Zweiman. September 15, 2026. P40, developed as Q46.

Decision

Proceed with this manuscript as a substantial research preprint. The originating assistant found no unresolved proof obligation in the stated theorem after the reviews recorded in REVIEW.md. The bounded current-literature comparison located an explicit predecessor frontier and no prior statement supplying this theorem. This is a reasoned internal admission under the revised research standard, contingent on verified publication; it is not independent certification of correctness, priority, citation potential or suitability for a particular journal.

The contribution is the sharp first-order logarithmic Sobolev asymptotic in every finite dimension for convex potentials with a smooth bounded domain of minima and nondegenerate quadratic normal growth, under the stated Laplacian growth condition. The selection formula includes the entire nonlinear extremizer set and the first Neumann eigensphere at equality. The smooth example, profile invariance and interval check are parts of this one paper.

Closest predecessor and precise difference

Aziz Ben Nejma, arXiv:2604.01785v1 proves the higher-dimensional Poincare asymptotic and the one-dimensional logarithmic Sobolev asymptotic. Remark 1.11 identifies nonlinear extremizers as an obstruction in higher dimensions. Its Theorem 3.12 already gives the spectral boundary coefficient. The version history was checked on September 15, 2026 and listed only v1, submitted April 2, 2026.

The comparison included the introduction, preliminaries, complete Section 4, and Theorem 3.12 with its proof; selected earlier Section 3 dependencies were also examined. P40 credits the formal boundary expansion, spectral result and one-dimensional strategy. The added work is the uniform exterior entropy estimate, including uniformity after division by vanishing energy, and the resulting nonlinear/spectral selection theorem. No claim that every auxiliary proof in the predecessor was independently verified is made.

Other relevant primary sources

• Chewi--Stromme, arXiv:2411.11415v1: unique-minimizer ballistic scaling and Gaussian comparison. The current version page listed only v1, November 18, 2024. The introduction, main hypotheses and Gaussian-comparison calculations were examined. P40's domain of minima and nonzero limiting constant require a different asymptotic. Its Gaussian inequality is classical, and the exterior comparison is derived in full.
• Bormann--von Renesse--Wang, arXiv:2401.00206v2: introduction and model, Section 4 Propositions 4.1--4.2 with proofs and endpoint discussion, and Section 5 Proposition 5.1 with proof were read. The version history lists v2, April 3, 2024. These are interior/boundary-mixture upper bounds, including entropy decomposition terms of logarithmic size in the mixture weight. They do not state a sharp first derivative or supply the unbounded Gibbs exterior estimate. The HTML's rendered date differs from its version header; the assessment uses the version history.
• Bormann, published 2026 article and arXiv:2409.19336v2: weighted versions of boundary-diffusion estimates. Both PDFs were archived; the published Section 3 and proofs were read. The latest arXiv version checked was April 23, 2026. Its bounds do not identify P40's sharp coefficient.
• Dolbeault--Kowalczyk, 2017: nonlinear Neumann rigidity and interpolation thresholds. The introductory problems, Theorems 1.1--1.2 and Section 6.2 were screened. P40 does not classify fixed-domain extremizers; this is related context, not a proof dependency.
• Dovetta--Serra, arXiv:2606.23225v1: the screened theorems concern Gaussian sharp stability, not a flat-domain Gibbs expansion. Capel et al., arXiv:2606.13453v1 was screened through its main statements and Theorems 5.1 and 6.1 concerning compact-manifold mixing and concentration. Neither is a claimed full-paper proof audit. Their version pages listed only v1 when checked.

The classical extremizer attribution to Rothaus follows Ben Nejma's account. Rothaus's 1981 full text was not obtained; P40 therefore supplies the required compactness and Euler-equation arguments instead of treating an unread theorem as a dependency. Bibliographic data were cross-checked, including the DOI and pages 102--109. Classical Sobolev extension, trace and compactness facts are stated explicitly; the specific Neumann operator-domain and uniform domain-extension statements are checked in the primary sources identified in REVIEW.md.

Why this meets the intended depth

The sharp constant is a supremum over an infinite-dimensional class. A fixed-test boundary expansion proves only a lower bound. The upper bound must rule out concentration in an unbounded exterior, retain all nonlinear maximizing limits, and remain accurate when energies vanish arbitrarily quickly compared with temperature. Sections 3--7 address those three issues. The proof needs neither a unique nonlinear optimizer nor a nondegenerate Hessian of the variational quotient. Proposition 8.1 demonstrates that the nonlinear alternative is necessary within the smooth strictly convex setting, rather than on an excluded singular example.

The reusable result is the uniform entropy expansion under only energy control. It can support future perturbation arguments after their own tail hypotheses are verified. The theorem also explains why knowledge of the first eigenvalue alone does not generally determine the low-temperature logarithmic Sobolev derivative. These are substantive analytic contributions with a plausible audience in functional inequalities, singular perturbations and diffusion theory. They justify proceeding at the user's intended research depth; they do not predict acceptance by Advances in Mathematics or Transactions of the AMS.

Scope and uncertainty

Convexity, a smooth bounded domain of minima, quadratic nondegenerate normal growth and a Laplacian growth bound are retained. Other exponents and degenerate boundary profiles remain open here. The coefficient is an implicit variational maximum; the paper does not calculate the limiting sharp constant or classify extremizers on a general domain. These restrictions limit breadth and should be explicit in any submission.

Searches used exact identifiers and combinations of log-Sobolev, low temperature, boundary asymptotics, flat minima, shape derivative, domain perturbation and extremizers. The dated query log and source archive manifest accompany this assessment. No bounded search can establish absence of all prior work, unpublished results or indexing-delayed revisions. An outside analyst's scrutiny of Proposition 3.2, the vanishing-energy argument and priority would be especially valuable. No researcher has been contacted and no outside endorsement is implied.