Priority, scope, and significance assessment

Henry Zweiman. September 16, 2026.

Provisional decision

Treat this connected theorem package as one prospective research preprint, P54, subject to verified publication. This is a provisional originating-assistant assessment. It does not certify historical priority, human expert agreement, suitability for a particular journal, or future citations.

The main result proves the spherical equal-perimeter Steklov comparison for every smooth proper geodesically convex domain in a dimensional interval immediately below maximal boundary measure, with equality only for balls. It also proves a weighted boundary-center theorem and a uniform quantitative endpoint estimate. The center, endpoint estimate, lune comparison, and compactness argument are components of this one contribution; they are not separately counted papers.

Relation to the open problem

The final Colbois–Girouard–Gordon–Sher survey, Revista Matemática Complutense 37 (2024), Open Question 4.27, asks for the convex fixed-perimeter comparison in spherical and hyperbolic space. The published question was inspected directly. The present theorem addresses a whole interval in the spherical direction, not the full question.

The recent Gao–Wei–Zhou preprint, arXiv:2608.11841v1, states Euclidean and hyperbolic results in its introduction and main theorems. Its official history still lists only v1 in the September 16 check. It does not state the spherical theorem in the inspected material.

The closest methodological precedents are the Euclidean moment and low-harmonic method of Gavitone–La Manna–Paoli–Trani, the weighted geometric inequalities of Kwong–Wei, ordinary spherical centroid results, and standard Crofton theory. These are credited. The proof does not claim a new Crofton formula or a new general compactness principle. The positive-curvature obstacles are the compatible harmonic center, uniform control at the degenerating hemispherical endpoint, and nonball limits with maximal perimeter.

The priority search included spherical Weinstock, convex spherical Steklov comparison, large perimeter, hemispherical stability, lune spectra, and weighted spherical boundary centroids. No matching theorem was found in the inspected primary sources. A finite search cannot exclude inaccessible, unindexed, or simultaneous work. Exact read scopes and access failures appear in source-audit.json.

Why this is more than a local adaptation

The earlier fixed-radius perturbative result by itself was not admitted. The current perimeter-only theorem makes no initial closeness assumption and covers all smooth convex shapes in its interval. In particular, perimeter approaching maximal area can leave a proper lune as a limit, so a local estimate around balls alone cannot establish this statement. The cut-and-join trial and the collapsed-volume bound resolve those alternatives.

The harmonic center is also a reusable geometric result: it simultaneously removes the trial functions' boundary means and puts the entire domain inside the centered hemisphere, for a class of center-dependent decreasing kernels. The quantitative estimate describes the deficit scale at the endpoint without imposing uniform curvature bounds on the competing domains.

This provides a plausible candidate for expert assessment at the requested Transactions-of-the-AMS-or-comparable level. That judgment is uncertain. A nonexplicit interval and the unresolved intermediate range limit the breadth of the result, and experts may regard a specialist journal as more appropriate. The full conjecture is a materially stronger objective. Expected citation counts are not used as evidence of significance.

Remaining limits

The complete final journal bodies of Kwong–Wei and Gavitone–La Manna–Paoli–Trani were not retrieved in this pass. Their official arXiv versions were compared, and publisher/author records were checked. The latter final abstract explicitly concerns Euclidean convex sets. These access limits are not evidence of absence. The present proof does not import an unverified theorem from either article. The final Castillon–Ruffini and Cabezas-Rivas–Scheuer texts were available, with the read scopes specified separately.

The result has not been independently reviewed. General nonsmooth initial domains, an explicit interval size, an optimal stability constant, and the comparison at every perimeter remain outside the claim. GitHub publication is public dissemination of an unreviewed preprint, not peer-reviewed publication.