Priority and significance assessment

Henry Zweiman. September 15, 2026. Developed as Q56.

Contribution and internal decision

Provisionally admit this as one substantial research preprint under the revised program standard, subject to completing the public artifact transaction. The manuscript proves strict W1p local minimality, modulo scaling, for radial generalized Neumann Hénon minimizers in the positive range of the known limiting Steklov threshold. It covers q close to p for every 2<p<n and all subcritical q when n>=4 and p>2 is close to 2. It also includes the elementary converse angular instability test and an analytic proof that the positive interval is nonempty.

The principal contribution is a nonlinear local-minimum theorem, supported by two ingredients not supplied by the source's directional positivity argument: a uniform gap in a common weighted space with compact boundary trace, and a convexity argument retaining concentrated perturbation energy. This addresses an explicit current open question and offers a mechanism reusable in degenerate variational problems. The coexistence of radial local minima and nonradial global minimizers gives a concrete variational consequence. These results, the analytic ODE observation and the negative test belong to one paper.

The target of Advances in Mathematics or Transactions of the AMS is a significance aspiration, not a prediction of acceptance. The paper is limited to a ball and the positive threshold range; the equality case remains open. Its relation to established weighted spectral and Morse theory could make editors judge the advance narrower than the program's target. Internal admission is revisable and is not independent verification, absolute priority certification or a citation forecast.

Primary problem and dependencies

Nazarov-Shcheglova, arXiv:2604.26286v2, introduction footnote on PDF page 2, expressly leaves local minimality open. The entire 21-page v2 and current history were inspected. The latest listed version remained May 11, 2026 on September 15. Their Lemma 1 and equation (25) already give the angular threshold used here; that threshold is not a novelty claim.

Shcheglova 2018, final publisher PDF, Proposition 2.1 and Theorems 2.1-2.2 supply radial existence, regularity and convergence. Their normalization is checked through the Euler multiplier rather than copied symbolically. Gazzini-Serra 2008, Section 3, proves the p=2 local-minimum result; it is credited as prior work.

The proof does not rely on the NS fixed-alpha trace replacement or on a joint-uniform unsaturated boundary expansion. Exact flux identities and a varying-alpha compactness argument replace those steps. This is a record of the dependency choice, not a claim that the source's main conclusions are false.

Close functional-analytic literature

Castorina-Esposito-Sciunzi 2011, introduction and Section 2, supplies the weighted spectral-space precedent. Cingolani's survey, introduction and Section 6, led to the final Cingolani-Degiovanni-Sciunzi 2024 article. The final article's introduction, all principal hypotheses and Section 2 were read. Its Dirichlet critical-group results use compactness for nearby regularized critical points; it does not directly establish this Neumann local minimum. General use of weighted compactness is not claimed as new.

Asselle-Cingolani-Starostka, arXiv:2504.19721v1, has a related journal DOI 10.1142/S0219199726500082. The primary author text's assumptions, Theorem 1.2 and Section 3 were compared. Its PDE theorem assumes positive regularization. Its abstract Theorem 3.7 needs a nearby directional Hessian estimate, which would still have to be proved in the present problem. The final publisher body was unavailable; the manuscript cites the inspected author version explicitly. This is background, not a proof dependency.

Current discovery and limitations

Eight arXiv API feeds returned all their listed results, totaling 225 unique entries, with raw XML, summaries, URLs and truncation flags retained. Both accented Hénon and unaccented Henon queries were necessary; the latter alone omitted the primary paper. Author queries and recent quasilinear/Morse searches were also used. Close primary records were inspected. Targeted web searches covered the exact title, local minimality, Neumann, Steklov and weighted coercivity. No matching resolution was identified. This finite search does not establish exhaustive historical priority.

The originating assistant performed the proof and source audit. No independent mathematician was contacted, and no human expert review is claimed. The manuscript is an unreviewed research preprint, with a complete proposed hand proof and explicit frontier.

Revision 1.1: sharp neighborhood scale

The earlier manuscript explicitly left a uniform Sobolev neighborhood unresolved. Section 7 now proves a uniform neighborhood for q at or below the critical trace exponent, and matching power bounds above it in the positive Steklov range. The radius is defined on the unit energy sphere, removing the scalar degeneracy. This answers an internal manuscript question; it is not presented as a separately named published conjecture or as completion of the exceptional-conjecture goal.

The boundary concentration mechanism and the trace threshold are prior ideas. The proposed extension is the matched radius scale, including a uniform nonlinear lower bound for the varying degenerate profiles. The 2026 source introduction and local-positivity statements were reread; Gazzini--Serra 2008 Proposition 3.6 and Theorem 3.8 were compared in the locally retained final PDF. Focused searches did not identify a matching radius theorem; this is a bounded comparison, not exhaustive priority certification. The new proof is internally audited and has not received independent expert review.