Revision 1.2: convex-domain ground-state uniqueness

Henry Zweiman. September 16, 2026.

The new Theorem 1.5 resolves the ground-state uniqueness and ground-state identification questions explicitly left open by Gallo, Mosconi and Squassina in Remark 1.6 of their April 2026 Mathematische Annalen article. It applies in every dimension to every bounded convex domain, without smoothness assumptions on the boundary. Every positive log-concave solution has negative logarithmic Hessian strictly above the Gaussian value and is the unique positive ground state.

The proof strengthens concavity through positive level sets and turns a strict local transport remainder in the logarithmic Sobolev inequality into a sharp variational inequality on all of H^1_0. The equality case is preserved by local remainders under strong approximation. These steps avoid unsupported boundary Hopf arguments, global quotient integrability and a uniform curvature gap. The internal review records the detailed checks; the source audit records the primary statements, erratum handling and later-work comparisons.

This is a broad extension of the same manuscript. It upgrades the deformation counterexamples to ground-state counterexamples. GMS existence, Caffarelli contraction, the Gaussian logarithmic Sobolev inequality and interior elliptic regularity are essential credited inputs. The contribution is the new PDE Hessian bound and its variational rigidity consequence on arbitrary bounded convex domains. Neither the classical transport inequality nor ball uniqueness is claimed as new.

The manuscript addresses an external open question beyond radial geometry and perturbations. It is an internally audited, AI-assisted preprint, not an independently refereed theorem, exhaustive priority certificate or assurance of journal acceptance. The prior assessment incorrectly treated this as sufficient to close the research goal. The clarified stopping condition requires solving a publicly well-known and significant conjecture, roughly at the level of having a Wikipedia page. No evidence here establishes that P46 meets that standard; it does not complete that goal. This scope correction changes no mathematical statement or proof.

Uniqueness among all positive solutions, the optimal ball power exponent's radius dependence, and the weaker small-rescaling half-log-concavity question remain open here.

Historical assessments of revisions 1.0 and 1.1

The following assessments describe the earlier versions and their then-open frontiers. Their statements about goal completion or lack of a general-domain theorem are historical.

Revision 1.1: contribution and limits

Henry Zweiman. September 16, 2026.

The revised paper answers its own published deformation question negatively. Near every ball in every dimension at least two, a smooth uniformly convex degree-three deformation produces a genuine positive solution whose normalized square-root-log transform is not concave. Failure occurs at arbitrarily small distances from the unique maximum, even though ordinary strict log-concavity persists. The proof combines the existing strong ball Hessian estimate, a classical weighted spectral bound, a boundary-compatible implicit-function theorem and a necessary cubic-jet condition.

This is a substantive extension of the same paper, not another distinct paper. It identifies a geometric instability of the exact normalization used in the ball theorem. The stronger external statement about all-dimensional balls was proved by the preceding revision; the new counterexample is not presented as a disproof of an unasserted general-domain GMS conjecture. The related scale-invariant half-log-concavity definition allows small rescaling, as explicitly explained by Ishige-Salani-Takatsu, Section 4.2. The present theorem does not disprove that weaker property.

The new source audit records the inspected primary passages and terminology distinction. Standard weighted Poincare, elliptic theory, shape differentiation and the implicit-function theorem are credited tools. No exhaustive priority or independent correctness certification has occurred. This one revised manuscript does not by itself meet the renewed extraordinary-conjecture completion condition; the broader goal remains active.

Global uniqueness on arbitrary convex domains, a classification of deformations preserving the normalized transform, and the explicit optimal positive-power exponent remain unresolved.

Original ball contribution and source assessment

Priority and significance assessment

Henry Zweiman. September 15, 2026. P46, developed as Q54.

Contribution and internal decision

The manuscript gives a complete proposed positive-series theorem for positive solutions of the logarithmic Schrodinger equation on balls, in every dimension and at every radius. Its global representation proves the negative-square-root-log concavity explicitly left open in Gallo--Mosconi--Squassina's final 2026 paper. It also answers their positive-power question, determines the sharp exponent among powers of the normalized negative logarithm, and gives a coefficientwise comparison of normalized profiles.

This is provisionally assessed as one substantial research preprint under the revised program standard. The argument has conceptual content beyond checking a special dimension: it identifies an infinite positive recurrence, proves that its convergence extends throughout the ball, and extracts global nonlinear concavity from a Hilbert-space norm. It connects radial ODE structure with geometric properties of nonlinear elliptic solutions. The explicit published question and the reusable representation support its significance. Possible uses in other nonlinearities, radial potentials, or deformation theory remain proposals, not established applications or predictions of citation counts.

The result is focused on radial geometry. An editor could judge the scope too narrow for the user's target journals despite the complete resolution of the stated question. This is an internal, revisable assessment; no independent correctness review, absolute priority certificate, or assurance of journal acceptance is claimed.

Closest primary source

Gallo, Mosconi and Squassina, Power law convergence and concavity for the logarithmic Schrodinger equation, Mathematische Annalen 395, article 21 (2026), final author-posted PDF, 60 pages. The source was published April 7, 2026. The current arXiv history lists v1 November 3, 2024 and v2 April 7, 2026.

The direct dependency is radial symmetry and radial decrease in Theorem 6.5, page 41; existence is supplied by Theorem 1.1 and the one-dimensional Theorem 6.6. The power-concavity question occurs on pages 41--42. The negative-square-root-log question is explicitly stated on page 44, after the dimension-one proof. These statements and the relevant surrounding arguments were read in the final source. The full sixty-page proof of the cited existence theorem was not independently re-proved here. The radial source explicitly treats the non-Lipschitz issue and invokes Dolbeault--Felmer's non-Lipschitz symmetry theorem; the manuscript does not substitute an inapplicable generic symmetry theorem.

Important prior-art correction

Ben Chrouda, EJQTDE 2022, no. 62, already proves strict first-zero ordering (Theorem 1.2), ball uniqueness (Corollary 1.3), and the Gaussian central-height obstruction (Corollary 1.5). His logarithmic example specializes with k=1 and beta=0; spatial scaling gives the factor-two normalization in P46. The 17-page primary PDF, especially its precise main hypotheses and conclusions on pages 2--3, was checked. These results are expressly credited as known in the manuscript. The recurrence offers another proof, not a new uniqueness theorem. Search metadata incorrectly suggested recent publication; the primary PDF establishes 2022.

Current later-work checks

Liu--Sun--Zou, arXiv:2606.19077, current v2 June 22, 2026, was downloaded and compared at Theorems 1.1--1.2 and its first-zero motion argument. The complete 36-page PDF was retained for the audit. It gives whole-space nodal shooting classification and asymptotic results, with strict zero motion already present. No concavity or positive-series theorem was identified in the checked text. P46 does not claim this shooting classification.

Gallo--Moraschi--Squassina, arXiv:2504.09494, current history v2 October 29, 2025, was compared with the current 41-page author manuscript. Its Theorem 1.1, Corollary 5.3 and logarithmic application were read. They concern logarithmic concavity in a parabolic setting, not the stronger normalized square-root transform on balls. The author file has updated organization relative to some cached web extracts; the audit uses the downloaded file's numbering. Its entire general concavity theory was not re-proved.

Troy's 2016 ground-state uniqueness article was checked at the primary publisher abstract as a historical lead; the present proof does not depend on it. The stronger relevant ball conclusions were located in the open 2022 source instead.

Searches combined logarithmic Schrodinger, radial, ball, concavity, square-root, power concavity, positive coefficients, absolute monotonicity, and the source authors. Current primary versions of the close matches were followed. These searches found no prior resolution of the targeted square-root question; unsuccessful searches do not establish absolute novelty. No institutional-access dependency remains for the manuscript's claimed results.

Original revision boundary (historical)

The original revision had no theorem for arbitrary convex domains, no sharp explicit value or radius-monotonicity result for the positive power exponent in dimensions at least two, and no new whole-space nodal classification. The product-domain observation is a corollary in this single paper. The method, its sharp transform, boundary lemma and comparison consequences count together as one contribution.