Priority and significance assessment

Henry Zweiman. September 15, 2026. P44 revision 4.1.

Current internal assessment

Revision 4.0 adds a complete written counterexample to Laugesen's Conjecture C: the second Robin eigenvalue need not be concave in the positive parameter, even for a convex polyhedron of prescribed diameter in any dimension at least four. The proof uses an asymmetric modification of the published transverse geometry and supplies its own ordered low-eigenvalue convergence argument. No numerical spectral experiment is used. The originating assistant found no remaining gap in the written argument; independent expert verification has not occurred.

Revision 3.0 adds a complete elementary proof that, for every fixed positive Robin parameter and fixed diameter, the infimum of the fundamental gap over convex polyhedra is zero in every dimension at least four. This disproves the universal positive-Robin fundamental-gap conjecture. A fixed explicit four-dimensional member also disproves the positive-parameter gap monotonicity conjecture recorded by Laugesen and attributed to Smits. The new proof is independent of the earlier corner and quasiconcavity regularity arguments.

The manuscript retains the discrete-exception polygonal log-concavity result and the full small-parameter polyhedral quasiconcavity proof from revision 2.0. The originating assistant found no remaining gap in the stated results. This is an internal assessment of an AI-assisted preprint, not independent expert verification, peer review, an absolute priority certificate, or a journal-acceptance prediction.

New in revision 4.1: perimeter-scaled triangular limits

Section 12 proves a positive-parameter limit alpha/2 on area-one thin triangles and an explicit strict triangle-versus-disk comparison at alpha=1. A positive polynomial on the disk satisfies its Robin boundary condition exactly and supplies a strict ground-state lower bound by integration by parts. Piecewise linear triangle trials include the base-boundary term; vertical slicing and the earlier elementary interval estimate give the matching lower limit. A separate exponential trial satisfies both sloping Robin conditions exactly for negative alpha and gives unboundedness below. There is no numerical eigenvalue computation, asymptotic corner theorem, or unproved spectral convergence input in this section.

The exact target is Laugesen, arXiv:1905.07658v1, Conjecture D, p.9, including its all-real-parameter quantifier and surrounding smooth-domain asymptotic discussion. Both the PDF wording and HTML were checked. The final Journal of Mathematical Physics DOI 10.1063/1.5116253, publication metadata, references and the author's publication list were inspected, but the full final journal text was not accessible. The result is therefore attributed to the exact preprint formulation, without asserting that the final journal version retains it.

There is a concrete source discrepancy: the preprint says that this minimization conjecture is also stated in Freitas--Laugesen's Szego-way paper. The complete Section 3, printed pp.1029--1031, of the final Canadian Journal of Mathematics article was read. Its Conjectures 1 and 2 concern second eigenvalues; its first-eigenvalue discussion states different unscaled optimization problems. It does not reproduce the first-eigenvalue perimeter-scaled minimization conjecture. This prevents treating that paper as corroboration of the precise target or a certificate that the target remained open after publication.

Focused September 15 searches covered the exact Conjecture D, perimeter-scaled Robin first eigenvalues, convex counterexamples, negative parameters and thin triangles. No competing result was found in the inspected primary statements. This bounded search and inaccessible final wording leave a stronger historical novelty claim uncertified. The mathematical comparison itself is fully written and internally checked. Its simplicity, and the absence of corroborated sustained study of this exact formulation, also mean that this revision does not satisfy the renewed goal's exceptional-significance threshold by itself. The goal remains active.

New in revision 4.0: second-eigenvalue concavity

The exact statement of Laugesen Conjecture C, its surrounding discussion and Theorem 2.3 were reread in the primary preprint. It concerns positive parameters and convex bounded domains in arbitrary dimension. The source observes that an odd second eigenfunction on a reflection-symmetric domain gives concavity by a mixed ground-state characterization; it proves strict concavity for rectangular boxes. The conjecture was recorded again in the 2020 sharp-eigenvalue problem list and the 2023 Oberwolfach Geometric Spectral Theory report, printed p.2099. The latter exact problem paragraph was checked. Negative-parameter numerical failures mentioned by Laugesen are distinct from the positive-parameter theorem here.

Section 11 treats a fixed convex four-dimensional polyhedron after choosing a sufficiently small geometric parameter. Its two end regions converge, after a common affine eigenvalue subtraction and rescaling, to half-line Neumann Airy operators with different slopes and a constant shift in one well. The two limiting ground branches exchange order near alpha=1, while the first excited well states remain separated. The second ordered eigenvalue therefore has a limiting upward corner. Convergence at three fixed parameters gives a strict midpoint failure for the actual fixed domain; no eigenvalue derivatives or actual crossings on the connected domain are assumed. Products and a direct interval estimate extend the result to all dimensions at least four, and scaling gives any diameter.

The proof retains transverse gradient coercivity from the earlier elementary estimate, builds a uniformly bi-Lipschitz radial fiber map from explicit gauges, proves tightness at infinity, preserves orthogonality in the lower-limit argument and supplies matching finite-dimensional upper trial spaces. Neumann/Dirichlet cut bracketing transfers all fixed ordered low eigenvalues to the original domain. The exterior Robin cap becomes a vanishing alpha h form term, yielding the correct Neumann endpoint. These steps are proved in the manuscript rather than delegated to an unverified thin-domain theorem.

A new primary comparison checked Grebenkov--Levitin--Polterovich, arXiv:2604.11526v1, April 13, 2026: complete introduction, Section 4.1 statements, Proposition 4.4 and Remark 4.5, and Theorem 4.15 with proof. Its concavity observation concerns the principal Dirichlet-to-Neumann eigenvalue as the Helmholtz parameter varies below the first Dirichlet level. Its Robin branch theorem concerns the negative-infinity limit. Neither settles positive-parameter second-Robin-eigenvalue concavity. The full 60-page survey was not independently audited and is not a proof dependency.

The search also surfaced a broad local-eigencurve concavity assertion in Bucur--Freitas--Kennedy, The Robin problem (2017), printed pp.82 and 87. The relevant indexed primary-source excerpts and publisher metadata were read; full PDF retrieval failed. That claim includes an exception at crossings and is not the exact global Conjecture C. We do not use it. In particular, the first-eigenvalue infimum-of-affine-functions argument cannot simply be applied to higher ordered eigenvalues, whose variational characterization is min--max. The present proof uses direct three-point inequalities and does not assume an eigencurve concavity principle.

Focused September 15 searches covered the exact conjecture, second Robin eigenvalue concavity, positive parameters, convex counterexamples, asymmetric wells, thin domains and 2026 work. No competing positive-parameter convex-domain counterexample was found in the inspected primary sources. This remains a bounded search, not an exhaustive historical priority certificate. The result is an existence theorem for small h, not an explicit numerical threshold or a smooth strictly convex example. Dimensions two and three remain open here.

Revision 4.0 is a distinct conjecture-level extension beyond revision 3.1, and together Sections 9--11 refute all three parameter conjectures A, B and C in the cited survey. The renewed goal's exceptional-significance requirement is still assessed separately from having a complete proof of a named conjecture. This publication records substantial verified manuscript progress and does not automatically close that stronger goal.

New in revision 3.1: spectral-ratio monotonicity

Laugesen, Conjecture B asserts that the ratio of the second to the first Robin eigenvalue decreases with the positive Robin parameter on every bounded Lipschitz domain. Its exact statement and the neighboring Conjectures A and C were read in the primary preprint. The 2020 conference list, Open Problem 7, also records the ratio question. The statement is distinct from gap monotonicity: a small absolute gap alone does not control its size relative to the first eigenvalue or to the Dirichlet ratio.

Section 10 supplies that missing comparison. A barrier supported in the central three quarters of the thin domain and the exact ground-state identity give a nested localization inequality. Repeated cutoffs give an exponential gap upper bound. Two cutoffs already give the explicit estimate Q_R-1 <= 2^23 epsilon^3 at alpha=1, epsilon=10^(-30). The established Andrews--Clutterbuck Dirichlet gap theorem, together with an inscribed-box eigenvalue upper bound, gives Q_D-1 > epsilon^2/200. A self-contained min--max argument proves the fixed-domain Robin-to-Dirichlet limit. Thus the ratio must increase between two positive parameters on this one domain. Product separation and uniform parameter estimates give every fixed diameter and positive starting parameter in all dimensions at least four.

The exact Corollary 1.4 and the convex-domain approximation paragraph in Andrews--Clutterbuck were checked in the complete introduction and main statements, pp.1--5 of arXiv v2. Its classical Dirichlet theorem is a proof dependency, not claimed as a new theorem or independently reconstructed here. The ratio counterexample is independent of Sections 2--8, but does use the Section 9 transverse estimates.

The current Dai--Sun preprint, arXiv:2511.20988v3, December 20, 2025, was checked against its version history. Its title is Upper bound estimation for the ratio of the first two eigenvalues of Robin Laplacian. The complete introduction and Theorems 1.1--1.3 were read. Theorem 1.3 proves ratio monotonicity on balls. Theorem 1.1 has an additional geometric condition for its sharp ratio bound; Theorem 1.2 gives another bound in the complementary case. None asserts ratio monotonicity on all bounded domains. Search results that retain the v1 title or theorem numbering were not used as the final comparison. The full 42-page proof is not a dependency and was not independently audited.

A focused September 15 search covered Robin eigenvalue ratios, positive-parameter monotonicity, Laugesen Conjecture B, convex counterexamples and 2025--2026 work. No competing general-domain counterexample was found in the inspected primary sources. This is a bounded priority review. The 2023 Oberwolfach Geometric Spectral Theory problem section was also read: it records gap monotonicity and second-eigenvalue concavity, but is not cited as another formulation of the ratio question.

This is a new conjecture-level manuscript result, distinct from revision 3.0. The originating assistant found no remaining logical gap. Independent proof review and priority certification have not occurred. The renewed user goal asks for another extraordinarily well-researched conjecture; the present sources establish a named, actively studied conjecture, but this record does not automatically equate that with the stronger exceptional-significance threshold. The renewed goal remains active while further manuscript frontiers are investigated.

Positive-Robin gap: precise statement and prior work

The exact universal conjecture was checked in ACH, final Section 10, p.307: convex domains of diameter D, convex potentials, and the same positive Robin parameter on the domain and comparison interval. Zero potential belongs to the conjectured class. Laugesen, Conjecture F records its zero-potential version and proves the rectangular-box case in Theorem 3.8. Kielty, Conjecture 1 states the same positive-parameter problem; Theorem 2 instead treats negative parameters on double cones. Their boundary sign convention agrees with this manuscript.

The present construction uses different geometry from those negative-parameter double cones. Three-dimensional cross-sections expand toward the middle while their surface-area-to-volume ratio increases. A central energy barrier results at positive parameter. The proof supplies its own segment-based Poincare inequality, divergence-based trace estimate, uniform transverse form bound, exact lateral facet factors, cap-inclusive trial bound, mass localization estimate, and ground-state identity. It requires neither numerical eigenvalues nor adiabatic spectral convergence. At alpha=1 and epsilon=10^(-30), the gap is less than 1/100 while the interval comparator has gap greater than one. Compact-parameter uniformity makes the diameter normalization rigorous.

The cross-section geometry exploits a previously known failure of Minkowski monotonicity for volume divided by surface area. Fradelizi--Giannopoulos--Meyer (2003) and the Fradelizi--Madiman--Meyer--Zvavitch introduction are credited for this phenomenon. The explicit formulas are rederived in the manuscript. Their geometric results are background, not unproved dependencies of the spectral argument.

Laugesen, Conjecture A records monotonicity of the fundamental gap with respect to the positive Robin parameter and attributes it to Smits, Michigan Mathematical Journal 43 (1996), Section 4. The conjecture and attribution were read in Laugesen's primary research paper. Attempts to retrieve Smits's original article from Project Euclid did not return its full text, so we do not claim a direct audit of the original wording. Corollary 9.4 disproves the exact convex-domain statement recorded by Laugesen. It uses a direct four-dimensional Poincare estimate and a min--max proof of continuity at zero; no sharp Poincare theorem is needed.

The April 2020 sharp-eigenvalue miniconference list, Open Problem 7, independently records both questions. The ACH interval paper proves the convex-potential interval case; its introduction and Theorems 1.1--1.4 were read. The full relevant negative-parameter argument of Kielty and the full rectangular-box proof of Laugesen were not independently reconstructed, because neither is a proof dependency. The complete Kielty introduction, main theorem, heuristic and literature section were read in corrected arXiv v2 (March 25, 2022).

Focused current searches covered positive Robin gap counterexamples, four-dimensional domains, diameter normalization, thin-domain localization, surface/volume ratios, Minkowski addition, and positive-parameter gap monotonicity. No competing positive-parameter gap degeneration or convex-domain monotonicity counterexample was located in the inspected primary sources. Ahrami--El Allali's 2022 lower-bound paper concerns interval problems; Bao--Huang's September 2026 preprint concerns Robin-to-Neumann differences, rather than the first-to-second Robin gap. Neither inspected statement supersedes the present result. These are bounded-search findings, not a guarantee against overlooked or unpublished work.

Exact originating conjectures and prior inputs

Andrews--Clutterbuck--Hauer, Cambridge Journal of Mathematics 8 (2020), 243-310, final Section 10, asks for eventual log-concavity on every bounded convex domain (Conjecture 1, printed p.306) and nonconvex ground-state superlevel sets at all sufficiently small positive parameters on every convex polyhedron of dimension at least three outside products of circumsolids (Conjecture 2, printed p.307). Both exact final statements were read. The arXiv history retains v2, February 20, 2018, as its latest listed version.

The current result matches Conjecture 2's quantifiers: every fixed domain in that class, every dimension at least three, and every parameter in a domain-dependent interval (0, alpha0). It imposes no cone angle condition, symmetry, simplicity of vertices, or assumed nonzero eligible corner mode. Its proof first handles the inconsistent-normal case by the existing ACH argument; only the remaining consistent-normal case uses the new energy argument.

The Robin perturbation, quadratic classification (final Corollaries 8.2--8.3), spherical Neumann lower bound and equality classification (Theorem 9.1), inconsistent-normal obstruction, and planar odd-mode argument are prior work and are credited. The local mode expansion is explicitly referenced to arXiv v2 Proposition 4.4 because numbering changed in the final paper. The full Section 9 arguments and the relevant spectral proof were read in the primary v2 text, with final statements and selected final passages checked. The complete 68-page final article was not independently re-proved.

The new edge lemma gives its own uniform Taylor and reflection argument. It does not rely on an unproved reading of the abbreviated ACH Theorem 9.4 after affine subtraction, and it proves quadraticity only for degree-two modes on wedge-times-Euclidean cones, where a direct tangential-derivative argument suffices.

Regularity sources

Maz'ya, Comptes Rendus Mathematique 347 (2009), 517-520 supplies the bounded-gradient estimate for homogeneous Neumann problems on bounded convex domains with forcing in Lq, q greater than dimension. The complete four-page final note and the complete eight-page arXiv v2 preprint were read. The final note states the theorem and applications but omits the full proof; the longer preprint contains the proof. Local copies and hashes are retained. The manuscript derives uniform gradient convergence of the cone expansion using this theorem, polynomial eigenfunction bounds, and an elementary eigenvalue-counting bound. It does not infer C1 from a W2,p estimate with an unavailable p greater than dimension.

Tolksdorf, Calculus of Variations and PDE 59 (2020), 154, final Section 4, recalls the scalar convex homogeneous-Neumann H2 estimate and convex approximation argument. This is the input used here; no Stokes theorem is used. Dauge's published wedge theorem remains the regularity input for the earlier polygonal analytic-coefficient result. These standard results are dependencies, not new claims of the present manuscript.

Current comparison and priority limits

Edelen--Li, arXiv:2608.17194v1, August 17, 2026, is a relevant recent comparison located during the current audit. Its complete introduction, main assertions, tensor cutoff argument (Lemma 2.5), product-cone argument (Lemma 3.3), and local regularity statement (Theorem 3.5) were read. The checked history lists v1 only. Its sharp spectral and C2 regularity theorem assumes non-obtuse dihedral angles; it does not state the general Robin quasiconcavity theorem proved here. Its bounded-tensor logarithmic-cutoff method and tangential product differentiation are closely related standard steps and are acknowledged. We do not use its sharp spectral theorem, and its full proof was not independently audited. The present higher-codimension estimate and use of the Hessian norm allow potential unboundedness there; quasiconcavity supplies edge regularity even for obtuse edges.

Crasta--Fragala, Cambridge Journal of Mathematics 9 (2021), 177-212, and Ye--Zhang, arXiv:2609.06223v1, September 5, 2026, establish eventual concavity for smooth uniformly convex domains, with the latter reaching C3,1 regularity. Their hypotheses exclude the polygonal counterexamples. The complete relevant preprint introductions and main statements were read. The final Crasta--Fragala full text was not successfully opened; institutional final metadata and abstract were checked. Their proofs are not dependencies of the present paper. No claim of a single smooth-domain large-parameter counterexample is made.

Targeted searches on September 15 covered Robin quasiconcavity, the exact conjecture, circumsolids, Neumann quadratic rigidity, polyhedral Hessian identities, corner singularities, analytic coefficient persistence, and current 2025--2026 work. No competing full quasiconcavity solution or prior analytic-persistence disproof was identified in the inspected sources. This is a bounded search; unindexed, unpublished, or overlooked work remains possible. The newly located Edelen--Li preprint is included precisely because a narrow search of Robin titles alone would miss related methods.

Publication and significance boundaries

All contributions remain one revised paper, P44. Revisions and corollaries do not increase the publication count. The earlier assessment of revision 2.0 deliberately left the broader research goal active: specialized concavity rigidity alone did not establish the requested exceptional conjecture-level outcome.

Revision 3.0 reaches the central spectral-gap question that motivated ACH's work and appears separately in Laugesen's survey, the 2020 eigenvalue problem list, and Kielty's gap-degeneration research. It also answers the related positive-parameter monotonicity conjecture traced by Laugesen to 1996. It rules out every positive diameter-only bound at each positive parameter in dimensions at least four, rather than merely adjusting a conjectured sharp constant. This breadth, the longstanding monotonicity question, and the documented sustained research on the gap supported completing the previous user goal at revision 3.0, at the manuscript-proof level. The renewed goal explicitly requires another result, so that previous completion is not reused. This is a reasoned internal significance assessment; external mathematical recognition and independent proof review have not occurred.

The universal conjectures are disproved by dimension four. Their restrictions to dimensions two and three remain unresolved here. The paper also leaves open the unrestricted local transverse-cone assertion and the exceptional polygonal parameters. No smooth strictly convex counterexample, matching exponential asymptotic, or new solution of the classical Dirichlet fundamental-gap theorem is claimed. Section 10 does prove an exponential upper bound. The second-eigenvalue concavity conjecture remains open here only in dimensions two and three; Section 11 disproves its universal version.

Publication readiness refers to the complete written argument, explicit attribution, reproducible build and verified artifacts. Formula parsing, PDF rendering and public byte checks are not mathematical proof or independent peer review. Any completion of the research task must be reported with that distinction intact.