P44: Robin ground states on convex polyhedra: concavity and vanishing spectral gaps

Henry Zweiman. September 15, 2026. Revision 4.1.

AI-assisted research preprint. Not peer reviewed or independently verified.

Section 9 disproves the universal positive-Robin fundamental-gap conjecture. For every dimension at least four, fixed diameter D, and fixed Robin parameter alpha > 0, the infimum of the first-to-second eigenvalue gap over bounded convex polyhedra is zero. Section 9 supplies a complete elementary variational proof and an explicit polytope with gap below 1/100 while the comparison interval has gap above one. The same fixed domain disproves the positive-parameter gap monotonicity conjecture attributed to Smits in Laugesen's survey.

Revision 3.1 adds a distinct counterexample to Laugesen's spectral-ratio monotonicity conjecture. Section 10 derives an exponential gap bound, places the Robin ratio below the Dirichlet ratio on the same convex polyhedron, and then uses the fixed-domain Robin-to-Dirichlet limit. The Dirichlet fundamental-gap theorem is an explicitly credited prior input. An explicit member works without numerical eigenvalue computation.

Revision 4.0 adds a counterexample to the second-eigenvalue concavity conjecture, Laugesen C. A convex asymmetric thin domain has two end regions with limiting ground energies that exchange order. Section 11 proves the required spectral convergence by forms, compactness and min--max, then obtains a strict midpoint violation on one fixed domain. This completes the manuscript's answers to the separate positive-parameter conjectures A, B and C in dimensions at least four.

Revision 4.1 adds a planar first-eigenvalue result for the parameter divided by perimeter. On area-one triangles of increasing aspect ratio the positive-parameter value tends to alpha/2, while the disk has a larger value for 0 < alpha <= 4 pi. An explicit triangle at alpha=1 has value below 123/175, while the disk exceeds 12/13. For each negative alpha the functional is unbounded below even among convex triangles. Section 12 disproves the exact Conjecture D formulation in Laugesen's arXiv v1. The final journal wording was not accessible and historical priority remains qualified explicitly.

The manuscript also retains its discrete-exception disproof of eventual polygonal log-concavity and full proof of the ACH small-parameter polyhedral quasiconcavity conjecture. The new gap proof is independent of those regularity arguments.

The gap proof derives uniform transverse estimates, includes the exact lateral-facet factors and end caps, bounds the ground-state mass in the middle, and uses an odd ground-state multiple to estimate the gap. Diameter normalization is exact. No numerical eigenvalues are used. Known Minkowski volume/surface phenomena, negative-Robin gap degeneration, interval and rectangular-box results are credited.

The positive-gap, gap-monotonicity and ratio-monotonicity conjectures in dimensions two and three remain unresolved here. Second-eigenvalue concavity in dimensions two and three, exceptional polygonal parameters and the unrestricted local transverse-cone question are also open here. This is a revision of P44, not a new paper. The assessment explains the significance judgment and the limits of internal review.

To rebuild, install Pandoc and Tectonic on PATH and run python3 build.py. The authoritative source is manuscript.md. Formula parsing and PDF inspection establish artifact integrity, not proof correctness or historical priority.