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Strict Neumann Pólya inequality on simply connected planar Lipschitz domains

Main theorem: for every bounded simply connected planar Lipschitz domain $\Omega$ , $|\Omega|\mu_j(\Omega)<4\pi j$ for all $j\ge1$ , with $0=\mu_0<\mu_1\le\cdots$ counted with multiplicity; equivalently $N_N(E)>|\Omega|E/(4\pi)$ for $E>0$ , where $N_N$ counts $\mu_j<E$ including $\mu_0$ . It contains the convex planar case, for which Filonov (CPAM 2025) had only a weaker constant, and the disk case of Filonov, Levitin, Polterovich and Sher (2023). Multiply connected domains, non-Lipschitz domains and higher dimensions remain open. The manuscript is not yet on arXiv; a draft is in the registered snapshot. The companion strict Dirichlet result is a separate entry.

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Oct 9, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-assisted. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Strict Neumann Pólya inequality on simply connected planar Lipschitz domains · Strict Neumann Pólya (simply connected)

Confidence: Not scored

Registry verification: lean checked · announcement · partial

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Quanyu Tang
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Zuoqin Wang
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VibeMathed record: Strict Neumann Pólya inequality on simply connected planar Lipschitz domains

specialist registry · passed

No independent mathematician has checked this yet. Read by this site on 15 October 2026, not rebuilt. The Palomar record PALOMAR-2026-10-09-000007 v1 was fetched from the registry's data API: registered commit f07f1c7, Comparator theorems strict_neumann_polya and strict_neumann_polya_count, checked with the Lean, nanoda and con-ron kernels, automated review neutral with no warnings, trust level high. Challenge.lean (Mathlib only, identical on main) was read: Lipschitz domain as open, connected and locally above a Lipschitz graph after a rigid motion; H1 by weak gradients in L2 tested against C_c^inf functions; mu_j by min-max over (j+1)-dimensional subspaces of L2 with no boundary condition, the standard Neumann eigenvalue indexed from mu_0 = 0. The registered snapshot (about 100,000 lines) was grepped: the only sorrys are the two Challenge statement holes; no admit, axiom declarations, native_decide, unsafe or implemented_by. The repository's Actions build passed on the registered commit, and its audit asserts only propext, Classical.choice and Quot.sound. A separate check written here on disks and squares (3,000 eigenvalues each) agrees with the strict inequality; it says nothing about general domains. The draft manuscript was not read. Lean-checked, not Lean-verified: the statement audit is this site's alone and nothing was rebuilt here.

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Strict Neumann Pólya inequality on simply connected planar Lipschitz domains

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PALOMAR-2026-10-09-000007 v1

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Draft manuscript in the registered snapshot (v3)

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