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Math-2268-A1 · revision r1 · on Math-2268

OpenAI claim summary for Math-2268

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: solved. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

Strict planar hot spots: The theorem gives nonvanishing interior gradient and strict boundary extrema for every first-positive Neumann eigenfunction on smooth bounded simply connected planar domains, including multiplicity. The catalogue also asks convex domains in n>=3 and a heat-flow formulation, so the planar clause alone is partial.

This page records partial support. Complete resolution of the catalogue question is not established by this record.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Hot spots: high-dimensional refutation, product rules, and heat-flow quantifiers
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Hot spots: high-dimensional refutation, product rules, and heat-flow quantifiers

OpenAI's claim

Strict planar hot spots: The manuscript claims that every nonzero eigenfunction at the first positive Neumann eigenvalue on a smooth bounded simply connected planar domain has no interior critical point. This includes multiple first positive eigenvalues and forces extrema to the boundary.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $\Omega\subset\mathbb R^2$ be nonempty, bounded, simply connected with smooth boundary, and let $\mu_1>0$ be the first positive Neumann eigenvalue. For every nonzero real $u$ in its entire eigenspace, the manuscript claims \[\nabla u(x)\ne0\quad(x\in\Omega),\qquad\min_{\partial\Omega}u<u(x)<\max_{\partial\Omega}u.\] The eigenspace may be multiple; the conclusion concerns every nonzero eigenfunction, not merely one chosen eigenfunction.

Outline of the argument

Strict planar hot spots: Conformal parametrization transfers the divergence--curl variational principle for gradient fields to boundary vector data. A positive boundary kernel expresses the energy of scalar-multiplied boundary gradients. Conditional negativity of a reciprocal kernel supplies multipliers that remove the obstructing sign, yielding a contradiction if the gradient vanished at an interior point.

Formal verification

Family 369 describes Lean coverage for gradient nonvanishing in the interior throughout the first positive Neumann eigenspace in the smooth simply connected planar setting.

Strict planar hot spots: The manuscript links HotSpots.lean.

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources