Strict hot spots and interior critical points

Source and verification

This article explains family 369 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

Overview

Heat flow with insulated boundaries is governed at long times by the first nonconstant Neumann modes. Family 369 claims that every such mode on a smooth simply connected planar domain has no interior critical point. This is stronger than requiring its hottest and coldest points to lie on the boundary.

🏷️ Preliminaries: the first positive Neumann eigenvalue

On a bounded connected domain $\Omega$, Neumann eigenfunctions solve

$$-\Delta u=\mu u,{\partial }_{\nu }u=0.$$

Constants have eigenvalue zero. The first positive eigenvalue is

$${\mu }_1=\underset{\begin{array}{c}v\in H^1(\Omega )\setminus \{0\}\\ {\int }_{\Omega }v=0\end{array}}{\inf }\frac{{\int }_{\Omega }|\nabla v{|}^2}{{\int }_{\Omega }|v{|}^2}.$$

The mean-zero condition removes constants. For heat evolution ${\partial }_{t}h=\Delta h$, an eigenmode decays as $e^{-\mu t}$. After subtracting the conserved mean, the first positive eigenspace gives the slowest possible decay. This explains the connection between a spectral question and late-time hot spots, provided the initial data have a nonzero projection onto that eigenspace.

🏷️ The manuscript’s strict claim

Family 369, as stated in the manuscript

On every nonempty bounded simply connected planar domain with smooth boundary, every nonzero real eigenfunction in the first positive Neumann eigenspace satisfies $\nabla u(x)\ne 0$ at every interior point. Consequently all global extrema occur on the boundary, with strict inequalities at every interior point. Eigenvalue multiplicity is allowed. (OpenAI, 2026)

An interior saddle point is a critical point without being an extremum. Thus the gradient assertion is stronger than the hot-spots inequality. The quantifier is also stronger than saying that some favorable eigenfunction has the property when the eigenspace is multidimensional.

No convexity or symmetry is assumed. Simple connectivity and smooth boundary remain part of the theorem; the manuscript does not assert the statement for multiply connected domains.

🏷️ From scalar eigenfunctions to tangent vector fields

The Neumann condition makes $\nabla u$ tangent to the boundary. The source uses a variational principle for tangent vector fields: a divergence–curl energy is at least ${\mu }_1$ times their squared $L^2$ norm, with equality characterized by gradients in the first eigenspace.

This supplies a nonnegative energy defect $E$. After a smooth conformal parametrization by the disk, let $g$ denote the boundary gradient of an eigenfunction. Since $E(g)=0$, multiplication by a scalar boundary function $b$ leads to an identity of the form

$$\begin{array}{rl}E(bg)=\frac{1}{2}\iint & N(s,t)(b(s)-b(t){)}^2\\ & \cdot (g(s)\cdot g(t))dsdt.\end{array}$$

Here $N$ is a positive boundary kernel. The dot product can have either sign, so positivity of $N$ does not make the integrand nonnegative. Choosing arbitrary multipliers does not solve this difficulty.

🏷️ The reciprocal-kernel mechanism

For an interior point $p$, write $K(p,s)$ for the corresponding evaluation kernel. The central manuscript claim is that

$$\begin{array}{c}{D}_p(s,t)=\frac{K(p,s)K(p,t)}{N(s,t)},\\ D_p(s,s)=0,\end{array}$$

is conditionally negative semidefinite: for finite coefficients with ${\sum }_i{a}_i=0$, one has ${\sum }_{i,j}{a}_i{a}_j{D}_p(s_i,s_j)\le 0$.

The elementary model is a squared Hilbert-space distance. If $D(s,t)=\Vert B(s)-B(t){\Vert }^2$, expanding the square gives

$$\begin{array}{c}\sum _{i,j}{a}_i{a}_{j}D(s_i,s_j)\\ =-2{\Vert \sum _i{a}_{i}B(s_i)\Vert }^2.\end{array}$$

The zero-sum condition cancels the two squared-norm terms. A representation theorem turns the manuscript’s negative-type kernel into such a distance. Using coordinates of $B$ as multipliers cancels the unwanted kernel factor when the energy identities are summed. The source obtains a sum of nonnegative energies equal to half the squared interior gradient.

At a hypothetical critical point, every energy in this sum would vanish. The equality characterization then supplies too many eigenfunction gradients, contradicting the planar multiplicity bound or boundary uniqueness.

The hard analytic part is proving the kernel property. The manuscript regularizes Green kernels, proves a reciprocal-matrix condition through finite updates, and passes through positive quadrature and boundary limits. The order of those limits matters because a boundary Poisson kernel need not have the integrability of an interior Green kernel. The simple squared-distance calculation explains the last mechanism but does not verify those limiting arguments.

🏷️ Key steps and proof outlines

The vector inequality includes an essential equality case

Let $\mu$ be the first positive Neumann eigenvalue and ${\lambda }_D$ the first Dirichlet eigenvalue. The source uses $\mu <{\lambda }_D$. For a tangent vector field $X$, solve

$$\Delta v=\mathrm{div}X,{\partial }_{\nu }v=0,{\int }_{\Omega }v=0,\Delta j=\mathrm{curl}X,j{|}_{\partial \Omega }=0.$$

The remainder $X-\nabla v-{\nabla }^{\perp }j$ has zero curl, divergence, and normal trace. Simple connectivity supplies a potential; testing its weak equation makes its gradient zero. The two remaining fields are orthogonal in $L^2$, so

$$\mathcal{Q}(X)\ge \mu \Vert X{\Vert }_2^2+({\lambda }_D-\mu )\Vert \nabla j{\Vert }_2^2.$$

Equality forces $j=0$ and $v$ to lie in the first positive Neumann eigenspace $\mathcal{V}$. This is stronger than a nonnegative energy: it identifies every zero-energy field as the gradient of another eigenfunction. That identification will turn a hypothetical critical point into too many eigenfunctions.

Reciprocal kernels are controlled by an inertia invariant

The kernel theorem considers a nonnegative disk potential $q$ satisfying $\int |\psi {|}^{2}dq\le d\int |\nabla \psi {|}^2$, with $d<1$. This ensures convergence of its Green resolvent. The key finite-dimensional invariant says that the entrywise reciprocal of a positive kernel matrix has at most one positive eigenvalue. This is not a statement about the matrix inverse.

Suppose a current positive kernel matrix $M$ has that property. Adding one quadrature node to its resolvent updates it by ${\tilde{M}}_{ij}=M_{ij}+c{M}_{iz}{M}_{zj}$, with $c\ge 0$. Normalize the reciprocal by the $z$-column and put

$$H_{ij}=\frac{M_{iz}{M}_{zj}}{M_{ij}},a=M_{zz},U=a11^T-H.$$

The $z$-row of $H$ is constant. A block congruence and the inertia hypothesis give $U⪰0$. The updated normalized reciprocal has entries $H_{ij}/(1+c{H}_{ij})$, and

$$\frac{a}{1+ca}-\frac{H_{ij}}{1+c{H}_{ij}}=\sum _{n\ge 0}\frac{c^n{U}_{ij}^{n+1}}{(1+ca{)}^{n+2}}.$$

Each entrywise power of a positive semidefinite matrix is positive semidefinite, by tensor products of Gram vectors. The updated reciprocal is therefore a positive rank-one matrix minus a positive semidefinite matrix and retains the invariant.

The singular Green diagonal prevents direct quadrature. The source first constructs bounded positive regularizations preserving the invariant, then refines quadrature, removes regularization, takes the boundary limit with compactly supported potential, and finally exhausts the full potential. Keeping this order avoids treating a boundary Poisson section as an $L^2(q)$ function without justification.

A bordered matrix becomes a squared Hilbert distance

At a fixed interior point $p$, the limiting bordered reciprocal matrix has boundary block $B_{ij}=1/N(s_i,s_j)$ and border ${\ell }_i=1/K(p,s_i)$, with zero diagonal. Its one-positive-eigenvalue property forces $x^{T}Bx\le 0$ whenever ${\ell }^{T}x=0$: a positive vector in that hyperplane, together with a suitably chosen border direction, would otherwise create a two-dimensional positive subspace.

Substitute $x_i=K(p,s_i)a_i$. The condition becomes $\sum a_i=0$, proving conditional negativity of

$$D_p(s,t)=\frac{K(p,s)K(p,t)}{N(s,t)}.$$

Anchoring at one boundary point turns this kernel into a positive semidefinite Gram kernel. Completing its formal vector space constructs a Hilbert-valued map $b$ with $\Vert b(s)-b(t){\Vert }^2=D_p(s,t)$. Positivity off the diagonal makes $b$ injective; the kernel bounds make it Lipschitz.

Zero multiplier energy contradicts multiplicity or boundary uniqueness

Let $g$ be the pulled-back boundary gradient of an eigenfunction. Apply the boundary multiplier identity to every real coordinate $b_j$ of $b$. The squared-distance identity and dominated convergence give

$$\sum _{j}E(b_{j}g)=\frac{1}{2}{\left|\int K(p,s)g(s)ds\right|}^2.$$

At a critical point the right side is zero. Every summand is nonnegative, so the vector equality case produces eigenfunctions with boundary gradients $b_{j}g$.

If $g\ne 0$ on a dense set, the injective image of the circle cannot lie in an affine line. Two coordinates therefore make $g,b_{j}g,b_{k}g$ linearly independent, contradicting the source’s multiplicity bound $\dim \mathcal{V}\le 2$. If $g$ vanishes on an open arc, one nonconstant coordinate yields two independent eigenfunctions whose gradients vanish there. A nonzero linear combination has both zero value and zero normal derivative on that arc. Extension by zero solves the constant-coefficient Helmholtz equation distributionally, and interior analyticity forces the combination to vanish. Both alternatives contradict the critical point. The reciprocal-kernel theorem and the eigenspace multiplicity bound are central source results in this chain. (OpenAI, 2026 s “A variational principle for tangent vector fields,” “Reciprocal Green functions and a squared-distance kernel,” and “Hilbert multipliers and the absence of critical points”)

See Also

• Family 365 also extracts interior information from boundary Green kernels.
• Family 071 introduces conformal parametrization as a way to reorganize a planar geometric problem.

🏷️ Manuscripts and verification scope

• Strict hot spots and absence of interior critical points on smooth simply connected planar domains — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The scope page describes nonvanishing interior gradients for every nonzero member of the first positive eigenspace, including multiplicity, and the strict boundary-extrema consequence. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026. Strict hot spots and absence of interior critical points on smooth simply connected planar domains.