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Math-2893 · mathematics · Probability

The thermodynamic pressure of every bounded Gaussian Ising perceptron activation

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For $M=\lfloor\alpha N\rfloor$ independent identically distributed standard Gaussian constraint rows $G_j$ and uniform Ising spins $\sigma$, and every $\alpha>0$ and bounded Borel function $f:\mathbb{R}\to\mathbb{R}$, does \[ N^{-1}\log\mathbb{E}_{\mathrm{uniform\text{-}spin}} \exp\left(\sum_{j\leq M}f\left(\frac{G_j\cdot\sigma}{\sqrt N}\right)\right) \] converge in mean and probability to the hierarchical variational value \[ \inf_q\{\alpha V_f(q)+S_I(q)\}, \] with $V_f$ and $S_I$ the explicitly defined cavity and Ising entropy functionals? No activation symmetry, concavity or smallness is assumed.

Short English statement

For every $\alpha>0$ and bounded Borel $f:\mathbb{R}\to\mathbb{R}$, with $M=\lfloor\alpha N\rfloor$ independent identically distributed standard Gaussian constraint rows $G_j$ and uniform Ising spins $\sigma$, does $N^{-1}\log\mathbb{E}_{\mathrm{uniform\text{-}spin}}\exp(\sum_{j\leq M}f(G_j\cdot\sigma/\sqrt N))$ converge in mean and probability to $\inf_q\{\alpha V_f(q)+S_I(q)\}$, where $V_f$ and $S_I$ are the explicitly defined cavity and Ising entropy functionals, without assuming activation symmetry, concavity or smallness?

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Math-2893-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2893

The Ising Gaussian perceptron is claimed to have an exact pressure formula for every bounded Borel log-potential at positive temperature and density. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

The thermodynamic pressure of every bounded Gaussian Ising perceptron activation

OpenAI's claim

OpenAI presents a proof in the selected manuscript. The Ising Gaussian perceptron is claimed to have an exact pressure formula for every bounded Borel log-potential at positive temperature and density.

Mathematical statement of the claimed result

Result 1.
Fix $\alpha>0$ and any bounded Borel $f:\mathbb R\to\mathbb R$. Let $M=\lfloor\alpha N\rfloor$, and take independent standard Gaussian vectors $g^1,\ldots,g^M\in\mathbb R^N$. With uniform spin probability, \[ p_N(f)=N^{-1}\log\left(2^{-N}\sum_{x\in\{-1,1\}^N} \exp\sum_{a=1}^Mf(g^a\cdot x/\sqrt N)\right). \] For $G\sim N(0,1)$ define \[ \mathcal T_{s,d}U(x)=\begin{cases}d^{-1}\log\mathbb Ee^{dU(x+\sqrt sG)},&d>0,\\ \mathbb EU(x+\sqrt sG),&d=0.\end{cases} \] Let $\mathcal Q$ contain nondecreasing paths $q:(0,1)\to[0,1]$. For a step path with values $q_i$ on $\zeta_i<u<\zeta_{i+1}$, $0=\zeta_0<\cdots<\zeta_k=1$, append $q_{-1}=0,q_k=1$, start $U_k=f$, and recurse $U_{i-1}=\mathcal T_{q_i-q_{i-1},\zeta_i}U_i$, $i=k,\ldots,0$. Define $V_f(q)=U_{-1}(0)$, using its continuous path extension for general $q$. For a finite nonnegative nondecreasing step field $h$, instead start $U_{k-1}(x)=\log\cosh x-h_{k-1}/2$, append $h_{-1}=0$, and recurse $U_{i-1}=\mathcal T_{h_i-h_{i-1},\zeta_i}U_i$. Put $\ell(h)=U_{-1}(0)$ and \[ S_{\mathrm I}(q)=\sup_h\left\{\ell(h)+\frac12\int_0^1h(u)q(u)\,du\right\}. \] The finite limiting value is \[ \lim_N\mathbb Ep_N(f)=\inf_{q\in\mathcal Q}\{\alpha V_f(q)+S_{\mathrm I}(q)\}=:\mathcal P_f(\alpha), \qquad p_N(f)\xrightarrow{\mathbb P}\mathcal P_f(\alpha). \] In particular $f=\beta\phi$ is allowed for every $\beta>0$ and bounded continuous $\phi$.

Outline of the argument

Hierarchical Gaussian enrichment and overlap identities produce a cascade description. Contact-point supersolution estimates for the pattern-density evolution supply the nonconvex upper bound, while cavity calculations give the matching lower bound and approximation treats merely Borel potentials.

Formal verification

For family 222, the spherical Gaussian perceptron pressure formula is included. The Ising companion entry only proves finite variational value for bounded continuous log-potentials; it neither gives its pressure convergence nor the bounded-Borel extension. A spherical-field supplement supplies a dual approximation formula. The listed Comparator files are IsingFiniteness.lean, PerceptronFreeEnergy.lean, SphericalField.lean. The inventory does not supply a named declaration for these Comparator entries. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

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