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Full support of every zero-temperature SK minimizing order parameter

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
If gamma minimizes the stated zero-temperature SK Parisi functional over its admissible class, must its Stieltjes measure have support [0,1), with gamma strictly increasing on every subinterval? Require the associated Parisi diffusion identities $E[u_\gamma (t,X_t)^2]=t$ and $E[Phi_{\gamma ,xx}(t,X_t)^2]=1$ for each $0\leq t<1$. The question does not assert minimizer existence.

Short English statement

If gamma minimizes the stated zero-temperature SK Parisi functional over its admissible class, must its Stieltjes measure have support [0,1), with gamma strictly increasing on every subinterval? Require the associated Parisi diffusion identities $E[u_\gamma (t,X_t)^2]=t$ and $E[Phi_{\gamma ,xx}(t,X_t)^2]=1$ for each $0\leq t<1$. The question does not assert minimizer existence.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-81-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-81

The full-support theorem applies to every integrable zero-temperature Parisi minimizer for pure zero-field SK, giving the target support throughout the relative interval below overlap one. The companion optimization corollary is an additional algorithmic application, with unit-cost real arithmetic rather than a bit-complexity guarantee.

Full support of every zero-temperature SK minimizing order parameter

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Every integrable zero-temperature Parisi minimizer for pure zero-field SK is claimed to have full relative Stieltjes support below overlap one.

Mathematical statement of the claimed result

Result 1.
Let $\mathcal U$ contain nonnegative, nondecreasing right-continuous $\gamma:[0,1)\to[0,\infty)$ with $\int_0^1\gamma<\infty$, and define $\mu_\gamma([0,t])=\gamma(t)$. For covariance $\xi(t)=t^2/2$, let \[ \Phi_t=-\tfrac12(\Phi_{xx}+\gamma(t)\Phi_x^2),\quad\Phi(1,x)=|x|, \qquad\mathcal P(\gamma)=\Phi_\gamma(0,0)-\tfrac12\int_0^1t\gamma(t)\,dt. \] Every minimizing $\gamma$ is claimed to satisfy \[ \operatorname{supp}\mu_\gamma=[0,1),\qquad \gamma(b)>\gamma(a)\quad(0\le a<b<1). \] For $dX_t=\gamma(t)\Phi_{\gamma,x}(t,X_t)dt+dB_t$, $X_0=0$, the source also gives $\mathbb E\Phi_{\gamma,x}(t,X_t)^2=t$ and $\mathbb E\Phi_{\gamma,xx}(t,X_t)^2=1$ for $t<1$. The optimization corollary, for $H_n(\sigma)=n^{-1/2}\sum_{i<j}J_{ij}\sigma_i\sigma_j$, gives a randomized $C(\varepsilon)n^2$-operation algorithm whose value is at least $\max H_n-\varepsilon n$ with probability tending to one. This operation model charges one unit for real sums/products and uses prescribed scalar functions; no separate finite-precision preprocessing bound is claimed.

Outline of the argument

Controlled diffusion treats integrable coefficients beyond finite step laws. Backward shape inequalities are proved first for step functions and extended by approximation; variational support conditions exclude any interval gap in the minimizing order parameter.

Formal verification

For family 281, the selected value identities express the SK ground energy through an admissible Parisi minimizer and diffusion, without asserting QAOA circuit convergence. A distinct full-support statement covers integrable zero-temperature minimizers. The listed Comparator files are SKFullSupport.lean, SKValue.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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