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Phys-81-A1 · revision r1 · on Phys-81

OpenAI claim summary for Phys-81

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: open. 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.

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.

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

Manuscript

Contents
  1. Full support of every zero-temperature SK minimizing order parameter
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Result 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

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.

Sources