OpenAI claim summary for Math-2914

A subexponential mixing bound from typical low-temperature SK starts

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Typical fixed equilibrium starts of low-temperature SK dynamics are claimed to mix by $\exp(n^{1-1/40000000})$ with high joint probability.

Mathematical statement of the claimed result

Result 1.

For independent standard Gaussian $J_{ij}$, $i<j$, with $J_{ji}=J_{ij},J_{ii}=0$, use the zero-field SK measure \[ \pi_{n,\beta}^J(x)=Z^{-1}\exp\left(\frac\beta{\sqrt n}\sum_{i<j}J_{ij}x_ix_j\right),\qquad x\in\{-1,1\}^n. \] Continuous heat-bath dynamics refreshes every site at rate one; a discrete step refreshes one uniformly selected site, including attempts that retain its spin. Fix $\beta>1$ and let $T_{n,\beta}^J(x)$ be the first continuous time when the transition law from fixed $x$ has total-variation distance at most $1/4$ from $\pi$. There exists $c_\beta\in(0,1)$ such that \[ \mathbb P_{J,x_0\sim\pi_{n,\beta}^J} \{T_{n,\beta}^J(x_0)\le\exp(n^{1-c_\beta})\}\longrightarrow1. \] The explicit choice $c_\beta=1/40000000$ works for every fixed $\beta>1$. This is a typical-Gibbs-start assertion, rather than a supremum over starting states.

Outline of the argument

Gaussian completion and an entropy--energy identity give a smoothed proposal built from noisy Gaussian queries. Reverse comparison relates its transitions to the heat-bath chain, and typical-state control avoids a worst-start barrier while producing the stated subexponential horizon.

Formal verification

For family 227, the selected statements include the high-temperature gap for $\beta<1$, cutoff ratios only for $\beta<1/2$, critical lower mixing scales with a subexponential upper bound, and low-temperature barriers. They do not provide the full critical exponent, the full cutoff location, or the newer autocorrelation limits. No Comparator is assigned to the selected manuscript here; the family page selects other papers or supporting statements. 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

• OpenAI, A typical-start upper bound for low-temperature SK Glauber dynamics, 2026-09-24; citation key OAI:A-typical-start-upper-bound-for-low-temperature-SK-Glauber-dynamics-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• Family 227: Lean scope.
• OpenAI catalogue entry.