OpenAI claim summary for Math-2921

Oracle-free computation of the limiting random 3-SAT threshold

OpenAI's claim

OpenAI presents a proof in the selected manuscript. The limiting random 3-SAT threshold is claimed computable to any requested accuracy by one finite deterministic procedure.

Mathematical statement of the claimed result

Result 1.

For uniform proper signed random 3-SAT clauses sampled independently with replacement, write $p(n,m)$ for satisfiability probability. The source asserts $\alpha_3\in(0,\infty)$ with \[ p(n,\lfloor an\rfloor)\to1\ (a<\alpha_3),\qquad p(n,\lfloor an\rfloor)\to0\ (a>\alpha_3). \] It also claims a single deterministic Turing machine that, on unary input $1^r$, halts with rational $q_r$ satisfying \[ |q_r-\alpha_3|\le2^{-r}\qquad(r\ge1). \] This machine has no oracle, advice, or noncomputable constant as input; no polynomial running-time bound is asserted. The clause-density parameter is restricted to $a\ge0$.

Outline of the argument

Deletion bounds certify rational lower brackets, and finite hierarchical soft-pressure trials certify upper brackets. A fair enumeration of both certificate families halts at every accuracy, using existence of the limiting threshold without needing an effective finite-size convergence rate.

Formal verification

For family 235, the selected entries establish threshold existence, threshold computability, linear variance for $k\geq4$, and the earlier $O(n\log n)$ bound for $k=3$. The later linear three-clause variance improvement is not selected. The listed Comparator files are FixedClauseThreshold.lean, SATComputability.lean, SATSharpness.lean, SATVariance.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

• OpenAI, Computing the Random 3-SAT Threshold, 2026-09-27; citation key OAI:Computing-the-Random-3-SAT-Threshold-September-27-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• Family 235: Lean scope.
• Comparator FixedClauseThreshold.json.
• Comparator FixedClauseThreshold.lean.
• Comparator SATComputability.json.
• Comparator SATComputability.lean.
• Comparator SATSharpness.json.
• Comparator SATSharpness.lean.
• Comparator SATVariance.json.
• Comparator SATVariance.lean.
• OpenAI catalogue entry.