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Computing the Random 3-SAT Threshold
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Abstract
The limiting satisfiability threshold of uniform random 3-SAT is a computable real. We credit Gaia Carenini [[4]](https://eccc.weizmann.ac.il/report/2026/229/) with priority for resolving the satisfiability conjecture, which establishes the threshold's existence. We prove that one finite deterministic machine can approximate it to any prescribed accuracy. A deletion estimate gives explicit lower certificates, while a finite hierarchical approximation of the soft pressure gives upper certificates at every larger rational density. A fair search through these certificates halts without requiring a computable rate of finite-size convergence.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Limiting random SAT thresholds, sharp variance and computability — Lean scope notes (lean/docs/235.md, October 2026). Describes a Lean statement that the random 3-SAT threshold $\alpha$ is a computable positive real (one machine outputs $q_r$ with $|q_r-\alpha|\le2^{-r}$), with no computable rate of finite-size convergence assumed in the statement (comparator
SATComputability.lean). [family-level]
Related prior literature mentioned by others
- Gaia Carenini, A polynomial scaling window for random k-SAT and a proof of the satisfiability conjecture (ECCC TR26-229, 5 October 2026). Its limiting-threshold theorem supplies the existence result distinguished from computability in this paper; the release and OfficeChai credit Carenini with priority for that existence result. [family-level]
Possibly relevant literature (found by the agent, not checked)
- F. Vorobyev, An upper bound of 4.268 for the random 3-SAT satisfiability threshold (arXiv:2610.10599, 2026-10-06). Claims that random 3-SAT at density 4.268 is unsatisfiable with high probability, via a finite rational certificate with a Lean 4 formalization, which is below the 4.4898 upper bound the paper quotes as bracketing $\alpha_3$; it does not mention the OpenAI papers.
News and blogs
- OfficeChai Team, OpenAI Releases Over 300 Mathematical Results Produced By An Internal Model (OfficeChai, 7 October 2026). Discusses the random SAT family specifically through its credit to Carenini for priority on threshold existence. [family-level]
Elsewhere
- Emergent Mind: family 235
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Computing-the-Random-3-SAT-Threshold-September-27-2026,
author = {{OpenAI}},
title = {{Computing the Random 3-SAT Threshold}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Computing-the-Random-3-SAT-Threshold-September-27-2026/article.pdf}{OAI:Computing-the-Random-3-SAT-Threshold-September-27-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/235.md. The Lean statements have not been checked by people against the paper's claims.
Family 235: Limiting random SAT thresholds, sharp variance and computability
For random k-SAT with independent uniformly signed proper clauses sampled with replacement, proves finite positive limiting thresholds and hitting-time variance $\Theta_k(n)$ for every fixed k ≥ 3, and computability of the 3-SAT threshold. We credit Gaia Carenini with priority for resolving the threshold-existence conjecture in her concurrent ECCC TR26-229, made public October 5, 2026; this family supplies another proof and the sharper variance and computability results.
Other papers in this family:
- A Limiting Satisfiability Threshold for Every Fixed Clause Size · September 25, 2026
- Linear Variance of the Random 3-SAT Hitting Time · October 5, 2026
- Variance of the Random k-SAT Hitting Time · September 27, 2026