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A Limiting Satisfiability Threshold for Every Fixed Clause Size
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Abstract
For every fixed integer k ≥ 3, random k-SAT with independent uniformly signed clauses on distinct variables, sampled with replacement, has a finite positive limiting satisfiability threshold. We credit Gaia Carenini [[5]](https://eccc.weizmann.ac.il/report/2026/229/) with priority for resolving the satisfiability conjecture. This paper gives an alternative proof, using concentration of a capped last satisfiable index and a comparison between different system sizes.
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Notes
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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 of the existence of a finite positive limiting threshold $\alpha_k$ for every fixed $k\ge3$, for independent proper uniformly signed clauses sampled with replacement (comparator
FixedClauseThreshold.lean), with no assertion at $c=\alpha_k$. [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). This concurrent work establishes limiting thresholds for every fixed $k\ge3$ through a polynomial window bound and the Abbe–Montanari criterion; the release and OfficeChai credit it with priority. [family-level]
Possibly relevant literature (found by the agent, not checked)
- W. Perkins, Searching for (sharp) thresholds in random structures: where are we now? (arXiv:2401.01800, 2024). A survey of thresholds and sharp thresholds in random structures, motivated in part by the Ding–Sly–Sun large-$k$ result; it gives background on the satisfiability conjecture that Theorem 1.1 addresses for every fixed $k\ge3$.
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
- VibeMathed: Problem record
- Emergent Mind: family 235
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026,
author = {{OpenAI}},
title = {{A Limiting Satisfiability Threshold for Every Fixed Clause Size}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/article.pdf}{OAI:A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026}},
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:
- Linear Variance of the Random 3-SAT Hitting Time · October 5, 2026
- Variance of the Random k-SAT Hitting Time · September 27, 2026
- Computing the Random 3-SAT Threshold · September 27, 2026