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Linear Variance of the Random 3-SAT Hitting Time
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Abstract
For random 3-SAT on n Boolean variables, with independent uniformly signed clauses on three distinct variables sampled with replacement, we prove that the first unsatisfiable prefix has variance $\Theta(n)$. The upper bound removes the logarithmic loss in the earlier variance estimate; the matching lower bound follows from Wilson's transition-width theorem.
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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). This paper is not among those listed in the scope notes; for $k=3$ the notes describe only the positive linear lower bound and the $O(n\log n)$ upper bound from the companion paper, not the $\Theta(n)$ upper bound claimed here. [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 $O(n^{1/2+1/k})$ scaling-window bound provides the concurrent comparison for this paper’s sharper 3-SAT fluctuation claim, and the release credits it with priority for threshold existence. [family-level]
Possibly relevant literature (found by the agent, not checked)
- B. Bollobás, C. Borgs, J. T. Chayes, J. H. Kim, D. B. Wilson, The Scaling Window of the 2-SAT Transition (Random Structures Algorithms 18 (2001); arXiv:math/9909031). This paper determines the 2-SAT scaling window as $1\pm\Theta(n^{-1/3})$ in density, which gives a point of comparison for the order-$\sqrt n$ 3-SAT clause window in the paper's Corollary (
cor:window).
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:Linear-Variance-of-the-Random-3-SAT-Hitting-Time-October-5-2026,
author = {{OpenAI}},
title = {{Linear Variance of the Random 3-SAT Hitting Time}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Linear-Variance-of-the-Random-3-SAT-Hitting-Time-October-5-2026/linear-variance-of-the-random-3-sat-hitting-time.pdf}{OAI:Linear-Variance-of-the-Random-3-SAT-Hitting-Time-October-5-2026}},
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
- Variance of the Random k-SAT Hitting Time · September 27, 2026
- Computing the Random 3-SAT Threshold · September 27, 2026