{"schema":"postcutoff/event@1","as_of":"2026-10-13T02:19:00+02:00","url":"https://postcutoff.com/e/2026-10-06-random-3-sat-threshold-upper-bound-4-268-llm/","md":"https://postcutoff.com/e/2026-10-06-random-3-sat-threshold-upper-bound-4-268-llm/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"Adam Bicz","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-06-random-3-sat-threshold-upper-bound-4-268-llm","date":"2026-10-06","date_precision":"day","short_title":"Preprint claims a 4.268 upper bound for the random 3-SAT threshold, down from 4.4898, with a Lean 4 proof; 'large language models drove the research'","deck":null,"takeaway":"Fedor Vorobyev's arXiv preprint 2610.10599 (Oct 6, 2026) claims that a random 3-SAT formula with 4.268n clauses is unsatisfiable with high probability.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"Vorobyev: An upper bound of 4.268 for the random 3-SAT satisfiability threshold (arXiv 2610.10599)","url":"https://arxiv.org/abs/2610.10599","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"Lean 4 formalization and certificate code (GitHub)","url":"https://github.com/yamelton/sat-upper-4268","type":"code","group":"primary","domain":"github.com"}],"official":2,"filed":"2026-10-13","updated":"2026-10-13","orgs":["Independent"],"title":"Preprint claims a 4.268 upper bound for the random 3-SAT threshold, down from 4.4898, with a Lean 4 proof; 'large language models drove the research'","summary":"Fedor Vorobyev's arXiv preprint 2610.10599 (Oct 6, 2026) claims that a random 3-SAT formula with 4.268n clauses is unsatisfiable with high probability. The best earlier rigorous upper bound was 4.4898, and the cavity method of statistical physics predicts a threshold near 4.267. The paper says \"Large language models drove the research leading to this paper\" and that the proof is formalized in Lean 4. It has no independent expert review yet.","key_facts":["Claim (abstract): 'a random 3-SAT formula with n variables and ⌊4.268n⌋ clauses is unsatisfiable with high probability'","Earlier state, as the abstract gives it: 'successive upper bounds reached 4.4898 in the work of Díaz, Kirousis, Mitsche, and Pérez-Giménez, while the cavity method predicts a value near 4.267'","Method (abstract): builds on interpolation methods, 'including the energetic approach of Achlioptas and Menchaca-Méndez'; bounds the smallest number of clauses any assignment must violate by comparison with a simpler system of constraints on single variables; ends in a finite calculation checked in exact rational arithmetic","AI disclosure (Section 9, verbatim): 'Large language models drove the research leading to this paper. Their contribution included developing the mathematical argument, generating and revising the Lean formalization and numerical code, and drafting and revising the manuscript. The author directed the work and worked with the models to improve the explanations and presentation.' The models are not named","Verification statement (verbatim): 'Lean checks the proof of the formal theorem; the correspondence between the formal definitions and the intended model, and between the formal proof and this exposition, requires human inspection. We do not claim an independent human verification of every step.'","16 pages, single author; Lean sources and certificate code at github.com/yamelton/sat-upper-4268 (Apache-2.0), archived on Zenodo (doi 10.5281/zenodo.23120101)","Reactions: none found. A Hacker News search on Oct 13 returned no thread; X was not searched in this quick run"],"key_numbers":[],"tags":["computer-science","random-3-sat","satisfiability-threshold","probability","lean","formal-verification","llm","preprint"],"science":{"field":"computer-science","subfield":"random constraint satisfaction / probabilistic combinatorics","problem":"Rigorous upper bound on the satisfiability threshold of random 3-SAT (conjectured value near 4.267)","result":"Claimed: random 3-SAT with ⌊4.268n⌋ clauses is unsatisfiable with high probability, improving the upper bound from 4.4898 to 4.268.","open_since":"","ai_system":["unnamed large language models"],"human_role":"AI-led by the author's own account: models developed the argument, the Lean formalization and the text; the author directed the work","verification":{"key":"lean","label":"Lean-verified","text":"Lean 4 formalization of the main theorem by the author; unrefereed preprint, no independent check of the formal statement against the intended model"},"status":"pending","shock":"A rigorous bound that had moved slowly for decades would drop almost to the physics prediction in one 16-page, model-written paper.","short":"Random 3-SAT threshold upper bound"},"body_md":"## What happened\n\nIn random 3-SAT, each clause picks three of n variables at random. As the ratio of clauses to variables grows, formulas go\nfrom almost always satisfiable to almost always unsatisfiable. Physicists' cavity method puts the switch near 4.267, but\nproofs lag: the rigorous upper bound stood at 4.4898.\n\nFedor Vorobyev's preprint claims 4.268, a hair above the predicted value. The argument compares the random formula with a\nsimpler system by interpolation and reduces the bound to a finite computation with rational numbers. The author states that\nlanguage models developed the mathematics, wrote the Lean 4 formalization and drafted the paper, and that he directed them.\n\n## Why it matters\n\nIf it holds, this closes almost all of the gap between what is proved and what physics predicts for one of the best-known\nproblems on random constraint satisfaction. It is also a test of a pattern that is becoming common: a single author, unnamed\nmodels, and a Lean proof in place of peer review.\n\nThe caution is the paper's own. Lean certifies the formal theorem, and a human has to confirm that the formal statement is\nthe real random 3-SAT model. Nobody outside has done that in public yet. The author's background was not checked, and the\nmodels used are not named. Treat the result as a claim until specialists respond.","disputed":[{"label":"Verification","text":"Lean 4 formalization of the main theorem by the author; unrefereed preprint, no independent check of the formal statement against the intended model","sources":[]}],"related":[{"id":"2026-10-06-openai-math-release-722-manuscripts","url":"https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems","deck":"Including quasi-Riemann, Unique Games, Hodge for CM abelian varieties and free group factors","takeaway":"If even a fraction of these results hold up, this is the largest single jump in mathematical knowledge on record, produced by an AI system in about six weeks.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":74,"official":13,"filed":"2026-10-07","updated":"2026-10-12","orgs":["OpenAI"]},{"id":"2026-09-30-ai-assisted-conjecture-wave-summer-2026","url":"https://postcutoff.com/e/2026-09-30-ai-assisted-conjecture-wave-summer-2026/","date":"2026-09-30","date_precision":"day","short_title":"Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help","deck":null,"takeaway":"This entry catalogues about 50 of them, with the AI role as the authors state it.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":9,"official":4,"filed":"2026-09-30","updated":"2026-10-12","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-13","type":"filed","text":"Created from the arXiv section of sweep 2026-10-13 (proof not assessed)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-13","run":null,"sources_read":"The arXiv section of sweep 2026-10-13 (proof not assessed)","updated":"2026-10-13","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":159,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":98,"in_training_data":false},{"model_id":"gemini-3-8-flash","name":"Gemini 3.8 Flash","cutoff":"2026-03","days_after":189,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":128,"in_training_data":false}],"short_url":null}