{"schema":"postcutoff/event@1","as_of":"2026-10-17T07:29:00+02:00","url":"https://postcutoff.com/e/2026-10-16-lee-loh-sudakov-directed-cuts-altifina-lean/","md":"https://postcutoff.com/e/2026-10-16-lee-loh-sudakov-directed-cuts-altifina-lean/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-16-lee-loh-sudakov-directed-cuts-altifina-lean","date":"2026-10-16","date_precision":"day","short_title":"Lee–Loh–Sudakov directed-cut conjecture proved with AI, Lean-verified","deck":null,"takeaway":"A 2013 conjecture on judicious partitions of digraphs, answering Scott's 2005 question for large graphs, was first proved by an AI research framework; the authors say the proofs are Lean-verified.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.21381: Optimally large cuts in both directions (Folz & Tyomkyn)","url":"https://arxiv.org/abs/2610.21381","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"arXiv 1305.2986: Judicious partitions of directed graphs (Lee, Loh & Sudakov)","url":"https://arxiv.org/abs/1305.2986","type":"paper","group":"primary","domain":"arxiv.org"}],"official":2,"filed":"2026-10-17","updated":"2026-10-17","orgs":["Altifina","Charles University","OpenAI","Anthropic"],"title":"Lee–Loh–Sudakov conjecture on cuts in both directions proved; proof first found by a multi-model AI framework (ChatGPT 6 Astra, 6.1 Sol, Claude Opus 5.5) and verified in Lean","summary":"arXiv 2610.21381 (v1 dated 15 Oct, announced 16 Oct 2026) by Frederic Folz (Altifina Discovery Lab) and Mykhaylo Tyomkyn (Charles University) proves that for every d ≥ 2 and all large m, every digraph with m edges and minimum outdegree ≥ d has a vertex bipartition with both directed cuts at least ⌈(d−1)m/(2(2d−1)) + d²/(2(2d−1))⌉, sharp for infinitely many m. This proves the Lee–Loh–Sudakov conjecture (2013) and answers Alex Scott's 2005 question for large graphs. The AI statement says the key proof was \"first discovered by the Altifina research framework\" combining ChatGPT 6 Astra, 6.1 Sol and Claude Opus 5.5, and that the proofs \"have been verified in Lean\". Unrefereed preprint.","key_facts":["Paper: 'Optimally large cuts in both directions', Frederic Folz & Mykhaylo Tyomkyn, arXiv 2610.21381 [math.CO]","Results: the sharp bound above for all large m (Theorem 1.1); ⌈m/6⌉ achievable for d = 2 and every m; a stability result describing near-extremal digraphs (Theorem 1.4)","Background: Scott (Surveys in Combinatorics 2005) asked for the largest c(d) such that every digraph with minimum outdegree ≥ d has a bipartition with both directed cuts ≥ c·m; Lee, Loh & Sudakov ('Judicious partitions of directed graphs', arXiv 1305.2986, RSA 2016) conjectured c_d = (d−1)/(2(2d−1)) and showed it cannot be raised","AI statement (quoted): 'The proof of Theorem 1.2 was first discovered by the Altifina research framework, which combines multiple AI models—in this work, OpenAI's ChatGPT 6 Astra and 6.1 Sol, as well as Anthropic's Claude Opus 5.5. The proof was revised and extended by the second author, aided by ChatGPT 6 Astra, leading to the main results of this paper, Theorems 1.1 and 1.4. The proofs have been verified in Lean, and a certificate is available upon request. The presentation is human-written and proofread with ChatGPT 6 Astra.'","Abstract: 'The proofs were obtained by LLMs, with human input. They have been revised by the authors and verified in Lean.' The Lean code is not public (certificate 'available upon request')","First author's affiliation is Altifina Discovery Lab (Überherrn, Germany), the company behind the framework; no Altifina announcement or Hacker News thread found by Oct 17"],"key_numbers":[],"tags":["math","combinatorics","graph-theory","judicious-partitions","lean","gpt-6-astra","gpt-6-1-sol","claude-opus-5-5","ai-for-math"],"science":{"field":"mathematics","subfield":"extremal graph theory (judicious partitions of digraphs)","problem":"Lee–Loh–Sudakov conjecture on bipartitions with both directed cuts large; Scott's question on c(d)","result":"Sharp bound ⌈(d−1)m/(2(2d−1)) + d²/(2(2d−1))⌉ for every d ≥ 2 and large m, ⌈m/6⌉ for d = 2 and all m, plus stability.","open_since":"2013","ai_system":["Altifina framework","GPT-6 Astra","GPT-6.1 Sol","Claude Opus 5.5"],"human_role":"AI-originated: the core proof (Theorem 1.2) was found by the multi-model framework; Tyomkyn revised and extended it with ChatGPT 6 Astra to the main theorems; humans wrote the paper","verification":{"key":"lean","label":"Lean-verified","text":"Unrefereed preprint; authors state Lean verification (certificate on request, not public)"},"status":"pending","shock":"A conjecture by Lee, Loh and Sudakov fell to a proof that an AI framework found first, with the claim backed by Lean."},"body_md":"## What happened\n\nFolz and Tyomkyn's preprint appeared in arXiv's 16 Oct 2026 announcement. A random-partition argument needs a deterministic\ncriterion (built from a partition of components into induced stars, going back to Erdős, Gyárfás and Kohayakawa) to guarantee\nboth directed cuts are large at once. The authors say that criterion's proof (Theorem 1.2) was first found by the Altifina framework,\nthen revised and extended by Tyomkyn with ChatGPT 6 Astra into the sharp theorem and the stability result.\n\n## Why it matters\n\nIt is one more named conjecture by well-known combinatorialists (Po-Shen Loh, Benny Sudakov) settled with an AI-first proof. The\nLean claim cannot be checked yet because the certificate is not public. The first author's affiliation shows a small company\nrunning a multi-model \"research framework\" that combines OpenAI and Anthropic models.","disputed":[{"label":"Verification","text":"Unrefereed preprint; authors state Lean verification (certificate on request, not public)","sources":[]}],"related":[{"id":"2026-10-17-ai-assisted-results-october-16-listing","url":"https://postcutoff.com/e/2026-10-17-ai-assisted-results-october-16-listing/","date":"2026-10-17","date_precision":"day","short_title":"arXiv results with disclosed AI help, 16 Oct 2026 announcement","deck":"Suffridge's conjecture, two Wahl conjectures, the 6-D Tarski problem and more","takeaway":"A second read of the 16 Oct arXiv announcement, through the HTML listings because the arXiv API was rate-limited, found three results for their own entries and about fifty more preprints with explicit AI statements, several settling named conjectures.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":15,"official":14,"filed":"2026-10-17","updated":"2026-10-17","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]},{"id":"2026-10-16-ai-assisted-results-october-15-listing","url":"https://postcutoff.com/e/2026-10-16-ai-assisted-results-october-15-listing/","date":"2026-10-16","date_precision":"day","short_title":"arXiv results with disclosed AI help, 15 Oct 2026 listing","deck":"Monodromy conjecture, Fried counterexamples, Godefroy–Lancien–Zizler question and more","takeaway":"One day's maths listing brought five results big enough for their own entries (Smale's mean value constant, Hopf with T³, class numbers ≤ 2000, the Delsarte rate, LLM-found quantum circuits) and about two dozen more with explicit AI statements.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":12,"official":12,"filed":"2026-10-16","updated":"2026-10-16","orgs":["OpenAI","Anthropic","various mathematicians"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-17","type":"filed","text":"Created (arXiv /list/math/new page for 16 Oct 2026, read because the arXiv API returned HTTP 429; PDF read with pdftotext)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-17","run":null,"sources_read":"ArXiv /list/math/new page for 16 Oct 2026, read because the arXiv API returned HTTP 429; PDF read with pdftotext","updated":"2026-10-17","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":169,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":108,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":199,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":138,"in_training_data":false}],"short_url":null}