{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2026-11-12-grid-minor-sixth-power-hajiaghayi-astra/","md":"https://postcutoff.com/e/2026-11-12-grid-minor-sixth-power-hajiaghayi-astra/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":"2026-11-12-grid-minor-sixth-power-hajiaghayi-astra","date":"2026-11-12","date_precision":"day","short_title":"Grid-minor exponent 6 in polynomial time (Hajiaghayi with Astra)","deck":null,"takeaway":"Robertson and Seymour's grid-minor theorem is central to graph minors and bidimensionality; Chuzhoy and Tan's g⁹ bound (2021) was the best. Two AI-assisted claims now go below it: Édouard Bonnet's Lean-checked g⁸·polylog with Codex, and Hajiaghayi's g⁶·polylog with a polynomial-time algorithm, developed with OpenAI Astra.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2611.16755: A Polynomial-Time Sixth-Power Excluded Grid Theorem (Hajiaghayi)","url":"https://arxiv.org/abs/2611.16755","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"Lax Lean Archive lax-17: An Exponent 8 Bound for the Grid-Minor Theorem (Bonnet, Codex 5.5, Codex 5.6)","url":"https://laxarchive.org/lax-17/","type":"code","group":"primary","domain":"laxarchive.org"},{"n":3,"title":"Lax Lean Archive (index of Lean-checked submissions)","url":"https://laxarchive.org/","type":"docs","group":"primary","domain":"laxarchive.org"}],"official":3,"filed":"2026-11-13","updated":"2026-11-13","orgs":["OpenAI","University of Maryland","various researchers"],"title":"Grid-minor theorem: Hajiaghayi claims treewidth g⁶·polylog forces a g×g grid minor, found in polynomial time, from 'extensive interaction' with OpenAI Astra; a Codex-formalized exponent-8 bound sits on the Lax Lean Archive","summary":"arXiv 2611.16755 (12 Nov 2026), Mohammad Hajiaghayi: every graph with treewidth at least g⁶ times a polylogarithmic factor has a g×g grid minor, found by a randomized polynomial-time algorithm with constant probability; this improves Chuzhoy–Tan's g⁹ existence bound and Chekuri–Chuzhoy's g⁹⁸ polynomial-time threshold. \"This work resulted from extensive interaction between the author and OpenAI Astra.\" It cites Bonnet, Codex 5.5 and Codex 5.6, \"An exponent 8 bound for the grid-minor theorem\", a Lean-checked submission (lax-17) on the Lax Lean Archive. Preprint, no formalization of the exponent-6 result.","key_facts":["Claim (abstract): 'Every graph whose treewidth is at least g⁶ times a fixed polylogarithmic factor contains a g×g grid minor that our randomized algorithm finds with constant probability. The algorithm outputs an explicit minor model, and the exponent in its running time is independent of g.'","Prior bounds: Robertson–Seymour (1986) exponential; Chekuri–Chuzhoy (2016) polynomial, g⁹⁸ threshold in their polynomial-time construction; Chuzhoy–Tan (2021) g⁹·polylog existence; known lower bound g² log g","AI Use Statement (verbatim): 'This work resulted from extensive interaction between the author and OpenAI Astra, with the author providing suggestions and critically checking Astra's responses through iterative exchanges. OpenAI Astra was also used in writing and revising the manuscript through detailed prompts from the author, with Gemini Argon assisting in some parts.' A GitHub link (hajiaghayi/Grid-Minor) is announced","Lax Lean Archive lax-17 (Édouard Bonnet, with Codex 5.5 and Codex 5.6 credited as formalizers): treewidth ≥ K·g⁸·(log g)^b forces a g×g grid minor, via 'a logarithmic-depth amortized controller' for Chuzhoy–Tan's recursive slicing; the abstract says the combinatorial producers, controller, numerical inequalities and final composition 'are checked in Lean'; running time not covered; registered 2 Aug 2026, no reviews or flags listed","Status: exponent 6 is an unrefereed preprint with no formal proof; exponent 8 is Lean-checked per the archive page; neither has expert commentary found by 13 Nov"],"key_numbers":[],"tags":["computer-science","graph-theory","graph-minors","treewidth","gpt-6-astra","codex","gemini-4-argon","lean","arxiv"],"science":{"field":"computer-science","subfield":"structural graph theory","problem":"Polynomial grid-minor theorem: smallest exponent c with treewidth ≥ g^c·polylog(g) forcing a g×g grid minor","result":"Claimed c = 6 with a randomized polynomial-time construction (Hajiaghayi); c = 8 Lean-checked (Bonnet with Codex)","open_since":"","ai_system":["GPT-6 Astra","Gemini 4 Argon","Codex"],"human_role":"Hajiaghayi directed and checked an extended exchange with Astra; Bonnet formalized with Codex","verification":{"key":"lean","label":"Lean-verified","text":"Preprint (exponent 6); Lean-checked on the Lax Lean Archive (exponent 8)"},"status":"pending","shock":""},"body_md":"## What happened\n\nOn 12 Nov 2026 Mohammad Hajiaghayi (University of Maryland) posted a preprint claiming a sixth-power excluded-grid theorem with a\npolynomial-time algorithm that outputs the grid minor. He states the work came from an extended exchange with OpenAI's Astra, which\nalso helped write the paper, with Gemini Argon in some parts. The paper cites an exponent-8 bound by Édouard Bonnet, formalized in\nLean with OpenAI's Codex and posted on the Lax Lean Archive, a site that lists Lean-checked submissions.\n\n## Why it matters\n\nQuantitative grid-minor bounds drive the running times of many algorithms on sparse graphs (bidimensionality). Going from g⁹ to g⁶\nwould be a large step, but it is an unrefereed claim; the exponent-8 result has a Lean check but no published human review.","disputed":[{"label":"Verification","text":"Preprint (exponent 6); Lean-checked on the Lax Lean Archive (exponent 8)","sources":[]}],"related":[{"id":"2026-11-13-ai-assisted-results-november-13-listing","url":"https://postcutoff.com/e/2026-11-13-ai-assisted-results-november-13-listing/","date":"2026-11-13","date_precision":"day","short_title":"AI-assisted results, 13 Nov","deck":null,"takeaway":"The arXiv listings of 12–13 Nov carried about 190 maths papers with an AI statement. Most disclose editing or literature search; about forty credit models with proofs, constructions or counterexamples. Seven got their own entries; the rest are catalogued here with the disclosure quoted. All are unrefereed preprints.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":27,"official":27,"filed":"2026-11-13","updated":"2026-11-13","orgs":["OpenAI","Anthropic","Google DeepMind","Axiom Math","various researchers"]},{"id":"2026-10-29-online-steiner-forest-greedy-log-k-loglog-k-gpt-astra-lean","url":"https://postcutoff.com/e/2026-10-29-online-steiner-forest-greedy-log-k-loglog-k-gpt-astra-lean/","date":"2026-10-29","date_precision":"day","short_title":"Online Steiner forest: greedy claimed Θ(log k log log k), with GPT Astra","deck":null,"takeaway":"In online Steiner forest, terminal pairs arrive one by one and must be connected at once. Awerbuch, Azar and Bartal proved in 1996 that the greedy algorithm is O(log² k)-competitive and conjectured O(log k). On 29 Oct a University of Maryland group led by MohammadTaghi Hajiaghayi posted matching bounds of Θ(log k log log k). If correct, that refutes the conjecture and separates greedy from the best online algorithm by a log log k factor. Some proofs 'were developed with the help of OpenAI's GPT Astra', with Gemini Argon assisting, and Lean files for the main theorem are on GitHub. It is a preprint, not refereed.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-30","updated":"2026-10-30","orgs":["OpenAI","Google DeepMind","University of Maryland"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-11-13","type":"filed","text":"Created (science agent; arXiv PDF and the lax-17 page read)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-11-13","run":null,"sources_read":"Science agent; arXiv PDF and the lax-17 page read","updated":"2026-11-13","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":196,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":135,"in_training_data":false},{"model_id":"gemini-4-hydrogen","name":"Gemini 4 Hydrogen","cutoff":"2026-03","days_after":226,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":165,"in_training_data":false}],"short_url":null}