{"schema":"postcutoff/event@1","as_of":"2026-10-15T23:34:00+02:00","url":"https://postcutoff.com/e/2026-10-06-doubleai-high-girth-regular-graphs-8-5/","md":"https://postcutoff.com/e/2026-10-06-doubleai-high-girth-regular-graphs-8-5/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":"2026-10-06-doubleai-high-girth-regular-graphs-8-5","date":"2026-10-06","date_precision":"day","short_title":"doubleAI's 'Double Agent' claims explicit regular graphs of girth (8/5)·log n, beating the 4/3 constant of Margulis and Lubotzky–Phillips–Sarnak (1988)","deck":null,"takeaway":"Amnon Shashua's start-up doubleAI posted a 40-page paper credited only to its AI 'Double Agent': for every prime power q, explicit (q³+1)-regular graphs with girth ≥ (8/5)·log_{q³} n − O(1), improving the 4/3 constant that had stood since 1988. A Lean proof is posted; the company says it ran autonomously for under $2,000 of compute. Not peer-reviewed.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"doubleAI: Advancing a decades-old problem in extremal graph theory","url":"https://www.doubleai.com/research/advancing-a-decades-old-problem-in-extremal-graph-theory","type":"official","group":"primary","domain":"doubleai.com"},{"n":2,"title":"Paper (PDF): d-Regular Graphs of Girth 8/5 log_{d−1} n","url":"https://www.doubleai.com/papers/girth85.pdf","type":"paper","group":"primary","domain":"doubleai.com"},{"n":3,"title":"Lean proof (zip)","url":"https://www.doubleai.com/papers/girth85-lean.zip","type":"code","group":"primary","domain":"doubleai.com"},{"n":4,"title":"Calcalist Tech: OpenAI is solving decades-old math problems. An Israeli startup is taking a different route (8 Oct)","url":"https://www.calcalistech.com/ctechnews/article/hswjujwr5","type":"press","group":"press","domain":"calcalistech.com"},{"n":5,"title":"Startup Fortune: An Israeli AI startup says it cracked a 38 year old graph theory bound for $2,000 (15 Oct)","url":"https://startupfortune.com/an-israeli-ai-startup-says-it-cracked-a-38-year-old-graph-theory-bound-for-2000","type":"press","group":"press","domain":"startupfortune.com"}],"official":3,"filed":"2026-10-15","updated":"2026-10-15","orgs":["doubleAI"],"title":"doubleAI's 'Double Agent' claims explicit regular graphs of girth (8/5)·log n, beating the 4/3 constant of Margulis and Lubotzky–Phillips–Sarnak (1988)","summary":"On 6 Oct 2026 doubleAI, the Israeli company founded by Mobileye co-founder Amnon Shashua, published \"d-Regular Graphs of Girth 8/5 log_{d−1} n\", authored by its AI system \"Double Agent\". For every prime power q it constructs explicit (q³+1)-regular graphs with girth at least (8/5)·log_{q³} n − O(1), improving the constant 4/3 of Margulis and of Lubotzky, Phillips and Sarnak, the best known since 1988 (the Moore bound caps it at 2). The graphs are quotients of a Bruhat–Tits tree of a special unitary group in four variables. A Lean proof is posted; per press reports the company says the run was autonomous and cost under $2,000. No independent expert assessment yet.","key_facts":["Paper: 'd-Regular Graphs of Girth 8/5 log_{d−1} n', Double Agent, doubleAI, October 2026, 40 pages (doubleai.com/papers/girth85.pdf); Lean proof as a zip (doubleai.com/papers/girth85-lean.zip); no arXiv version found","Abstract: 'For every prime power q we construct an explicit infinite family of (q³+1)-regular graphs whose girth satisfies girth ≥ 8/5 log_{q³} n − O(1) … This improves the constant 4/3 of Margulis and of Lubotzky, Phillips and Sarnak, which has been the best known since 1988.' Degrees: 9, 28, 65, … (q = 2, 3, 4, …)","Method: quotients of the Bruhat–Tits tree of SU in four variables of Witt index one, by congruence subgroups of an arithmetic group from a degree-4 division algebra with an involution of the second kind; the girth bound 'rests on a certificate' bounding det(w − I) for congruence elements","For one explicit family of 28-regular graphs the paper also proves a matching upper bound: girth = (8/5)·log₂₇ n + O(1)","Company claims (Calcalist, 8 Oct; Startup Fortune, 15 Oct): 'the first improvement in 38 years on a central problem in extremal graph theory'; produced autonomously, 'No human steering the math', for under $2,000 of compute. The paper lists no human authors and does not describe the process","Startup Fortune notes that Lean checks logical consistency, not whether the theorem is stated as the field intends or whether the construction is new, and that the result is not peer-reviewed; Calcalist presents it as a company claim","Reach: an HN submission of the PDF (7 Oct) drew 1 point; no expert reactions found as of 15 Oct"],"key_numbers":[],"tags":["math","combinatorics","graph-theory","extremal-graph-theory","girth","expander-graphs","lean","israel"],"science":{"field":"mathematics","subfield":"extremal graph theory / expander graphs","problem":"Largest girth of explicit d-regular graphs on n vertices: girth ≥ c·log_{d−1} n with c = 4/3 (Margulis; Lubotzky–Phillips–Sarnak, 1988); Moore bound gives c ≤ 2","result":"Claimed: c = 8/5 for explicit families of (q³+1)-regular graphs, q any prime power; exact girth (8/5)·log₂₇ n + O(1) for one 28-regular family.","open_since":"1988","ai_system":["Double Agent (doubleAI)"],"human_role":"Company says autonomous (no human steering); the paper names no human authors and gives no process details","verification":{"key":"peer-review","label":"Peer-reviewed","text":"Lean proof published by the company (not independently built by us); not peer-reviewed; no expert assessment yet"},"status":"pending","shock":"The 4/3 constant from the Ramanujan-graph constructions of 1988 had been the record for explicit high-girth graphs for 38 years."},"body_md":"## What happened\n\ndoubleAI, which so far had been known for its WarpSpeed GPU-kernel optimizer and the FormulaOne benchmark, posted a 40-page\npaper whose only listed author is its AI system, \"Double Agent\". The paper builds explicit infinite families of regular\ngraphs whose shortest cycle (girth) grows like (8/5)·log n. The previous record for explicit constructions, 4/3, came from\nthe Ramanujan graphs of Margulis and of Lubotzky, Phillips and Sarnak in 1988. A Lean formalization is offered as a\ndownload. The company told Calcalist that the system worked autonomously for under $2,000 of compute.\n\nWe found the result through a 15 Oct Startup Fortune article in the news sweep, nine days after the announcement. There was\nno arXiv preprint, the HN post got no traction, and the company's page has no human authors to search for.\n\n## Why it matters\n\nHigh-girth regular graphs are used in coding theory (LDPC codes) and in expander constructions, and the 4/3 constant was a\nlong-standing benchmark. If experts confirm the result, it is a solo AI result on a classic problem from a small company,\noutside the big US labs. Until then it remains a company claim, formally checked by the company's own Lean files.","disputed":[{"label":"Verification","text":"Lean proof published by the company (not independently built by us); not peer-reviewed; no expert assessment yet","sources":[]}],"related":[{"id":"2026-10-14-ai-assisted-results-mid-october-2026","url":"https://postcutoff.com/e/2026-10-14-ai-assisted-results-mid-october-2026/","date":"2026-10-14","date_precision":"day","short_title":"Mid-October 2026: arXiv results with disclosed AI help","deck":null,"takeaway":"The 13 Oct arXiv listing alone brought a human write-up of GPT-6 Astra's rational exponents proof, Ulam's Lie group question answered with an Astra construction, a better-than-2 k-median algorithm, and about a dozen smaller results whose AI statements range from 'suggested by ChatGPT' to 'GPT 6 Pro was used … to obtain all other new results'.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":20,"official":19,"filed":"2026-10-14","updated":"2026-10-15","orgs":["OpenAI","Anthropic","various mathematicians"]},{"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":84,"official":13,"filed":"2026-10-07","updated":"2026-10-15","orgs":["OpenAI"]},{"id":"2026-09-15-erdos-108-high-girth-problem-disproved-conjectures-io","url":"https://postcutoff.com/e/2026-09-15-erdos-108-high-girth-problem-disproved-conjectures-io/","date":"2026-09-15","date_precision":"day","short_title":"Erdős–Hajnal high-girth problem (Erdős #108) disproved with ChatGPT/Codex help and a Lean proof, via the Conjectures.io bounty; experts sharpen it with GPT-6 Astra","deck":null,"takeaway":"This is a well-known problem from Erdős's list, and the counterexample comes with a machine-checked proof.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":7,"official":7,"filed":"2026-10-05","updated":"2026-10-07","orgs":["Conjectures.io","Purdue University","OpenAI"]}],"people":[{"id":"amnon-shashua","name":"Amnon Shashua","url":"https://postcutoff.com/person/amnon-shashua/"}],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-15","type":"filed","text":"Created (Startup Fortune; read the doubleAI paper PDF, research page and Calcalist)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-15","run":null,"sources_read":"Startup Fortune; read the doubleAI paper PDF, research page and Calcalist","updated":"2026-10-15","human_review":null,"version":{"date":"2026-10-15"}},"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-4-argon","name":"Gemini 4 Argon","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}