{"schema":"postcutoff/event@1","as_of":"2026-10-19T09:28:00+02:00","url":"https://postcutoff.com/e/2026-10-16-lonely-runner-constant-gap-hu-jing-chatgpt/","md":"https://postcutoff.com/e/2026-10-16-lonely-runner-constant-gap-hu-jing-chatgpt/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-lonely-runner-constant-gap-hu-jing-chatgpt","date":"2026-10-16","date_precision":"day","short_title":"Lonely runner: (1+c)/2n bound with ChatGPT 6","deck":null,"takeaway":"A human-led result on the 1967 lonely runner problem in which ChatGPT 6 Pro proposed the GCD-sum route, pointed to Green–Walker and fixed gaps, but 'the models were unable to improve the lonely runner bound on their own'.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.23790: Near optimal gap in the Lonely Runner Conjecture (Hu, Jing)","url":"https://arxiv.org/abs/2610.23790","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"Bedert, Riesz products and the Lonely Runner Conjecture (arXiv 2511.16636, prior bound)","url":"https://arxiv.org/abs/2511.16636","type":"paper","group":"primary","domain":"arxiv.org"},{"n":3,"title":"Wikipedia: Lonely runner conjecture","url":"https://en.wikipedia.org/wiki/Lonely_runner_conjecture","type":"discussion","group":"reactions","domain":"en.wikipedia.org"}],"official":2,"filed":"2026-10-19","updated":"2026-10-19","orgs":["OpenAI","University of Chinese Academy of Sciences","Ohio State University"],"title":"Lonely Runner Conjecture: Hu and Jing get the first constant-factor gain over the union bound, (1+c)/2n, after 'many rounds of discussion' with ChatGPT 6 Pro and Astra","summary":"Xiyu Hu (UCAS) and Yifan Jing (Ohio State) posted arXiv 2610.23790 on 16 Oct 2026. They prove that for every set V of n ≥ 2 distinct positive speeds there is a time when all runners are more than (1+c)/(2n) from the stationary one, for an absolute c > 0. This is the first improvement over the trivial union bound 1/(2n) by a constant factor. Earlier gains, by Tao and Bedert, shrink to zero as n grows. The conjecture predicts 1/(n+1). ChatGPT 6 Pro and ChatGPT 6 Astra suggested the GCD-sum approach, pointed to the Green–Walker refinement, fixed gaps in a proof and helped strengthen Green–Walker. The authors stress that \"the models were unable to improve the lonely runner bound on their own\".","key_facts":["Paper: Xiyu Hu and Yifan Jing, 'Near optimal gap in the Lonely Runner Conjecture', arXiv 2610.23790 [math.NT], submitted 16 Oct 2026, 24 pp.","Theorem 1.1: there is an absolute c > 0 such that ML(V) > (1+c)/(2n) for every set V of n ≥ 2 distinct positive integers; in fact the set of good times has measure at least an absolute κ0, and c = κ0/2","Previous bounds (paper): Tao, 1/(2n) + c·log n/(n²(log log n)²); Bedert (2025), 1/(2n) + n^(−5/3−o(1)); Bedert, communicated, n^(−3/2−o(1)). 'All these bounds still give a relative improvement tending to zero'","Corollary: a lower bound P(ε) ≥ (1+c)/(2ε) for the rotation-and-dilation pyjama problem (Kravitz–Leng equivalence)","Method: a refinement of Green and Walker's large-GCD argument plus a new 'physical spillover inequality' for interval multiplicities (Proposition 4.4)","AI statement: 'The main strategy and mathematical framework of the paper were developed by the authors. ChatGPT 6 Astra and ChatGPT 6 Pro were used primarily to verify arguments, turn preliminary or informal proofs into rigorous proofs, and explore possible refinements… the models were unable to improve the lonely runner bound on their own'","AI contributions named: 'ChatGPT 6 Pro suggested introducing suitable GCD sum estimates'; 'the AI tools pointed the authors toward the Green–Walker refinement'; 'Some small gaps in the first version of his proof were later fixed by ChatGPT 6 Pro'; ChatGPT 6 Pro 'assisted the second author in refining the Green–Walker argument'; Astra used for language polishing","No Hacker News thread or high-reach X post found on 19 Oct morning"],"key_numbers":[],"tags":["math","number-theory","diophantine-approximation","lonely-runner","chatgpt","gpt-6-astra","human-ai-collaboration"],"science":{"field":"mathematics","subfield":"number theory (Diophantine approximation, view-obstruction)","problem":"Lonely Runner Conjecture (Wills 1967, Cusick 1973): every runner is at some time at distance ≥ 1/(n+1) from all others","result":"ML(V) > (1+c)/(2n) for an absolute constant c > 0, the first constant-factor improvement over the union bound 1/(2n)","open_since":"1967","ai_system":["ChatGPT 6 Pro","GPT-6 Astra"],"human_role":"Human-led: authors developed the strategy; ChatGPT 6 Pro/Astra proposed the GCD-sum ingredient, pointed to Green–Walker, fixed gaps and turned informal proofs into rigorous ones","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed arXiv preprint"},"status":"pending","shock":""},"body_md":"## What happened\n\nIn the lonely runner problem, n+1 runners with distinct constant speeds start together on a unit circular track. The conjecture says each runner is at some moment at least 1/(n+1) away from all the others. Equivalently, for n speeds there is a time t when every v·t is at least 1/(n+1) from the nearest integer. A simple union bound gives 1/(2n), and decades of work had only added lower-order terms.\n\nHu and Jing push the bound to (1+c)/(2n). Their AI statement gives an unusually detailed history: Hu's partial results under modularity conditions, ChatGPT 6 Pro's suggestion of GCD sums to combine information across primes, the models pointing to Green–Walker, Jing's spillover inequality (with small gaps fixed by ChatGPT 6 Pro), and a strengthened Green–Walker theorem worked out with the model. \"These discussions did not produce any uniform results\" until the human spillover idea arrived.\n\n## Why it matters\n\nIt is a substantial result on a well-known problem, and its disclosure is a clear record of the division of labour. The models acted as a knowledgeable collaborator that supplies tools and fixes proofs, while the decisive idea came from the humans. The authors put it as \"an interesting example of the importance of human mathematician input\".","disputed":[{"label":"Verification","text":"Unrefereed arXiv preprint","sources":[]}],"related":[{"id":"2026-10-19-ai-assisted-results-october-19-listing","url":"https://postcutoff.com/e/2026-10-19-ai-assisted-results-october-19-listing/","date":"2026-10-19","date_precision":"day","short_title":"AI-assisted results in the 19 Oct 2026 arXiv listing","deck":"Experts rework OpenAI's release papers (Deligne–Drinfeld, Hénon–Lane–Emden, Navier–Stokes), Wright's harmonic-analysis problems disproved with Codex, an Aaronson–Ambainis lower bound, and DeepMind's Busy Beaver agent","takeaway":"Monday's arXiv announcement brought the first wave of human follow-up papers on OpenAI's October maths release, alongside a dozen routine AI disclosures. The strongest items have their own entries: Brennan, Smith–Toda, lonely runner and tournament Hamiltonian cycles.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":14,"official":14,"filed":"2026-10-19","updated":"2026-10-19","orgs":["OpenAI","Anthropic","Google DeepMind","Harmonic","various mathematicians"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-19","type":"filed","text":"Created (PDF read)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-19","run":null,"sources_read":"PDF read","updated":"2026-10-19","human_review":null,"version":{"date":"2026-10-19"}},"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":"https://postcutoff.com/s/lonely-runner-1-c"}