{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2026-09-14-k-server-conjecture-proved/","md":"https://postcutoff.com/e/2026-09-14-k-server-conjecture-proved/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":"2026-09-14-k-server-conjecture-proved","date":"2026-09-14","date_precision":"day","short_title":"The k-server conjecture, the 'holy grail' of online algorithms, is proved at Oxford","deck":"ChatGPT 6 Astra generalised the authors' k = 3 proof to all k","takeaway":"Christian Coester, Elias Koutsoupias and Marek Zbysiński (Oxford) posted a proof of the k-server conjecture (Manasse–McGeoch–Sleator, 1988): the work function algorithm is k-competitive on every metric space (arXiv 2609.15979, 14 Sep 2026).","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2609.15979: The k-server conjecture is true (Coester, Koutsoupias, Zbysiński)","url":"https://arxiv.org/abs/2609.15979","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"Hacker News discussion","url":"https://news.ycombinator.com/item?id=49709129","type":"discussion","group":"reactions","domain":"news.ycombinator.com"}],"official":1,"filed":"2026-09-30","updated":"2026-09-30","orgs":["University of Oxford","Christian Coester","Elias Koutsoupias","Marek Zbysiński","OpenAI"],"title":"The k-server conjecture, the 'holy grail' of online algorithms, is proved at Oxford; ChatGPT 6 Astra generalised the authors' k = 3 proof to all k","summary":"Christian Coester, Elias Koutsoupias and Marek Zbysiński (Oxford) posted a proof of the k-server conjecture (Manasse–McGeoch–Sleator, 1988): the work function algorithm is k-competitive on every metric space (arXiv 2609.15979, 14 Sep 2026). The authors designed the potential function and proved k = 3 without AI. ChatGPT 6 Astra then 'derived an algebraic proof of correctness for any k', which the authors adapted and revised.","key_facts":["The k-server problem was introduced by Manasse, McGeoch and Sleator in 1988; the paper notes it has repeatedly been called the 'holy grail' of competitive analysis","Previous best: the work function algorithm is (2k−1)-competitive (Koutsoupias and Papadimitriou, 1995); the conjecture asks for ratio k","New proof: represent the work function algebraically as a matrix whose column determinants encode work-function values; the amortised analysis uses a potential defined on a larger matrix of coordinate pairs","AI role (acknowledgments): the k = 3 potential and 'initial proof for three servers … was obtained without AI assistance'; discussions with ChatGPT 5.5 Pro and Gemini 3.1 Pro gave 'a deeper understanding of the potential'; 'ChatGPT 6 Astra subsequently derived an algebraic proof of correctness for any k'; the published proof adapts it, and Astra helped draft some sections","Discussed on Hacker News (106 points, 15 Sep 2026)","Status: preprint, not peer-reviewed"],"key_numbers":[],"tags":["computer-science","online-algorithms","competitive-analysis","theoretical-cs","astra"],"science":{"field":"computer-science","subfield":"online algorithms / competitive analysis","problem":"k-server conjecture","result":"Deterministic k-competitiveness of the work function algorithm on every metric space.","open_since":"1988","ai_system":["GPT-6 Astra (ChatGPT)","ChatGPT 5.5 Pro","Gemini 3.1 Pro"],"human_role":"Human-designed potential and k = 3 proof; the AI generalised it to all k; humans adapted and verified the final proof","verification":{"key":"preprint","label":"Preprint only","text":"Preprint; not peer-reviewed"},"status":"pending","shock":"A 38-year-old central conjecture of theoretical computer science was finished when a model generalised a human proof from 3 servers to k."},"body_md":"## What happened\n\nThe Oxford team first found a new potential function that proved the open three-server case by hand. After refining it in conversations with ChatGPT and Gemini, they gave the reformulation to ChatGPT 6 Astra, which produced an algebraic proof for all k. The authors then recast that proof in a more natural matrix representation.\n\n## Why it matters\n\nThe k-server conjecture is one of the best-known open problems in algorithms, and Koutsoupias co-proved the previous best bound in 1995. Here the AI's contribution is the generalisation step that turned a special case into the full theorem.","disputed":[{"label":"Verification","text":"Preprint; not peer-reviewed","sources":[]}],"related":[{"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":11,"official":5,"filed":"2026-09-30","updated":"2026-10-19","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]},{"id":"2026-09-03-gpt-6-astra","url":"https://postcutoff.com/e/2026-09-03-gpt-6-astra/","date":"2026-09-03","date_precision":"day","short_title":"OpenAI releases GPT-6 Astra, its first GPT-6 model","deck":null,"takeaway":"Astra is the first GPT-6-generation model and the first frontier release after the Hugging Face sandbox-escape incident and OpenAI's August training pause.","category":"model-release","category_label":"Model releases","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":20,"official":11,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-09-30","type":"filed","text":"Created"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-09-30","run":null,"sources_read":null,"updated":"2026-09-30","human_review":null,"version":{"date":"2026-10-04"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":137,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":76,"in_training_data":false},{"model_id":"gemini-4-hydrogen","name":"Gemini 4 Hydrogen","cutoff":"2026-03","days_after":167,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":106,"in_training_data":false}],"short_url":"https://postcutoff.com/s/k-server-conjecture-holy-grail"}