{"schema":"postcutoff/event@1","as_of":"2026-10-19T22:14:00+02:00","url":"https://postcutoff.com/e/2026-07-22-dinitz-garg-goemans-conjecture-disproved-gpt-5-6-pro/","md":"https://postcutoff.com/e/2026-07-22-dinitz-garg-goemans-conjecture-disproved-gpt-5-6-pro/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":"2026-07-22-dinitz-garg-goemans-conjecture-disproved-gpt-5-6-pro","date":"2026-07-22","date_precision":"day","short_title":"Dmitry Rybin posts a GPT-5.6 Pro counterexample to the ~30-year-old Dinitz–Garg–Goemans (Goemans cost) conjecture on unsplittable flows","deck":null,"takeaway":null,"category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"DataCamp: GPT-5.6 and the Dinitz-Garg-Goemans conjecture","url":"https://www.datacamp.com/blog/gpt-5-6-dinitz-garg-goemans-conjecture","type":"press","group":"press","domain":"datacamp.com"},{"n":2,"title":"36Kr: GPT-5.6 disproves 30-year graph theory conjecture; Peking University alumnus solves six Erdős problems in five days (24 July)","url":"https://eu.36kr.com/en/p/3908956256605577","type":"press","group":"press","domain":"eu.36kr.com"},{"n":3,"title":"Dmitry Rybin on X: Dinitz-Garg-Goemans conjecture is false (22 July 2026)","url":"https://x.com/DmitryRybin1/status/2079904005652893709","type":"discussion","group":"reactions","domain":"x.com"},{"n":4,"title":"ChatGPT conversation with GPT-5.6 Pro (shared by Rybin)","url":"https://chatgpt.com/share/6a60b2eb-0b64-83ee-9c76-7931ca1de063","type":"discussion","group":"reactions","domain":"chatgpt.com"}],"official":0,"filed":"2026-10-19","updated":"2026-10-19","orgs":["OpenAI"],"title":"Dmitry Rybin posts a GPT-5.6 Pro counterexample to the ~30-year-old Dinitz–Garg–Goemans (Goemans cost) conjecture on unsplittable flows","summary":"On 22 July 2026 Dmitry Rybin posted on X that the Dinitz–Garg–Goemans conjecture, open for about 30 years, is false, with a small graph found in a chat with GPT-5.6 Pro: a fractional flow of cost 58 for which every unsplittable flow with capacity violation at most 15 costs at least 60. He shared the full chat. The post had ~4.9M views. No paper or peer review has followed as far as we found.","key_facts":["Rybin on X (22 July 2026, ~4.88M views by Oct 19, fxtwitter): 'Dinitz-Garg-Goemans conjecture is false. This graph theory problem was open for ~30 years. The graph below has fractional flow cost 58. Any unsplittable flow (with capacity violation <=15) has cost at least 60.' The post links the ChatGPT conversation with GPT-5.6 Pro","What is refuted (DataCamp): the cost version, Goemans' conjecture that a cheap splittable flow can be turned into an unsplittable one keeping both congestion and cost bounded; the Dinitz–Garg–Goemans congestion theorem itself is not in question","Construction (as reported by 36Kr and DataCamp): one source, three terminals with demands 15, 10 and 15; fractional cost 58 vs ≥ 60 for any unsplittable routing within the congestion budget. DataCamp could not reproduce the exact graph from a primary source","Checks: DataCamp says several people checked the arithmetic and found it consistent and one researcher placed the instance in an infinite three-parameter family; QbitAI/36Kr says the deliverables included proof certificates, an exhaustive enumeration program and LaTeX source. No named expert statement found","Seen by us only on 19 Oct 2026, through Fireship's video 'Summer math fell to the machines' (Aug 19); Chinese tech press (36Kr, 24 July) covered it the same week together with Shouqiao Wang's six Erdős problems"],"key_numbers":[],"tags":["math","combinatorial-optimization","network-flows","counterexample","gpt-5-6","ai-generated-proof"],"science":{"field":"mathematics","subfield":"combinatorial optimization / network flows","problem":"Dinitz–Garg–Goemans (Goemans) conjecture on the cost of unsplittable flows","result":"Claimed counterexample: a small single-source, three-terminal instance with fractional flow cost 58 where every unsplittable flow with capacity violation ≤ 15 costs at least 60.","open_since":"1999","ai_system":["GPT-5.6 Pro"],"human_role":"Rybin prompted GPT-5.6 Pro with a few short prompts; the model found the instance; Rybin published the chat","verification":{"key":"expert","label":"Expert-checked","text":"Unverified claim (X post with chat log); arithmetic reportedly checked by others; no paper or peer review found"},"status":"pending","shock":"A decades-old optimization conjecture fell to a 7-node graph found in a public chatbot conversation.","short":"Goemans cost conjecture refuted","verified_by":["unverified"]},"body_md":"## What happened\n\nRybin asked GPT-5.6 Pro for a complete counterexample to the conjecture and posted the resulting graph and the conversation. The claim spread\nwidely on X (one of the most viewed AI-maths posts of the summer) and in Chinese tech media. We read the X post (via fxtwitter), DataCamp and\n36Kr; the open-since year is DataCamp's \"roughly 1999\", which fits Rybin's \"~30 years\". The construction details are secondary; verify against\nthe chat before relying on them.\n\n## Why it matters\n\nIt is one of several AI-found counterexamples posted in July 2026 (see the Jacobian, Hessian and Rota-flats entries of the same week) and was\nmissed by this dataset for three months because it lived only on X and in a chat log, with no arXiv paper.","disputed":[{"label":"Verification","text":"Unverified claim (X post with chat log); arithmetic reportedly checked by others; no paper or peer review found","sources":[]}],"related":[{"id":"2026-07-25-sharp-thin-shell-inequality-gpt-5-6-pro","url":"https://postcutoff.com/e/2026-07-25-sharp-thin-shell-inequality-gpt-5-6-pro/","date":"2026-07-25","date_precision":"day","short_title":"Sharp thin-shell inequality","deck":"Var(|X|²) ≤ 8n for isotropic log-concave vectors, with a proof 'found by GPT-5.6 Pro' (Chen & Klartag)","takeaway":"This is a sharp result in a mature area, signed by people who built that area, with the model credited for the proof in the abstract.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":3,"official":2,"filed":"2026-10-13","updated":"2026-10-13","orgs":["ETH Zurich","Weizmann Institute of Science","OpenAI"]},{"id":"2026-07-24-hessian-conjecture-counterexample","url":"https://postcutoff.com/e/2026-07-24-hessian-conjecture-counterexample/","date":"2026-07-24","date_precision":"day","short_title":"Hessian conjecture refuted in five variables, derived from Claude-found Jacobian counterexample","deck":null,"takeaway":"It shows how AI-found results feed quickly into human follow-up work.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":2,"official":1,"filed":"2026-09-29","updated":"2026-10-19","orgs":["Independent researchers"]},{"id":"2026-07-22-shouqiao-wang-six-erdos-problems-gpt-5-6-sol-codex","url":"https://postcutoff.com/e/2026-07-22-shouqiao-wang-six-erdos-problems-gpt-5-6-sol-codex/","date":"2026-07-22","date_precision":"day","short_title":"Columbia PhD student Shouqiao Wang says he solved six open Erdős problems in five days with GPT-5.6 Sol in Codex, and publishes his workflow","deck":null,"takeaway":"It is an early public recipe for turning a coding agent into a problem-solving loop on open maths problems, published months before the Erdős-problem wave of September–October 2026.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":0,"filed":"2026-10-19","updated":"2026-10-19","orgs":["OpenAI","Columbia University"]},{"id":"2026-07-20-jacobian-conjecture-counterexample","url":"https://postcutoff.com/e/2026-07-20-jacobian-conjecture-counterexample/","date":"2026-07-20","date_precision":"day","short_title":"Claude Fable 5 finds a counterexample to the Jacobian conjecture in dimension 3","deck":null,"takeaway":"The Jacobian conjecture is one of the most famous open problems in algebra.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":4,"official":1,"filed":"2026-09-29","updated":"2026-10-19","orgs":["Anthropic"]}],"people":[],"posts":[{"id":"x-dmitryrybin1-2079904005652893709","title":"“Dinitz-Garg-Goemans conjecture is false. This graph theory problem was open for ~30…”","url":"https://postcutoff.com/p/x-dmitryrybin1-2079904005652893709/"}],"videos":[],"models":[],"changes":[{"date":"2026-10-19","type":"filed","text":"Created. Late from the 2026-10-19 lead (Fireship video); X post read via fxtwitter"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-19","run":null,"sources_read":"Late from the 2026-10-19 lead (Fireship video); X post read via fxtwitter","updated":"2026-10-19","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":83,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":22,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":113,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":52,"in_training_data":false}],"short_url":null}