{"schema":"postcutoff/event@1","as_of":"2026-10-08T23:45:00+02:00","url":"https://postcutoff.com/e/2026-10-07-multiple-unicast-conjecture-false-gpt-6/","md":"https://postcutoff.com/e/2026-10-07-multiple-unicast-conjecture-false-gpt-6/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"Adam Bicz","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-07-multiple-unicast-conjecture-false-gpt-6","date":"2026-10-07","date_precision":"day","short_title":"Braverman & He refute the 2004 undirected multiple-unicast (network coding) conjecture with a GPT-6-found counterexample on PG(2, 9)","deck":null,"takeaway":"A 22-year-old network coding conjecture is false: on undirected networks, coding can beat routing. The counterexample came from GPT-6, and the Princeton authors checked it with a standalone symbolic verifier.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":[{"n":1,"title":"Braverman & He: The Multiple Unicast Conjecture is False (arXiv 2610.10108)","url":"https://arxiv.org/abs/2610.10108","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"OpenAI math catalogue (CONTENTS.md: Fourier circuits and integer multiplication)","url":"https://github.com/openai/math/blob/main/CONTENTS.md","type":"paper","group":"primary","domain":"github.com"},{"n":3,"title":"On the Multiple-Unicast Conjecture: Beyond Cut Metrics (arXiv 2608.06070, background)","url":"https://arxiv.org/abs/2608.06070","type":"paper","group":"primary","domain":"arxiv.org"}],"official":3,"filed":"2026-10-08","updated":"2026-10-08","orgs":["Princeton University","OpenAI"],"title":"Braverman & He refute the 2004 undirected multiple-unicast (network coding) conjecture with a GPT-6-found counterexample on PG(2, 9)","summary":"On 7 Oct 2026 Mark Braverman and Zhongtian He (Princeton) posted arXiv 2610.10108, \"The Multiple Unicast Conjecture is False\". The conjecture (Li & Li and Harvey–Kleinberg–Lehman, 2004) says network coding gives no throughput advantage over fractional multicommodity flow in undirected networks. They give a linear network code over 𝔽₉ on a 182-vertex subgraph of the point–line incidence graph of PG(2, 9), carrying 157 unicast sessions at rate ≥ 1 where every flow has rate ≤ 147/157. By earlier amplification results this gives coding gaps of Ω((log n)^ε). They write that they \"use GPT-6 to find a nondeterministic counterexample\" and thank ChatGPT \"for coming up with the counterexample and drafting the proofs\".","key_facts":["Result: a deterministic linear network code over 𝔽₉ on a 182-vertex bipartite subgraph of the incidence graph of PG(2, 9): 157 independent unicast sessions at common rate ≥ 1, while every fractional multicommodity flow has common rate ≤ 147/157; amplification (Braverman–Garg–Schvartzman 2017) gives a family with coding gap Ω((log n)^ε)","Method: the authors first posed a stronger 'nondeterministic' version of the conjecture (edges carry certificate bits checked locally); GPT-6 found a counterexample using a new choice of Reed–Solomon local codes in the high-girth-graph framework of Braverman–He 2025; an edge orientation plus local search turned it into a causal code","AI use statement (verbatim): 'We thank ChatGPT for coming up with the counterexample and drafting the proofs. … The authors checked the proof and take responsibility for the correctness of the results.'","Verification: 'The accompanying repository provides complete data, a standalone symbolic verifier, and an executable simulator with automated tests'; no Lean proof","Stakes: a positive answer would have implied lower bounds for external-memory and oblivious computation and an Ω(n log n) lower bound for constant-degree Boolean circuits for integer multiplication (Afshani et al. 2019)","Concurrent work: the authors note that OpenAI's sub-(n log n) exact Fourier circuits in its 6 Oct catalogue also imply a refutation, via the Afshani et al. cyclic-shift argument; their construction is an explicit 182-vertex example"],"key_numbers":[],"tags":["computer-science","information-theory","network-coding","counterexample","gpt-6","chatgpt","complexity"],"science":{"field":"computer-science","subfield":"information theory / network coding","problem":"Undirected multiple-unicast conjecture (Li & Li 2004; Harvey, Kleinberg & Lehman 2004)","result":"Disproved: an explicit linear network code over 𝔽₉ on a 182-vertex PG(2, 9) incidence subgraph beats fractional multicommodity flow (rate 1 vs ≤ 147/157), giving Ω((log n)^ε) coding gaps by amplification.","open_since":"2004","ai_system":["GPT-6 (ChatGPT)"],"human_role":"AI found the counterexample to a strengthened version the authors formulated; humans steered the search, converted it to a causal code and checked the proof","verification":{"key":"preprint","label":"Preprint only","text":"Computer-verified construction (symbolic verifier, simulator); unrefereed preprint"},"status":"pending","shock":"Network coding versus routing in undirected graphs was a long-standing open question tied to circuit lower bounds for multiplication; the decisive example came from a chatbot.","short":"Multiple-unicast network coding conjecture"},"body_md":"## What happened\n\nIn directed networks, coding at intermediate nodes is known to beat routing (the classic butterfly example). For undirected\nnetworks with many source–sink pairs, Li and Li and, independently, Harvey, Kleinberg and Lehman conjectured in 2004 that\ncoding gives no advantage over fractional multicommodity flow. The conjecture mattered beyond networking. A proof would have\ngiven unconditional lower bounds for external-memory algorithms and an Ω(n log n) circuit lower bound for multiplying\nintegers.\n\nMark Braverman and Zhongtian He had refuted stronger variants of the conjecture in 2025. This time they posed an even stronger\n\"nondeterministic\" version to GPT-6, in which bits simply appear on edges and each vertex accepts or rejects by local checks.\nGPT-6 found a counterexample using Reed–Solomon codes on the projective plane over 𝔽₉. The authors then oriented the edges\nand ran a local search, and the example became a genuine causal code, which disproves the conjecture itself. The paper ships\na standalone verifier and simulator.\n\nThe authors also note that OpenAI's catalogue, released the day before, implies the same refutation indirectly: its claimed\nsub-n log n Fourier circuits, combined with the Afshani et al. argument, would yield a coding advantage.\n\n## Why it matters\n\nThis is a clean, checkable AI-found counterexample to a 22-year-old conjecture in information theory, by a leading\ntheoretical computer scientist who describes exactly how the model was steered. Together with OpenAI's claimed sub-n log n\nmultiplication, it shifts a whole cluster of beliefs about lower bounds in the same week.","disputed":[{"label":"Verification","text":"Computer-verified construction (symbolic verifier, simulator); unrefereed preprint","sources":[]}],"related":[{"id":"2026-10-07-colkitt-codex-integer-multiplication-exponent","url":"https://postcutoff.com/e/2026-10-07-colkitt-codex-integer-multiplication-exponent/","date":"2026-10-07","date_precision":"day","short_title":"Outside researchers using AI crowdsource a tighter exponent in OpenAI's integer-multiplication result #109, from 2^-182 to above 2^-14 (conditional)","deck":null,"takeaway":"In a parallel repository Swapnil Jain (@SJ_Swapnil_Jain, with Claude) posted seven rounds ending at κ ≈ 6.40×10^-5 (> 2^-14) on the evening of 8 Oct, the highest claimed value.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":20,"official":6,"filed":"2026-10-07","updated":"2026-10-08","orgs":["Independent researchers","OpenAI"]},{"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":42,"official":12,"filed":"2026-10-07","updated":"2026-10-08","orgs":["OpenAI"]},{"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":9,"official":4,"filed":"2026-09-30","updated":"2026-10-08","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-08","type":"filed","text":"Created from the arXiv PDF (abstract, introduction, AI use statement)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-08","run":null,"sources_read":"The arXiv PDF (abstract, introduction, AI use statement)","updated":"2026-10-08","human_review":null,"version":{"date":"2026-10-08"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":160,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":99,"in_training_data":false},{"model_id":"gemini-3-8-flash","name":"Gemini 3.8 Flash","cutoff":"2026-03","days_after":190,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":129,"in_training_data":false}],"short_url":"https://postcutoff.com/s/braverman-he-refute-2004"}