{"schema":"postcutoff/event@1","as_of":"2026-10-18T07:35:00+02:00","url":"https://postcutoff.com/e/2026-10-16-yang-mills-claude-balaban-program-alpoge-sussman/","md":"https://postcutoff.com/e/2026-10-16-yang-mills-claude-balaban-program-alpoge-sussman/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-yang-mills-claude-balaban-program-alpoge-sussman","date":"2026-10-16","date_precision":"day","short_title":"Alpöge and Sussman say Claude has an (unreadable) existence argument for 4D Yang–Mills on ℝ⁴, by finishing Bałaban's 1980s program","deck":null,"takeaway":"A public repo of raw Claude output claims a construction of four-dimensional SU(2) Yang–Mills theory, the existence half of the Millennium problem but not the mass gap; its authors call it 'slop in progress' that nobody can yet read or check.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"GitHub: ethanwsussman/Yang--Mills-LLM-experiment (README, zeroth draft, raw Claude output)","url":"https://github.com/ethanwsussman/Yang--Mills-LLM-experiment","type":"code","group":"primary","domain":"github.com"},{"n":2,"title":"README.pdf: proposal for an LLM-aided attack on YM4 (6 Oct 2026)","url":"https://github.com/ethanwsussman/Yang--Mills-LLM-experiment/blob/main/README.pdf","type":"code","group":"primary","domain":"github.com"},{"n":3,"title":"ym4_manuscript_zeroth_draft.pdf (153 pages)","url":"https://github.com/ethanwsussman/Yang--Mills-LLM-experiment/blob/main/ym4_manuscript_zeroth_draft.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":4,"title":"GitHub: scottnarmstrong/balaban-lean (Lean 4, Claude and Codex agents)","url":"https://github.com/scottnarmstrong/balaban-lean","type":"code","group":"primary","domain":"github.com"},{"n":5,"title":"tech-insider.org: Yang-Mills 'Solved Next Before 2045?' odds jump 20 points on Kalshi (17 Oct)","url":"https://tech-insider.org/yang-mills-solved-next-before-2045-kalshi-odds/","type":"press","group":"press","domain":"tech-insider.org"},{"n":6,"title":"Erdős Problems forum, 'AI Contributions 3' (George Coote's post, 16 Oct)","url":"https://www.erdosproblems.com/forum/thread/AI%20Contributions%203","type":"discussion","group":"reactions","domain":"erdosproblems.com"}],"official":4,"filed":"2026-10-18","updated":"2026-10-18","orgs":["Anthropic","Northwestern University"],"title":"Alpöge and Sussman say Claude has an (unreadable) existence argument for 4D Yang–Mills on ℝ⁴, by finishing Bałaban's 1980s program","summary":"Northwestern mathematician Ethan Sussman and Anthropic's Levent Alpöge run a public GitHub repo (since 6 Oct 2026) in which Claude, steered by Alpöge, tries to complete Tadeusz Bałaban's renormalisation-group program for constructing quantum Yang–Mills theory in four dimensions. On 16 Oct Alpöge added to the README: \"At the moment we do (appear to) have a YM4 (i.e. ℝ⁴, not just in finite volume) existence argument / slop, but it is extremely long and basically not understandable as is.\" The repo holds a 153-page \"zeroth draft\" (SU(2), continuum limits on ℝ⁴ with Osterwalder–Schrader properties, conditional on listed inputs; no mass gap) and about 26 MB of raw LaTeX. Unverified; the authors say there are \"certainly gaps and mistakes\".","key_facts":["Repo: github.com/ethanwsussman/Yang--Mills-LLM-experiment, 'an attempt with Levent Alpoge to get Claude to finish Balaban's program to construct Yang--Mills'; created 6 Oct 2026, last upload 17 Oct; 47 stars on 18 Oct","README (16 Oct, signed 'LA'): 'we do (appear to) have a YM4 (i.e. \\R^4, not just in finite volume) existence argument / slop, but it is extremely long and basically not understandable as is'","Same note: 'we have been made aware of emails going around from groups now working on the exact same thing, making sudden progress off of the arguments above … Please, we need to not incentivize rushing out 10,000-page slop'","Plan: start with QED3 and YM3 (3-D, 'where the UV situation is friendlier'), show 'some of the coolest YM4 arguments (e.g. winning O(4)-invariance via an RP trick)' there, write a human exposition and 'formalize what we can'","Zeroth draft (PDF dated 6 Oct, 153 pp.): 'SU(2) Yang–Mills theory in four dimensions via Bałaban's renormalisation group: construction, axioms and non-triviality'; it claims continuum limits on tori and on ℝ⁴ with OS regularity, reflection positivity and a Hilbert space with vacuum and non-negative Hamiltonian, plus non-triviality, 'assuming the proof inputs declared clause by clause in Section 9'","Not claimed in the draft: 'uniqueness of the infinite-volume limit …, the cluster property, a mass gap'; 'the present paper does not address the mass gap'. Full O(4) invariance rests on a displayed hypothesis","Origin (README.pdf, 6 Oct): Sussman first asked Codex to reproduce Dimock's QED3 work along Bałaban's lines; 'After several days, the model (5.6 Sol) reported success', then about two more weeks of compute for infinite volume and the axioms; 'I do not know whether the manuscript is correct'. He then emailed Alpöge 'whom I recognized as having an excellent track record of getting the models to produce correct results'","README.pdf on the Claude output: 'raw LLM outputs, not manuscripts or anything meant to be read by a human … There are certainly gaps and mistakes. I cannot tell how reliable the reported progress is'; it proposes a working group 'for the production, formalization, digestion, and rewriting of an AI-written manuscript on Yang–Mills'","Spin-off: Scott Armstrong's Lean 4 project scottnarmstrong/balaban-lean (created 13 Oct) formalizes Bałaban's related O(N) spin-model papers; ~800,000 Lean lines by 18 Oct, 'most of the Lean code and design notes are written by AI agents (Anthropic's Claude and OpenAI's Codex models)'; its acknowledgements say the Sussman–Alpöge account 'suggested the route we now follow for the large-field gas'","Context: Kalshi's 'Yang-Mills solved next before 2045?' contract rose about 20 points to ~20% on 17 Oct (tech-insider.org), which the article attributes to AI-maths attention and a cluster of self-published Yang–Mills preprints; it does not mention this repo"],"key_numbers":[],"tags":["math","mathematical-physics","quantum-field-theory","yang-mills","millennium-prize","claude","codex","gpt-5-6-sol","anthropic","lean","unverified"],"science":{"field":"physics","subfield":"mathematical physics / constructive quantum field theory","problem":"Construction of four-dimensional quantum Yang–Mills theory (existence part of the Yang–Mills existence and mass gap Millennium Prize Problem), via Bałaban's renormalisation-group program (1985–1989)","result":"Claimed, not established: a very long Claude-generated argument for the existence of SU(2) Yang–Mills on ℝ⁴ (continuum limit with Osterwalder–Schrader properties and non-triviality), conditional on listed inputs; no mass gap, no uniqueness of the infinite-volume limit.","open_since":"","ai_system":["Claude","Codex","GPT-5.6 Sol"],"human_role":"Alpöge 'coaxing Claude' with 'some small amount of direction'; Sussman proposed the program and ran the earlier Codex QED3 experiment; both say they cannot yet tell how reliable the output is","verification":{"key":"preprint","label":"Preprint only","text":"Unverified raw LLM output on GitHub; no preprint, no Lean proof of the YM4 claim; authors plan exposition and partial formalization"},"status":"pending","shock":"A Millennium-problem-adjacent construction that experts had shelved as too much work for humans is being attempted in public by an AI lab's mathematician, with the claim of an argument that exists but cannot yet be read.","short":"Yang–Mills existence (YM4)"},"body_md":"## What happened\n\nBuilding four-dimensional quantum Yang–Mills theory rigorously is the existence half of the Clay Millennium problem \"Yang–Mills\nexistence and mass gap\" (the other half, the mass gap, is considered harder). In 1985–1989 Tadeusz Bałaban published a long\nrenormalisation-group program that is widely thought to contain the seeds of a construction in finite volume, but it was never\nfinished.\n\nOn 6 Oct 2026 Ethan Sussman, a Boas Assistant Professor at Northwestern, opened the public repository\n[Yang--Mills-LLM-experiment](https://github.com/ethanwsussman/Yang--Mills-LLM-experiment). Its `README.pdf` explains the idea.\nBałaban's approach \"would require superhuman levels of work\", and LLMs could supply that work. As a first test, Sussman asked\nCodex to redo Dimock's papers on QED in three dimensions. \"After several days, the model (5.6 Sol) reported success.\" After about\ntwo more weeks of compute it also reported the infinite-volume limit and the axioms. He writes \"I do not know whether the\nmanuscript is correct.\" He then emailed Levent Alpöge (Anthropic), who has been \"coaxing Claude\" along Bałaban's line for YM4.\n\nThe repo holds a 153-page \"zeroth draft\" dated 6 Oct and three raw LaTeX files of about 26 MB in total (\"theorem_0_complete_account_raw\").\nThe draft states a conditional theorem for SU(2) with Wilson's lattice action. It claims continuum limits on tori and on ℝ⁴ with\nOsterwalder–Schrader regularity and reflection positivity, a Hilbert space with a vacuum, and non-triviality. These rest on\n\"proof inputs declared clause by clause in Section 9\", and on some supporting statements \"that we do not claim as proved, two of\nthem not yet written\". The draft says plainly that \"the present paper does not address the mass gap\".\n\nOn 16 Oct Alpöge added a note to the README: \"At the moment we do (appear to) have a YM4 (i.e. ℝ⁴, not just in finite volume)\nexistence argument / slop, but it is extremely long and basically not understandable as is.\" The plan is to show the key\narguments first in the easier 3-D theories (QED3, YM3), to write a human exposition and to formalize what they can. He added that\nother groups were \"now working on the exact same thing, making sudden progress off of the arguments above\", and asked people not\nto \"incentivize rushing out 10,000-page slop\". A new upload followed on 17 Oct.\n\nScott Armstrong's Lean project [balaban-lean](https://github.com/scottnarmstrong/balaban-lean), started on 13 Oct, formalizes\nBałaban's related papers on the O(N) spin model. It is written mostly by Claude and Codex agents, has about 800,000 lines so far\nand has not proved its two main theorems. It credits the Sussman–Alpöge account with the route it now follows for the\nlarge-field gas.\n\n## Why it matters\n\nThis is the first public claim by a frontier-lab mathematician that a model has produced an existence argument for 4-D Yang–Mills\non all of ℝ⁴. It comes with strong warnings from its own authors: it is not a proof anyone has checked, the mass gap is not\naddressed, and parts are conditional. It also shows a new way of working. A human-abandoned program is handed to models at scale,\nand the human job becomes digesting the output and formalizing it in Lean (Armstrong's project). It follows the OpenAI\nNavier–Stokes controversy, and Alpöge himself raises the risk of a race to post very long unreadable manuscripts.\n\n## Reactions\n\n- On the Erdős Problems forum, George Coote posted it on 16 Oct as \"Alpoge claims possible construction of a 4D Yang-Mills theory\n  on SU(2)… He confesses the proof is not yet human readable.\" Another user replied that prediction markets were \"not fully\n  convinced, yet\".\n- Kalshi's \"Yang-Mills solved next before 2045?\" contract rose about 20 points to roughly 20% on 17 Oct\n  ([tech-insider.org](https://tech-insider.org/yang-mills-solved-next-before-2045-kalshi-odds/)). The article puts the move down to\n  general AI-maths attention and several self-published Yang–Mills \"solutions\" on Zenodo. It does not mention this repo, so the\n  two are not linked here.\n- We found no press coverage, HN thread or expert assessment by 18 Oct. Alpöge's and Sussman's X timelines could not be read\n  through fxtwitter.","disputed":[{"label":"Verification","text":"Unverified raw LLM output on GitHub; no preprint, no Lean proof of the YM4 claim; authors plan exposition and partial formalization","sources":[]}],"related":[{"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":"2026-09-25-claude-nine-loop-amplitude-n4-sym","url":"https://postcutoff.com/e/2026-09-25-claude-nine-loop-amplitude-n4-sym/","date":"2026-09-25","date_precision":"day","short_title":"Claude (Fable 5.1 in Claude Science) computes the nine-loop six-gluon amplitude in planar N=4 super-Yang-Mills, answering a physicist's public challenge","deck":null,"takeaway":"It is a frontier-level computation in theoretical physics done almost autonomously by an AI agent on a modest budget.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":6,"official":5,"filed":"2026-09-30","updated":"2026-10-07","orgs":["Anthropic"]}],"people":[{"id":"levent-alpoge","name":"Levent Alpöge","url":"https://postcutoff.com/person/levent-alpoge/"}],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-18","type":"filed","text":"Created (repo, README.pdf and zeroth-draft PDF read; balaban-lean README read; found through the Erdős Problems forum)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-18","run":null,"sources_read":"Repo, README.pdf and zeroth-draft PDF read; balaban-lean README read; found through the Erdős Problems forum","updated":"2026-10-18","human_review":null,"version":null},"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/alpoge-sussman-claude-unreadable"}