Alpöge and Sussman say Claude has an (unreadable) existence argument for 4D Yang–Mills on ℝ⁴, by finishing Bałaban’s 1980s program

Building four-dimensional quantum Yang–Mills theory rigorously is the existence half of the Clay Millennium problem “Yang–Mills existence and mass gap” (the other half, the mass gap, is considered harder). In 1985–1989 Tadeusz Bałaban published a long renormalisation-group program that is widely thought to contain the seeds of a construction in finite volume, but it was never finished.

On 6 Oct 2026 Ethan Sussman, a Boas Assistant Professor at Northwestern, opened the public repository Yang--Mills-LLM-experiment1. Its README.pdf explains the idea. Bałaban’s approach “would require superhuman levels of work”, and LLMs could supply that work. As a first test, Sussman asked Codex to redo Dimock’s papers on QED in three dimensions. “After several days, the model (5.6 Sol) reported success.” After about two more weeks of compute it also reported the infinite-volume limit and the axioms. He writes “I do not know whether the manuscript is correct.” He then emailed Levent Alpöge (Anthropic), who has been “coaxing Claude” along Bałaban’s line for YM4.

The 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”). The draft states a conditional theorem for SU(2) with Wilson’s lattice action. It claims continuum limits on tori and on ℝ⁴ with Osterwalder–Schrader regularity and reflection positivity, a Hilbert space with a vacuum, and non-triviality. These rest on “proof inputs declared clause by clause in Section 9”, and on some supporting statements “that we do not claim as proved, two of them not yet written”. The draft says plainly that “the present paper does not address the mass gap”.

On 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) existence argument / slop, but it is extremely long and basically not understandable as is.” The plan is to show the key arguments first in the easier 3-D theories (QED3, YM3), to write a human exposition and to formalize what they can. He added that other groups were “now working on the exact same thing, making sudden progress off of the arguments above”, and asked people not to “incentivize rushing out 10,000-page slop”. A new upload followed on 17 Oct.

Scott Armstrong’s Lean project balaban-lean4, started on 13 Oct, formalizes Bałaban’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 and has not proved its two main theorems. It credits the Sussman–Alpöge account with the route it now follows for the large-field gas.

This is the first public claim by a frontier-lab mathematician that a model has produced an existence argument for 4-D Yang–Mills on 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 addressed, and parts are conditional. It also shows a new way of working. A human-abandoned program is handed to models at scale, and the human job becomes digesting the output and formalizing it in Lean (Armstrong’s project). It follows the OpenAI Navier–Stokes controversy, and Alpöge himself raises the risk of a race to post very long unreadable manuscripts.

• On the Erdős Problems forum, George Coote posted it on 16 Oct as “Alpoge claims possible construction of a 4D Yang-Mills theory on SU(2)… He confesses the proof is not yet human readable.” Another user replied that prediction markets were “not fully convinced, yet”.
• Kalshi’s “Yang-Mills solved next before 2045?” contract rose about 20 points to roughly 20% on 17 Oct (tech-insider.org5). The article puts the move down to general AI-maths attention and several self-published Yang–Mills “solutions” on Zenodo. It does not mention this repo, so the two are not linked here.
• We found no press coverage, HN thread or expert assessment by 18 Oct. Alpöge’s and Sussman’s X timelines could not be read through fxtwitter.