{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-26-three-dimensional-su2-yang-mills-continuum-limit-aronow-lopatto-ai/","md":"https://postcutoff.com/e/2026-10-26-three-dimensional-su2-yang-mills-continuum-limit-aronow-lopatto-ai/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":"2026-10-26-three-dimensional-su2-yang-mills-continuum-limit-aronow-lopatto-ai","date":"2026-10-26","date_precision":"day","short_title":"3D SU(2) Yang–Mills continuum limit claimed in five AI-written papers","deck":null,"takeaway":"Building Yang–Mills theory as a rigorous continuum measure is open even in three dimensions, where the theory is superrenormalizable. On 26 Oct P. M. Aronow (Yale) and Patrick Lopatto (UNC Chapel Hill) posted five Hexagon papers. They claim that flowed Wilson loops of 3D SU(2) lattice Yang–Mills on a torus converge as the lattice spacing goes to zero, to a unique measure on the Cao–Chatterjee state space. The proof is a Bałaban-style multiscale renormalization-group analysis. The provenance statement says proof discovery, verification and writing were done by Opus 5.5, GPT 6 Astra and GPT 6.1 Sol, with only 'limited background formalizations' in Lean. Hexagon records human understanding as 'no parts'. It is not the four-dimensional Clay problem. We found no expert check as of 29 Oct.","category":"science","category_label":"Science & math","importance":4,"confidence":"low","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"Hexagon 2610.00506: The continuum limit of three-dimensional SU(2) lattice Yang–Mills theory on a torus I","url":"https://hexagonmath.org/abs/2610.00506v1","type":"paper","group":"primary","domain":"hexagonmath.org"},{"n":2,"title":"Hexagon 2610.00509: … II: Fluctuation integration and local reconstruction","url":"https://hexagonmath.org/abs/2610.00509v1","type":"paper","group":"primary","domain":"hexagonmath.org"},{"n":3,"title":"Hexagon 2610.00510: … III: Regional renormalization and large-field control","url":"https://hexagonmath.org/abs/2610.00510v1","type":"paper","group":"primary","domain":"hexagonmath.org"},{"n":4,"title":"Hexagon 2610.00511: … IV: Wilson flow and continuum measure","url":"https://hexagonmath.org/abs/2610.00511v1","type":"paper","group":"primary","domain":"hexagonmath.org"},{"n":5,"title":"Hexagon 2610.00512: … V: Short-distance and weak-coupling asymptotics","url":"https://hexagonmath.org/abs/2610.00512v1","type":"paper","group":"primary","domain":"hexagonmath.org"},{"n":6,"title":"Hexagon 2610.00402: Glauber dynamics from typical states of the low-temperature SK model (Aronow, Lopatto)","url":"https://hexagonmath.org/abs/2610.00402v1","type":"paper","group":"primary","domain":"hexagonmath.org"}],"official":6,"filed":"2026-10-29","updated":"2026-10-29","orgs":["Anthropic","OpenAI","Yale University","University of North Carolina at Chapel Hill"],"title":"A five-part, AI-written construction of the continuum limit of three-dimensional SU(2) lattice Yang–Mills theory on a torus, posted on Hexagon by P. M. Aronow and Patrick Lopatto with 'human understanding: no parts'","summary":"\"The continuum limit of three-dimensional SU(2) lattice Yang–Mills theory on a torus I–V\" (Hexagon 2610.00506, 00509, 00510, 00511, 00512; contributed by P. M. Aronow and Patrick Lopatto, 26 Oct 2026) claims that, at every fixed coupling and with bare coupling g² = g₃²a, the joint laws of finitely many Wilson-flowed Wilson loops of the SU(2) Wilson lattice measure on a fixed 3D torus converge as a → 0 along refinement sequences, to a unique probability measure on gauge orbits of Yang–Mills heat-flow trajectories (a version of the Cao–Chatterjee state space). It also gives short-distance Gaussian-free-field asymptotics and the weak-coupling concentration on flat connections. All five papers are labelled \"Primarily AI-generated text\" and \"Human understanding: no parts\".","key_facts":["Claim (abstract shared by the five papers, verbatim): 'We prove that, at every fixed value of the coupling, the joint laws of finitely many flowed Wilson loops converge as the lattice spacing tends to zero along the sequences of lattice sides nM^k, for fixed odd M ≥ 3 and fixed positive n divisible by four. For each such refinement class, the limits are realized by a unique probability measure on the space of gauge orbits of Yang–Mills heat-flow trajectories at positive times, a version of the state space introduced by Cao and Chatterjee.'","Method (abstract): 'a multiscale renormalization group analysis in the tradition of Bałaban, carried out on a small-field event whose complement we show to have small probability, uniformly in the lattice spacing, using reflection positivity and chessboard estimates', with stochastic estimates for the lattice heat flow and a deterministic analysis of the Wilson flow for rough initial data","Further claims: rescaled deficits (1 − W)/(g₃²ε) of small flowed Wilson loops converge to explicit quadratic functions of the heat-smoothed Gaussian free field; as the coupling tends to zero the measure concentrates on flat connections, with Gaussian fluctuations near regular commuting holonomies","Limits of the claim: SU(2) only, a fixed-size torus, convergence along refinement classes nM^k (not every lattice sequence), observables are flowed Wilson loops at positive flow time; nothing about the infinite-volume limit, a mass gap or four dimensions","Provenance statement (verbatim): 'Proof discovery and verification were done with a combination of Opus 5.5, GPT 6 Astra, and GPT 6.1 Sol. Opus 5.5 and GPT Sol 6.1 performed limited background formalizations of some claims and proof architecture in Lean 4 as part of verification. The writing was done by Opus 5.5, GPT 6 Astra, and GPT 6.1 Sol.' Contributors: P. M. Aronow (Yale), Patrick Lopatto (UNC Chapel Hill)","Same pair, 20 Oct: Hexagon 2610.00402 on Glauber dynamics of the low-temperature Sherrington–Kirkpatrick model from typical states, A_β n^{7/30} < log T ≤ n^{1−ω}, with the same models and Lean formalizations 'as part of verification'","Context: OpenAI's 26 Oct physics claim of 4D Yang–Mills and QCD existence has no public artifacts (entry 2026-10-26-openai-yang-mills-qcd-existence-claim); Anthropic's Alpöge and Sussman run a public attempt to finish Bałaban's 4D programme with Claude","Not checked: no referee, no expert comment found as of 29 Oct"],"key_numbers":[],"tags":["math","mathematical-physics","quantum-field-theory","yang-mills","constructive-qft","renormalization-group","hexagon","claude-opus-5-5","gpt-6-astra","gpt-6-1-sol","unverified"],"science":{"field":"physics","subfield":"mathematical physics (constructive quantum field theory, lattice gauge theory)","problem":"Rigorous continuum limit of three-dimensional Yang–Mills theory (here SU(2) on a torus, via Wilson-flowed observables)","result":"Claimed convergence of flowed Wilson-loop laws of 3D SU(2) lattice Yang–Mills on a torus to a unique continuum measure on the Cao–Chatterjee state space, with short-distance and weak-coupling asymptotics","open_since":"","ai_system":["Claude Opus 5.5","GPT-6 Astra","GPT-6.1 Sol"],"human_role":"Contributors posted the work; by the Hexagon label no part is understood by a human; proof discovery, verification and writing by the models","verification":{"key":"lean","label":"Lean-verified","text":"Unrefereed Hexagon preprints; only limited Lean formalization of background claims"},"status":"pending","shock":"A constructive-QFT result that would cap decades of work on lattice gauge theory is posted with an explicit statement that no human understands any part of it."},"body_md":"## What happened\n\nYang–Mills theory underlies the strong and weak forces, but nobody has built it as a mathematically rigorous continuum theory. In three\ndimensions the problem is easier because the coupling has positive dimension. Bałaban's renormalization-group work of the 1980s gave\nultraviolet bounds, and Cao and Chatterjee proposed a state space of gauge orbits of heat-flow trajectories on which a limit could live.\nA full construction has not been published.\n\nOn 26 Oct P. M. Aronow and Patrick Lopatto posted five linked papers on Hexagon. They claim the continuum limit for SU(2) on a fixed\nthree-dimensional torus, for Wilson-flowed observables, along refinement sequences of lattice sizes. The papers say the models found the\nproofs, checked them and wrote them, with partial Lean formalization of background claims. Hexagon's form lets contributors state how much\nthey understand, and they answered \"no parts\".\n\n## Why it matters\n\nIt is one of the largest AI-generated constructions in mathematical physics so far, and it targets a problem with a long human history.\nThe authors make no claim of human understanding, so the result counts as unverified until specialists read it. It also arrived on the\nday OpenAI announced its own, still private, four-dimensional Yang–Mills existence claim.","disputed":[{"label":"Verification","text":"Unrefereed Hexagon preprints; only limited Lean formalization of background claims","sources":[]}],"related":[{"id":"2026-10-29-ai-assisted-results-october-29-listing-2","url":"https://postcutoff.com/e/2026-10-29-ai-assisted-results-october-29-listing-2/","date":"2026-10-29","date_precision":"day","short_title":"AI-assisted results, 29 Oct 2026 (part 2)","deck":"Claude 'largely carried out' Lipshitz's Khovanov research, ChatGPT found Soundararajan's proof and wrote Etingof's and Mathew's papers, George Elliott's free-product classification 'carried out using ChatGPT', and a first read of Hexagon's backlog","takeaway":"A second pass over the 28 Oct arXiv announcement and a first full read of Hexagon (hexagonmath.org), which the sweep had never covered, found results with large AI roles signed by senior mathematicians. Robert Lipshitz writes that his Khovanov paper's research 'was largely carried out by various Claude models'. Kannan Soundararajan's proof of a problem from Montgomery's CBMS book 'was found by ChatGPT 6'. ChatGPT 6 Pro wrote Pavel Etingof's answer to a 1999 question and produced Akhil Mathew's answer to a question of Drinfeld. George Elliott's classification of reduced free products of nuclear C*-algebras 'was carried out using ChatGPT'. Eight items have their own entries: the unforced 2D Euler non-uniqueness claims, the Erdős–Lovász covering-number bound, UPP^cc under intersection, Kisin's Serre conjecture, Linnik's constant ≤ 3.99, 3D Yang–Mills on a torus, random k-SAT and Chenglong Ma's Hovey–Strickland provenance claim.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":30,"official":29,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI","Anthropic","Google DeepMind","various researchers"]},{"id":"2026-10-26-openai-yang-mills-qcd-existence-claim","url":"https://postcutoff.com/e/2026-10-26-openai-yang-mills-qcd-existence-claim/","date":"2026-10-26","date_precision":"day","short_title":"OpenAI claims 4D Yang–Mills and QCD existence","deck":null,"takeaway":"The physics third of OpenAI's 26 Oct announcement claims that certain interacting 4D quantum field theories exist mathematically, among them pure Yang–Mills and quantum chromodynamics. That is the existence half of the Yang–Mills Millennium problem, not the mass gap. A group led by David Gross, with Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov, is reviewing it. At the group's request the AI-produced material is not public while they write their own account. Nothing can be checked yet.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":13,"official":2,"filed":"2026-10-27","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-hexagon-repository-llm-assisted-math","url":"https://postcutoff.com/e/2026-10-06-hexagon-repository-llm-assisted-math/","date":"2026-10-06","date_precision":"day","short_title":"Hexagon launches: a new arXiv-style repository for LLM-assisted mathematics papers","deck":null,"takeaway":"It is the first dedicated infrastructure for AI-generated mathematics backed by leading mathematicians, and it accepts work that no human claims to understand.","category":"research","category_label":"Research","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":2,"official":2,"filed":"2026-10-07","updated":"2026-10-29","orgs":["Hexagon Mathematics Foundation"]}],"people":[{"id":"levent-alpoge","name":"Levent Alpöge","url":"https://postcutoff.com/person/levent-alpoge/"}],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-29","type":"filed","text":"Created (Hexagon records and API metadata of the five papers and of 2610.00402; found in a first full read of the Hexagon API)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-29","run":null,"sources_read":"Hexagon records and API metadata of the five papers and of 2610.00402; found in a first full read of the Hexagon API","updated":"2026-10-29","human_review":null,"version":{"date":"2026-10-29"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":179,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":118,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":209,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":148,"in_training_data":false}],"short_url":"https://postcutoff.com/s/3d-su-2-yang"}