{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2026-09-07-anandkumar-euler-singularity-r3/","md":"https://postcutoff.com/e/2026-09-07-anandkumar-euler-singularity-r3/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":"2026-09-07-anandkumar-euler-singularity-r3","date":"2026-09-07","date_precision":"day","short_title":"Caltech team (Anandkumar) reports a stable self-similar singularity candidate for the unforced 3D Euler equations on R³, found with PINNs and LLM help","deck":null,"takeaway":"On 7 Sep 2026, the evening before OpenAI's Navier–Stokes announcement, Anima Anandkumar's Caltech group posted a self-similar singular profile for the unforced incompressible 3D Euler equations on all of R³.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2609.10867: Self-Similar Singularity of the Euler Equations on R³","url":"https://arxiv.org/abs/2609.10867","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"arXiv 2609.10860: Stability Framework for the Singularity of the Euler Equations on R³","url":"https://arxiv.org/abs/2609.10860","type":"paper","group":"primary","domain":"arxiv.org"},{"n":3,"title":"Anandkumar group page on the Euler result","url":"https://tensorlab.cms.caltech.edu/users/anima/euler.html","type":"official","group":"primary","domain":"tensorlab.cms.caltech.edu"},{"n":4,"title":"Anima Anandkumar (guest post on Tao's blog): Stable singularity of the Euler equations on R³","url":"https://terrytao.wordpress.com/2026/09/10/stable-singularity-of-the-euler-equations-on-r3/","type":"discussion","group":"reactions","domain":"terrytao.wordpress.com"}],"official":3,"filed":"2026-09-29","updated":"2026-09-29","orgs":["Caltech"],"title":"Caltech team (Anandkumar) reports a stable self-similar singularity candidate for the unforced 3D Euler equations on R³, found with PINNs and LLM help","summary":"On 7 Sep 2026, the evening before OpenAI's Navier–Stokes announcement, Anima Anandkumar's Caltech group posted a self-similar singular profile for the unforced incompressible 3D Euler equations on all of R³. Physics-informed neural networks found it, and LLMs helped simplify the bounds and formalise derivations in Lean. The arXiv papers (2609.10867, 2609.10860) describe \"evidence\" and a stability framework that is conditional on certifying explicit constants, so this is not yet a complete proof.","key_facts":["Authors: Adarsh Ganeshram, Valentin Duruisseaux, Anima Anandkumar (+ Robert J. George on the stability paper)","Setting: incompressible Euler on unbounded R³, no forcing; axisymmetric self-similar ansatz at blow-up rate 0.5 (matching a prediction by Constantin et al., arXiv 2602.17570)","Method: PINN finds approximate profile; second-order optimisers (SS-eSOAP, SS-Broyden); certified via spline representation with interval arithmetic","AI use (guest post): 'we used the OpenAI and other models extensively to simplify our bounds as well as formalize the derivations in Lean'","arXiv 2609.10867 (111 pp.) abstract: 'We provide evidence of a finite-time singularity'; 2609.10860 (113 pp.): stability proof closes 'conditional on rigorous certification of the estimates and constants'","The authors complain that mainstream media followed OpenAI's press release and did not acknowledge their work"],"key_numbers":[],"tags":["math","pde","euler-equations","fluid-dynamics","pinn","lean","navier-stokes"],"science":{"field":"mathematics","subfield":"partial differential equations / fluid dynamics","problem":"Finite-time singularity for the unforced 3D incompressible Euler equations on R³ from smooth initial data","result":"Numerically certified self-similar singular profile plus a (conditional) framework for its nonlinear stability; full rigorous blow-up proof not yet complete.","open_since":"","ai_system":["physics-informed neural networks","OpenAI models and other LLMs"],"human_role":"Human-led; AI (PINNs) discovered the candidate, LLMs simplified bounds and helped formalise derivations","verification":{"key":"none","label":"Unverified claim","text":"Interval-arithmetic certification of the profile; partial Lean formalisation; stability conditional"},"status":"pending","shock":"Unlike OpenAI's and Buckmaster–Alpöge's results, it targets the unforced problem on the whole space, which is closer to what physicists care about."},"body_md":"## What happened\n\nIn the same week as the Buckmaster–Alpöge forced blow-up results (7 Sep) and OpenAI's forced Navier–Stokes claim (8 Sep), a third group posted a singularity for the *unforced* Euler equations on the whole space. They used AI-driven numerical discovery followed by computer-assisted proof techniques. Their guest post on Tao's blog stresses AI as a \"complementary\" tool, \"built to propose solutions that did not compete with humans\".\n\n## Why it matters\n\nUnforced Euler blow-up on R³ is a famous open problem in its own right, and it is a stepping stone toward the unforced Navier–Stokes question. The claim is still partly conditional, so its status should be tracked.","disputed":[{"label":"Verification","text":"Interval-arithmetic certification of the profile; partial Lean formalisation; stability conditional","sources":[]}],"related":[{"id":"2026-09-08-openai-navier-stokes-blowup","url":"https://postcutoff.com/e/2026-09-08-openai-navier-stokes-blowup/","date":"2026-09-08","date_precision":"day","short_title":"OpenAI claims a Millennium Prize problem","deck":"10,000 AI agents prove forced Navier–Stokes blow-up; priority dispute erupts","takeaway":"It is the first credible AI claim on a Clay Millennium Prize problem, even if only a technically permitted variant.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"disputed","labels":["Disputed"]},"sources":42,"official":4,"filed":"2026-09-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2025-09-17-deepmind-unstable-singularities-fluids","url":"https://postcutoff.com/e/2025-09-17-deepmind-unstable-singularities-fluids/","date":"2025-09-17","date_precision":"day","short_title":"DeepMind and mathematicians use neural networks to find new unstable singularities in fluid equations","deck":null,"takeaway":"It was the groundwork of the AI-plus-computer-assisted-proof approach to fluid blow-up that culminated in the disputed 2026 Navier–Stokes claims.","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":["Google DeepMind","New York University","Stanford University","Brown University"]}],"people":[{"id":"terence-tao","name":"Terence Tao","url":"https://postcutoff.com/person/terence-tao/"},{"id":"levent-alpoge","name":"Levent Alpöge","url":"https://postcutoff.com/person/levent-alpoge/"}],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-09-29","type":"filed","text":"Created. Marked pending because the arXiv abstracts describe the stability proof as conditional"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-09-29","run":null,"sources_read":null,"updated":"2026-09-29","human_review":null,"version":{"date":"2026-10-04"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":130,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":69,"in_training_data":false},{"model_id":"gemini-4-hydrogen","name":"Gemini 4 Hydrogen","cutoff":"2026-03","days_after":160,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":99,"in_training_data":false}],"short_url":null}