{"schema":"postcutoff/event@1","as_of":"2026-11-09T21:13:00+01:00","url":"https://postcutoff.com/e/2026-11-06-navier-stokes-unforced-leray-hopf-nonuniqueness-buckmaster-coiculescu-li/","md":"https://postcutoff.com/e/2026-11-06-navier-stokes-unforced-leray-hopf-nonuniqueness-buckmaster-coiculescu-li/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":"2026-11-06-navier-stokes-unforced-leray-hopf-nonuniqueness-buckmaster-coiculescu-li","date":"2026-11-06","date_precision":"day","short_title":"Navier–Stokes: unforced Leray–Hopf non-uniqueness, AI-written","deck":null,"takeaway":"On 6 Nov 2026 three NYU mathematicians deposited on Hexagon an 80-page proof, written mainly by Claude Opus and Codex, that the unforced 3D Navier–Stokes equations have a continuum of Leray–Hopf solutions with the same initial data, following the Jia–Šverák programme. The authors say they understand the broad steps; a Lean formalization is announced but not yet public.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"Hexagon 2611.00143: Non-Uniqueness of Leray-Hopf Solutions to the Unforced Three-Dimensional Navier-Stokes Equations","url":"https://hexagonmath.org/abs/2611.00143","type":"paper","group":"primary","domain":"hexagonmath.org"},{"n":2,"title":"Announced Lean repository (forthcoming)","url":"https://github.com/mcoiculescu/unforced-leray-hopf-public","type":"code","group":"primary","domain":"github.com"},{"n":3,"title":"Hou, Wang, Yang: computer-assisted non-uniqueness for the unforced equations (arXiv 2509.25116)","url":"https://arxiv.org/abs/2509.25116","type":"paper","group":"primary","domain":"arxiv.org"},{"n":4,"title":"Albritton, Brué, Colombo: forced case (arXiv 2112.03116)","url":"https://arxiv.org/abs/2112.03116","type":"paper","group":"primary","domain":"arxiv.org"}],"official":4,"filed":"2026-11-07","updated":"2026-11-08","orgs":["New York University","Anthropic","OpenAI"],"title":"Buckmaster, Coiculescu and Li post a 'primarily AI-generated' proof that Leray–Hopf solutions of the unforced 3D Navier–Stokes equations are not unique","summary":"Hexagon 2611.00143 (Tristan Buckmaster, Matei Coiculescu, Chen Li, NYU; tools Claude Opus, Codex, Lean) claims that the incompressible Navier–Stokes equations on R³ without external force admit a continuum of distinct suitable Leray–Hopf weak solutions with the same compactly supported, axisymmetric no-swirl datum in L^{3,∞}. The construction uses a discretely self-similar background built from a train of thin vortex rings. Provenance: \"Primarily AI-generated\"; some parts checked by a human. Hou, Wang and Yang had claimed the unforced case earlier by a computer-assisted proof (arXiv 2509.25116).","key_facts":["Reactions by 8 Nov: no response from Thomas Hou (whose computer-assisted proof of the unforced case, arXiv 2509.25116, came first) or other experts found by web search","Claim (abstract): a continuum of distinct suitable weak solutions in the Leray–Hopf class with the same datum; solutions smooth at positive times, satisfying local and global energy equalities, separating in L² at rate t^{1/4+σ}","Method (abstract): 'The proof follows the program of Jia and Šverák, with a discretely self-similar background in place of a self-similar one', built 'from a train of thin vortex rings' obtained by lifting a periodic row of planar vortices; a growing Floquet mode is constructed from the angular-two instability of the planar vortex","AI disclosure (Hexagon provenance, verbatim): 'Text declaration: Primarily AI-generated. … Some parts of this contribution, but not all, have been checked and understood by a human. … The broad steps of the construction have been understood by the authors.' Tools: Claude Opus, Codex, Lean","Lean: 'Lean formalization forthcoming at the public repository at https://github.com/mcoiculescu/unforced-leray-hopf-public'; the authors 'used an adaptation of Scott Armstrong's skill files for the Lean formalization process'","Prior work: Albritton–Brué–Colombo proved non-uniqueness for the forced equations (Annals 2022); Hou, Wang and Yang claimed the unforced case with a computer-assisted proof of an unstable self-similar profile (arXiv 2509.25116, v2 March 2026). The Hexagon paper's bibliography cites Albritton–Brué–Colombo and Guillod–Šverák but, in the parsed list, not Hou–Wang–Yang","Status: unrefereed deposit, 80 pages, v1 6 Nov; no expert reaction found as of 7 Nov (HN, web search)"],"key_numbers":[],"tags":["math","pde","navier-stokes","fluid-dynamics","claude","codex","lean","hexagon"],"science":{"field":"mathematics","subfield":"partial differential equations (fluid dynamics)","problem":"Uniqueness of Leray–Hopf weak solutions of the unforced 3D incompressible Navier–Stokes equations (Leray 1934; Jia–Šverák programme 2014–15)","result":"Claimed continuum of distinct suitable Leray–Hopf solutions from one compactly supported axisymmetric datum in L^{3,∞}, without force","open_since":"1934","ai_system":["Claude Opus","Codex"],"human_role":"AI-generated text and proofs; authors understood the broad steps of the construction","verification":{"key":"lean-unreviewed","label":"Lean proof, not yet reviewed","text":"Unrefereed Hexagon deposit; Lean formalization announced, not yet public"},"status":"pending","shock":""},"body_md":"## What happened\n\nTristan Buckmaster, Matei Coiculescu and Chen Li deposited the paper on Hexagon on 6 Nov, labelled \"no human author claimed\" and \"primarily AI-generated\". It carries out the Jia–Šverák strategy: find a background solution whose linearization has a growing mode, then build a second solution that branches off it. Here the background is discretely self-similar and is constructed explicitly, so the argument does not depend on numerically certified profiles.\n\nThe question was already reported answered: Hou, Wang and Yang (arXiv 2509.25116) gave a computer-assisted proof in 2025–26. The new deposit claims a different and more explicit route, written mainly by AI. The authors say a Lean formalization will follow.\n\n## Why it matters\n\nUniqueness of Leray–Hopf solutions is one of the central questions about the Navier–Stokes equations, alongside (and separate from) the Millennium regularity problem. Buckmaster is a leading figure in fluid non-uniqueness, so the deposit carries weight, but it is unrefereed and only partly checked by its authors.","disputed":[{"label":"Verification","text":"Unrefereed Hexagon deposit; Lean formalization announced, not yet public","sources":[]}],"related":[{"id":"2026-10-17-2d-euler-unforced-nonuniqueness-lp-vorticity-ai","url":"https://postcutoff.com/e/2026-10-17-2d-euler-unforced-nonuniqueness-lp-vorticity-ai/","date":"2026-10-17","date_precision":"day","short_title":"2D Euler non-uniqueness without force claimed three times, all with AI","deck":null,"takeaway":"Uniqueness of 2D Euler solutions is known for bounded vorticity (Yudovich, 1963). With a force, Vishik showed in 2018 that it fails for vorticity in L^p, p > 2. Without a force this stayed open. Within ten days, three groups posted proofs that it fails there as well. On 17 Oct Umberto Pappalettera and Massimo Sorella posted on Hexagon a manuscript 'developed autonomously by ChatGPT 6.1 Sol'. They claim no authorship and run a reading group to check it. On 22 Oct Matei Coiculescu (NYU) posted Part I of a two-part proof whose details, code and text were 'largely produced with Claude Opus 5.5'. His repository has a Lean 4 formalization of the whole proof, with no sorry. On 27 Oct Hyungjun Choi (Princeton) posted an 86-page construction, with ChatGPT completing technical steps. As of 29 Oct we found no expert assessment of any of the three.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":9,"official":6,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI","Anthropic","Hexagon Mathematics Foundation"]},{"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"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-11-08","type":"edit","text":"Re-checked reactions incl. Hou's: none found"},{"date":"2026-11-07","type":"filed","text":"Created (science agent; Hexagon record and provenance read; prior Hou–Wang–Yang claim found by web search)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-11-07","run":null,"sources_read":"Science agent; Hexagon record and provenance read; prior Hou–Wang–Yang claim found by web search","updated":"2026-11-08","human_review":null,"version":{"date":"2026-11-08"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":190,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":129,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":220,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":159,"in_training_data":false}],"short_url":"https://postcutoff.com/s/navier-stokes-unforced-leray"}