{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-17-2d-euler-unforced-nonuniqueness-lp-vorticity-ai/","md":"https://postcutoff.com/e/2026-10-17-2d-euler-unforced-nonuniqueness-lp-vorticity-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":"2026-10-17-2d-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":[{"n":1,"title":"Hexagon 2610.00314: Non-unique vanishing-viscosity solutions of 2D Euler with vorticity in L^p (Pappalettera, Sorella; ChatGPT 6.1 Sol)","url":"https://hexagonmath.org/abs/2610.00314v1","type":"paper","group":"primary","domain":"hexagonmath.org"},{"n":2,"title":"arXiv 2610.30655: Coiculescu, Non-uniqueness of weak solutions to the unforced 2D Euler equations with vorticity in L¹ ∩ L^p, Part I","url":"https://arxiv.org/abs/2610.30655","type":"paper","group":"primary","domain":"arxiv.org"},{"n":3,"title":"mcoiculescu/NonUniqueEuler: manuscript Parts I–II, Lean 4 formalization and Arb certificate","url":"https://github.com/mcoiculescu/NonUniqueEuler","type":"code","group":"primary","domain":"github.com"},{"n":4,"title":"arXiv 2610.37653: Choi, Nonuniqueness of solutions to 2D incompressible Euler equations with unbounded vorticity","url":"https://arxiv.org/abs/2610.37653","type":"paper","group":"primary","domain":"arxiv.org"},{"n":5,"title":"Background: Bruè, Colombo, Kumar, Flexibility of two-dimensional Euler flows with integrable vorticity (arXiv 2408.07934)","url":"https://arxiv.org/abs/2408.07934","type":"paper","group":"primary","domain":"arxiv.org"},{"n":6,"title":"Background: Mengual, Non-uniqueness of admissible solutions for the 2D Euler equation with L^p vortex data (arXiv 2304.09578)","url":"https://arxiv.org/abs/2304.09578","type":"paper","group":"primary","domain":"arxiv.org"},{"n":7,"title":"2D Euler reading-group site: manuscripts, chats and source material (nonuniqueness and Delort's question)","url":"https://2deuler.github.io/2deuler/","type":"discussion","group":"reactions","domain":"2deuler.github.io"},{"n":8,"title":"2deuler: source material (prompts, models, dates, 'Human contribution: None')","url":"https://2deuler.github.io/2deuler/sources.html","type":"discussion","group":"reactions","domain":"2deuler.github.io"},{"n":9,"title":"2deuler: Delort's question, sign-changing vortex-sheet data (GPT-6 Astra manuscript)","url":"https://2deuler.github.io/2deuler/delort.html","type":"discussion","group":"reactions","domain":"2deuler.github.io"}],"official":6,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI","Anthropic","Hexagon Mathematics Foundation"],"title":"Three AI-assisted proofs in ten days claim non-uniqueness for the unforced 2D Euler equations with L^p vorticity: a fully GPT-6.1 Sol manuscript on Hexagon, Coiculescu's Lean-formalized proof written largely with Claude Opus 5.5, and Choi's quadrupole cascade","summary":"Three independent preprints claim that the two-dimensional incompressible Euler equations without external force have two distinct weak solutions with the same initial vorticity in L^p for large finite p, the regime where Vishik proved non-uniqueness only with a force: Pappalettera and Sorella (Hexagon 2610.00314, 17 Oct; manuscript \"fully AI generated\" by ChatGPT 6.1 Sol, vanishing-viscosity solutions, every 2 < p < ∞), Coiculescu (arXiv 2610.30655, 22 Oct, Part I of two; proofs, code and text \"largely produced with Claude Opus 5.5\"; Lean 4 formalization of the full proof, including the interval-arithmetic certificate) and Choi (arXiv 2610.37653, 27 Oct; vorticity in every finite L^p; ChatGPT 5.6, 6.0 and 6.1 completed technical details). The same reading-group site also hosts a GPT-6 Astra manuscript claiming Delort's 1990 question on sign-changing vortex sheets. None has been refereed or publicly assessed by experts.","key_facts":["The problem, as Choi states it from Mengual (CMP 2024) and the Princeton book on Vishik's theorem (Albritton, Bruè, Colombo, De Lellis, Giri, Janisch, Kwon, 2024): 'For each 2 < p < ∞, there exist initial velocity and vorticity … from which the unforced Euler equation admits distinct weak solutions with u ∈ C([0, T]; L²), ω ∈ L∞([0, T]; L¹ ∩ L^p). The regularity is required throughout the evolution, not only initially.'","Prior state: Yudovich (1963) proved uniqueness for bounded integrable vorticity; Vishik (2018) proved non-uniqueness for the forced equation with L^p vorticity; Mengual (2023) obtained infinitely many admissible solutions from L¹ ∩ L^p initial vorticity (control only at the initial time); Bruè, Colombo and Kumar (2024) got unforced non-uniqueness with vorticity in L^p 'for some p > 1' by convex integration","Pappalettera–Sorella (Hexagon 2610.00314v1, 17 Oct; both listed with the University of Basel): 'For every finite p > 2, we construct one real compactly supported zero-circulation initial vorticity in L¹(ℝ²) ∩ L^p(ℝ²) whose unforced Navier–Stokes evolutions have two distinct strong vanishing-viscosity accumulation paths. Both limits are renormalized unforced Euler solutions with the same strong initial trace.' Hexagon labels: 'Primarily AI-generated text', 'Human understanding: no parts'","Their provenance statement: 'The manuscript was developed autonomously by ChatGPT 6.1 Sol and the manuscript is fully AI generated. The contributors did not edit the resulting PDF and do not claim authorship of the manuscript’s text or proofs. We are sharing the manuscript here because we would like the mathematics behind this result to be prioritized over a counterproductive race to upload AI-generated proofs to arXiv.' The source page dates the research request to 8 Oct (model 'OpenAI GPT-6.1 Sol (Ultra)', 'Human contribution: None') and publishes the chat, prompts and manuscript versions","Coiculescu (arXiv 2610.30655, 22 Oct, Part I; Part II only in the repository): two distinct renormalized weak solutions from a radial, nonnegative, unbounded datum in L¹ ∩ L^q for every q < ∞, one stationary and one 3-fold symmetric that leaves it in finite time. AI statement: 'the overall strategy of the proof are due to the author. Within this strategy, the detailed proofs and estimates, the code for the computer-assisted verification in Part II and the text were largely produced with Claude Opus 5.5 (Anthropic), under the direction of the author. The author has checked all proofs by hand and takes full responsibility for the paper.' He thanks Tristan Buckmaster","Coiculescu's repository (github.com/mcoiculescu/NonUniqueEuler): a Lean 4 + Mathlib formalization of 'the whole proof, including the algorithms of the computer-assisted part', with 'No sorry, only the standard axioms'; the Lean-verified certificate checker ran 8.75 hours on 22 Oct and printed true on all 261 cells (106,680 Evans-function enclosures); the same certificate is checked by a Python/Arb program. Nobody else has reported rebuilding it","Choi (arXiv 2610.37653, 27 Oct, 86 pages; Princeton PhD student of Peter Constantin and Alexandru Ionescu): 'We construct two distinct solutions of the two-dimensional incompressible Euler equation with the same initial vorticity in ⋂_{1≤p<∞} L^p(ℝ²). The vorticities are compactly supported in space and have mean zero, so the corresponding velocities have finite energy.' Mechanism: 'a cascade of transitions in quadrupole dynamics across infinitely many scales'. Lean formalization 'coming soon'","Choi's AI statement: 'ChatGPT 5.6, 6.0, and 6.1 were used under the author’s guidance to realize the cascade strategy through a sequence of quadrupole patches. In particular, they helped complete the technical details of replacing point vortices by patches, mainly in Section 3. … The main ideas of the paper were generated by the author, who takes full responsibility for the correctness of the proof.' He adds that 'while completing this work' he learned of Coiculescu's preprint and the Hexagon preprint","Second claim on the same site (2deuler.github.io): 'Global weak Euler flows with finite signed initial vorticity', a GPT-6 Astra Ultra run of 8 Oct prompted by Josef Demmel ('Human contribution: None'), answering Delort's 1990 question with infinitely many weak solutions for every finite signed measure in H⁻¹ by convex integration; the FAU Erlangen and Basel team (Emil Wiedemann, Josef Demmel, Alessandro Violini) says it wants to 'examine the mathematical ideas behind this work, understand their scope and limitations'","Not checked yet: no referee report, no expert comment, no Hacker News or press coverage found as of 29 Oct (web, HN Algolia, Hexagon). The three constructions differ (vanishing-viscosity limits from Navier–Stokes; a stationary radial datum with an unstable ring; a quadrupole cascade), so they do not depend on each other"],"key_numbers":[],"tags":["math","pde","fluid-dynamics","euler-equations","non-uniqueness","yudovich","hexagon","lean","computer-assisted-proof","chatgpt","gpt-6-1-sol","gpt-6-astra","claude-opus-5-5"],"science":{"field":"mathematics","subfield":"partial differential equations (incompressible fluids)","problem":"Non-uniqueness of weak solutions of the unforced 2D Euler equations with vorticity in L^p, 2 < p < ∞ (beyond Yudovich's uniqueness class; the unforced form of Vishik's theorem)","result":"Three independent claimed constructions of two distinct solutions with the same initial vorticity: vanishing-viscosity solutions for every p > 2 (Pappalettera–Sorella, fully AI-generated), renormalized weak solutions in L¹ ∩ L^q for every q < ∞ (Coiculescu, Lean-formalized), and finite-energy solutions in every finite L^p (Choi)","open_since":"1963","ai_system":["GPT-6.1 Sol","Claude Opus 5.5","ChatGPT 5.6","ChatGPT 6.0","ChatGPT 6.1","GPT-6 Astra"],"human_role":"Ranges from none (Pappalettera–Sorella: prompted only, no edits, no claim of authorship) to human strategy with AI-written details (Coiculescu: idea and strategy his, proofs, code and text largely by Claude; Choi: main ideas his, ChatGPT completed technical details)","verification":{"key":"lean","label":"Lean-verified","text":"Unrefereed preprints (Hexagon, arXiv); Coiculescu reports a complete Lean 4 formalization with a Lean-verified interval-arithmetic checker, not independently rebuilt"},"status":"pending","shock":"A problem that the Vishik literature treated as the next big open question in 2D fluid dynamics drew three unrelated AI-assisted proofs within ten days, one of them generated with no human contribution."},"body_md":"## What happened\n\nYudovich proved in 1963 that a 2D ideal fluid has a unique evolution when its vorticity is bounded. Whether this survives for\nunbounded vorticity in L^p is a central question in mathematical fluid dynamics. Vishik answered it negatively in 2018, but only with a\nspecially built external force. Without a force, earlier constructions either controlled the vorticity only at the initial time\n(Mengual) or reached only L^p for p close to 1 (Bruè, Colombo and Kumar).\n\nIn October three groups claimed the unforced case:\n\n- **17 Oct, Hexagon.** Umberto Pappalettera and Massimo Sorella deposited a manuscript produced by ChatGPT 6.1 Sol from a research request\n  of 8 Oct. They did not edit it and do not claim it as theirs. They posted it on Hexagon rather than arXiv, to avoid \"a counterproductive\n  race to upload AI-generated proofs to arXiv\", and set up a reading group. Its site publishes the chats and prompts. It also hosts a\n  second machine-generated manuscript, by GPT-6 Astra, on Delort's 1990 vortex-sheet question, which a team from FAU Erlangen and Basel\n  is examining.\n- **22 Oct, arXiv.** Matei Coiculescu (NYU, a Simons Junior Fellow and earlier co-author of Tristan Buckmaster) posted Part I of a two-part proof\n  and the whole proof in a public repository. He credits himself with the strategy and Claude Opus 5.5 with most of the detailed proofs,\n  the code and the text. He says he checked every proof by hand, and that the repository holds a complete Lean formalization.\n- **27 Oct, arXiv.** Hyungjun Choi (Princeton) posted an 86-page construction from a cascade of rotating vortex quadrupoles. ChatGPT\n  helped turn point vortices into patches. He writes that he learned of the other two works while finishing his own.\n\n## Why it matters\n\nIf any of the three is right, the gap between Yudovich's uniqueness theorem and non-uniqueness closes completely, without a force.\nThe three papers show the range of AI involvement on one problem: a proof with no human contribution, posted with a public reading group;\na human strategy with a model-written proof and a machine-checked formalization; and a human construction with model help on the details.\nThe Lean formalization is the strongest evidence offered so far, but nobody has rebuilt it or checked that its statements match the\ntheorem. We found no expert assessment of any of the three as of 29 Oct.","disputed":[{"label":"Verification","text":"Unrefereed preprints (Hexagon, arXiv); Coiculescu reports a complete Lean 4 formalization with a Lean-verified interval-arithmetic checker, not independently rebuilt","sources":[]}],"related":[{"id":"2026-10-29-ai-assisted-results-october-29-listing","url":"https://postcutoff.com/e/2026-10-29-ai-assisted-results-october-29-listing/","date":"2026-10-29","date_precision":"day","short_title":"AI-assisted results in the 29 Oct 2026 sweep","deck":"Counterexamples to Wilking's 2007 question found by GPT-6 Astra, an LLM-written Plateau paper by De Lellis and Glaudo, explicit Reed–Solomon codes with optimal list size (Guruswami et al.), rank-four quiver mutation decided by GPT Astra, and a dozen named conjectures","takeaway":"The 29 Oct sweep covered the arXiv announcement of 28 Oct, which the previous run had read only through keyword queries. This time 43 papers with an AI statement surfaced, and a PDF scan of a further 405 new and cross-listed submissions in math, cs.CC, cs.DS and cs.DM found about 180 more with an AI statement, most of them describing more than editing help. Four results have their own entries: OpenAI's abundance claim and Jihao Liu's priority remark, OpenAI's Lean-checked L = RL = BPL, R(3,3,3,3) ≤ 61 and the sharp Banach–Mazur exponent. This page lists the rest. GPT-6 Astra found the counterexamples to Wilking's 2007 question 'with minimal human input' and discovered the main results of a rank-four quiver paper. Claude found examples in opinion dynamics and proposed a minimal-surface construction. An LLM supplied 'the main idea' of a paper by Camillo De Lellis, a member of the IAS advisory group on AI and mathematics, and wrote it.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":23,"official":23,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI","Anthropic","various researchers"]},{"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"]},{"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":"2026-10-29","type":"filed","text":"Created (Hexagon record and API, 2deuler.github.io pages, arXiv PDFs of 2610.30655 and 2610.37653, the NonUniqueEuler README, arXiv abstracts of 2408.07934 and 2304.09578, HN Algolia and web search for reactions). Found through Choi's paper in the 28 Oct arXiv listing; our automated search did not read Hexagon and missed Coiculescu's preprint (its AI note is in the arXiv comments field and the PDF, not the abstract)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-29","run":null,"sources_read":"Hexagon record and API, 2deuler.github.io pages, arXiv PDFs of 2610.30655 and 2610.37653, the NonUniqueEuler README, arXiv abstracts of 2408.07934 and 2304.09578, HN Algolia and web search for reactions).","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":170,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":109,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":200,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":139,"in_training_data":false}],"short_url":"https://postcutoff.com/s/2d-euler-non-uniqueness-without"}