2D Euler non-uniqueness without force claimed three times, all with AI

Your AI and this story

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