{"payload":{"schema_version":"research-record-v0","publication":{"version":9,"sha256":"1883b4c256f264536335a7f29901c121e0292036b6d36b1a78a72d75553b51fd","published_at":"2026-10-21T13:14:56.516136+00:00","reason":"Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.","withdrawn":false,"draft":{"reader_guide":{"reading":[{"entry_id":"euler-regularity:entry:bkm:v1","role":"introduction","note":"Read the setting and notation before comparing the results."},{"entry_id":"euler-regularity:entry:paper:v1","role":"proof","note":"Read the mathematical argument; inspect its Claim for assessment and scope."},{"entry_id":"euler-regularity:entry:lean-nonexistence:v2","role":"formal_source","note":"Inspect the formal statement, assumptions and reported checks."}],"definitions":[{"id":"euler-regularity-definition","term":"Incompressible and unforced","explanation":"The velocity has zero divergence and the equation has no external force. 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Forced, bounded-domain and lower-regularity results have different hypotheses.","assessment":"Independently assess the reported smooth-data construction and its formal correspondence.","source_entry_ids":["euler-regularity:entry:paper:v1","fluid-regularity:entry:announcement:v1","fluid-regularity:entry:lean:v2","euler-regularity:entry:bkm:v1","euler-regularity:entry:lean-singularity:v2","euler-regularity:entry:lean-nonexistence:v2"]}},"representations":[{"hash_profile":"rfc8785","schema_version":"visual-representation-v1","id":"euler-state-ladder","version":1,"subject":{"kind":"problem","id":"problem-euler-regularity","sha256":"ee6e0dcd152c3818d574f4e343ee6fa3d778859795e8d7a912d44bbe0488abd9"},"purpose":"setup","title":"How close earlier results came","caption":"From the question at the top, through singularities proved in nearby settings, to what holds for every smooth solution. 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it is not an existence construction.","claims":[{"id":"claim-euler-vorticity-breakdown-criterion","version":1}],"source_entry_ids":["euler-regularity:entry:bkm:v1","euler-regularity:entry:paper:v1"],"independent_review":"pending"},{"id":"euler-regularity-context","date":null,"date_basis":"publication","category":"foundation","title":"Regularity and boundary conditions matter","summary":"Earlier constructions discussed in the manuscript use lower-regularity data, forcing or different domains.","limitation":"Their conclusions must not be relabeled as smooth compact-data whole-space unforced blowup.","claims":[],"source_entry_ids":["euler-regularity:entry:paper:v1"],"independent_review":"pending"},{"id":"euler-regularity-release","date":{"value":"2026-09-08"},"date_basis":"announcement","category":"proposal","title":"A smooth unforced whole-space construction is released","summary":"OpenAI releases the Euler manuscript and formal source alongside the viscous work.","limitation":"The announcement credits separate concurrent forced-Euler work to Alpöge and Buckmaster; 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the 2026 manuscript’s §1.1 for earlier singularity results.","formal_status":"The pinned project reports both a detailed compact-data finite-lifespan theorem and a separate global bounded-energy nonexistence theorem, with zero sorry and standard logical axioms. 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In particular, a finite value of this integral allows smooth continuation past that time.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"For an unforced three-dimensional Euler solution in the classical Sobolev class "},{"kind":"math","text":"H^s(ℝ³)","latex":"H^s(\\mathbb R^3)"},{"kind":"text","text":", "},{"kind":"math","text":"s > 5/2","latex":"s>5/2"},{"kind":"text","text":", a finite maximal smooth lifespan T* requires "},{"kind":"math","text":"∫₀ᵀ* ‖curl u(t)‖∞ dt = ∞","latex":"\\int_0^{T_*}\\|\\operatorname{curl}u(t)\\|_{\\infty}\\,dt=\\infty"},{"kind":"text","text":". In particular, a finite value of this integral allows smooth continuation past that time."}]},"id":"claim-euler-vorticity-breakdown-criterion","slug":"euler-vorticity-breakdown-criterion","version":1,"kind":"theorem","title":"Finite-time smooth breakdown requires divergent integrated vorticity","assumptions":["Classical divergence-free Euler solution on ℝ³ in H^s with s > 5/2.","The solution is smooth up to each earlier time and the continuation criterion applies."],"scope":"Beale–Kato–Majda is a conditional criterion; it does not itself construct a singularity.","source_entry_ids":["euler-regularity:entry:bkm:v1","euler-regularity:entry:paper:v1"],"formal_declarations":[]},{"statement":"For sufficiently small α > 0 there is a C^{1,α} axisymmetric velocity without swirl on ℝ³ whose incompressible Euler solution forms a singularity in finite time.","id":"claim-euler-elgindi-c1alpha","slug":"euler-elgindi-c1alpha","version":1,"kind":"theorem","title":"Blowup from C^{1,α} data in ℝ³","assumptions":["Three-dimensional incompressible Euler equations.","Reported in the 2026 manuscript’s historical context."],"scope":"Data less regular than smooth; the smooth-data question in ℝ³ is not settled by it.","source_entry_ids":["euler-regularity:entry:history:v1"],"formal_declarations":[]},{"statement":"For smooth initial data in an axially periodic cylinder with an impermeable wall, the three-dimensional incompressible Euler equations form a singularity in finite time.","id":"claim-euler-chen-hou-boundary","slug":"euler-chen-hou-boundary","version":1,"kind":"theorem","title":"Smooth-data blowup with a boundary","assumptions":["Three-dimensional incompressible Euler equations.","Reported in the 2026 manuscript’s historical context."],"scope":"A bounded domain with a wall, proved with analysis and rigorous numerics; not compactly supported data in all of ℝ³.","source_entry_ids":["euler-regularity:entry:history:v1"],"formal_declarations":[]},{"statement":"There are finite-energy solutions of the forced incompressible Euler equations on ℝ³, with velocity in C^{3,1/2} before blowup and a force uniformly bounded in C^{1,1/2−ε} ∩ L², that form a singularity in finite time.","id":"claim-euler-forced-blowup","slug":"euler-forced-blowup","version":1,"kind":"theorem","title":"Blowup with external forcing","assumptions":["Three-dimensional incompressible Euler equations.","Reported in the 2026 manuscript’s historical context."],"scope":"A force is applied; the question asks about unforced evolution.","source_entry_ids":["euler-regularity:entry:history:v1"],"formal_declarations":[]}],"problem":{"resolution_scope":{"resolution":"unresolved","extent":"general_case","summary":"OpenAI reports finite-time breakdown from smooth compactly supported unforced Euler data, with a detailed formal theorem. 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Lower-regularity data, bounded domains and forced Euler are different scopes.","scope_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"Velocity u and pressure p satisfy "},{"kind":"math","text":"∂ₜu + (u·∇)u = −∇p","latex":"\\partial_t u+(u\\cdot\\nabla)u=-\\nabla p"},{"kind":"text","text":" and div u = 0 on ℝ³. There is no viscosity and no external force. The selected question uses smooth divergence-free compactly supported initial velocity, the maximal smooth lifespan T*, and growth of velocity derivatives/vorticity. Lower-regularity data, bounded domains and forced Euler are different scopes."}]},"aliases":["Unforced Euler blowup","Euler global regularity","Smooth 3D Euler singularity"],"fields":["Partial differential equations","Mathematical physics"],"msc":["35Q31","76B03"],"topics":[],"background":"specialist","scale":"major conjecture","depth":"maintained","source_entry_ids":["euler-regularity:entry:paper:v1","fluid-regularity:entry:announcement:v1","fluid-regularity:entry:lean:v2","euler-regularity:entry:bkm:v1","euler-regularity:entry:lean-singularity:v2","euler-regularity:entry:lean-nonexistence:v2","euler-regularity:entry:history:v1"],"status_report_ids":["euler-regularity:report:singularity:v1"],"editorial":{"collection_id":"openmath-pilot","collection_title":"OpenMath pilot collection","reviewer":"OpenMath source curation","state":"under_assessment","scope":"Source comparison and reader curation; independent mathematical assessment remains pending.","rationale":"OpenAI reports finite-time breakdown from smooth compactly supported unforced Euler data, with a detailed formal theorem. The new proof and statement correspondence are under assessment.","decided_on":"2026-10-13","review_due_on":"2027-01-13","report_ids":["euler-regularity:report:singularity:v1"],"independent_mathematical_review":"pending"},"literature":[{"title":"Finite Time Blowup for the Euler Equation","url":"https://cdn.openai.com/pdf/315b36cd-ec98-4023-8342-93345194ece1/euler.pdf","note":"Theorem 1.1 and Section 1.1"},{"title":"On the Navier–Stokes Millennium Prize Problem","url":"https://openai.com/index/navier-stokes-solution/","note":"Our process; concurrent work; Euler and Navier–Stokes scope"},{"title":"Navier–Stokes and Euler formal proof sources","url":"https://github.com/openai/NavierStokesAndEuler/tree/f9e8bc5b38b6e212696e8a30e3e91517af887bbd","note":"formalization.yaml; exact main declarations and Comparator instructions"},{"title":"Remarks on the Breakdown of Smooth Solutions for the 3-D Euler Equations","url":"https://doi.org/10.1007/BF01212349","note":"Communications in Mathematical Physics 94 (1984), 61–66; criterion also recalled in the 2026 manuscript Section 1.1"}],"formalizations":[{"title":"Lean: compact smooth Euler singularity","url":"https://github.com/openai/NavierStokesAndEuler/blob/f9e8bc5b38b6e212696e8a30e3e91517af887bbd/Euler/Solution.lean","note":"This formal theorem supplies compact support, a maximal lifespan 0 < T* ≤ 1 in the all-order Sobolev class, bounded energy before breakdown, infinite C¹ limsup and divergent integrated vorticity. The project reports only standard logical axioms and self-assessed correspondence. It is not an OpenMath execution.","target":{"id":"claim-euler-smooth-unforced-singularity","version":3},"proof_status":"reported_complete","source_entry_ids":["euler-regularity:entry:lean-singularity:v2","fluid-regularity:entry:lean:v2"],"declaration":"Euler.exists_compact_smooth_euler_singularity","source_commit":"f9e8bc5b38b6e212696e8a30e3e91517af887bbd","alignment":"staff_linked","checked_on":"2026-10-13"},{"title":"Lean: unforced whole-space nonexistence","url":"https://github.com/openai/NavierStokesAndEuler/blob/f9e8bc5b38b6e212696e8a30e3e91517af887bbd/Euler/Solution.lean","note":"This formal theorem excludes a global smooth uniformly bounded-energy solution. 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