{"schema_version":"source-reader-v1","record":{"source_id":"fluid-regularity-lean","versions":[{"work":{"authors":["OpenAI"],"kind":"research_project","publication":"project","publication_note":"Public formal source; execution and statement correspondence are separate checks.","ai_role":"self_reported","ai_note":"The project metadata names GPT-6 Astra with Codex for formalization and marks alignment review as self-assessed. Lean, Mathlib and independent statement/Comparator infrastructure have separate supporting roles.","artifacts":[{"kind":"formal_code","title":"Navier–Stokes and Euler formal proof sources — pinned source","url":"https://github.com/openai/NavierStokesAndEuler/tree/f9e8bc5b38b6e212696e8a30e3e91517af887bbd","version":"f9e8bc5b38b6e212696e8a30e3e91517af887bbd","assessment":"not_executed","note":"Exact source inspected; OpenMath has not executed this formal proof project.","sha256":"d9866d5c1fdc16832f098e5f2affa96baf0b156a5f34b9be015d793e1abe2fdd"}],"published":{"value":"2026-09-08"}},"id":"fluid-regularity:source:lean:v1","source_id":"fluid-regularity-lean","title":"Navier–Stokes and Euler formal proof sources","owner":"OpenAI","url":"https://github.com/openai/NavierStokesAndEuler/tree/f9e8bc5b38b6e212696e8a30e3e91517af887bbd","subject_scope":"The project reports complete main-result formalizations, zero sorry counts and only standard logical axioms at the listed endpoints.","revision":"f9e8bc5b38b6e212696e8a30e3e91517af887bbd","content_sha256":"d9866d5c1fdc16832f098e5f2affa96baf0b156a5f34b9be015d793e1abe2fdd","update_method":"Compare the exact source and publish a new immutable curation revision.","identifier_policy":"Stable source identity with exact versioned passages.","observed_on":"2026-10-13","source_updated_on":null,"rights":"metadata-link","license":"Original source license and document rights apply.","terms_url":"https://github.com/openai/NavierStokesAndEuler/tree/f9e8bc5b38b6e212696e8a30e3e91517af887bbd","attribution":"OpenAI","rights_note":"OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.","check_interval_days":90}],"entries":[{"id":"euler-regularity:entry:lean-nonexistence:v1","source_version_id":"fluid-regularity:source:lean:v1","item_id":"lean-nonexistence","url":"https://github.com/openai/NavierStokesAndEuler/blob/f9e8bc5b38b6e212696e8a30e3e91517af887bbd/Euler/Solution.lean","wording":"A separate endpoint excludes a global smooth uniformly bounded-energy solution for decaying unforced initial data.","wording_kind":"paraphrase","locator":"Euler/Solution.lean: euler_breakdown_R3","supersedes":null},{"id":"euler-regularity:entry:lean-singularity:v1","source_version_id":"fluid-regularity:source:lean:v1","item_id":"lean-singularity","url":"https://github.com/openai/NavierStokesAndEuler/blob/f9e8bc5b38b6e212696e8a30e3e91517af887bbd/Euler/Solution.lean","wording":"The exact endpoint gives compact smooth nonzero data, a finite positive maximal Sobolev lifespan, bounded energy, infinite C¹ limsup and divergent integrated vorticity.","wording_kind":"paraphrase","locator":"Euler/Solution.lean: exists_compact_smooth_euler_singularity","supersedes":null},{"id":"fluid-regularity:entry:lean:v1","source_version_id":"fluid-regularity:source:lean:v1","item_id":"lean","url":"https://github.com/openai/NavierStokesAndEuler/tree/f9e8bc5b38b6e212696e8a30e3e91517af887bbd","wording":"The project reports complete main-result formalizations, zero sorry counts and only standard logical axioms at the listed endpoints.","wording_kind":"paraphrase","locator":"formalization.yaml; exact main declarations and Comparator instructions","supersedes":null},{"id":"navier-stokes:entry:lean-periodic:v1","source_version_id":"fluid-regularity:source:lean:v1","item_id":"lean-periodic","url":"https://github.com/openai/NavierStokesAndEuler/blob/f9e8bc5b38b6e212696e8a30e3e91517af887bbd/NavierStokes/ComparatorSolution.lean","wording":"The pinned comparator endpoint has every positive viscosity, periodic initial data and force, and no global smooth periodic solution.","wording_kind":"paraphrase","locator":"NavierStokes/ComparatorSolution.lean: navier_stokes_breakdown_periodic","supersedes":null},{"id":"navier-stokes:entry:lean-r3:v1","source_version_id":"fluid-regularity:source:lean:v1","item_id":"lean-r3","url":"https://github.com/openai/NavierStokesAndEuler/blob/f9e8bc5b38b6e212696e8a30e3e91517af887bbd/NavierStokes/ComparatorSolution.lean","wording":"The pinned comparator endpoint has every positive viscosity, decaying initial data and force, and no global smooth uniformly bounded-energy solution.","wording_kind":"paraphrase","locator":"NavierStokes/ComparatorSolution.lean: navier_stokes_breakdown_R3","supersedes":null}],"problems":[{"slug":"euler-regularity","title":"Three-dimensional Euler regularity","publication_version":3},{"slug":"navier-stokes","title":"Navier–Stokes existence and smoothness","publication_version":1}],"selected_version_id":"fluid-regularity:source:lean:v1","available_versions":[{"id":"fluid-regularity:source:lean:v2","revision":"f9e8bc5b38b6e212696e8a30e3e91517af887bbd","observed_on":"2026-10-13","title":"Navier–Stokes and Euler formal proof sources","publication":"project","content_sha256":null,"used_in_current_publications":true},{"id":"fluid-regularity:source:lean:v1","revision":"f9e8bc5b38b6e212696e8a30e3e91517af887bbd","observed_on":"2026-10-13","title":"Navier–Stokes and Euler formal proof sources","publication":"project","content_sha256":"d9866d5c1fdc16832f098e5f2affa96baf0b156a5f34b9be015d793e1abe2fdd","used_in_current_publications":false}],"usages":[{"id":"source-use-a08ddb485fb4d0849125","entry_id":"euler-regularity:entry:lean-nonexistence:v1","problem":{"slug":"euler-regularity","title":"Three-dimensional Euler regularity","publication_version":3},"kind":"assessment","title":"Global smooth finite-energy nonexistence reported","claim":{"id":"claim-euler-no-global-smooth-energy-solution","version":1,"slug":"euler-no-global-smooth-energy-solution","title":"No global smooth bounded-energy solution for selected data"},"fragment":"claim-euler-no-global-smooth-energy-solution","assessment":{"id":"euler-regularity:report:nonexistence:v1","entry_id":"euler-regularity:entry:lean-nonexistence:v1","state":"proposed_solution","source_label":"Global smooth finite-energy nonexistence reported","scope":"The distinct, less detailed formal endpoint.","source_date":"2026-09-08","observed_on":"2026-10-13","evidence_url":"https://github.com/openai/NavierStokesAndEuler/blob/f9e8bc5b38b6e212696e8a30e3e91517af887bbd/Euler/Solution.lean","supersedes":null},"superseded":false,"formalization":null,"update":null},{"id":"source-use-60bb6823c91abb8baf7e","entry_id":"euler-regularity:entry:lean-nonexistence:v1","problem":{"slug":"euler-regularity","title":"Three-dimensional Euler regularity","publication_version":3},"kind":"formalization","title":"Lean: unforced whole-space nonexistence","claim":{"id":"claim-euler-no-global-smooth-energy-solution","version":1,"slug":"euler-no-global-smooth-energy-solution","title":"No global smooth bounded-energy solution for selected data"},"fragment":"claim-euler-no-global-smooth-energy-solution","assessment":null,"superseded":false,"formalization":{"title":"Lean: unforced whole-space nonexistence","url":"https://github.com/openai/NavierStokesAndEuler/blob/f9e8bc5b38b6e212696e8a30e3e91517af887bbd/Euler/Solution.lean","note":"This endpoint excludes a global smooth uniformly bounded-energy solution. It is kept separate from the stronger compact-data finite-lifespan endpoint, so the former is not used alone to claim all of the latter’s details.","target":{"id":"claim-euler-no-global-smooth-energy-solution","version":1},"proof_status":"reported_complete","source_entry_ids":["euler-regularity:entry:lean-nonexistence:v1","fluid-regularity:entry:lean:v1"],"declaration":"Euler.euler_breakdown_R3","source_commit":"f9e8bc5b38b6e212696e8a30e3e91517af887bbd","alignment":"staff_linked","checked_on":"2026-10-13"},"update":null},{"id":"source-use-02f3a67f4def7e737a4f","entry_id":"euler-regularity:entry:lean-nonexistence:v1","problem":{"slug":"euler-regularity","title":"Three-dimensional Euler regularity","publication_version":3},"kind":"milestone","title":"Two scoped Euler proof endpoints","claim":null,"fragment":"history","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-cf123206ca624ee61d8b","entry_id":"euler-regularity:entry:lean-nonexistence:v1","problem":{"slug":"euler-regularity","title":"Three-dimensional Euler regularity","publication_version":3},"kind":"question","title":"Three-dimensional Euler regularity","claim":null,"fragment":"question","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-772dbf317d3123561d8b","entry_id":"euler-regularity:entry:lean-nonexistence:v1","problem":{"slug":"euler-regularity","title":"Three-dimensional Euler regularity","publication_version":3},"kind":"statement","title":"No global smooth bounded-energy solution for selected data","claim":{"id":"claim-euler-no-global-smooth-energy-solution","version":1,"slug":"euler-no-global-smooth-energy-solution","title":"No global smooth bounded-energy solution for selected data"},"fragment":"claim-euler-no-global-smooth-energy-solution","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-98ff5a9148b70a63ecf6","entry_id":"euler-regularity:entry:lean-singularity:v1","problem":{"slug":"euler-regularity","title":"Three-dimensional Euler regularity","publication_version":3},"kind":"formalization","title":"Lean: compact smooth Euler singularity","claim":{"id":"claim-euler-smooth-unforced-singularity","version":1,"slug":"euler-smooth-unforced-singularity","title":"Smooth unforced data with finite-time breakdown"},"fragment":"claim-euler-smooth-unforced-singularity","assessment":null,"superseded":false,"formalization":{"title":"Lean: compact smooth Euler singularity","url":"https://github.com/openai/NavierStokesAndEuler/blob/f9e8bc5b38b6e212696e8a30e3e91517af887bbd/Euler/Solution.lean","note":"This endpoint 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":1},"proof_status":"reported_complete","source_entry_ids":["euler-regularity:entry:lean-singularity:v1","fluid-regularity:entry:lean:v1"],"declaration":"Euler.exists_compact_smooth_euler_singularity","source_commit":"f9e8bc5b38b6e212696e8a30e3e91517af887bbd","alignment":"staff_linked","checked_on":"2026-10-13"},"update":null},{"id":"source-use-0d1b9f1839a483a4db7b","entry_id":"euler-regularity:entry:lean-singularity:v1","problem":{"slug":"euler-regularity","title":"Three-dimensional Euler regularity","publication_version":3},"kind":"milestone","title":"Two scoped Euler proof endpoints","claim":null,"fragment":"history","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-ca5457f2e6359d57c767","entry_id":"euler-regularity:entry:lean-singularity:v1","problem":{"slug":"euler-regularity","title":"Three-dimensional Euler regularity","publication_version":3},"kind":"question","title":"Three-dimensional Euler regularity","claim":null,"fragment":"question","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-0d8060a7568d3356686f","entry_id":"euler-regularity:entry:lean-singularity:v1","problem":{"slug":"euler-regularity","title":"Three-dimensional Euler regularity","publication_version":3},"kind":"statement","title":"Smooth unforced data with finite-time breakdown","claim":{"id":"claim-euler-smooth-unforced-singularity","version":1,"slug":"euler-smooth-unforced-singularity","title":"Smooth unforced data with finite-time breakdown"},"fragment":"claim-euler-smooth-unforced-singularity","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-68d26400ccdf46b0bad6","entry_id":"fluid-regularity:entry:lean:v1","problem":{"slug":"euler-regularity","title":"Three-dimensional Euler regularity","publication_version":3},"kind":"formalization","title":"Lean: compact smooth Euler singularity","claim":{"id":"claim-euler-smooth-unforced-singularity","version":1,"slug":"euler-smooth-unforced-singularity","title":"Smooth unforced data with finite-time breakdown"},"fragment":"claim-euler-smooth-unforced-singularity","assessment":null,"superseded":false,"formalization":{"title":"Lean: compact smooth Euler singularity","url":"https://github.com/openai/NavierStokesAndEuler/blob/f9e8bc5b38b6e212696e8a30e3e91517af887bbd/Euler/Solution.lean","note":"This endpoint 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":1},"proof_status":"reported_complete","source_entry_ids":["euler-regularity:entry:lean-singularity:v1","fluid-regularity:entry:lean:v1"],"declaration":"Euler.exists_compact_smooth_euler_singularity","source_commit":"f9e8bc5b38b6e212696e8a30e3e91517af887bbd","alignment":"staff_linked","checked_on":"2026-10-13"},"update":null},{"id":"source-use-96eafd108e0a0eedda6a","entry_id":"fluid-regularity:entry:lean:v1","problem":{"slug":"euler-regularity","title":"Three-dimensional Euler regularity","publication_version":3},"kind":"formalization","title":"Lean: unforced whole-space nonexistence","claim":{"id":"claim-euler-no-global-smooth-energy-solution","version":1,"slug":"euler-no-global-smooth-energy-solution","title":"No global smooth bounded-energy solution for selected data"},"fragment":"claim-euler-no-global-smooth-energy-solution","assessment":null,"superseded":false,"formalization":{"title":"Lean: unforced whole-space nonexistence","url":"https://github.com/openai/NavierStokesAndEuler/blob/f9e8bc5b38b6e212696e8a30e3e91517af887bbd/Euler/Solution.lean","note":"This endpoint excludes a global smooth uniformly bounded-energy solution. It is kept separate from the stronger compact-data finite-lifespan endpoint, so the former is not used alone to claim all of the latter’s details.","target":{"id":"claim-euler-no-global-smooth-energy-solution","version":1},"proof_status":"reported_complete","source_entry_ids":["euler-regularity:entry:lean-nonexistence:v1","fluid-regularity:entry:lean:v1"],"declaration":"Euler.euler_breakdown_R3","source_commit":"f9e8bc5b38b6e212696e8a30e3e91517af887bbd","alignment":"staff_linked","checked_on":"2026-10-13"},"update":null},{"id":"source-use-4eb9052434174d8a0b15","entry_id":"fluid-regularity:entry:lean:v1","problem":{"slug":"euler-regularity","title":"Three-dimensional Euler regularity","publication_version":3},"kind":"milestone","title":"Two scoped Euler proof endpoints","claim":null,"fragment":"history","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-9e5c3548fe505754df08","entry_id":"fluid-regularity:entry:lean:v1","problem":{"slug":"euler-regularity","title":"Three-dimensional Euler regularity","publication_version":3},"kind":"question","title":"Three-dimensional Euler regularity","claim":null,"fragment":"question","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-6e5f5ebac7cf5d58521d","entry_id":"fluid-regularity:entry:lean:v1","problem":{"slug":"euler-regularity","title":"Three-dimensional Euler regularity","publication_version":3},"kind":"statement","title":"Smooth unforced data with finite-time breakdown","claim":{"id":"claim-euler-smooth-unforced-singularity","version":1,"slug":"euler-smooth-unforced-singularity","title":"Smooth unforced data with finite-time breakdown"},"fragment":"claim-euler-smooth-unforced-singularity","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-8c5944208eea74e8288c","entry_id":"fluid-regularity:entry:lean:v1","problem":{"slug":"euler-regularity","title":"Three-dimensional Euler regularity","publication_version":3},"kind":"statement","title":"No global smooth bounded-energy solution for selected data","claim":{"id":"claim-euler-no-global-smooth-energy-solution","version":1,"slug":"euler-no-global-smooth-energy-solution","title":"No global smooth bounded-energy solution for selected data"},"fragment":"claim-euler-no-global-smooth-energy-solution","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-48c9c224ec773af17a95","entry_id":"fluid-regularity:entry:lean:v1","problem":{"slug":"navier-stokes","title":"Navier–Stokes existence and smoothness","publication_version":1},"kind":"activity","title":"Compare manuscript, comparator statement and proof dependencies","claim":null,"fragment":"activity","assessment":null,"superseded":false,"formalization":null,"update":{"id":"navier-stokes-alignment","title":"Compare manuscript, comparator statement and proof dependencies","kind":"formalization","source_entry_ids":["navier-stokes:entry:clay:v1","fluid-regularity:entry:lean:v1","navier-stokes:entry:lean-r3:v1","navier-stokes:entry:lean-periodic:v1"],"summary":"Check the official decay/periodic predicates, force regularity, energy condition and imported proof adapters at the pinned endpoints.","limitation":"The inspected metadata is self-assessed; a local build and independent correspondence check are not attached.","related_claim_ids":["claim-navier-stokes-forced-r3","claim-navier-stokes-forced-periodic"]}},{"id":"source-use-4c79dc96e4f7a2ee4dbd","entry_id":"fluid-regularity:entry:lean:v1","problem":{"slug":"navier-stokes","title":"Navier–Stokes existence and smoothness","publication_version":1},"kind":"formalization","title":"Lean: forced periodic breakdown","claim":{"id":"claim-navier-stokes-forced-periodic","version":1,"slug":"navier-stokes-forced-periodic","title":"Forced breakdown on the three-torus"},"fragment":"claim-navier-stokes-forced-periodic","assessment":null,"superseded":false,"formalization":{"title":"Lean: forced periodic breakdown","url":"https://github.com/openai/NavierStokesAndEuler/blob/f9e8bc5b38b6e212696e8a30e3e91517af887bbd/NavierStokes/ComparatorSolution.lean","note":"Exact periodic nonexistence endpoint for every positive viscosity. The comparator reference definitions and independent execution are separate checks; no unforced theorem is linked to this declaration.","target":{"id":"claim-navier-stokes-forced-periodic","version":1},"proof_status":"reported_complete","source_entry_ids":["navier-stokes:entry:lean-periodic:v1","fluid-regularity:entry:lean:v1"],"declaration":"NavierStokes.Comparator.navier_stokes_breakdown_periodic","source_commit":"f9e8bc5b38b6e212696e8a30e3e91517af887bbd","alignment":"staff_linked","checked_on":"2026-10-13"},"update":null},{"id":"source-use-cf64c1b8a041b63425c6","entry_id":"fluid-regularity:entry:lean:v1","problem":{"slug":"navier-stokes","title":"Navier–Stokes existence and smoothness","publication_version":1},"kind":"formalization","title":"Lean: forced whole-space breakdown","claim":{"id":"claim-navier-stokes-forced-r3","version":1,"slug":"navier-stokes-forced-r3","title":"Forced whole-space nonexistence"},"fragment":"claim-navier-stokes-forced-r3","assessment":null,"superseded":false,"formalization":{"title":"Lean: forced whole-space breakdown","url":"https://github.com/openai/NavierStokesAndEuler/blob/f9e8bc5b38b6e212696e8a30e3e91517af887bbd/NavierStokes/ComparatorSolution.lean","note":"The endpoint matches official alternative C with explicit decay/force/energy predicates. The stronger from-rest compact singular-flow Claim is separate and has no complete formalization asserted by this link. Metadata reports zero sorry and standard logical axioms only.","target":{"id":"claim-navier-stokes-forced-r3","version":1},"proof_status":"reported_complete","source_entry_ids":["navier-stokes:entry:lean-r3:v1","fluid-regularity:entry:lean:v1"],"declaration":"NavierStokes.Comparator.navier_stokes_breakdown_R3","source_commit":"f9e8bc5b38b6e212696e8a30e3e91517af887bbd","alignment":"staff_linked","checked_on":"2026-10-13"},"update":null},{"id":"source-use-b6f05704dedae0b26e33","entry_id":"fluid-regularity:entry:lean:v1","problem":{"slug":"navier-stokes","title":"Navier–Stokes existence and smoothness","publication_version":1},"kind":"milestone","title":"Pinned comparator proof revision","claim":null,"fragment":"history","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-083bfdc0e6038bd35204","entry_id":"fluid-regularity:entry:lean:v1","problem":{"slug":"navier-stokes","title":"Navier–Stokes existence and smoothness","publication_version":1},"kind":"question","title":"Navier–Stokes existence and smoothness","claim":null,"fragment":"question","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-2300d4637f70eb7d1035","entry_id":"fluid-regularity:entry:lean:v1","problem":{"slug":"navier-stokes","title":"Navier–Stokes existence and smoothness","publication_version":1},"kind":"statement","title":"Forced breakdown on the three-torus","claim":{"id":"claim-navier-stokes-forced-periodic","version":1,"slug":"navier-stokes-forced-periodic","title":"Forced breakdown on the three-torus"},"fragment":"claim-navier-stokes-forced-periodic","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-3d2db4a6ed165e3ba911","entry_id":"fluid-regularity:entry:lean:v1","problem":{"slug":"navier-stokes","title":"Navier–Stokes existence and smoothness","publication_version":1},"kind":"statement","title":"Forced whole-space nonexistence","claim":{"id":"claim-navier-stokes-forced-r3","version":1,"slug":"navier-stokes-forced-r3","title":"Forced whole-space nonexistence"},"fragment":"claim-navier-stokes-forced-r3","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-c4f15837a2edc0aac59a","entry_id":"navier-stokes:entry:lean-periodic:v1","problem":{"slug":"navier-stokes","title":"Navier–Stokes existence and smoothness","publication_version":1},"kind":"activity","title":"Compare manuscript, comparator statement and proof dependencies","claim":null,"fragment":"activity","assessment":null,"superseded":false,"formalization":null,"update":{"id":"navier-stokes-alignment","title":"Compare manuscript, comparator statement and proof dependencies","kind":"formalization","source_entry_ids":["navier-stokes:entry:clay:v1","fluid-regularity:entry:lean:v1","navier-stokes:entry:lean-r3:v1","navier-stokes:entry:lean-periodic:v1"],"summary":"Check the official decay/periodic predicates, force regularity, energy condition and imported proof adapters at the pinned endpoints.","limitation":"The inspected metadata is self-assessed; a local build and independent correspondence check are not attached.","related_claim_ids":["claim-navier-stokes-forced-r3","claim-navier-stokes-forced-periodic"]}},{"id":"source-use-14a599cfa39c1d01d2a3","entry_id":"navier-stokes:entry:lean-periodic:v1","problem":{"slug":"navier-stokes","title":"Navier–Stokes existence and smoothness","publication_version":1},"kind":"assessment","title":"Forced periodic breakdown reported","claim":{"id":"claim-navier-stokes-forced-periodic","version":1,"slug":"navier-stokes-forced-periodic","title":"Forced breakdown on the three-torus"},"fragment":"claim-navier-stokes-forced-periodic","assessment":{"id":"navier-stokes:report:forced-periodic:v1","entry_id":"navier-stokes:entry:lean-periodic:v1","state":"proposed_solution","source_label":"Forced periodic breakdown reported","scope":"Every positive viscosity; smooth periodic data and forcing.","source_date":"2026-09-08","observed_on":"2026-10-13","evidence_url":"https://github.com/openai/NavierStokesAndEuler/blob/f9e8bc5b38b6e212696e8a30e3e91517af887bbd/NavierStokes/ComparatorSolution.lean","supersedes":null},"superseded":false,"formalization":null,"update":null},{"id":"source-use-35e50ad8b6eb22270399","entry_id":"navier-stokes:entry:lean-periodic:v1","problem":{"slug":"navier-stokes","title":"Navier–Stokes existence and smoothness","publication_version":1},"kind":"formalization","title":"Lean: forced periodic breakdown","claim":{"id":"claim-navier-stokes-forced-periodic","version":1,"slug":"navier-stokes-forced-periodic","title":"Forced breakdown on the three-torus"},"fragment":"claim-navier-stokes-forced-periodic","assessment":null,"superseded":false,"formalization":{"title":"Lean: forced periodic breakdown","url":"https://github.com/openai/NavierStokesAndEuler/blob/f9e8bc5b38b6e212696e8a30e3e91517af887bbd/NavierStokes/ComparatorSolution.lean","note":"Exact periodic nonexistence endpoint for every positive viscosity. The comparator reference definitions and independent execution are separate checks; no unforced theorem is linked to this declaration.","target":{"id":"claim-navier-stokes-forced-periodic","version":1},"proof_status":"reported_complete","source_entry_ids":["navier-stokes:entry:lean-periodic:v1","fluid-regularity:entry:lean:v1"],"declaration":"NavierStokes.Comparator.navier_stokes_breakdown_periodic","source_commit":"f9e8bc5b38b6e212696e8a30e3e91517af887bbd","alignment":"staff_linked","checked_on":"2026-10-13"},"update":null},{"id":"source-use-f4cd771537c1099d4390","entry_id":"navier-stokes:entry:lean-periodic:v1","problem":{"slug":"navier-stokes","title":"Navier–Stokes existence and smoothness","publication_version":1},"kind":"milestone","title":"Pinned comparator proof revision","claim":null,"fragment":"history","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-77b5c12a276dac15b920","entry_id":"navier-stokes:entry:lean-periodic:v1","problem":{"slug":"navier-stokes","title":"Navier–Stokes existence and smoothness","publication_version":1},"kind":"question","title":"Navier–Stokes existence and smoothness","claim":null,"fragment":"question","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-c0cbdfff4ed4cfabb9a7","entry_id":"navier-stokes:entry:lean-periodic:v1","problem":{"slug":"navier-stokes","title":"Navier–Stokes existence and smoothness","publication_version":1},"kind":"statement","title":"Forced breakdown on the three-torus","claim":{"id":"claim-navier-stokes-forced-periodic","version":1,"slug":"navier-stokes-forced-periodic","title":"Forced breakdown on the three-torus"},"fragment":"claim-navier-stokes-forced-periodic","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-170e8a2e3451b21dc500","entry_id":"navier-stokes:entry:lean-r3:v1","problem":{"slug":"navier-stokes","title":"Navier–Stokes existence and smoothness","publication_version":1},"kind":"activity","title":"Compare manuscript, comparator statement and proof dependencies","claim":null,"fragment":"activity","assessment":null,"superseded":false,"formalization":null,"update":{"id":"navier-stokes-alignment","title":"Compare manuscript, comparator statement and proof dependencies","kind":"formalization","source_entry_ids":["navier-stokes:entry:clay:v1","fluid-regularity:entry:lean:v1","navier-stokes:entry:lean-r3:v1","navier-stokes:entry:lean-periodic:v1"],"summary":"Check the official decay/periodic predicates, force regularity, energy condition and imported proof adapters at the pinned endpoints.","limitation":"The inspected metadata is self-assessed; a local build and independent correspondence check are not attached.","related_claim_ids":["claim-navier-stokes-forced-r3","claim-navier-stokes-forced-periodic"]}},{"id":"source-use-d4fe0a344c99eec6377d","entry_id":"navier-stokes:entry:lean-r3:v1","problem":{"slug":"navier-stokes","title":"Navier–Stokes existence and smoothness","publication_version":1},"kind":"formalization","title":"Lean: forced whole-space breakdown","claim":{"id":"claim-navier-stokes-forced-r3","version":1,"slug":"navier-stokes-forced-r3","title":"Forced whole-space nonexistence"},"fragment":"claim-navier-stokes-forced-r3","assessment":null,"superseded":false,"formalization":{"title":"Lean: forced whole-space breakdown","url":"https://github.com/openai/NavierStokesAndEuler/blob/f9e8bc5b38b6e212696e8a30e3e91517af887bbd/NavierStokes/ComparatorSolution.lean","note":"The endpoint matches official alternative C with explicit decay/force/energy predicates. The stronger from-rest compact singular-flow Claim is separate and has no complete formalization asserted by this link. Metadata reports zero sorry and standard logical axioms only.","target":{"id":"claim-navier-stokes-forced-r3","version":1},"proof_status":"reported_complete","source_entry_ids":["navier-stokes:entry:lean-r3:v1","fluid-regularity:entry:lean:v1"],"declaration":"NavierStokes.Comparator.navier_stokes_breakdown_R3","source_commit":"f9e8bc5b38b6e212696e8a30e3e91517af887bbd","alignment":"staff_linked","checked_on":"2026-10-13"},"update":null},{"id":"source-use-ab277db75260746b34a5","entry_id":"navier-stokes:entry:lean-r3:v1","problem":{"slug":"navier-stokes","title":"Navier–Stokes existence and smoothness","publication_version":1},"kind":"milestone","title":"Pinned comparator proof revision","claim":null,"fragment":"history","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-b75ecaac2923863ca51c","entry_id":"navier-stokes:entry:lean-r3:v1","problem":{"slug":"navier-stokes","title":"Navier–Stokes existence and smoothness","publication_version":1},"kind":"question","title":"Navier–Stokes existence and smoothness","claim":null,"fragment":"question","assessment":null,"superseded":false,"formalization":null,"update":null},{"id":"source-use-fb15b0a691b0372ffff1","entry_id":"navier-stokes:entry:lean-r3:v1","problem":{"slug":"navier-stokes","title":"Navier–Stokes existence and smoothness","publication_version":1},"kind":"statement","title":"Forced whole-space nonexistence","claim":{"id":"claim-navier-stokes-forced-r3","version":1,"slug":"navier-stokes-forced-r3","title":"Forced whole-space nonexistence"},"fragment":"claim-navier-stokes-forced-r3","assessment":null,"superseded":false,"formalization":null,"update":null}],"linked_entries":[{"id":"euler-regularity:entry:lean-nonexistence:v1","source_version_id":"fluid-regularity:source:lean:v1","item_id":"lean-nonexistence","url":"https://github.com/openai/NavierStokesAndEuler/blob/f9e8bc5b38b6e212696e8a30e3e91517af887bbd/Euler/Solution.lean","wording":"A separate endpoint excludes a global smooth uniformly bounded-energy solution for decaying unforced initial data.","wording_kind":"paraphrase","locator":"Euler/Solution.lean: euler_breakdown_R3","supersedes":null},{"id":"euler-regularity:entry:lean-singularity:v1","source_version_id":"fluid-regularity:source:lean:v1","item_id":"lean-singularity","url":"https://github.com/openai/NavierStokesAndEuler/blob/f9e8bc5b38b6e212696e8a30e3e91517af887bbd/Euler/Solution.lean","wording":"The exact endpoint gives compact smooth nonzero data, a finite positive maximal Sobolev lifespan, bounded energy, infinite C¹ limsup and divergent integrated vorticity.","wording_kind":"paraphrase","locator":"Euler/Solution.lean: exists_compact_smooth_euler_singularity","supersedes":null},{"id":"fluid-regularity:entry:lean:v1","source_version_id":"fluid-regularity:source:lean:v1","item_id":"lean","url":"https://github.com/openai/NavierStokesAndEuler/tree/f9e8bc5b38b6e212696e8a30e3e91517af887bbd","wording":"The project reports complete main-result formalizations, zero sorry counts and only standard logical axioms at the listed endpoints.","wording_kind":"paraphrase","locator":"formalization.yaml; exact main declarations and Comparator instructions","supersedes":null},{"id":"navier-stokes:entry:lean-periodic:v1","source_version_id":"fluid-regularity:source:lean:v1","item_id":"lean-periodic","url":"https://github.com/openai/NavierStokesAndEuler/blob/f9e8bc5b38b6e212696e8a30e3e91517af887bbd/NavierStokes/ComparatorSolution.lean","wording":"The pinned comparator endpoint has every positive viscosity, periodic initial data and force, and no global smooth periodic solution.","wording_kind":"paraphrase","locator":"NavierStokes/ComparatorSolution.lean: navier_stokes_breakdown_periodic","supersedes":null},{"id":"navier-stokes:entry:lean-r3:v1","source_version_id":"fluid-regularity:source:lean:v1","item_id":"lean-r3","url":"https://github.com/openai/NavierStokesAndEuler/blob/f9e8bc5b38b6e212696e8a30e3e91517af887bbd/NavierStokes/ComparatorSolution.lean","wording":"The pinned comparator endpoint has every positive viscosity, decaying initial data and force, and no global smooth uniformly bounded-energy solution.","wording_kind":"paraphrase","locator":"NavierStokes/ComparatorSolution.lean: navier_stokes_breakdown_R3","supersedes":null}],"linked_sources":[{"work":{"authors":["OpenAI"],"kind":"research_project","publication":"project","publication_note":"Public formal source; execution and statement correspondence are separate checks.","ai_role":"self_reported","ai_note":"The project metadata names GPT-6 Astra with Codex for formalization and marks alignment review as self-assessed. Lean, Mathlib and independent statement/Comparator infrastructure have separate supporting roles.","artifacts":[{"kind":"formal_code","title":"Navier–Stokes and Euler formal proof sources — pinned source","url":"https://github.com/openai/NavierStokesAndEuler/tree/f9e8bc5b38b6e212696e8a30e3e91517af887bbd","version":"f9e8bc5b38b6e212696e8a30e3e91517af887bbd","assessment":"not_executed","note":"Exact source inspected; OpenMath has not executed this formal proof project.","sha256":"d9866d5c1fdc16832f098e5f2affa96baf0b156a5f34b9be015d793e1abe2fdd"}],"published":{"value":"2026-09-08"}},"id":"fluid-regularity:source:lean:v1","source_id":"fluid-regularity-lean","title":"Navier–Stokes and Euler formal proof sources","owner":"OpenAI","url":"https://github.com/openai/NavierStokesAndEuler/tree/f9e8bc5b38b6e212696e8a30e3e91517af887bbd","subject_scope":"The project reports complete main-result formalizations, zero sorry counts and only standard logical axioms at the listed endpoints.","revision":"f9e8bc5b38b6e212696e8a30e3e91517af887bbd","content_sha256":"d9866d5c1fdc16832f098e5f2affa96baf0b156a5f34b9be015d793e1abe2fdd","update_method":"Compare the exact source and publish a new immutable curation revision.","identifier_policy":"Stable source identity with exact versioned passages.","observed_on":"2026-10-13","source_updated_on":null,"rights":"metadata-link","license":"Original source license and document rights apply.","terms_url":"https://github.com/openai/NavierStokesAndEuler/tree/f9e8bc5b38b6e212696e8a30e3e91517af887bbd","attribution":"OpenAI","rights_note":"OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.","check_interval_days":90}]}}