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The project reports complete main-result formalizations, zero sorry counts and only standard logical axioms at the listed endpoints.
Public formal source; execution and statement correspondence are separate checks.
5 cited passages ↓2 problems ↓Record f9e8bc5b38b6e212696e8a30e3e91517af887bbd ↓
The exact passages cited in OpenMath. Each passage identifies how its wording was recorded.
A separate endpoint excludes a global smooth uniformly bounded-energy solution for decaying unforced initial data.
Passage ID: euler-regularity:entry:lean-nonexistence:v1
Source item: lean-nonexistence
Source record: fluid-regularity:source:lean:v1
The exact endpoint gives compact smooth nonzero data, a finite positive maximal Sobolev lifespan, bounded energy, infinite C¹ limsup and divergent integrated vorticity.
Passage ID: euler-regularity:entry:lean-singularity:v1
Source item: lean-singularity
Source record: fluid-regularity:source:lean:v1
The project reports complete main-result formalizations, zero sorry counts and only standard logical axioms at the listed endpoints.
Passage ID: fluid-regularity:entry:lean:v1
Source item: lean
Source record: fluid-regularity:source:lean:v1
The pinned comparator endpoint has every positive viscosity, periodic initial data and force, and no global smooth periodic solution.
Passage ID: navier-stokes:entry:lean-periodic:v1
Source item: lean-periodic
Source record: fluid-regularity:source:lean:v1
The pinned comparator endpoint has every positive viscosity, decaying initial data and force, and no global smooth uniformly bounded-energy solution.
Passage ID: navier-stokes:entry:lean-r3:v1
Source item: lean-r3
Source record: fluid-regularity:source:lean:v1
Explicit citations in versioned problem publications. A citation records a use; it does not confer proof or independent verification.
Problem publication v3
Source assessment
Proposed
Global smooth finite-energy nonexistence reported
The distinct, less detailed formal endpoint.
Source date Sep 8, 2026 · Observed Oct 13, 2026
Formal declaration
Lean: unforced whole-space nonexistence
Formal declaration
Lean: compact smooth Euler singularity
History
Two scoped Euler proof endpoints
Problem publication v1
Formal declaration
Lean: forced periodic breakdown
Statement
Source assessment
Proposed
Forced periodic breakdown reported
Every positive viscosity; smooth periodic data and forcing.
Source date Sep 8, 2026 · Observed Oct 13, 2026
Formal declaration
Lean: forced whole-space breakdown
Research activity
Compare manuscript, comparator statement and proof dependencies
Check the official decay/periodic predicates, force regularity, energy condition and imported proof adapters at the pinned endpoints.
The inspected metadata is self-assessed; a local build and independent correspondence check are not attached.
History
Pinned comparator proof revision
Linked manuscripts, code and data. External proof reports, statement correspondence and OpenMath execution are separate checks.
Formal code
No OpenMath execution recorded
Exact source inspected; OpenMath has not executed this formal proof project.
Resource revision: f9e8bc5b38b6e212696e8a30e3e91517af887bbd
SHA-256 for this linked file. A matching checksum identifies the same bytes; it does not assess the mathematics.
d9866d5c1fdc16832f098e5f2affa96baf0b156a5f34b9be015d793e1abe2fddComplete formal proof reported externally
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.
Statement match: Linked by OpenMath · independent review separate · checked Oct 13, 2026
Euler.euler_breakdown_R3Commit f9e8bc5b38b6e212696e8a30e3e91517af887bbd
Complete formal proof reported externally
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.
Statement match: Linked by OpenMath · independent review separate · checked Oct 13, 2026
Euler.exists_compact_smooth_euler_singularityCommit f9e8bc5b38b6e212696e8a30e3e91517af887bbd
Complete formal proof reported externally
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.
Statement match: Linked by OpenMath · independent review separate · checked Oct 13, 2026
NavierStokes.Comparator.navier_stokes_breakdown_periodicCommit f9e8bc5b38b6e212696e8a30e3e91517af887bbd
Complete formal proof reported externally
The endpoint matches official alternative C with explicit decay/force/energy predicates. The stronger from-rest compact singular-flow statement is separate and has no complete formalization asserted by this link. Metadata reports zero sorry and standard logical axioms only.
Statement match: Linked by OpenMath · independent review separate · checked Oct 13, 2026
NavierStokes.Comparator.navier_stokes_breakdown_R3Commit f9e8bc5b38b6e212696e8a30e3e91517af887bbd
Recorded authorship
OpenAI
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.
OpenMath source records preserve exact citations. Observation dates order these records; they do not establish the latest upstream paper or proof.
Navier–Stokes and Euler formal proof sources
Navier–Stokes and Euler formal proof sources
OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.
Source terms ↗