Reading source record f9e8bc5b38b6e212696e8a30e3e91517af887bbd. View default cited record →
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.
Replaces an earlier passage. View versioned records →
Passage ID: euler-regularity:entry:lean-nonexistence:v2
Source item: lean-nonexistence
Source record: fluid-regularity:source:lean:v2
Replaces passage: euler-regularity:entry:lean-nonexistence:v1
The exact endpoint gives compact smooth nonzero data, a finite positive maximal Sobolev lifespan, bounded energy, infinite C¹ limsup and divergent integrated vorticity.
Replaces an earlier passage. View versioned records →
Passage ID: euler-regularity:entry:lean-singularity:v2
Source item: lean-singularity
Source record: fluid-regularity:source:lean:v2
Replaces passage: euler-regularity:entry:lean-singularity:v1
The project reports complete main-result formalizations, zero sorry counts and only standard logical axioms at the listed endpoints.
Replaces an earlier passage. View versioned records →
Passage ID: fluid-regularity:entry:lean:v2
Source item: lean
Source record: fluid-regularity:source:lean:v2
Replaces passage: fluid-regularity:entry:lean:v1
The pinned comparator endpoint has every positive viscosity, periodic initial data and force, and no global smooth periodic solution.
Replaces an earlier passage. View versioned records →
Passage ID: navier-stokes:entry:lean-periodic:v2
Source item: lean-periodic
Source record: fluid-regularity:source:lean:v2
Replaces passage: navier-stokes:entry:lean-periodic:v1
The pinned comparator endpoint has every positive viscosity, decaying initial data and force, and no global smooth uniformly bounded-energy solution.
Replaces an earlier passage. View versioned records →
Passage ID: navier-stokes:entry:lean-r3:v2
Source item: lean-r3
Source record: fluid-regularity:source:lean:v2
Replaces passage: navier-stokes:entry:lean-r3:v1
Explicit citations in versioned problem publications. A citation records a use; it does not confer proof or independent verification.
Problem publication v9
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
Scope
No global smooth bounded-energy solution for selected data
Formal declaration
Lean: compact smooth Euler singularity
Scope
Smooth unforced data with finite-time breakdown
History
Two distinct Euler formal theorems
Question
Three-dimensional Euler regularity
Further reading
Inspect the formal statement, assumptions and reported checks.
Problem publication v8
Formal declaration
Lean: forced periodic breakdown
Scope
Forced breakdown on the three-torus
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
Scope
Forced whole-space nonexistence
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 theorems.
The inspected metadata is self-assessed; a local build and independent correspondence check are not attached.
History
Pinned formal proof revision
Question
Navier–Stokes existence and smoothness
Further reading
Inspect the formal statement, assumptions and reported checks.
Linked manuscripts, code and data. External proof reports, statement correspondence and OpenMath execution are separate checks.
Formal code
No OpenMath execution recorded
Exact proof source inspected. This digest identifies this file only, not a project build, theorem correspondence or independent reproduction.
Resource revision: f9e8bc5b38b6e212696e8a30e3e91517af887bbd
SHA-256 for this linked file. A matching checksum identifies the same bytes; it does not assess the mathematics.
52950d5d618a8d34c9bfbdb16641c81c276e97b6d7fd2a76c08577353f0b0227Formal code
No OpenMath execution recorded
Exact proof source inspected. This digest identifies this file only, not a project build, theorem correspondence or independent reproduction.
Resource revision: f9e8bc5b38b6e212696e8a30e3e91517af887bbd
SHA-256 for this linked file. A matching checksum identifies the same bytes; it does not assess the mathematics.
5dced6caecbfd85d952ff9537e420a85d4f38386e2c655a9c569a21b7139ea4fComplete formal proof reported externally
This formal theorem excludes a global smooth uniformly bounded-energy solution. It is kept separate from the stronger compact-data finite-lifespan theorem, 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 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.
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 theorem for every positive viscosity. The 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 formal theorem matches official alternative C with explicit decay/force and energy assumptions. 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 ↗