Proposed
Forced periodic breakdown reported ↗
Every positive viscosity; smooth periodic data and forcing.
Source dated Sep 8, 2026 · Observed Oct 13, 2026
Theorem · version 1
Forced breakdown on the three-torus
Earlier version 1. View current version 2 →
Statement
For every ν > 0 there are smooth divergence-free periodic initial data and smooth periodic forcing for which no global smooth periodic solution of the incompressible Navier–Stokes equations exists on ℝ³/ℤ³.
Assumptions
- Positive viscosity and smooth periodic data/force.
- The spatial domain is the unit three-torus.
Scope
Corollary 10.6 and official alternative D; no unforced conclusion is asserted.
This source presents a theorem with a proposed proof. Its wording does not establish acceptance or independent mathematical review.
Evidence
Source reports
Formal proofs
Lean: forced periodic breakdown ↗
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
Declaration & revision
NavierStokes.Comparator.navier_stokes_breakdown_periodicCommit f9e8bc5b38b6e212696e8a30e3e91517af887bbd
Connections
In this problem · publication v1
Navier–Stokes existence and smoothness →
Different statement · related variant · Proposed
Relationship not independently reviewed. This concerns its use in this problem.
Sources
Existence and Smoothness of the Navier–Stokes Equation →
- Equations (1)–(9) and alternatives (A)–(D) → · Original passage ↗
Recorded wording
The official problem specifies alternatives A/B for unforced regularity and C/D for forced breakdown, with separate decay, periodicity and energy conditions.
paraphrase · OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.
Charles L. Fefferman · Published · Publication date 2000
Revision & attribution
Source revision: 2000. Observed Oct 13, 2026.
No specific AI contribution is attributed by this source record.
Finite Time Blowup for Navier–Stokes →
- Theorem 1.1; Corollary 10.6; Section 1.1 → · Original passage ↗
Recorded wording
The manuscript states forced breakdown from rest with compactly supported smooth forcing, bounded kinetic energy and unbounded velocity; it also states a periodic corollary.
paraphrase · OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.
OpenAI · Preprint · Publication date Sep 8, 2026
Revision & attribution
Source revision: 2026-09-08. Observed Oct 13, 2026.
OpenAI attributes the argument to its internal research model. Bibliographic authorship is printed as OpenAI; individual human preparation and checking roles are not inferred.
Navier–Stokes and Euler formal proof sources →
- formalization.yaml; exact main declarations and Comparator instructions → · Original passage ↗
Recorded wording
The project reports complete main-result formalizations, zero sorry counts and only standard logical axioms at the listed endpoints.
paraphrase · OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.
- NavierStokes/ComparatorSolution.lean: navier_stokes_breakdown_periodic → · Original passage ↗
Recorded wording
The pinned comparator endpoint has every positive viscosity, periodic initial data and force, and no global smooth periodic solution.
paraphrase · OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.
OpenAI · Project documentation · Publication date Sep 8, 2026
Public formal source; execution and statement correspondence are separate checks.
Navier–Stokes and Euler formal proof sources — pinned source ↗No OpenMath execution recorded
Exact source inspected; OpenMath has not executed this formal proof project.
Revision & attribution
Source revision: f9e8bc5b38b6e212696e8a30e3e91517af887bbd. Observed Oct 13, 2026.
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.
Navier–Stokes and Euler formal proof sources — pinned source: f9e8bc5b38b6e212696e8a30e3e91517af887bbd.
History
Statement versions
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessment history · 1
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proposed
Forced periodic breakdown reported ↗
Every positive viscosity; smooth periodic data and forcing.
Source dated Sep 8, 2026 · Observed Oct 13, 2026 · Source passage →
Record details
- Claim ID
- claim-navier-stokes-forced-periodic
- Statement version
- 1
- SHA-256
- cd1df99f66e9fef0fb604037a3e3094dfb60286f5cc1c2e5ad4b62dc70a68e9b