Theorem · version 2
Earlier version 2. View current version 3 →
For every ν > 0 there exist smooth divergence-free rapidly decaying initial velocity and a smooth force whose space-time derivatives decay faster than every inverse power, for which no global smooth incompressible Navier–Stokes velocity and pressure with uniformly bounded kinetic energy exist on ℝ³.
Theorem 1.1 and official alternative C. The manuscript and full formal endpoint are source-reported; mathematical assessment remains pending.
This source presents a theorem with a proposed proof. Its wording does not establish acceptance or independent mathematical review.
Proposed
Every positive viscosity; compact smooth force, zero initial velocity, bounded kinetic energy.
Finite Time Blowup for Navier–Stokes · Theorem 1.1; Corollary 10.6; Section 1.1 →
Source dated Sep 8, 2026 · Observed Oct 13, 2026
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
In this problem · publication v2
Counterexample · Proposed
Relationship not independently reviewed. This concerns its use in this problem.
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
Source revision: 2000. Observed Oct 13, 2026.
No specific AI contribution is attributed by this source record.
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
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.
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.
The pinned comparator endpoint has every positive viscosity, decaying initial data and force, and no global smooth uniformly bounded-energy 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.
Exact proof source inspected. This digest identifies this file only, not a project build, theorem correspondence or independent reproduction.
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 C/D comparator proofs: f9e8bc5b38b6e212696e8a30e3e91517af887bbd.
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proposed
Every positive viscosity; compact smooth force, zero initial velocity, bounded kinetic energy.
Source dated Sep 8, 2026 · Observed Oct 13, 2026 · Source passage →