A Lean 4 Formalization of the Classical Continuation Chain for Periodic Navier–Stokes on 𝕋³: Local Theory, BKM Criterion, and a Conditional Reduction of Statement (B)
Year: 2026
Type: preprint
Abstract: A machine-checked Lean 4/Mathlib formalization (repository BenFrohman/periodic-ns-classical-continuation, commit 9e43113 (Lean audit as of parent 36fa434)) of the classical continuation theory for smooth incompressible Navier–Stokes on the periodic torus, modelled as ℝ³ with ℤ³-periodicity and unit-cell integrals. It is proved using only the standard axioms (propext, Classical.choice, Quot.sound), with 1048 theorems, 285 definitions and no sorry. The results are: local existence for ν ≥ 0, energy-estimate uniqueness, maximal solutions, the Hˢ blow-up alternative for s > 5/2, H³ Beale–Kato–Majda control, and the BKM criterion. It is further proved that PeriodicBKMIntegrability ⇔ PeriodicGlobalLifespan ⇒ Clay_B_exact. PeriodicBKMIntegrability is the sole open input and is equivalent to the full Millennium-problem content. This work does not solve the Clay problem or prove global regularity of 3D Navier–Stokes. Includes an axiom-audit appendix.
Author: Frohman, Benjamin Stanley
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225011779