Grade Decomposition and Gevrey Regularity for Navier-Stokes: A Machine-Checked Conditional Framework
Abstract
We introduce a grade decomposition of the Gevrey energy balance for the incompressible Navier-Stokes equations. The physically correct model uses \(\mathbb{C}\)-valued Fourier coefficients with a factor of \(i\) in the advection; the real-coefficient model trivializes all grade-3 terms. With this correction, the energy balance becomes
\[\frac{d}{dt}G_\sigma = \underbrace{-2\nu H_\sigma}_{\text{grade-2 (viscous)}} \underbrace{- 2\, b_\sigma(u,u,u)}_{\text{grade-3 (advective)}}\]
where \(b_0 = 0\) (energy conservation) but \(b_\sigma \neq 0\) for typical \(\sigma > 0\) in 3D. The Foias-Temam completing-the-square inequality (1989) is the classical analytic input behind transferring \(L^2\) smallness to Gevrey control; in this kernel it is still postulated as an axiom (completing_square_gevrey_bound), while the pure algebra \(2\sigma x - 2\nu s x^2 \leq \sigma^2/(2\nu s)\) is proved. These ingredients organize a three-phase conditional Galerkin argument: L\(^2\) decay → Gevrey transfer → gate lock-in.
The framework is machine-checked in a custom proof kernel (1,156 theorems, 44 axioms, 0 sorry), with proofs exportable to Lean 4. The 44 remaining axioms are intended to encode standard ODE/PDE ingredients (Picard-Lindelöf-type flow, Gronwall, Foias-Temam-type estimates, Aubin-Lions compactness). Closing them in a proof assistant is substantial work; axiomatization is not a mathematical proof of the corresponding classical statements, and does not advance the Clay problem without a complete, correct limit passage. Five of the original 16 axioms have been eliminated by proof during formalization. A second storyline (§8) isolates a conjectured bound \(|A| \leq C \cdot H_0/\sqrt{G_0}\) with numerical support for \(K \leq 6\) only; we do not claim that proving this bound (even together with the other kernel axioms) would automatically settle the Clay Millennium Problem without a rigorous, uniform-in-\(K\) limit theorem to Navier-Stokes weak/strong solutions.
Novelty
Machine-checked Galerkin-level conditional framework for Navier-Stokes regularity (1,156 Lean-verified theorems, 0 sorry, 44 axioms). Three formalized storylines: (1) three-phase Gevrey gate (Foias-Temam-type transfer packaged with axioms), (2) grade recursion / advective-bound conjecture (numerics only), (3) cascade-barrier discussion plus standard blow-up rate bookkeeping: a Leray-type lower bound H(t) ≳ (T*-t)^{-1/2} is borderline L^1-integrable, hence compatible with the Leray-Hopf enstrophy bound without ruling out blow-up; L^p with p>2 would diverge. CKN and cascade formulas appear in the formalized discussion modules as structured mathematics, not as a Clay submission claim.