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Clay Millennium Problem
Navier–Stokes Global Regularity
Do smooth initial conditions always give smooth global solutions to the 3D incompressible Navier–Stokes equations?
Progress
60%Conditional framework verified
Current approach
Grade decomposition of Gevrey energy balance; conditional regularity via machine-checked Sobolev embedding chains.
Status notes
Full proof architecture Lean-checked. Remaining gap: unconditional Gevrey-class bootstrap from grade-2 decay.
Direct contributions
2 papers
Formal Verification
Working Paper
Lean
DOI
Flagship
Grade Decomposition and Gevrey Regularity for Navier-Stokes: A Machine-Checked Conditional Framework
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.
Physics
Working Paper
Lean
Turbulence Scaling Laws from the Grade Equation: Kolmogorov Spectrum and Intermittency from Analyticity
We derive the Kolmogorov energy spectrum $E(k) \sim \varepsilon^{2/3} k^{-5/3}$ and anomalous intermittency corrections to the structure function exponents $\zeta_p$ from the Grade Equation — a universal structural decomposition theorem for analytic dynamical systems. The derivation proceeds in three steps.