3D Incompressible Navier-Stokes II: Continuum Enstrophy Integration by Parts, Exact Vortex-Stretching Box Integrals, Constantin-Fefferman Direction Depletion, and Ladyzhenskaya-Prodi-Serrin Conditional Regularity Closure

We formalize the continuum enstrophy balance, exact three-dimensional periodic box integrals, geometric vorticity-direction depletion, and conditional Prodi–Serrin regularity closures for the 3D incompressible Navier–Stokes equations on [0, 2π]³, building on Paper I (JesseMath.Unified.NavierStokesContinuumGronwall, DOI: 10.5281/zenodo.23696796). First, we establish pointwise enstrophy transport and viscous Laplacian identities together with 1D coordinate-slice periodic integration by parts via the Fundamental Theorem of Calculus. Second, we evaluate iterated 3D [0, 2π]³ box integrals for the viscous Arnold–Beltrami–Childress (ABC) eigenflow and the non-Beltrami 3D Taylor–Green vortex, proving zero net vortex stretching and Beltrami energy-enstrophy equivalence (E = Ω for λ = 1) for ABC flow, alongside the exact Taylor–Green palinstrophy-to-enstrophy wavenumber ratio P₀/Ω₀ = (9/8)/(3/8) = 3. Third, we formalize the Constantin–Fefferman geometric direction-depletion criterion and its conditional Grönwall enstrophy closure. Fourth, we prove weighted Young absorption and Ladyzhenskaya–Prodi–Serrin conditional regularity closures (including the (2, ∞) BKM and (8, 4) quartic closures) and contrast them against the unclosed 3D cubic enstrophy barrier.

All 33 theorems and 24 domain declarations in JesseMath/Unified/NavierStokesEnstrophyDepletion.lean compile in Lean 4 (leanprover/lean4:v4.34.1, Mathlib d13f23b723b8) with zero sorry declarations and only Lean’s standard axioms (#assert_zfc_axioms), paired with a 60-assertion deterministic Python verification oracle.