3D Incompressible Navier-Stokes IV: Iterated 1D Slice Ladyzhenskaya Interpolation on [0, 2pi]^3, Explicit (3/4, 3/4) Vortex-Stretching Constant, and Galerkin Enstrophy Closure

We formalize in Lean 4 the concrete N-mode divergence-free Fourier–Galerkin velocity space VN (GalerkinField3D), the 3D finite-trigonometric Parseval identity (torus_3d_real_trig_parseval), and the iterated 1D slice interpolation closing the (3/4, 3/4) vortex-stretching estimate on the 3-torus 𝕋3 = [0, 2π]3, building on Paper I (JesseMath.Unified.NavierStokesContinuumGronwall, DOI: 10.5281/zenodo.23696796), Paper II (JesseMath.Unified.NavierStokesEnstrophyDepletion, DOI: 10.5281/zenodo.23716627), and Paper III (JesseMath.Unified.NavierStokesLeibnizEnstrophyBalance, DOI: 10.5281/zenodo.23716675). Adopting a single raw-integral convention throughout (ΩN = ½‖ωN‖L22, PN = ‖∇ωN‖L22, and SN = ∫ ωN · (ωN · ∇)uN dx, so ΩN′(t) = SN(t) − νPN(t)), we prove across 12 theorems, 2 structures, 62 definitions, and 82 lemmas (158 declarations in total): (I) the 1D periodic slice inequality directly on h(s) = |ω(s)|2 with its (2π)−1 slice-mean term, iterated 2D and 3D slice factorization of ‖ωN‖L44, the 3D finite-trigonometric Parseval Biot–Savart isometry ‖∇uN‖L22 = ‖ωN‖L22 = 2ΩN and zero-mean torus Poincaré inequality 2ΩN ≤ PN, the 3D torus Ladyzhenskaya L4 interpolation with explicit constant CLad = (2 + 1/(2π))3 ≈ 10.065873, and the unconditional (3/4, 3/4) vortex-stretching bound |SN(F)| ≤ C ΩN(F)3/4 PN(F)3/4 for every N ∈ ℕ and F ∈ VN with N-independent constant C = 23/4√CLad ≈ 5.335784; and (II) Galerkin projection drop-out, vanishing advection work, viscous integration by parts, the exact (2π)3 bridge to the box-averaged constant Cbox = (2π)3/2C ≈ 84.036511, small-initial-data global enstrophy decay for C√ΩN(0) < ν/2 via a compact first-exit-time bootstrap on [0, T], the large-data cubic Riccati differential inequality ΩN′(t) ≤ (27C4/(256ν3))ΩN(t)3, and unconditional 3D Arnold–Beltrami–Childress (ABC) and Taylor–Green Poincaré verifications. All 12 theorems, 2 structures, 62 definitions, and 82 lemmas in JesseMath/Unified/NavierStokesTorusFourierSobolev.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 40-assertion deterministic Python verification oracle and a 20-seed cold-start numerical sanity check across N ∈ {4, 8, 16, 32}.