3D Incompressible Navier-Stokes III: Parametric Leibniz Differentiation of 3D Torus Box Integrals, Derived Box Enstrophy Balance, and Conditional Gronwall and Riccati Comparison Bounds

We formalize parametric Leibniz differentiation under the 3D torus box integral on [0, 2π]³ and derive the 3D box enstrophy balance, exact 3D flow benchmarks, and conditional Grönwall and Riccati comparison bounds for the 3D incompressible Navier–Stokes equations, building on Paper I (JesseMath.Unified.NavierStokesContinuumGronwall, DOI: 10.5281/zenodo.23696796) and Paper II (JesseMath.Unified.NavierStokesEnstrophyDepletion, DOI: 10.5281/zenodo.23716627). First, using Mathlib’s intervalIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le, we prove 1D, 2D, and 3D iterated parametric Leibniz differentiation on [0, 2π]³ and derive the 3D box enstrophy derivative dΩ_box/dt = S_box − 2νP_box (with P_box ≥ 0) from the pointwise vorticity PDE and two explicit integration-by-parts hypotheses (hadv_zero, hvisc_ibp), replacing Paper II’s assumed scalar derivative hypothesis (H1). Second, we prove unconditional 3D [0, 2π]³ box enstrophy, palinstrophy, and Laplacian work identities for the 3D viscous Arnold–Beltrami–Childress (ABC) flow and the 3D Taylor–Green vortex. Third, invoking the Leibniz-derived box enstrophy balance, we derive conditional Constantin–Fefferman, Ladyzhenskaya, Beale–Kato–Majda, and Prodi–Serrin (8, 4) Grönwall closures together with a conditional cubic Riccati comparison bound over [0, T_ricc).

All 13 theorems and 13 definitions in JesseMath/Unified/NavierStokesLeibnizEnstrophyBalance.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 including symbolic dimensional analysis.