3D Incompressible Navier-Stokes: Continuum Energy-Vorticity Identities, Beltrami and Taylor-Green Flows, Scaling Exponents, and Differential Gronwall Bounds

We study the analytical boundary separating globally tame fluid flows from the supercritical three-dimensional regularity regime for the incompressible Navier–Stokes equations. First, we show that kinetic-energy conservation in periodic domains is an integral divergence phenomenon rather than a pointwise identity: while the antisymmetric spin quadratic form u⊤Ωu vanishes identically, the trace-free rate-of-strain work u · ((u · ∇)u) = u⊤S u is generally non-zero and cancels only after integrating ∇ · (½|u|2u + p u) over [0, 2π]3 via the Fundamental Theorem of Calculus. Second, using the Lamb-vector decomposition (u · ∇)u = ∇(½|u|2) + (∇ × u) × u, we contrast Beltrami eigenflows (∇ × u = λu), whose rotational nonlinearity vanishes so the dynamics reduce to the linear heat equation ∂tu = νΔu, against two-mode cross-helical triads and planar 2D flows where vortex stretching (ω · ∇)u vanishes identically. Third, we establish full continuum derivative, C∞ smoothness, periodicity, and iterated [0, 2π]3 box-integral proofs for two classic 3D benchmarks: the viscous Arnold–Beltrami–Childress (ABC) flow with mean kinetic energy E(t) = ½(A2 + B2 + C2)e−2νt, and the non-Beltrami 3D Taylor–Green vortex (E0 = 1/8, enstrophy Ω0 = 3/8, palinstrophy P0 = 9/8, and non-zero vortex stretching). Finally, we quantify the +1/2 supercritical scaling deficit of 3D Leray–Hopf bounds relative to the Prodi–Serrin critical line 2/p + 3/q = 1, and prove the differential Grönwall inequality from the Mean Value Theorem and Fundamental Theorem of Calculus to derive BKM-type enstrophy bounds, Poincaré exponential decay, and the small-data cubic regularity threshold. All 42 theorems are formalized and machine-checked in Lean 4 (JesseMath/Unified/NavierStokesContinuumGronwall.lean) with zero sorry declarations and only Lean’s standard axioms, paired with a 57-assertion deterministic verification oracle.