Solenoidal Vorticity Duality, Zero-Trace Spectral Constraint, and Intrinsic Regularity Criteria in the 3D Navier-Stokes Equations_V1

AbstractThis paper investigates the geometric, spectral, and functional constraints governing the evolution of enstrophy andconditioning the exclusion of finite-time singularities (finite-time blow-up) in the classical three-dimensional incompressibleNavier-Stokes equations on R3. Without invoking external regularizations in the proofs or imposing artificial confinementgeometries in flat slabs, we establish an analytical framework based on the simultaneous enforcement of the system’stwo divergence-free laws: the zero-trace spectral constraint (∇ · u = tr(D) = 0) and the solenoidal condition of vorticity(∇·ω =0).We prove seven main mathematical results in R3: (1) the pointwise orthogonal decomposition of enstrophy dissipation|∇ω|2 = |∇|ω||2 +|ω|2|∇ξ|2 over the active support Ω+(t) (Lemma 2.1); (2) the logarithmic transport identity ∇ · ξ =−(ξ · ∇)ln|ω|, which guarantees a pointwise lower bound with a factor of 3/2 along the longitudinal direction of vorticity,|∇ω|2 ≥ |∇⊥|ω||2 + 32|(ξ · ∇)|ω||2 (Lemma 3.1); (3) the global directional flux neutrality ∮︁∂Ωλ(ξ · nλ)dS = 0 acrossregular C3 boundaries of compact superlevel sets (Lemma 3.2); (4) a sufficient and conditional regularity criterion validfor superlevel sets of arbitrary geometry (Lemma 5.1 and Theorem 5.1), where by defining the effective positivestretching rate q(x,t) := max{0,ξ · D · ξ}, the optimal Sobolev embedding ˙H1(R3) ↪→ L6(R3) and the intrinsic volumetricscale ℓΛ0(t) = CS|ΩΛ0(t)|1/3 allow closing, under the sub-viscous hypothesis M(t)ℓΛ0(t)2 ≤ ν/2, the linear differentialestimate E′(t) ≤ 6Λ0E(t) that excludes singularity at T∗ < ∞ via the Ladyzhenskaya-Prodi-Serrin criterion; (5) thedirectional cancellation bound in the singular Biot-Savart-Constantin-Fefferman representation (Theorem 5.2); (6) theTheorem of Double Spherical Cancellation in the Core and Explicit Estimation of the Regularized LocalContribution (Theorem 5.3), which proves that the first-order linear contribution of ∇ξ at constant amplitude vanishesidentically by odd spherical parity, whereas the quadratic remainder and variable-amplitude cross terms satisfy explicitlocally integrable bounds of order O(ρ2) and O(ρ3) in terms of the local constants Aρ,Kρ,Hρ; and (7) the ConditionalBiot-Savart–Sobolev Coupling Theorem with Fixed Scale ρ∗ := √︁ν/(2Λ0) = √︁µ/(2ρ0Λ0) (Theorem 6.1), whichuniformly bounds the far field, establishes an explicit sufficient condition for the geometric inclusion Bρ∗(x) ⊂ Ω+(t)throughout the superlevel set ΩΛ0(t), and proves that when ΩΛ0(t) ̸ = ∅ and under said sufficient inclusion condition, thejoint boundedness of the local curvature functional Σρ∗(x) and the far field implies the sub-viscous hypothesis of Theorem5.1.Keywords: Navier-Stokes Equations, Solenoidal Vorticity, Longitudinal Dissipative Bonus, Sobolev Embedding onSuperlevel Sets, Spherical Biot-Savart Cancellation, Fixed Viscous-Inertial Scale, Conditional Serrin Regularity.MSC 2020 Classification: 35Q30, 76D05, 76D03, 35B65.