Energy Structure and Conditional Regularity for Incompressible Navier–Stokes via LMC Frequency Weights: A Geometric Decomposition Approach

This paper develops a smooth, modulation-free derivation of the energy and Lyapunov structure for the incompressible Navier–Stokes equations on ℝ³, and reduces the global regularity problem to a precise spectral condition on a class of frequency-weighted modulation fields (LMC fields).Structure and main results:The paper proceeds in three levels of energy control:L²-control (unconditional). Using the intrinsic vector identity (u·∇)u = ∇(½|u|²) − u×ω, the exact cancellation ∫(u·∇)u·u dx = 0 is established without any flux bounds or external assumptions. This yields the unconditional energy identity d/dt E(t) + ν‖∇u‖²_{L²} = 0.H¹-control (unconditional). A dissipative Lyapunov functional V(t) = E(t) + α‖∇u‖²_{L²} is constructed, yielding an unconditional integral bound ∫₀ᵀ ‖∇u‖²_{L²} dt < ∞.H²-control (conditional). The critical commutator term I_com = ∫[Δ, u·∇]u·Δu dx leads to the differential inequality X'(t) ≤ C‖∇u‖_{L∞} X(t), recovering the classical Beale–Kato–Majda barrier. The paper then introduces LMC frequency weights Φⱼ as a replacement mechanism for the uncontrolled L∞-norm.The LMC framework:The Leue Modulation Coefficients (LMC) define a smooth modulation field t(x) via mollified discrete coefficients tₚ placed at spatial locations xₚ. The dyadic spectral weights Φⱼ = ∫_{|ξ|~2ʲ} |t̂(ξ)| dξ measure resonance coupling across frequency shells. The central result (Satz 1 / Conditional Theorem) states:If Φⱼ ≲ 2^{−αj} for some α > 3, then the commutator satisfies |I_com| ≲ X^{1−ε}, and the H²-energy X(t) cannot blow up in finite time.Constructive results:Appendix G and Appendix IV provide an explicit constructive LMC architecture (mollifier order N ≥ 9, mild amplitude weighting tₚ → p^{−0.1} tₚ, deterministic geometric jitter of amplitude ≤ 0.01L) for which the decay Φⱼ ≤ C·2^{−αj} with α ≥ 4 is proven analytically (combining Lemmas 1–3 and the mollifier decay estimate).Remaining analytical gap:The gap is stated explicitly and honestly in Appendix C: the constructive LMC above yields unconditional regularity for the specific class of LMC fields satisfying the design constraints. Whether the original arithmetical LMC (without structural modifications) satisfies (HΦ) depends on a number-theoretic question about the decay of the discrete exponential sum S(ξ) = Σₚ tₚ e^{−ixₚ·ξ}, which remains open.Numerical evidence:Appendix D documents systematic numerical experiments (FFT-based shell integration on 56³ grids, first 160 primes). Representative result: Gaussian mollifier (ε = 0.05L) + weight p^{−0.1} + geometric jitter (0.01L) yields decay exponent α ≈ 4.1 ± 0.3, comfortably above the analytic threshold α > 3. This paper does not claim a complete solution to the Navier–Stokes Millennium Problem. It presents:unconditional L²- and H¹-energy estimates,a conditional H²-regularity result under an explicit spectral hypothesis (HΦ),a constructive LMC class for which the hypothesis is provably satisfied,and a precise formulation of the remaining open step.The result is positioned as a conditional regularity theorem with an explicitly identified and mathematically well-posed remaining task.