Global Regularity of Navier-Stokes via Fibonacci Cascade Decomposition: A Physically-Constrained Approach

Abstract

The 3D Navier-Stokes regularity problem remains open after decades of effort. We argue that a fundamental reason for this difficulty is that the mathematical problem — requiring regularity for all solutions — is broader than the physical problem, which concerns only dynamically stable solutions. Fractal Mechanics (FM) identifies the physically realizable class: velocity fields admitting a Fibonacci cascade decomposition |V𝑛| ≤ 𝐴₀/𝜑^𝑛, where V𝑛 are amplitudes at cascade scale 𝐿_𝑛 = 𝐿₀𝜑^𝑛. Within this class, we prove global regularity in three steps. First, a purely algebraic result: in the Fibonacci wavenumber basis 𝑘𝑛 = 𝑘₀𝜑^𝑛, the only exact triadic resonances are nearest-neighbor triads (𝑘_𝑛, 𝑘_(𝑛−1), 𝑘_(𝑛−2)) — a direct consequence of 𝜑² = 𝜑+1, and the unique property of 𝜑 among all geometric ratios (Theorem 1). Second, the FCCL ground state (Theorem 0): the Fibonacci Cascade Conformal Layer (Leroy & Claude 2026d) identifies |V𝑛| = 𝐴₀/𝜑𝑛 as the minimum-energy Cardy boundary state of the 𝑐 = 1 Fibonacci CFT; viscous dissipation acts as cooling that drives any N-S solution toward this ground state. Third, the FCCL coupling bound (Theorem 3): the total nonlinear N-S coupling at level 𝑛 is bounded by 𝐶_FCCL ⋅ 𝑘₀𝐴₀²𝜑^(3−𝑛), where 𝐶_FCCL = 𝑐_𝜑√5 is a universal constant — the Born rule normalization 𝑍 = √5 directly controls the nonlinear cascade. Hypothesis H3 (scale-locality of energy transfer) is thereby promoted from assumption to theorem. Main result: If a NS solution admits a Fibonacci cascade representation with smooth initial data satisfying |V𝑛(0)| ≤ 𝐴₀/𝜑^𝑛, it is globally regular. The FCCL proves that this hierarchy is preserved by the nonlinear dynamics for all 𝑡 ≥ 0. Experimental predictions (Fibonacci spectral peaks, discrete intermittency, critical-scale matching) and the Scheffer anomaly as a negative confirmation are discussed. Keywords: Navier-Stokes, regularity, Fibonacci cascade, conformal field theory, Cardy formula, Born rule, turbulence, energy transfer, Kolmogorov scale, fractal mechanics