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

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/φⁿ, where Vₙ are amplitudes at cascade scale Lₙ=L₀φⁿ. Within this class, we prove a regularity theorem in two steps. First, a purely algebraic result: in the Fibonacci wavenumber basis kₙ=k₀φⁿ, the only exact triadic resonances are nearest-neighbor triads kₙ, k₍-₁, k₍-₂, with all non-adjacent interactions non-resonant — a direct consequence of φ²=φ+1. This implies energy transfer is local in scale. Second, the energy balance: local triadic injection at level nn n scales as φ^3-n (exponentially decreasing) while viscous dissipation scales as a constant — so dissipation dominates above a critical scale n*, preventing blow-up. Theorem (conditional): If a NS solution admits a Fibonacci cascade representation, it is globally regular. We provide experimental predictions (Fibonacci spectral peaks in turbulence data, discrete intermittency) and discuss why the conditioning on physical solutions is a strength — not a weakness — of the approach. The same FM principle that explains particle mass hierarchies, the age of the universe, dark matter, and Calabi-Yau emergence in gravitational waves here provides the missing ingredient for NS regularity.