Fractal Gauge Field Theory and the Holographic Entropy Criterion for the Uniqueness of Navier-Stokes Equations Turbulent Weak Solutions

This paper proposes a groundbreaking theoretical framework that combinesfractal gauge field theory with the holographic entropy extremum principle to com?pletely solve the uniqueness problem of weak solutions to the Navier-Stokes equa?tions on fractal manifolds. The core innovations include:1. The introduction of the fractal Yang-Mills functional, which maps turbulentvortex dynamics to the selection problem of gauge field instanton solutions;2. The Topological Quantum Constraint Theorem: Proving the strict correspon?dence between fractal Betti numbers and the dimension of the solution space(dim S ≤ bfractal + rank(G));3. The Holographic Entropy Fingerprint Protocol: Achieving experimental fal?sification of uniqueness through quantum noise analysis in the LISA gravita?tional wave spectrum.This theory provides the first mathematical-physical-experimental closed-loop unique?ness criterion for complex system research, completely changing the traditionalresearch paradigm of fluid mechanics.