Yuuki Theory: A Finite-Pool and Interference-Field Framework for Global Regularity of the 3D Navier–Stokes Equation

This preprint presents an integrated theoretical framework, referred to as Yuuki Theory, for analyzing the global regularity problem of the three-dimensional incompressible Navier–Stokes equations.The central idea is to reinterpret the Navier–Stokes equations as governing only effective degrees of freedom within an interference field, constrained by a finite information (energy) pool. Within this scope, the nonlinear term acts purely as a redistribution operator, while dissipation and scale-selective relaxation prevent unbounded concentration of energy.A three-layer hierarchy of degrees of freedom is introduced:Latent degrees of freedom: existing but non-interacting, without direct physical observability.Effective degrees of freedom (interference field): activated, mutually interacting, and dynamically governed by the Navier–Stokes equations.Structural degrees of freedom: stable and fixed configurations corresponding to persistent physical structures.The paper argues that classical Navier–Stokes dynamics operate exclusively at the second layer and do not describe the activation of latent degrees of freedom or the fixation of structural ones. From this perspective, finite-time blow-up is structurally excluded under the finite-pool and interference-field assumptions.To support this interpretation, a scale-selective relaxation mechanism associated with a critical granularity scale ℓ_c is introduced, enforcing symmetry breaking and energy relaxation before singular behavior can occur. The framework is further illustrated using a qualitative spectral monitoring scheme (DC-1), demonstrating stable behavior under extreme redistribution parameters.This work does not claim a formal resolution of the Clay Millennium Problem. Instead, it clarifies the physical and structural domain in which the Navier–Stokes equations are valid and provides a coherent explanation for why singularity formation cannot occur within that domain.Keywords:Navier–Stokes equations, global regularity, turbulence, finite energy, degrees of freedom, interference field, dissipation, symmetry breaking