Yuuki Theory: A Finite-Pool and Symmetry-Breaking Framework for Global Regularity of 3D Navier–Stokes

This record presents the DC-1 formal proof module of Yuuki Theory, a structural framework addressing the global regularity of the three-dimensional incompressible Navier–Stokes equations. The core idea is that the dynamics of Navier–Stokes evolution are governed by a finite initial energy (ψ²_pool), within which the nonlinear term acts solely as a redistribution operator and cannot generate additional information density. A scale-selective relaxation mechanism ℛ_{ℓ_c}, motivated by boundary relaxation and a water-phase analogy, dissipates high-frequency concentration before singular behavior can develop. By combining a finite energy pool, symmetry breaking at critical information density, and high-k relaxation, the framework demonstrates that finite-time blow-up is structurally suppressed. The result is formulated as a DC-1 certificate of global smoothness under clearly stated assumptions, without asserting a direct solution to the Millennium Prize Problem. This document is intended as a theoretical framework and formal regularity certificate, suitable as a reference for further analytical, numerical, or physical investigations of Navier–Stokes dynamics.