The Logic of Fluids: Coherence and Regularity in the Navier–Stokes System

We present a structural proof of global regularity for the three-dimensional incompressible Navier–Stokes equations. Introducing the Coherence Manifold Σ = L2(R3) ∩ L3(R3), we identify it as the minimal functional space in which velocity, pressure, and energy remain mutually definable under the dynamics. Within this framework, a finite-time singularity corresponds not merely to blow-up, but to a breakdown of internal coherence—specifically, the loss of definability in the pressure–velocity coupling governed by the system’s elliptic structure. For Leray–Hopf weak solutions with initial data in Σ, the energy inequality ensures time-integrability of the critical L3 norm. This integrability yields uniform-in-time L3 bounds, satisfying the Escauriaza–Seregin–Šverák criterion for global regularity. Regularity, while analytically proven via known estimates, reflects an internal coherence constraint when viewed structurally.