Unconditional Global Regularity of 3D Navier-Stokes Equations via Scale-Matching Energy Accounting and Geometric Deficit

This paper establishes the unconditional global regularity of 3D incompressible Navier-Stokes equations by reducing the blow-up problem to a non-compatible accounting contradiction between nonlinear energy injection and geometric dissipation. We introduce a "Minimum Active Volume" V_\theta(t)—the minimum spatial volume required to carry a fixed fraction \theta of the total vorticity energy \mathcal{L}(t)—to compress potential blow-up mechanisms into two mutually exclusive regimes: extreme concentration (Pole A) and infinite diffusion (Pole B). The closure of the proof consists of two main pillars: 1. The Concentration Pole (A): By identifying a structural "Geometric Deficit" \eta arising from the topological divergence-free constraint \nabla \cdot \omega = 0, we prove that the nonlinear term is strictly prohibited from reaching the scalar Gagliardo-Nirenberg extremizer. Consequently, as V_\theta(t) \to 0, the dissipation cost D(t) grows faster than any possible nonlinear energy injection, forcing the energy derivative \mathcal{L}'(t) to become negative. 2. The Diffusion Pole (B): We demonstrate that the temporal growth of the active volume V_\theta(t) is polynomially bounded within any finite time interval T, effectively blocking the "escape by spreading" route. Ultimately, we show that any hypothetical singularity would require an infinite cumulative "Refresh Cost" that contradicts the finite global energy budget provided by the Leray-Hopf inequality (the "Bank"). The combination of these constraints confirms that the 3D Navier-Stokes equations remain regular for all time.