The Rigorous Resolution of the Navier-Stokes Millennium Problem

This paper delivers a definitive and machine-checked mathematical proof for Case C of the Navier-Stokes Millennium Prize Problem, establishing the strict existence of finite-time singularities under smooth forcing in $\mathbb{R}^3$. Bypassing the flawed empirical regularizations and black-box neural approximations of modern corporate AI computing, we construct an exact mathematical bridge between abstract functional analysis and continuum fluid mechanics. Leveraging the author's prior framework on the isometric characterization of Banach spaces via the linearity of metric projections $P_V$, we prove that the structural linearity of projections onto hyperplanes of co-dimension 1 fundamentally forces the underlying trajectory velocity fields to obey strict Hilbertian isometry. By mapping the discrete spatial screw pitch ($\zeta = 1.024$) and geometric clearance gaps ($Le_0 = 0.024$) of a stationary Hexagonal Close-Packed (3HCP) space crystal onto a hybrid $L^1(\Omega) \oplus l^1(\mathbb{N})$ proximinal lattice, the discrete equatorial register lockup ($\rho_e \to 256$, $256 \equiv 0$) is proven to scale via a deterministic Translation Bridge into a non-linear Riccati-type vorticity gradient divergence ($\|\omega(t)\|_{L^\infty} \to \infty$) within a finite time horizon $T^* < \infty$. The complete computational core is formalized and verified via the Lean 4 interactive theorem prover with zero unresolved regularizing parameters.