The Rigorous Resolution of the Navier-Stokes Millennium Problem (Case C): Finite-Time Singularity via Phase-Inversion and Peripheral Boundary Recirculation over 3HCP Space Crystal Lattices

This research monograph delivers a complete analytical and machine-checked proof for Case C of the Navier-Stokes Millennium Prize Problem, establishing the strict formation of finite-time singularities under smooth forcing fields in three-dimensional Euclidean space (\mathbb{R}^3). Bypassing the empirical regularizations, black-box approximations, and artificial smoothing constraints typical of contemporary corporate AI models, this framework constructs an exact mathematical bridge between non-continuous discrete topologies and continuous fluid mechanics mechanics. We model the continuum limit (h \to 0) of a stationary, rigid Hexagonal Close-Packed (3HCP) space crystal operating under finite register bounds (\mathbb{Z}/256\mathbb{Z}) derived entirely from first principles. The paper demonstrates that intensive cumulative hydrostatic confinement triggers a localized register phase inversion (\rho_e \to 256, 256 \equiv 0), acting as a deterministic electro-mechanical breaker that completely locks horizontal displacements (L_{xx}, L_{yy} \to 0) and vents volumetric stress exclusively through vertical polar channels. Upon impacting adjacent lattice shells, this high-velocity polar jet generates an exact, non-linear peripheral wrap flow returning along the cells' outer boundaries. This closed feedback recirculation loop acts as an autocatalytic process that concentrates kinetic energy and drives a localized Riccati-type vorticity gradient divergence (\|\omega(t)\|_{L^\infty} \to \infty) within a finite time horizon T^* < \infty. Crucially, the entire mathematical architecture, layer translation matrices, and bounding inequalities are fully formalized and verified via the Lean 4 interactive theorem prover with zero unresolved or non-computable parameter. Keywords: Navier-Stokes, Millennium Problem, Case C Singularity, Phase Inversion, 3HCP Space Crystal, Lean 4 Formalization, Finite-Time Blow-up, Peripheral Recirculation. Lean Source (Markov-Navier-Stokes.lean.txt) licensed:GNU Affero General Public License v3.0 (AGPL-3.0)