On the Non-Computability of Continuous Navier-Stokes Blow-Ups in Dependent Type Theory and the Logical Inevitability of Modular Register Inversion

This technical preprint establishes a rigorous type-theoretic critique of recent claims regarding the formal verification of finite-time singularities (blow-ups) in the continuous Navier-Stokes equations within dependent type theory proof assistants (such as the Lean Kernel). From the perspective of constructive logic and the Calculus of Inductive Constructions (CIC), any computable function must satisfy the totality constraint—it must be uniquely defined for all inputs within its domain. We demonstrate that inside an infinitely divisible real continuum (R³), a velocity field collapsing into a finite-time singularity inherently violates this constraint, rendering the resulting proof terms non-computable. Consequently, any successful compilation pass of a continuous blow-up configuration in a formal verifier inadvertently validates artificial smoothing regularizers (e.g., epsilon-adjustments in denominators) rather than the authentic partial differential equations. To resolve this computational deadlock, we present the mathematical and logical inevitability of a discrete spatial matrix architecture. Operating over the finite integer modular ring Z/256Z and governed by a hardware-level Link Inversion Operator (I_lock), this framework eliminates infinite divergences entirely. The overflow of localized volumetric charge density triggers a deterministic state transition into a rigid structural null-state (v_rest = 0), ensuring that the underlying kinematic functions remain total, constructive, and natively computable within the Lean Kernel without numerical regularizers or classical logic overrides. Keywords: Navier-Stokes Equations, Millennium Prize Problems, Formal Verification, Lean Kernel, Type Theory, Modular Arithmetic, Discrete Kinematics, Finite Registers. Creative Commons Attribution 4.0 International