This Complete Formalism (CF10) specifies a VDM “fluid sector” built from lattice hydrodynamics (lattice Boltzmann / discrete walkers / multi–relaxation-time kernels) expressed

in metriplectic form. The document has two tightly separated goals: (i) a discrete-level

contract—state space, generators, invariants/entropy, and a gate suite for global stability

and hydrodynamic consistency; and (ii) a conditional regularity program for incompressible

three-dimensional Navier–Stokes. The regularity component is intentionally framed as a

falsifiable hypothesis lattice rather than an analytic proof: the only genuinely hard PDE

step is isolated as an explicit A8-style conjecture requiring time-uniform geometric decay

of dyadic-shell enstrophy together with scale-wise dominance of dissipation over stretching.

Under this conjecture, a short lemma chain (Littlewood–Paley + Bernstein bounds →

Beale–Kato–Majda criterion) yields global smoothness. The deliverable is a publishable

program specification plus implementation-ready gates (Taylor–Green viscosity recovery,

cavity incompressibility, spectrum-tail decay, and refinement collapse) for a companion

notebook CFN10.