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Recognition Science (RS) derives spatial dimension D=3 and the golden ratio φ = (1+√5)/2 from a single functional equation. A structural consequence is that physical space is a discrete Z3 lattice with voxel scale ℓ₀, and the energy cascade in turbulent flow proceeds through a finite sequence of φ-rungs terminating at the lattice cutoff. We formulate the incompressible Navier–Stokes equations on this lattice and prove five results: (1) the discrete system has unique global solutions for all time with bounded vorticity; (2) the φ-ladder cascade terminates after at most Nd = ⌊3/4 ln Re / ln φ⌋ rungs, giving explicit vorticity bounds; (3) the J-cost functional J[ω] = ∑x J(|ω(x)|/ωrms) is monotonically non-increasing under the discrete evolution, proved via the Recognition Composition Law pair budget absorbed by viscosity; (4) below the Kolmogorov microscale, a sub-Kolmogorov viscous domination estimate yields vorticity bounds independent of the lattice spacing h; (5) the h-independent bound opens a standard compactness route to the continuum limit. Taken together, these yield an unconditional global regularity theorem for the physical RS Navier–Stokes system on each fixed voxel lattice. All discrete results are machine-verified in the Lean 4 proof assistant with zero sorry across fourteen formalization modules. The three formerly open operator estimates are now closed by citing published results: the weighted-Laplacian Hessian from [7], the exact cost identity from [5], and the coherent comparison framework from [12]. For the continuum Navier–Stokes equations, we isolate a single sharp remaining classical gap: the 8-tick η-block reciprocal-ratio energy theorem, which asks for non-increase of Kolmogorov-block reciprocal-ratio energy over one full recognition window uniformly in the lattice refinement. We also record two broader sufficient conjectures—φ-structured spectral decay and J-cost monotonicity—as alternative routes to the same continuum goal. The RS framework reframes the Millennium Prize question: the continuum equation is an approximation to the physically fundamental discrete system, on which regularity is a theorem.