Navier–Stokes regularity from face saturation on the pentachoric lattice

The existence and smoothness problem for three-dimensional Navier-Stokes (Clay Millennium Problem #4) asks whether the vorticity of an incompressible fluid can diverge in finite time. We reformulate the question on its physical substrate: the discrete K₅ pentachoric network, where spacetime is fundamentally combinatorial, time advances in ticks τ₀, and the continuum equations are a large-scale approximation. On this lattice, vorticity is a face excitation (depth 2) and enstrophy is the saturation density ρ of those faces. When ρ exceeds the critical threshold ρ* = 4 ln 2, the face saturates: the causal flux F (ρ) = ρ exp (−α*ρ) drops below its maximum, propagation efficiency collapses, and the stretching cascade halts. This is the same mechanism that produces the Higgs boson (P2), black hole horizons (P15), and the cooperative-individual crossover in liquid water at ~40°C (P7). Three results are established. (i) The face-adjacency graph of K₅ — the Johnson graph J (5, 3) — has spectral ratio λₘin/λₘax = −1/3, the same ratio as the K₄ adjacency matrix used in the Yang-Mills mass gap (P5). The 10×10 transfer matrix on the full face lattice has spectral gap Δ₁₀ = 0. 589 > 0, uniform in ρ, with minimum at ρ* T1. (ii) The Pauli exclusion principle bounds the face density: ρf ≤ 3 for all faces, all times, unconditionally. Since blow-up requiresρ → ∞ and the maximum is 3, blow-up is structurally impossible T1. (iii) Beyond the auto-stabilisation threshold ρcrit = 2ρ* = 8 ln 2, compression reduces the gravitational effect instead of amplifying it. The Lyapunov functional Φ (ρ) = −ln F (ρ) has a unique minimum at ρ*; the gel state is thermodynamically unstable and defrosts T1. Together: no value of ρ exists at which enstrophy is simultaneously high and inter-face coupling is strong. The vortex stretching tensor lives in End (V₁) (dim 9, the same 9 Hilbert-Schmidt-orthogonal channels that govern the Cooper-pair mass anomaly in P16) ; only 5 of 9 channels carry the physical strain. On the pentachoric lattice, blow-up is impossible. The continuum Navier-Stokes equation is a mathematical approximation of this discrete dynamics. The postulated singularity is an artefact of the approximation failing at the scale where face saturation intervenes. The burden of proof is reversed: it is the hypothesis of continuous spacetime, not its discreteness, that requires justification. Companion script: P17companionᵥ1. py (88 tests, 0 failures).