VCML Paper 116: Wave Coherence Crossover -- Single-Level to Multi-Level at L* = 317 (Hardy's Prime)

We derive a closed-form critical system size L* = 200*sqrt(4*pi/5) = 317.07 for the Viability-Constrained Map Lattice (VCML), below which the wave field achieves global coherence (Gamma > 1 wave per cell per step) and above which coherence breaks into local patches, producing spontaneous multi-level stratification without any external coupling. The four ingredients entering L* --- living fraction lf=4/5, natural length scale 40, disc kernel area pi*r_w^2 with r_w=5, and the budget identity beta+beta^2=1 --- are all forced by the system architecture. The result coincides with the prime 317 to within 0.021%. G.H. Hardy chose 317 in A Mathematician's Apology (1940) as his canonical example of a prime whose truth is independent of human thought. VCML wave physics arrives at the same prime by independent physical reasoning. The physical result (wave coherence crossover) is the primary claim; the arithmetic coincidence is discussed separately. Architecture implication: one VCML unit = L < 317. Multi-level VCML = single run at L >> 317, with hierarchy depth ~ log(L/317).