Theorem: Swift-Hohenberg roll-amplitude saturation A^2 = 4r/3 (via Lyapunov variational minimum)

Theorem (Swift-Hohenberg roll-saturation canonical): the Lyapunov functional F[A]/L = -r A^2/4 + 3 A^4/32 (per unit length on q_c = 1 rolls) has extrema at dF/dA = 0 giving A = 0 (max) or A^2 = 4r/3 (min). The positive saturated amplitude…