Universal $φ$^(N+1) Criticality in Stochastic $φ⁴$ and Allen-Cahn Systems: A Numerical Verification of Fractal Mechanics

We test the Fractal Mechanics (FM) prediction R = φ^{N+1 for the ratio R ≡ τ_pol/τ_rev of stable polarity epoch duration to reversal duration in bistable dissipative systems, where φ = (1+√{5)/2 is the golden ratio and N is the Pöschl-Teller order of the reversal soliton. For the geomagnetic field, four active poloidal modes (ℓ = 1–4, N = 4) predict R = φ⁵ ≈ 11.09. Analysis of the paleomagnetic catalogue and solar polar field data [Clement 2004; Gradstein et al. 2012] yields R_obs ≈ 11.1, confirmed at 4.1𝜎 significance. The FM cascade T_n = t_P · φⁿ simultaneously predicts seven major geoclimatic cycles within a contiguous 13-level cluster (n = 254–267) at sub-percent accuracy, with zero adjustable parameters. The N=1 case (R = φ² ≈ 2.618) is validated numerically via two complementary stochastic models: a 0D overdamped Langevin equation in a φ⁴ double-well, yielding R = φ² at 𝜎* = 0.389; and a 1D Allen-Cahn field equation, yielding R = φ² at 𝜎* = 0.817 via kink–antikink nucleation. The kink width is measured at δ_kink = 1.411, confirming δ_th = √{2κ/Λ} = √2 to 0.3%. The ratio 𝜎*(1D)/𝜎*(0D) = 2.10 is accounted for by Kramers–Langer theory to within 8.2% and carries no FM content. The azimuthal extension is validated by a 2D Allen-Cahn simulation on a cylinder (N_φ = 3 active modes, L_x = L_φ = 10), yielding R* = 7.046 ± 0.53 at 𝜎*_2D = 0.85, consistent with φ⁴ = 6.854 to 0.4𝜎 and revealing an emergent FM staircase R ≈ φ^{4+(0.85-𝜎)/0.01} in the critical region. FM universality resides in the exponent φ^{N+1}, invariant across spatial dimension and reversal mechanism.