Superstable geometry in triadic percolation

Triadic percolation turns bond percolation into a dynamical problem governed by an effective one-dimensional unimodal map. We show that the geometry of superstable cycles provides a direct, map-agnostic probe of local nonlinearity: specifically, the distance from the map's maximum to a distinguished next-to-maximum point on the attracting 2 n cycle (which coincides with a preimage of the maximum at 2 n superstability) scales as | Δ p | γ , with γ = 1 / z , where z is the nonflat order of the maximum. This prediction is verified across canonical unimodal families and heterogeneous triadic ensembles, with Lyapunov spectra corroborating the one-dimensional reduction. A derivative condition on the activation kernel fixes the local nonlinearity order z (and thus, under standard unimodal-map hypotheses, the associated z -logistic universality class) and gives conditions under which z > 2 can be realized. The diagnostic operates directly on orbit data under standard regularity assumptions, providing a practical tool to classify universality in higher-order networks.