Finite-size scaling and cutoff functions of critical avalanches under cubic saturation

Power-law avalanche statistics characterize the branching bulk but leave open how finite systems terminate their largest events. Here we determine both the finite-size cutoff and the full crossover function connecting the branching regime to the large-event tail for an exchangeable absorbing event process motivated by Cauchy-stable recurrent dynamics. The Cauchy dynamical mean-field theory yields a linear-cubic low-activity drift, while a repeatable-event lift supplies absorbing branching fluctuations. Their competition at criticality generates a distinct finite-size edge controlled by cubic saturation. For avalanches initiated by a fixed microscopic seed, the rescaled edge measure connects the universal branching power law continuously to an exponential large-area tail. This full crossover admits an analytic spectral representation in terms of the boundary spectrum of a quartic Schrödinger operator on the half-line: high modes reconstruct the branching singularity, while the ground state fixes the far-tail decay. Exact finite-size calculations and Monte Carlo simulations confirm both the predicted cutoff behavior and the full spectral crossover.