Crossing the Functional Desert: Critical Cascades and a Feasibility Transition for the Emergence of Life

Abstract: The origin of life poses a problem of combinatorial feasibility: How can persistent functional organization arise in exponentially branching assembly spaces when unguided exploration behaves as a memoryless random walk? We show that nonlinear threshold-cascade dynamics in connected interaction networks provide a minimal, substrate-agnostic mechanism that can soften this obstruction. Below a critical connectivity threshold, cascades die out locally and structured input-output response mappings remain sparse and transient-a "functional desert" in which accumulation is dynamically unsupported. Near the critical percolation threshold, system-spanning cascades emerge, enabling persistent and discriminative functional responses. We illustrate this transition using a minimal toy model and generalize the argument to arbitrary networked systems. Also near criticality, cascades introduce structural and functional persistence, directional bias, and weak dynamical path-dependence into otherwise memoryless exploration, allowing biased accumulation over short coherence timescales. This connectivity-driven transition-functional percolation-requires only generic ingredients: interacting units, nonlinear thresholds, influence transmission, and non-zero coherence times. The mechanism does not explain specific biochemical pathways, but it identifies a necessary dynamical regime in which structured functional organization can emerge and persist, providing a physical foundation for how combinatorial feasibility barriers can be crossed through network dynamics alone.