Discrete Stochastic Cascade Dynamics on Finite Graphs, Part I: Construction, Guarantees, and Phase Structure

We construct a new class of discrete stochastic dynamical systems on finite graphs that generate rich, transient structure from irreversible resource consumption alone. Two fundamentally asymmetric excitation species, distinguished by intrinsic frequency and self-interaction rules, interact stochastically while drawing from a strictly non-renewable local energy field. The cross-species interaction efficiency is modulated by a phase-interference factor with beat period 2π/|ω_X − ω_Y|, so that frequency mismatch between the two species directly controls the dynamics; same-species channels are always phase-aligned and unmodulated. Four coupled state variables (excitation counts, capacity energy, structural state, and a ripple measure) evolve through layered update rules that produce three dynamical regimes: quiescent, dissipative leakage, and explosive pair creation. Three rigorous guarantees are established. Energy Monotonicity: the total capacity energy is a non-negative supermartingale that decreases strictly in expectation whenever the system is active, providing a built-in thermodynamic arrow. Finite Activity Bound: the cumulative number of interactions across all space and time is almost surely finite, bounded by the ratio of initial energy to unit interaction cost. Almost Sure Absorption: the system reaches a frozen absorbing configuration in finite time with probability one, without requiring fine-tuning to a critical point. Beyond these guarantees, a transient complexity regime is identified: structural diversity, initially zero, becomes strictly positive in the cascade regime and eventually freezes at a constant value; simulation consistently shows it passes through a peak before levelling off, though a proof of the peak is not claimed. A phase diagram over initial energy density and interaction strength is derived. The framework applies to any finite-resource system with two asymmetric interacting classes, including cosmological dark energy dynamics, biological pattern formation, and information cascade processes.