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

This paper introduces Discrete Stochastic Cascade Dynamics (DSCD), a class of finite-resource stochastic dynamical systems defined on arbitrary finite graphs. The model contains two fundamentally asymmetric excitation species with distinct intrinsic frequencies and interaction rules. Cross-species interaction efficiency is modulated by their phase alignment, allowing the frequency mismatch to generate a dynamical beat that cannot be absorbed into a simple rescaling of the coupling constants. The system evolves through four coupled state variables: excitation populations, local capacity energy, accumulated structural state, and ripple intensity. All activity draws from a strictly non-renewable energy field. The ripple measure governs transitions among three local dynamical regimes: quiescent evolution, dissipative leakage, and explosive pair creation. Bond formation is determined by a local Landau-type free-energy criterion, producing a structural hierarchy of: excitation → co-located pair → bond → clump Three principal mathematical guarantees are established. First, total capacity energy is a non-negative supermartingale and decreases strictly in expectation whenever the system is active, providing an intrinsic thermodynamic arrow. Second, the cumulative number of energy-consuming events over the complete history of the system is almost surely finite and bounded by the initial energy endowment. Third, every realisation reaches a frozen absorbing configuration in finite time with probability one, without requiring fine-tuning to a critical point. A quantitative upper bound on the expected absorption time is also derived. The paper identifies three qualitative regions of behaviour determined by initial excitation co-density and energy density: immediate absorption, dissipative decay, and a transient cascade regime. In the cascade regime, spatial structural heterogeneity emerges and ultimately freezes when activity terminates. Simulations consistently exhibit a transient complexity peak, with structure developing during explosive cascades before activity declines as the finite energy reserve is exhausted. The resulting frozen structural field retains a history-dependent record of the stochastic trajectory. The computational implementation scales linearly with graph size for bounded-degree graphs, with per-step complexity (O(dP)) and expected full-run complexity controlled by the initial total energy. This is Part I of the DSCD programme. It presents the model construction, rigorous guarantees, phase structure, computational analysis, numerical illustrations, and emergent structural properties. The associated source code and reproducible simulations are available through the project repository.