Structural Absorption Theory: A Meta-Framework for Viability Dynamics

Existence as Synchronized Expansion II: Regularity, Boundary Inaccessibility, and Quantified Metastability Description This paper presents a technical completion and structural refinement of Existence as Synchronized Expansion. The original formulation introduced a nonlinear dynamical framework in which viability arises from the synchronized interaction between internal structural complexity and boundary constraints. While the first paper established the conceptual and structural core of the model, the present work provides full mathematical clarification of regularity assumptions, transfer structure, stochastic boundary behavior, and metastability characterization. The deterministic core system is rigorously treated under explicit local Lipschitz and linear growth conditions, ensuring global existence and boundedness of solutions within the positive viability domain. Transfer mechanisms are formalized through bounded Laplacian-type operators, clarifying the role of conservative and dissipative propagation in networked slack dynamics. A central contribution of this work is the complete stochastic extension. Using Itô calculus and Feller boundary classification, we demonstrate that the viability boundary at zero is inaccessible from the interior under both constant and state-dependent diffusion regimes. While the technical distinction between entrance and natural boundary types remains, both regimes are operationally equivalent in ensuring almost-sure preservation of positivity. Finite-horizon underflow probabilities are further quantified through Lyapunov-based estimates. The critical parameter θ=β−γ\theta = \beta - \gammaθ=β−γ is reinterpreted not as a bifurcation of equilibrium stability, but as a structural index governing the dominance relationship between boundary amplification and nonlinear dissipation. This resolves prior ambiguities and situates crisis dynamics within a rigorous exponent-competition framework. Metastability is characterized through linearization near invariant sets and Lyapunov exponent estimates, yielding bounded oscillatory regimes under dissipative dominance. This work completes the technical foundation of the synchronized expansion framework, transforming a conceptual viability model into a mathematically closed nonlinear dynamical system with deterministic and stochastic coherence. Keywords Nonlinear dynamical systems; stochastic differential equations; viability theory; boundary inaccessibility; metastability; Lyapunov methods; Feller classification; network dynamics.