Dynamics of Admissible Systems: A Unified Operator-Theoretic Framework for Offensive, Defensive, and Constraint Dynamics

This manuscript introduces a generalized, operator-theoretic framework that unifies seemingly disparate domains—such as physical forces, economic markets, combat strategy, psychology, and biological adaptation—under a single mathematical structure known as "Admissible Systems." The theory models all interacting systems through three fundamental components: state-generating (offensive) dynamics, state-preserving (defensive) dynamics, and environmental constraints. By framing system evolution this way, the core mathematical problem becomes determining the maximum amount of expansive perturbation a system can generate before violating its structural or environmental boundaries. The paper derives universal evolution equations, Lyapunov stability inequalities, offense-defense duality structures, and Floquet admissibility cycles for periodically driven systems. It also utilizes regularized Fredholm determinants to establish rigorous criteria for structural collapse, proving that systemic rupture occurs exactly when the combined offensive-defensive operator loses invertibility. Ultimately, the framework culminates in the Universal Admissibility Principle: sustained evolutionary systems must maximize their expansive and offensive capacities strictly subject to the preservation of their structural coherence. This provides a unified, mathematically closed foundation for understanding resilience, dominance, and collapse across physics, biology, markets, and strategic environments.