Continuity Strain Theory (CST-Ω): Pre-Stability Limits and the Domain of Stabilizability

This foundational work introduces Continuity Strain Theory (CST-Ω) as a rigorous framework to define the intrinsic limits of stabilizability in dynamical control systems. Departing from classical Lyapunov stability, CST-Ω formalizes three key constructs: the continuity strain field (ψ), continuity curvature (κ), and accumulated irrecoverability (ΔS). These collectively characterize both instantaneous and cumulative deviations from stable behavior. Central contributions include: ΔS Threshold Theorem:* Demonstrates a critical bound of irrecoverability beyond which stabilization is mathematically impossible. Stabilizability Totality Impossibility Theorem: Proves that no system with non-zero strain can be globally stabilizable across its entire state space. Semantic Refusal Outcome: Introduces a refusal operator to formally define when states exceed the domain of stabilizability. The paper’s definitions-first approach eschews heuristics for strict formalism, establishing CST-Ω as a powerful tool to map partial stabilizability and state-space boundaries with semantic precision.