Viability-Preserving Dynamics under Distortion, Compensation, and Reinvestment

Viability-Preserving Dynamics under Distortion, Compensation, and Reinvestment Description This paper introduces a deterministic dynamical framework extending classical state-only viability systems. The proposed class of systems incorporates five structural mechanisms absent from standard stability formalisms: Separation between true state and perceived state via a differentiable perception map. Virtual compensation dynamics acting on perceived coordinates. Iterative boundary reinforcement under unrecognized vulnerability. Reinvestment–capture macro feedback linking distortion and information asymmetry. Non-sacrificial cascade suppression through isolation and mediation operators. We prove that this class strictly extends state-only viability dynamics. In particular, we construct explicit trajectories where perceived entropy decreases while true boundary risk increases—behavior that cannot be represented in any state-only model without auxiliary coordinates. The framework also establishes: A reinforcement theorem showing linear expected boundary contraction under repeated unrecognized trauma. A truth-cost functional quantifying the short-term entropy increase required for repair proximity. A two-dimensional macro-stability condition under reinvestment feedback, yielding explicit determinant and trace criteria. A cascade suppression theorem demonstrating long-run entropy control without sacrificial boundary violation. A Grand Viability Theorem providing sufficient conditions for sustained structural survival above a minimal energy threshold. The theory does not claim universality, but provides a structurally coherent and internally consistent extension of classical nonlinear stability theory within deterministic ODE form. Keywords Viability theory, nonlinear dynamics, distortion dynamics, reinforcement, stability analysis, compensation–repair tradeoff, cascade suppression, structural survival.