Survival Dynamics and Irreversible Non-Closure: A Mathematical Framework for Reopenable Structures

Survival Dynamics and Irreversible Non-Closure: A Mathematical Framework for Reopenable Structures This paper develops Survival Dynamics, an axiomatic framework for systems whose persistence depends on maintaining reopenable structure under irreversible constraints. The central object is the Irreversible Completion Obstruction (ICO), an invariant measuring the minimal inevitable loss of restart-possibility induced by any admissible completion operator. The ICO formalizes irreversible loss not as total collapse, but as structural reduction of future choice. The framework is built on four base axioms independent of the ICO, followed by a single extended axiom (non-escalation). From these, the paper establishes: Five structural properties of the ICO invariant: nonnegativity monotonicity under class refinement composition lower bound conjugation invariance upper semicontinuity Cost Equivalence Theorem:The cost of maintaining choice, the cost of avoiding damage accumulation, and the cost of preventing absorption are identical and equal to the ICO. Breathing Necessity Theorem:In regions where the ICO is strictly positive, no fixed completion policy preserves restart-possibility indefinitely; survival requires alternation between structurally distinct completion operators. Two canonical models are provided: A single-gate system exhibiting asymptotic vanishing despite zero ICO. A two-gate system that survives indefinitely through alternation. The framework is domain-independent and abstract. It does not claim to model any specific physical or biological system, but instead provides a structural theory of persistence under irreversible constraints. The results are fully derived from the stated axioms, and explicit models demonstrate consistency of the system. This work extends and formalizes key structural principles of Yang Jihoon Dynamics into a self-contained mathematical framework.