Yang Jihoonism: Structural Survival Systems and the Dual Threshold Law

Yang Jihoonism: Structural Survival Systems and the Dual Threshold Law This paper develops a rigorous mathematical theory of dynamical systems defined on constrained admissible domains.Within the research program of Yang Jihoonism, we formalize the interaction between boundary-singular distortion and boundary-dominant stabilization and derive precise structural conditions for global invariance. The core problem addressed in this work is the following:When a system evolves under constraints, under what structural conditions does it remain inside its admissible domain, and when does it collapse in finite time? The paper establishes three main results: Finite-Time Escape vs. Global InvarianceIf distortion becomes singular near the boundary and no sufficient stabilization exists, trajectories reach the boundary in finite time.Conversely, boundary-dominant stabilization guarantees global invariance. Entropy Barrier DynamicsA logarithmic entropy barrier functional is introduced.Finite-time boundary approach corresponds to entropy blow-up, while stabilization ensures bounded entropy growth. Dual Threshold LawUnder explicit boundary scaling assumptions, a normalization threshold for the associated partition function is derived.The same structural singularity also induces a quantified logical form in safety synthesis problems (∃S ∀x), revealing a separation between verification and synthesis complexity. The results are developed in a deterministic and measure-theoretic framework, with fully explicit derivations of finite-time escape, entropy growth, and normalization conditions.The theory is presented as a standalone mathematical contribution while situated within the broader structural research program of Yang Jihoonism.