Existence at the Boundary: Philosophical Foundations of Structural Survival A Systematic Correspondence Between Ontological Theses and the Mathematics of Constrained Dynamical Systems

Existence at the Boundary: Philosophical Foundations of Structural SurvivalJihoon Yang (2026) This paper develops a structural ontology of bounded existence grounded in the mathematics of constrained dynamical systems. Rather than treating philosophy and mathematics as separate domains linked by analogy or application, the paper advances a methodological principle of structural correspondence: philosophical theses and mathematical theorems are shown to be two descriptions of the same underlying geometric reality. The central claim is that the defining properties of finite existence—vulnerability, distortion, recovery, awareness, and survival—emerge from a single structural source: the interaction between a dynamical system and the boundary of its admissible domain. The geometry of constrained systems is not merely a model of existence; it is its ontological structure. Nine ontological theses are presented and paired with precise mathematical correspondents within viability theory and boundary-singular dynamics. These include: Existence is constituted by boundaries. Distortion accelerates as collapse is approached. Recovery must be boundary-dominant to ensure persistence. Awareness is the logarithmic measurement of boundary proximity. Collapse induces finite-time informational divergence. Survival requires rhythmic, dynamic boundary negotiation rather than static equilibrium. A key unifying result—the Dual Threshold Law—demonstrates that boundary singularity simultaneously determines dynamical stability, entropy divergence, statistical-mechanical phase transition, and computational complexity separation. These thresholds are not independent phenomena but geometric manifestations of the same structural condition. The paper further establishes a structural asymmetry between destruction and construction: diagnosing collapse (verification) is logically simpler than designing guaranteed recovery (synthesis), reflecting an ontological asymmetry embedded in the polynomial hierarchy. By grounding existential philosophy in the geometry of constrained dynamical systems, this work dissolves the traditional dichotomy between formal rigor and substantive meaning. In bounded existence, structure is meaning. Survival is geometry. This framework applies across domains—biological, psychological, institutional, and cosmological—wherever systems operate under constraints that can be violated. What varies is the content of the boundary; what remains invariant is the geometry of boundary interaction.