Existence as Synchronized Expansion: Boundary Vector Reversal and Metastable Survival

Existence as Synchronized Expansion: Boundary Vector Reversal and Metastable Survival This paper develops a unified geometric theory of structural survival under singular boundary pressure. Existence is formalized as synchronized expansion between internal structural complexity and the evolving boundary constraints that surround it. Crisis is identified not as termination, but as a critical amplification that reverses the boundary vector field — transforming a repelling boundary into an attracting one. The paper establishes a complete deterministic and stochastic framework for boundary-driven collapse and survival: A refined singular ODE model with asymmetric exponent structure A sharp finite-time collapse bound under dominant boundary attraction A crisis bifurcation theorem determined by exponent crossing A network slack redistribution inequality A Lyapunov forward-invariance theorem under exponent dominance and strong connectivity A stochastic hitting-time exclusion theorem proving zero boundary probability A global metastable attractor geometry sustained by transferable slack The central structural parameter is the exponent gap θi=βi−γi,\theta_i = \beta_i - \gamma_i,θi=βi−γi, which governs boundary behavior: $\theta_i > 0$ → boundary repelling $\theta_i < 0$ → boundary attracting (collapse regime) $\theta_i = 0$ → critical metastable regime Thus, crisis is rigorously characterized as a structural phase transition. Survival emerges not as equilibrium and not as strength, but as a geometric property of slack propagation within a strongly connected network. Metastability is shown to form a forward-invariant attractor sustained by transferable structural margin. This work extends the Yang Jihoon Dynamics program by: Introducing singular boundary geometry into viability theory Quantifying collapse time explicitly Proving stochastic boundary exclusion under exponent dominance Identifying crisis as a bifurcation of boundary orientation The result is a structural survival theory applicable across physical, biological, economic, and systemic domains. Existence is synchronized expansion.Crisis is boundary vector reversal.Survival is transferable slack within connected structure.