Second-Order Regime Validity

This paper introduces a regime-level framework for mathematical existence when admissibility is not guaranteed. It shows that bounded first-order quantities such as energy do not ensure regime persistence, and that regime validity is instead governed by second-order geometric coherence. When this coherence fails, geometry-attached time terminates and a buffer interval arises in which no regime is valid. The paper formalizes how new regimes may emerge after collapse via minimal rebirth paths selected by second-order structure, allows for multiplicity of successor regimes, and proves an exact trichotomy of observational outcomes. The framework is equation-independent and operates upstream of dynamics, clarifying when mathematical descriptions apply and how meaning can be lost and regained.