Universal Regime Transition, Transgression-Invariant–Based Stack Detector

This paper introduces a universal, geometry-first framework for analyzing regime transitions in complex systems, addressing failures that occur without classical instabilities, unbounded energy, or breakdown of governing equations. By treating regime validity and admissibility as primary objects, it unifies five complementary tools—second-order geometric incoherence, vector-valued distance-to-admissibility, finite buffer time, second-order rebirth selection, and transgression invariants—into a single analytical stack capable of detecting regime degradation, certifying collapse, resolving regime absence, constructing post-collapse regimes, and rigorously comparing distinct regimes. Granular materials are presented as a canonical Case II application, demonstrating structured regime switching with non-zero transgression, but the framework itself is domain-agnostic and unit-independent. The result is a general theory that determines not how systems evolve within a regime, but when regimes exist, fail, re-emerge, and differ—providing a foundational tool for physics, engineering, autonomous systems, finance, and AI.