Saturation and Collapse in Dynamical Systems:Emergent Irreversibility and Threshold Phenomena

Abstract

This paper presents a saturation-collapse framework for algebraic structure in which algebraic coherence, reversibility, and composability deteriorate as functions of an intrinsic saturation functional. In contrast to classical algebra, where axioms are assumed to hold universally, the proposed framework
permits axioms to remain valid only within specified regimes and to fail in a structured and derivable manner beyond them. Collapse is not imposed as a primitive notion but arises naturally through the degeneration of admissible endomorphisms. The framework establishes a unified formal language for irreversibility, threshold effects, and structural degradation, and is readily applicable to groups, semigroups, and dynamical algebraic systems.