Emergence as Regime Formation II: Topological Defects, Anomalous Phase Transitions, and Algebraic Stability

Phase transitions are often presented as clean reorganizations between macroscopic regimes, with persistent structure treated as a secondary imperfection. In practice, regime entry routinely leaves behind vortices, domain walls, frozen correlations, and other topological defects, as well as anomalous transitions that bypass or only partially realize intermediate phases. Building on Emergence as Regime Formation I, where emergent laws were reframed as residues left behind when a system exhausts its freedom to continue in qualitatively different ways, the present work shows that non-uniform exhaustion generically deposits persistent structure when regime entry cannot be globally smoothed. Defects and anomalous transitions are thus reinterpreted as encoding the mechanism of regime formation itself rather than deviations from it. The central contribution is an operator-level framework for generative exhaustion. Distinct modes of constraint act as schematic operators on a domain-relative generative capacity Ω. When constraint operators fail to commute, regime entry admits no globally consistent completion and algebraic obstruction is retained as stable defect structure, motivating an Algebraic Stability Criterion. Within this subtractive framework, topological defects, finite-rate quench scaling (Kibble–Zurek), anomalous phase transitions, and persistent heterogeneity are unified as signatures of non-commuting generative constraints. Illustrations include quantized vortices in rotating Bose–Einstein condensates, cosmic strings, and a structural interpretation of cosmological parameter tensions. This work is structural and diagnostic rather than predictive, providing a unifying algebraic language for understanding why persistent structure is generically produced during non-uniform regime formation.