Causal Update Cosmology IX: Existence Conditions for Central Regimes in Causal Update Systems

Causal Update Cosmology IX (CUC IX) is the final installment of the Causal Update Cosmology (CUC) series. CUC investigates how temporal structure, irreversibility, and law-like behavior can emerge from minimal causal update dynamics, without presupposing physical time, spacetime geometry, energy principles, or thermodynamic postulates. Across CUC I–VIII, a robust three-regime organization—initial, central, and terminal—was established, and the central regime was identified as the unique structural band in which multi-path causal competition, irreversible history accumulation, self-averaging of effective degrees of freedom, and entropy-like growth of history indistinguishability coexist. While CUC VIII completed the structural definition of the central regime, one foundational question remained open:under what conditions does a causal update system admit a central regime at all? CUC IX answers this question by providing: Structural necessary conditions for the existence of a central regime Candidate sufficient conditions, formulated to be weak and broadly applicable Explicit counterexamples and boundary cases, demonstrating that definability alone does not guarantee existence By elevating the central regime from a definitional concept to one characterized by existence conditions, this work completes the core CUC framework. No appeal is made to relativistic physics, quantum theory, empirical cosmology, or numerical parameter tuning; the analysis is purely structural and applies to a general class of causal update systems. With CUC IX, the series CUC I–IX forms a closed and self-contained theoretical framework. Any further developments (e.g., connections to existing physical theories or applications) lie outside the scope of the core CUC program. Author: TI Note on AI Use This work also explores the broader question of whether new physical theories can emerge through collaboration with AI systems. ChatGPT (OpenAI) was used extensively in the generation and structuring of theoretical ideas and manuscript text, beyond simple editing or polishing. All numerical experiments and scientific validation were performed by the human author.