Causal Update Cosmology IV (CUC V4): Minimal Invariants, Velocity-Free Causal Cones, and the Meaning of Front Thickness

Description(for Zenodo update) Causal Update Cosmology IV (CUC V4) extends the previous stages (V1–V3) by shifting the focus from numerical fitting to structural and invariant analysis of causal update dynamics. While CUC I–III demonstrated that finite causal propagation speed, cone-like structures, and front thickness emerge numerically from minimal causal update rules, V4 asks a deeper question: what, if anything, remains invariant when definitions and parameterizations are stripped away? In this work, we address three core questions: What is the minimal invariant that can be defined in CUC? Can a causal cone be defined without invoking velocity or distance? What does front thickness represent in informational terms? Key findings of CUC V4 include: No global numerical invariant is supported once quantile-based tautologies are removed. A robust three-regime organization (early / central / late) consistently appears across update rules and random seeds. The central regime exhibits quasi-invariant behavior, remaining stable under changes in neighborhood structure and stochastic initialization. A causal cone can be defined without velocity, as a band of stable reachability (accessibility cone) emerging only within the central regime. Front thickness is interpreted as an informational signature of unresolved causal competition, naturally encoding irreversibility without assuming external time. CUC V4 establishes that the fundamental invariant of causal update cosmology is structural rather than numerical, and that time-like and cone-like behavior arise only in specific regimes of the update process. This work marks a transition of CUC from numerical observation to structural theory, and sets the stage for CUC V5, which will investigate whether the three-regime structure is a necessary consequence of causality itself. 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.