Causal Priority Theory Part-8 Coarse-Graining Classes, Stable Order, and Admissible Partitions of Derived Time

This paper clarifies when time comparisons are physically meaningful in Causal Priority Theory (CPT), introducing a structured notion of comparability. In CPT, time is defined as the accumulation of a causal update quantity along a history, and is therefore inherently dependent on event partitioning. As a result, time is not globally unique, raising the question of when different histories can be meaningfully compared. To address this, the paper introduces three key structures: coarse-graining classes (describing how distinguishability is reduced), stable order (identifying when temporal ordering is preserved), and admissible partition classes (defining domains of robust comparison). These lead to a central conclusion: comparability is not absolute but physically constrained by information loss and redistribution. The monotonicity of the update quantity under CPTP maps induces a contraction structure that organizes derived time into distinct comparability regimes. CPT thus replaces the assumption of globally comparable time with a notion of layered comparability, consisting of history-dependent time, coarse-graining-dependent comparison, and robust comparison within admissible classes. This framework establishes the physical scope of time comparison and sets the stage for Part-9, where the necessity of quantum relative entropy is justified. --- Update (v2): Added description for clarity and discoverability.