Causal Priority Theory Part-10 On the Refinement Limit of Derived Time

This paper is Part-10 of the Causal Priority Theory (CPT) series. The unified formulation of CPT (covering Parts 0–6) has been established separately; the present series (Parts 7 onward) constructs the theorem layer on that foundation. In CPT, time is not a fundamental physical variable but a derived quantity constructed from causally ordered state updates, quantified by quantum relative entropy. The present paper studies the refinement-limit problem for derived time: whether a finite derived-time quantity is obtained automatically when event partitions are successively refined. The main result is that refinement-limit derived time is not automatic but conditional. Three logically distinct outcomes are identified and organized into a structural classification: (1) the undefined case, in which finite derived time ceases to be assignable due to support failure along the refining sequence; (2) the trivialized case, in which the refinement-limit derived time exists but collapses to zero; and (3) the conditionally existent case, in which a finite limit is assigned only when an additional convergence condition is imposed. For the trivialized case, a restricted zero-collapse theorem is proved: for path-generated refining sequences induced by commuting C² state paths with fixed support and vanishing-mesh nested partitions, the refinement-limit derived time exists and equals zero. For the conditionally existent case, a Cauchy criterion is formulated as the minimal sufficient condition for finite assignability. These results establish that refinement-limit derived time in CPT is a conditional history quantity whose existence, finiteness, and nontriviality depend on admissibility and convergence structure beyond refinement alone. The relation of this refinement-limit problem to continuous state descriptions is left to Part-11.