Causal Priority Theory Part-12 Layered Breakdown of Derived-Time Availability and Comparison

This paper, Part-12 of the Causal Priority Theory (CPT) series, develops a systematic classification of breakdown regimes of derived time. In CPT, time is constructed from causally ordered quantum state updates quantified by quantum relative entropy. While previous parts focused on the construction of derived time, this paper analyzes its structural failure modes, where temporal availability and comparability cease to be automatically guaranteed. Key Contributions: • Four-Layer Classification of Breakdown: The paper formalizes four logically distinct failure regimes of derived time:• Type-D (Definability): Failure of finite event-wise assignability due to support obstruction.• Type-R (Refinement-level): Non-convergence or trivialization of derived time under partition refinement (reorganizing Part-10).• Type-A (Asymptotic-assignment): Non-canonicity of temporal scales even under continuous state descriptions (reorganizing Part-11).• Type-C (Comparability): Instability and non-comparability of derived times, including a constructive theorem demonstrating order reversal (Type-C1). • Observational Boundary: A protocol-relative interface is identified at which these breakdown layers become operationally manifest. • Conditional Recovery: Each breakdown layer admits specific sufficient conditions (e.g., support-regularity, contraction-ratio separation) under which partial restoration of derived-time structure is possible. This work advances the core objective of the CPT program—the separation of time from causal structure—by showing that causal ordering alone does not determine a stable, unique, or canonical temporal metric.