Causal Priority Theory Part-13 Non-Comparable Histories, Partition Closure, and the Non-Existence of a Process-Only Canonical Derived-Time Selector

This paper, Part-13 of the Causal Priority Theory (CPT) series, establishes the final structural limits of derived-time assignment and comparability. While previous parts organized the breakdown and dependence of time, this paper provides the definitive proof that even under physical or operational admissibility restrictions, a unique or total temporal structure cannot be canonically recovered from causal-process data alone. Key Contributions: • Explicit Non-Totality of Comparison: Introduces a "strong class-relative comparison" relation and constructs an explicit classical commuting example to prove that admissibility-relative non-comparability occurs within the framework, demonstrating that restricted history classes do not restore a total order. • No-Go Theorem for Canonical Selectors: Proves that under a singleton-admissibility hypothesis, no "process-only canonical derived-time selector" exists. This confirms that primitive causal order and overall-process data are fundamentally insufficient to determine a unique, partition-independent derived-time value. • Closure Persistence: Distinguishes between refinement closure and T-image closure, showing that strong comparison, when it holds, persists at the T-closure level. Conclusion: This concluding work mathematically formalizes the "separation of time from causal structure". It demonstrates that admissible partition structure does not eliminate non-comparability and that causal-process data alone do not fix a canonical temporal metric. Consequently, history-dependent partition structure is established as an ineliminable and fundamental feature of the time-assignment problem in the Causal Priority framework.