Causal Priority Theory part-3 On the Validity Conditions and Limits of the ΔC Conservation Law

Abstract: In Causal Priority Theory (CPT), the ΔC conservation law has previously been treated as a fundamental axiom. This paper (v3) provides a rigorous examination of this law, explicitly identifying the classes of Completely Positive Trace-Preserving (CPTP) maps for which this conservation law holds and those for which it does not. The paper presents: 1. Counterexamples: Four specific cases demonstrating that ΔC conservation does not hold for general CPTP maps. 2. Necessary Conditions: Derivation of four essential conditions required for the conservation law to be valid. 3. Main Theorem: A formal proof that in a closed quantum system on a finite-dimensional Hilbert space, the total ΔC is conserved under unitary evolution. 4. Application to Black Holes: A structural establishment of ΔC_env = 0 using Hilbert space decomposition, upgrading the axiomatic assumptions of the previous papers (v1 and v2) to conditional theorems. This work serves as a critical bridge between the heuristic descriptions of CPT and its formal quantum mechanical foundation, setting the stage for the quantum extension via von Neumann entropy in v4 and relativ