Causal Priority Theory part-6 Causal Construction of ρ_BH(t) and Proof of Eq.(8'') — Re-derivation of 0.646 via Relative Entropy

[Overview]This paper is the sixth installment in the Causal Priority Theory (CPT) series and serves as the rigorous culmination of the framework. It provides the formal proof for the "redistribution principle" of information distance introduced in Part-5 and completes the non-approximated derivation of the Page curve. By utilizing the chain rule for relative entropy and the newly proposed Causal Maximum Entropy Principle, this work demonstrates that the numerical prediction of Page time (≈ 0.646 t_evap) is a fundamental consequence of the causal structure itself. [Key Contributions of Part-6]1. Formal Proof of the Redistribution Equality: Rigorously derives the relationship between the relative entropy of the black hole, the radiation, and their mutual information distance, resolving the "unfinished business" from Part-5.2. Definition of the Vacuum Reference State: Establishes the maximally mixed state (ρ_vac) as the objective reference point for causal updates in black hole systems.3. Introduction of the Causal Maximum Entropy Principle (Theorem 6): Proposes a new fundamental principle that governs how information distance is distributed across causal boundaries.4. Rigorous Re-derivation of 0.646: Re-calculates the Page time using the chain rule for relative entropy. This confirms that the 0.646 prediction from Part-2 is not a result of semi-classical approximations but is an exact requirement of the CPT causal kinematics.5. Distinction from Island Formula: Clarifies how CPT's structural approach to information conservation differs from the geometric "island" prescription, providing testable distinctions. [Position in the Series]• Previous (Part-5): Redefined ΔC via relative entropy and derived the arrow of time.• Current (Part-6): Completes the mathematical proof of the Page curve prediction (0.646) using the relative entropy framework.• Future (v7 and beyond): The incorporation of the "Will operator (W)" and its impact on the causal structure is explicitly identified as the next frontier. [Series Abstract]Causal Priority Theory (CPT) is a comprehensive theoretical framework that redefines the fundamental structure of the universe by prioritizing "Causal Change" (ΔC) over "Time." This series demonstrates how conventional physical laws—including relativity, quantum mechanics, and the arrow of time—emerge as derived properties from a single fundamental axiom (Axiom 0: Primacy of Causal Updates).