This paper formulates "Event Partition Dependence" in Causal Priority Theory (CPT)—the principle that the flow of derived time depends on how an observer partitions events. • Partition Dependence: Formalizes time as a quantity dependent on the resolution of observation. • Stability of Order: Identifies the physical conditions under which the time order remains stable despite information loss (coarse-graining). • Conclusion: Proves that the "arrow of time" is robustly preserved under specific uniformity conditions, regardless of the partition. This work extends the unified framework (Episodes 0–6) to address the hierarchical nature of physical observation.
This paper formulates "Event Partition Dependence" in Causal Priority Theory (CPT)—the principle that the flow of derived time depends on how an observer partitions events. • Partition Dependence: Formalizes time as a quantity dependent on the resolution of observation. • Stability of Order: Identifies the physical conditions under which the time order remains stable despite information loss (coarse-graining). • Conclusion: Proves that the "arrow of time" is robustly preserved under specific uniformity conditions, regardless of the partition. This work extends the unified framework (Episodes 0–6) to address the hierarchical nature of physical observation.
This paper clarifies when time comparisons are physically meaningful in Causal Priority Theory (CPT), introducing a structured notion of comparability. In CPT, time is defined as the accumulation of a causal update quantity along a history, and is therefore inherently dependent on event partitioning. As a result, time is not globally unique, raising the question of when different histories can be meaningfully compared. To address this, the paper introduces three key structures: coarse-graining classes (describing how distinguishability is reduced), stable order (identifying when temporal ordering is preserved), and admissible partition classes (defining domains of robust comparison). These lead to a central conclusion: comparability is not absolute but physically constrained by information loss and redistribution. The monotonicity of the update quantity under CPTP maps induces a contraction structure that organizes derived time into distinct comparability regimes. CPT thus replaces the assumption of globally comparable time with a notion of layered comparability, consisting of history-dependent time, coarse-graining-dependent comparison, and robust comparison within admissible classes. This framework establishes the physical scope of time comparison and sets the stage for Part-9, where the necessity of quantum relative entropy is justified. --- Update (v2): Added description for clarity and discoverability.
This paper investigates the relation between continuous state descriptions and derived time within the framework of Causal Priority Theory (CPT). In CPT, time is treated as a history-dependent quantity constructed from causally ordered quantum state updates quantified by quantum relative entropy. Key contributions: • Structural non-automaticity: The paper shows that the existence of a continuous state description does not by itself guarantee a canonically assigned continuous derived time. • Two levels of non-automaticity: • Existence level: Continuity does not ensure finite or convergent partition-sum derived time. • Asymptotic-assignment level: The leading asymptotic time scale is not determined by the state image alone in general, as shown by an explicit same-image counterexample. • Conditional existence: The paper formulates conditions under which continuous derived-time candidates exist, while showing that such candidates remain description-dependent. This part (Part-11) provides the continuous-description layer of the broader CPT program and prepares the transition to the subsequent structural analysis of breakdown regimes and non-comparability.
Title: Causal Priority Theory part-4: Quantum Extension of \bm{\Delta C} via von Neumann Entropy - A Basis-Independent Formulation Description: [Overview] This paper is the fourth installment in the Causal Priority Theory (CPT) series. While previous papers (v1-v3) utilized Shannon information entropy to define causal change (\bm{\Delta C}), this work addresses the fundamental limitations of that approach—specifically basis-dependence and observer-dependence—by extending the theory into the quantum domain using von Neumann entropy. [Key Contributions of Part 4] 1. Identification of Shannon Entropy Limitations: Quantitatively demonstrates how Shannon-based \bm{\Delta C} fails to serve as a fundamental physical variable due to its dependence on the choice of basis and coarse-graining. 2. Quantum Redefinition of \bm{\Delta C}: Formally redefines causal update quantity (\bm{\Delta C}) using the von Neumann entropy \bm{S(\rho) = -\text{Tr}(\rho \log \rho)}. 3. Proof of Basis Independence: Provides a mathematical proof that the new definition is invariant under unitary transformations, ensuring that "causal change" is an objective property of the quantum state itself. 4. Resolution of the Coarse-Graining Problem: Establishes an operationally defined quantum framework that eliminates the ambiguity inherent in classical information-theoretic descriptions. 5. Consistency Check: Confirms that the results of Part 1 through Part 3 remain valid under this more rigorous quantum formulation. [Position in the Series] • Previous (Part 3): Examined the validity conditions of \bm{\Delta C} conservation in CPTP maps. • Current (Part 4): Completes the transition of the CPT framework from classical information theory to a robust quantum information-theoretic foundation. • Next (Part 5): Further elevates \bm{\Delta C} from an "entropy difference" to a "relative entropy (distance)," embedding causal directionality (the arrow of time) into the heart of the theory.
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.
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.
This paper establishes the necessity and minimality of quantum relative entropy as the causal update quantity in Causal Priority Theory (CPT). In CPT, time is defined as the accumulation of an information-theoretic update along causal histories. However, the choice of this update quantity has not previously been given a systematic justification. This paper addresses that gap. A set of structural requirements is introduced, including non-negativity, identity vanishing, direction sensitivity, compatibility with product composition, monotonicity under CPTP maps, basis independence, consistency with the classical Kullback–Leibler limit, and compatibility with history accumulation and correlation redistribution. Representative alternative candidates—such as entropy difference, trace distance, fidelity, and Bures distance—are shown to fail to satisfy these requirements or to support the central structural roles required in CPT. Quantum relative entropy (Umegaki form) is then identified as the canonical minimal choice: not uniquely required, but the standard candidate that satisfies the full requirement set with the fewest additional assumptions while maintaining coherence with the existing CPT framework. The result does not provide a new numerical prediction but instead imposes a theoretical constraint on the class of admissible causal update quantities in CPT. It strengthens the foundations of derived time, comparability, and coarse-graining stability developed in earlier parts of the series. Questions concerning the behavior of derived time under refinement limits and its connection to continuous dynamical descriptions are not treated here, and are instead addressed in subsequent parts of this series.