Causal Pregeometry II: Reachability and Kinetic Selection

Building upon the local algebra on dimensionless discrete graphs established in the preceding work (Causal Pregeometry I), this paper addresses the problem of how macroscopic global relaxation may emerge from simple microscopic local rules. We examine the nonimplication that the mere existence of a local relaxation path (constructive reachability) does not by itself determine the probabilistic accessibility of global target states (typical accessibility), thereby providing a mathematical foundation for an intermediate hierarchy connecting local algebra to macroscopic dynamical descriptions. [Key Highlights] Separation of the Kinetic Selection Hierarchy and Macroscopic Observables: We mathematically distinguish transition possibilities defined by local algebra from their actual selection probabilities under specified kinetic rules, and formulate quantities such as the eligible-edge stock (eJ) and unconditional expected Join relief (JJ) as distinct macroscopic bridging variables. Proof of Constructive Reachability: Using only the local graph transition rules of CCA and baseline Join, we constructively prove the existence of a finite legal transition sequence from the specified class of initial states to a low-burden boundary state under the stated assumptions. Derivation of the Central Nonimplication: By separating constructive path existence from probability under the full study-level measure, we derive that reachability alone does not determine the probability magnitude, system-size dependence, or asymptotic law of the target event: Reachability ⇒ Typical\ accessibility. [Abstract] How does the macroscopic flow of relaxation across an entire system emerge from simple rules governing its microscopic parts? Addressing the problem that the realization of global relaxation cannot be deduced solely from the existence of a local relaxation path, this paper derives that a nonimplication relation holds between "Constructive Reachability" and "Typical Accessibility." Using the local algebra on dimensionless graphs established in previous work as a starting point, we formulate a kinetic selection hierarchy from local events to global target events. Through the evaluation of independent evidence layers, it is derived that absolute macroscopic relaxation trajectories cannot be determined solely by local candidate enrichment or the relative advantage of realized relief. Furthermore, while the existence of a finite legal transition sequence from a specified initial state class to a low-burden state is constructively proved, it is derived that this existence proposition alone cannot determine the probability magnitude, size dependence, or asymptotic law of the target event under the full study-level measure. That is, Reachability ⇒ Typical\ accessibility is derived as the central nonimplication relation of this paper. The scope of this paper is strictly limited to clarifying the nonimplication structure that holds among local algebra, kinetic selection, constructive reachability, and typical accessibility. The connection to physical phenomenology is left for future research. [Computational Verification DevKit]An executable core specification designed for LLM-based verification of this framework is available here:https://buymeacoffee.com/az_laboratory/extras