HγC W1: From τ to Emergent Effective Time in the HγC Framework

This paper investigates the conditions under which an effective macroscopic time parameter can emerge within the HγC framework from a microscopic substrate organized only by pre-temporal relational update ordering. The microscopic update parameter τ is treated not as physical time, but as an ordering parameter for successive local relational state transitions. The paper argues that temporal directionality arises prior to clock-like time through irreversible accumulation of retained relational traces and the resulting asymmetry in admissible future transitions. On this basis, W1 identifies the regime conditions under which a continuous effective time variable teff becomes meaningful after coarse-graining. Effective time is treated as a regime-dependent mesoscopic descriptor rather than a universal primitive background coordinate. The paper introduces a minimal regime-level ansatz relating dteff/dτ to update density, trace persistence, admissibility-bias stability, and the realized coarse-graining regime. The scope of the paper is deliberately limited. It does not construct a full relativistic theory of time, a final theory of emergent temporality, a path-summing formalism, or a quantum-effective dynamics. Instead, W1 establishes the temporal threshold that later time-dependent, fluctuation-level, and restricted quantum-effective developments of the HγC programme must first respect.