Within the Projective Dynamic Logo (PDL) framework, physical reality is reconstructed from four axioms on finite signed graphs without presupposing spacetime, particles, or fields. This paper develops the relational foundations of time and metric structure within PDL. Three interlocking constructions are presented. First, proper time is reinterpreted as a discrete counting of internal coherence cycles: for a stationary (4,6) closure (electron prototype), each complete two-step pulsation constitutes a minimal logical clock, and the proper time elapsed along a worldline is defined as the number of such cycles completed. The standard relation E = ħω is thereby recast as a statement about action per coherence cycle, establishing compatibility with standard relativistic proper time without presupposing it. Second, a notion of relational density ρ is introduced as the number of active relations per unit of logical volume. The proper-time element dτ experienced by a closure in a region of density ρ is related to a reference pulsation increment by a factor that decreases in regions of high relational density, reproducing the qualitative structure of gravitational time dilation from purely combinatorial considerations. Third, a coherence-cost pseudometric is defined on the space of closures: the distance between two configurations is the minimal additional coherence leakage required to embed them compatibly into a common relational environment. This pseudometric generates a coexistence topology on configuration space whose neighbourhoods are determined by logical compatibility, not by any a priori geometric proximity. In suitable coarse-grained regimes, the coherence-cost pseudometric is approximated by an effective Lorentzian metric, with the temporal component encoding the dependence of proper time on relational density in direct analogy with gravitational potentials. These constructions provide the relational underpinning of the emergent metric structure developed in subsequent PDL documents and connect the combinatorial architecture of the proton to the large-scale dynamics explored in the Hubble tension resolution.
Cédric Laubscher (Mon,) studied this question.