Two Clocks on the Light Cone: links, numbers, and the de Sitter causal set

This paper develops a quantitative investigation of how discrete causal structure encodes geometric and curvature information in four-dimensional de Sitter spacetime. Within Poisson-sprinkled causal set theory, it analyzes the link valency of a single element using exact causal-diamond geometry, controlled second-order asymptotic expansions, and independent numerical verification. The analysis identifies a scale-independent contribution of −12 links per affine e-fold, derives finite-density curvature corrections with explicit remainder estimates, and establishes an analytic closed-form expression for the apex constant. These results clarify the complementary roles of causal order and spacetime volume in determining geometric observables, particularly where leading-order null-curvature responses are insensitive to pure-trace Ricci curvature. The resulting asymptotic structure provides a rigorous mathematical setting for exploring connections between causal-set discreteness, cosmological curvature, and modular timescales. The proposed interpretation of scale-independent link statistics as a candidate microscopic clock is distinguished explicitly from the established geometric and asymptotic results.