Causal-set links see light-cone focusing

One of the simplest observables in a causal set is the number of links ending at a given element. For Poisson sprinklings of Minkowski space, Rideout and Zohren showed that the number of links to the tip of a causal diamond grows like the area of the diamond's edge in units of the discreteness scale, and left curved spacetime open. We give the answer for small causal diamonds in curved spacetime, in every dimension d ≥ 3. At leading order in the sprinkling density, the links of the tip lie in a thin collar along its past light cone, and their expected number is an explicit integral of Jacobi-field data along the cone's null generators. In four dimensions the result is the edge area plus null focusing. The first curvature correction is a positively weighted integral of R_kk along each generator, and unlike the four-dimensional edge area it depends on the observer's frame. In vacuum the leading correction comes from shear: for a diamond of proper height T, the angle-averaged relative excess is (17/50400) T⁴ (E_ij E^ij + B_ij B^ij), proportional to the Bel–Robinson density. The same light-cone integral governs every statistic of interval sizes whose Poisson transform decays like z^(−p) with p > 2/d, including scale-averaged links obtained by random thinning, up to one Mellin constant per statistic. For the Benincasa–Dowker–Glaser kernels this constant vanishes. Finally, link counts self-average: in flat four-dimensional diamonds their variance is O(√n) for n expected elements, and in small curved diamonds their relative fluctuations still vanish. The file code.zip contains Python scripts that reproduce every number in Sections 2–7 and Figures 1–2, together with the output of each run (see README.md inside the archive).