Reconstruction of the Riemann tensor from causal order and element counts

A causal set is a locally finite partially ordered set. Causal set theory proposes such discrete structures as a basis for spacetime physics. Connecting this description with continuum physics requires recovering the geometric quantities used in general relativity and controlling the errors introduced by discreteness. Here, with dimension and sampling density specified, we construct approximate normal coordinates, estimate parallel transport, and recover the full Riemann tensor from causal order and element counts. For samples embedded in a fixed smooth spacetime and satisfying stated uniform counting assumptions, we prove a uniform curvature error bound that tends to zero as the sampling density increases.