RZS Series: Projection-Stable Regge Geometry from Relational Graphs

This paper asks a simple question: if we begin not with space, but with a network of relations, when is it legitimate to say that geometry has been recovered? I study this question through the RZS-ELCL benchmark pipeline. The pipeline starts from weighted relational graphs, filters candidate geometric edges, builds local clique and simplicial structure, and then measures curvature with Regge-style operators. The goal is not to prove that spacetime emerges from a relational zero state. The goal is narrower and more testable: to identify when a relational graph can be projected into a stable discrete geometry, and when that projection fails. The results are deliberately presented as controlled benchmarks rather than as an asymptotic theorem. In low- and medium-damage regimes, explicit projectors recover the intended geometric graph and reject several families of synthetic contaminants. The error analysis separates two effects that are often mixed together: compression error from the spectral estimator, and discretization error from the Regge operator itself. This makes it possible to see whether a failure comes from the estimator or from the underlying discrete geometry. The paper also records the limits of the method. Early 3D reconstruction fails under shortcut contamination, the 4D Regge operator depends on the correct internal-dihedral-angle convention, and high-damage repair succeeds only when scaffold incidence information is supplied. The main contribution is therefore not a claim that relational geometry has been solved, but a falsifiable framework for testing how much geometry can be recovered from relational data.