From Lorentzian Regge Defects to Finite Holonomy: Causal-Sector-Resolved Curvature in an Oriented 1-to-5 Simplicial Star.

Abstract

This paper develops a finite, independently reproducible Lorentzian Regge–holonomy correspondence on an oriented four-dimensional 1-to-5 simplicial star. Starting from signed squared-edge data, local Lorentzian simplex frames are reconstructed and adjacent simplices are connected by proper, orthochronous Lorentz transformations across non-null shared tetrahedral facets. Closed products of these facet-attachment maps define the holonomy around each interior triangular hinge. Independently, signed Regge defect changes are computed from outward facet normals. The construction explicitly resolves the causal type of each hinge. Lorentzian-signature hinges produce Euclidean normal planes and elliptic \(SO(2)\) holonomy, while spacelike hinges produce Lorentzian normal planes and hyperbolic \(SO^+(1,1)\) holonomy. The hyperbolic branch is fixed using the oriented normal-plane prescription\[\operatorname{sign}(n_1\!\cdot n_2)\operatorname{asinh}(w).\] Across all ten interior hinges and six signed perturbations of one edge capacity, all 60 signed Regge–holonomy comparisons pass at a tolerance of \(10^{-10}\). The author-executed evidence chain gives a maximum mismatch of \(1.5221\times10^{-14}\), while the included public-safe reproducibility implementation gives \(1.0951\times10^{-14}\). Loop reversal correctly reverses the signed holonomy, the flat control is identity to approximately \(10^{-14}\), and all ten hinges pass the declared first-order scaling test. The contribution is not a claim that the classical deficit–holonomy relation itself is new. The new result is an explicit causal-sector, orientation, sign, and branch lock together with an independently reproducible finite verification package. The result does not establish a Lorentzian continuum limit, Einstein dynamics, universality for arbitrary triangulations, or native microscopic RIQG transport.