Total-Variation Stability and Adversarial Fragility of a Two-Dimensional Jensen–Shannon–Regge Curvature Estimator

This manuscript studies a two-dimensional Regge-type curvature estimator in which edge lengths are obtained from Jensen–Shannon distances between states assigned to the vertices of a triangulation. The work proves a deterministic total-variation stability estimate for the raw atomic angular-deficit curvature measure under unstructured edge-length perturbations on fixed, uniformly nondegenerate triangulations. The main bound shows that the perturbation of the curvature measure is controlled by Cσε/ε3Cσ_ε/ε^3Cσε/ε3, where σεσ_εσε is the uniform edge-length error. Consequently, σε=o(ε3)σ_ε=o(ε^3)σε=o(ε3) is a sufficient condition for preservation of the weak limit of the estimator, although it is not claimed to be necessary under structured or geometrically compatible perturbations. The manuscript also gives adversarial sharpness constructions, including a flat hexagonal star and a periodic flat-torus construction, showing that the total-variation scale is essentially optimal for unstructured perturbations. In the classical finite-alphabet setting with uniformly positive support, it derives a local estimate converting L1L^1L1 state error into Jensen–Shannon edge-length error and discusses the resulting sufficient empirical sampling regime. The conclusions emphasize the statistical and adversarial fragility of the unsmoothed raw estimator rather than asserting general robustness.