Kalimuthu Degenerate Triangle Calculus (K-DTC) as a UV Regulator Tool for Regge Quantum Gravity: Bridging Validated Degenerated Spherical Triangles with Planck-Scale Discreteness

Source abstract

Background: In validation articles published in Annals of Mathematics and Physics [Kalimuthu 2024, 2025], we showed that Girard’s formula Area = (A+B+C-π) R² remains valid under Type I (side→0) and Type II (angle→π) degeneration. The tri-rectangular triangle (Sum=3π/2, E=π/2, Area=1/8·4πR²) was identified as a fundamental brick. Methods: We develop Kalimuthu Degenerate Triangle Calculus (K-DTC) as a computational regulator for Regge quantum gravity. The validated excess E_deg = lim(Sum_deg-π) is identified rigorously as integrated Gaussian curvature/holonomy defect Φ_h concentrated at the hinge, corresponding to U(1) holonomy. We define K_eff = Φ_h / A*_h. Results: K-DTC provides a finite limit even when A*_h→0. Maximal excess E_max=2π gives area bound 4lp² and curvature bound K_max= (π/2)/lp². Numerical tests on S² and S³ show convergence O (N⁻¹) and exact Gauss-Bonnet ΣΦ_i=4π. Conclusion: Validated degenerate triangles act as UV regulator, eliminating singularities while preserving continuum limit.