We compute the complete spectrum of the Regge Hessian for the equilateral 4-simplex using symmetry methods and the Bose–Mesner algebra of the triangular scheme T(5) . The full eigenvalue structure is derived analytically and verified numerically. We extend the analysis to multi-simplex systems, obtain the exact dispersing sector via a cubic characteristic equation, and derive the corresponding propagator. In the long-wavelength limit, the shape-mode sector exhibits a 1/p² behavior consistent with the linearised Einstein action. These results provide a fully analytic spectral foundation for simplicial quantum gravity.
Mukka Srivatsav (2026) studied this question.