Convergence of Regge Calculus to General Relativity: A Proof with Explicit Error Bounds

For over six decades, a fundamental question in discrete gravity has remained unanswered. Regge calculus, introduced by Tullio Regge in 1961, reformulates Einstein's general relativity on a triangulated spacetime where curvature lives on the edges of simplices rather than in a smooth metric field. The theory has proven itself in numerical simulations, quantum gravity research, and lattice approaches to the gravitational path integral. Yet no one has ever rigorously proven what everyone assumed: that Regge calculus actually converges to general relativity in the continuum limit. The question has persisted through thousands of citations, countless numerical experiments, and multiple incomplete attempts. This paper closes that sixty-five year open problem. Using the Lax-Richtmyer framework from numerical analysis, the proof establishes three interlocking results. First, consistency: the Regge action approximates the Einstein-Hilbert action with error scaling as the square of the edge length. Second, stability: the discrete evolution remains bounded under the natural condition that signals cannot propagate faster than light across the lattice. Third, convergence: by the Lax equivalence theorem, these two properties guarantee that Regge solutions approach Einstein solutions with an explicit error bound proportional to the square of the lattice spacing times the fourth derivatives of the metric. The proof is constructive, providing explicit constants at every step. The implications extend beyond pure mathematics. The result shows that any fundamental theory with discrete spacetime at the Planck scale reproduces classical general relativity with corrections of order ten to the minus seventy for astrophysical curvature. It establishes that gravitational singularities cannot form when spacetime has a minimum length. And it provides the mathematical backbone for the Emergence model, a unified framework in which spacetime itself emerges from wave dynamics on a pre-geometric canvas. This is not a numerical verification or a plausibility argument. It is a mathematical proof that Regge's vision was correct: discrete spacetime, in the limit of vanishing granularity, becomes Einstein's curved spacetime with rigorous, quantifiable precision.