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.六十余年来，离散引力（discrete gravity）领域始终存在一个悬而未决的基础问题。1961年由图利奥·雷奇（Tullio Regge）提出的雷奇微积分（Regge calculus），将爱因斯坦广义相对论重新表述于三角化时空框架下：此时曲率驻留于单形（simplex，复数形式为simplices）的边之上，而非光滑度量场之中。该理论已在数值模拟、量子引力研究以及引力路径积分的格点方法中得到广泛验证。然而，学界长期默认的核心结论——雷奇微积分在连续极限下确实收敛于广义相对论——却从未被严格证明。这一问题历经数千次引用、无数数值实验以及多次未竟的尝试，始终悬而未决。本文解决了这一延续六十五载的公开难题。借助数值分析领域的拉克斯-里希特迈尔框架（Lax-Richtmyer framework），本证明确立了三个相互关联的核心结论。其一为一致性（consistency）：雷奇作用量可对爱因斯坦-希尔伯特作用量（Einstein-Hilbert action）进行近似，其误差随边长按平方缩放。其二为稳定性（stability）：在"信号在格点上的传播速度不超过光速"这一自然条件下，离散演化过程始终保持有界。其三为收敛性（convergence）：根据拉克斯等价定理（Lax equivalence theorem），上述两项性质可保证雷奇解趋近于爱因斯坦解，且显式误差界与晶格间距的平方以及度量的四阶导数成正比。本证明为构造性证明，每一步均给出了显式常数。该结论的意义远超纯数学范畴。研究表明，任何在普朗克尺度下具备离散时空结构的基础理论，在天体物理曲率场景下，其对经典广义相对论的修正量级可达10的负70次方级别。该成果还证明，当时空存在最小长度时，无法形成引力奇点。同时，它为涌现模型（Emergence model）提供了数学支撑——这是一个统一框架，时空本身源于前几何画布上的波动力学。这并非数值验证或似真性论证，而是严格的数学证明：雷奇的构想是正确的——即当离散时空的粒度趋近于零时，其将以严格且可量化的精度收敛为爱因斯坦的弯曲时空。