Resolution of the Regge Calculus Problem: How the Canvas Model Provides the Physical Foundation for Discrete Quantum Gravity

Regge calculus, proposed in 1961, formulates general relativity on a discrete spacetime triangulated into simplices. For over sixty years, Regge calculus has been recognized as a promising approach to quantum gravity, but three fundamental problems have prevented it from becoming a complete theory: the triangulation dependence (which triangulation should be used?), the measure problem (what is the correct measure for summing over geometries?), and the continuum limit (how is the limit of vanishing lattice spacing taken?). This paper shows that the canvas model resolves all three problems by providing the physical origin of the triangulation. In the canvas model, spacetime is a discrete voxel lattice generated by wave intersections on a pre-geometric canvas. The triangulation is not an arbitrary discretization—it is the physical structure of spacetime, determined by the threshold condition (Pillar II). The edge lengths are not continuous free variables—they are quantized by the wave amplitudes, which satisfy the Unified Wave Equation (Pillar I). The continuum limit is not a mathematical artifact—it is the limit of large scales compared to the physical Planck length \ell_P. Three convergence theorems are proved: · Theorem 1 (Action Convergence): The Regge action on the voxel lattice approximates the Einstein-Hilbert action with error O(\ell_P^2). The full step-by-step proof from edge lengths to the Ricci scalar is provided.· Theorem 2 (Stability): The linearized Regge equations form a stable discrete hyperbolic system under the Courant-Friedrichs-Lewy condition c\Delta t \leq \Delta x/\sqrt{3}. Von Neumann stability analysis establishes marginal stability.· Theorem 3 (Convergence): Discrete solutions converge to continuum solutions with error \|g^{(\ell_P)} - g\|_{C^0} \leq C \ell_P^2 \|\nabla^4 g\|_\infty. The proof uses the Lax-Richtmyer framework with energy estimates and Sobolev embedding. The resolution of the three Regge problems: · Triangulation dependence: Eliminated. The triangulation is the physical voxel lattice generated by wave intersections. It is not chosen by hand.· Measure problem: Solved. The measure is inherited from the wave functional integral on the discrete canvas. Edge lengths are determined by wave amplitudes.· Continuum limit: Resolved. The limit is L \gg \ell_P, not \ell_P \to 0. The discrete theory is the fundamental theory; the continuum is an approximation. Quantum gravity on the voxel lattice: The discrete path integral Z = \int \mathcal{D}\Phi \, e^{iS_{\text{Regge}}[\Phi]/\hbar} is well-defined, finite, and non-perturbative. The UV cutoff is physical (\ell_P), not a regularization artifact. The classical limit recovers general relativity. Quantum corrections are suppressed by powers of \ell_P/L. Why this matters: The Regge calculus problem was not purely mathematical. It required a physical theory to determine the correct triangulation, measure, and continuum limit. The canvas model is that theory. With this foundation, Regge calculus becomes a complete, finite theory of quantum gravity. Keywords: Regge calculus, canvas model, discrete quantum gravity, voxel lattice, convergence proof, triangulation dependence, measure problem, continuum limit, path integral, quantum gravity