Emergence of the Einstein Field Equations from the Discrete Voxel Lattice

We derive the Einstein field equations of general relativity from the discrete voxel lattice that emerges from wave intersections in the canvas framework. The premises are: (i) spacetime is a discrete lattice of voxels formed when space and time waves intersect above a threshold intensity; (ii) the local lattice spacing L(x) is determined by the inverse product of the space and time wave amplitudes; (iii) the geometry of the lattice is described by Regge calculus, in which curvature is concentrated on 2D hinges and measured by deficit angles; (iv) the discrete action is the sum over hinges of the deficit angle times the hinge area; and (v) the continuum limit L \to 0 recovers the Einstein-Hilbert action with the stress-energy tensor sourced by the energy-momentum of the canvas fields. What this paper does: · Derives the continuum limit of the Regge action for a hypercubic lattice with slowly varying spacing· Shows that the limit yields the Einstein-Hilbert action \frac{1}{16\pi G} \int R \sqrt{-g} \, d^4x, with Newton's constant G = \ell_P^2 in natural units· Varies the action to obtain the Einstein field equations G_{\mu\nu} = 8\pi G T_{\mu\nu}· Computes the stress-energy tensor for a scalar field on the emergent manifold· Identifies five specific open computations required to make the derivation fully rigorous What this paper does not do: · Derive the functional form of L(x) from the threshold condition (identified as an open computation)· Extend the derivation to gauge fields, fermions, or the Higgs field (the scalar field case is treated as a representative example)· Compute quantum corrections or the cosmological constant The five open computations: 1. Explicit deficit angle computation — verify the leading-order expression \epsilon_{\mu\nu} = -L^2 \partial_\mu \partial_\nu \ln L for the varying-spacing hypercubic lattice2. Rigorous continuum limit of the Regge action — including all derivative terms, without the conformally flat approximation3. Explicit variation of the Regge action — showing the discrete equations reduce to G_{\mu\nu} = 8\pi G T_{\mu\nu}4. Derivation of L(x) from the threshold condition — the critical missing link between the canvas premises and the lattice geometry5. Stress-energy tensor for all Standard Model fields — gauge fields, fermions, and the Higgs field Why this matters: General relativity describes gravity as the curvature of spacetime. In the canvas framework, spacetime is not a fundamental smooth manifold — it is a discrete lattice of voxels formed by wave intersections. This paper shows that the Einstein equations emerge as the continuum limit of the discrete lattice dynamics. When the five open computations are completed, the canvas framework will contain a unified description of gauge interactions, wavefunction collapse, and gravity, all derived from the same set of premises. Keywords: canvas model, emergent gravity, Regge calculus, Einstein field equations, discrete spacetime, voxel lattice, general relativity, quantum gravity, continuum limit