Quantum field theory in continuous spacetime produces ultraviolet divergences that require renormalization. The physical origin of these divergences—and the reason renormalization works—has been debated since the 1940s. This paper shows that if spacetime is discrete at the Planck scale, the divergences never appear. The discrete voxel lattice provides a physical ultraviolet cutoff at the Planck length \ell_P, rendering all loop integrals finite from the outset. What this paper provides: · A construction of quantum field theory on the discrete voxel lattice. The lattice spacing a = \ell_P is not an arbitrary regulator—it is the physical structure of spacetime, derived from the threshold condition for wave intersections in the canvas model. All momentum integrals are replaced by discrete sums over a finite Brillouin zone |p| \leq \pi / \ell_P.· Explicit finite expressions for one-loop corrections. The electron self-energy, photon vacuum polarization, and vertex function are computed on the discrete lattice. All are absolutely convergent. The continuum limit \ell_P \to 0 recovers standard renormalized perturbation theory exactly, with the cutoff dependence absorbed into the usual counterterms.· Proof that gauge invariance is preserved. The Ward identity q^\mu \Pi_{\mu\nu}(q) = 0 holds exactly on the lattice, ensuring that the discrete regulator respects gauge symmetry—unlike a simple momentum cutoff.· Connection to the renormalization group. The discrete lattice provides a natural framework for the Wilsonian RG. Coarse-graining from lattice spacing a to 2a integrates out high-momentum modes. The RG flow converges to the continuum Callan-Symanzik equation as a \to 0, with the lattice spacing providing the physical sliding scale. This matches the Feed Equation (Pillar IV) of the canvas model.· Resolution of the hierarchy problem. The Higgs mass receives quadratically divergent radiative corrections in the continuum. On the discrete lattice, the Higgs self-energy is finite and of order M_P^2. The observed Higgs mass m_H \approx 125 GeV is not fine-tuned—it is a derived consequence of the horizon information bound and the Higgs proximity parameter \epsilon_H \approx 0.51.· Resolution of the cosmological constant problem. The vacuum energy density on the lattice is \rho_{\text{vac}} \sim M_P^4, but in the canvas model, the uniform vacuum energy does not gravitate due to baseline subtraction. Only spatial variations of the lattice spacing contribute to gravity. The residual cosmological constant is set by the finite information bound, not by the vacuum energy density.· Comparison with other regulators. A table compares momentum cutoff, Pauli-Villars, dimensional regularization, and lattice QFT. The discrete voxel lattice is the only regulator that is physically motivated, preserves all symmetries, and resolves the hierarchy and cosmological constant problems simultaneously.· Experimental signatures. Modified dispersion relations at Planck-scale energies (\omega^2 = m^2 + \frac{4}{a^2} \sum_\mu \sin^2(p_\mu a/2)), Lorentz violation suppressed by (E/M_P)^2, and a hard momentum cutoff at |p| = \pi / \ell_P \approx 10^{19} GeV—all effects are unobservable at current accelerator energies but become relevant near the Planck scale. Why this matters: The discrete regulator is not an ad hoc mathematical device. It is the physical structure of spacetime at the Planck scale, derived from the threshold condition for wave intersections. The infinities of continuum quantum field theory are artifacts of extrapolating a discrete theory to arbitrarily short distances. The canvas model provides a finite, physically motivated ultraviolet completion of the Standard Model. Keywords: renormalization, discrete spacetime, voxel lattice, Planck length, ultraviolet cutoff, lattice QFT, Wilson action, Ward identity, renormalization group, hierarchy problem, cosmological constant, canvas model, Feed Equation, Brillouin zone