Lattice Quantum Canvas: A Complete Formulation of Quantum Mechanics on the Discrete Spacetime Lattice

The canvas model establishes that spacetime is a discrete voxel lattice with minimum spacing \ell_P = \sqrt{\hbar G / c^3} \approx 1.6 \times 10^{-35} m. Quantum mechanics emerges from wave dynamics on this lattice. This paper provides the complete, rigorous formulation of quantum mechanics directly on the discrete spacetime lattice—the Lattice Quantum Canvas (LQC). What this paper provides: · Discrete Hilbert space. The space \ell^2(\mathbb{Z}^3) is a separable Hilbert space. The discrete Laplacian is bounded and everywhere-defined, hence self-adjoint. The spectral theorem guarantees a complete orthonormal basis of eigenfunctions. Explicit formulas for the momentum representation and the dispersion relation \omega(\mathbf{k}) = \frac{2c}{\ell_P} \sqrt{\sum_i \sin^2(k_i \ell_P/2)} are provided. Lattice corrections to the continuum dispersion are of order (\ell_P/\lambda)^2 \sim 10^{-60} for atomic scales—completely negligible for all observable phenomena.· Discrete path integral. The Feynman path integral is a finite sum over voxel paths. Using the Trotter product formula and the Baker-Campbell-Hausdorff expansion, we prove convergence to the continuum path integral with explicit error bound O(\ell_P^2/\lambda^2 + \ell_P/(cT)). The discrete formulation eliminates the need for renormalization—all sums are finite, all integrals are discrete. The finite Hilbert space for any finite region implies the total number of degrees of freedom in the observable universe is I_{\text{max}} = 4\pi R_H^2 / \ell_P^2 \sim 10^{122} bits, consistent with the holographic bound.· Discrete Schrödinger equation. From the discrete path integral via Legendre transform, we derive the discrete-time Schrödinger equation. The continuum Schrödinger equation is recovered in the limit \ell_P \to 0 with corrections O(\ell_P^2 \nabla^4 \psi). We solve the discrete harmonic oscillator analytically and the tunneling problem exactly on the lattice. The harmonic oscillator eigenvalues are E_n = \hbar\omega(n+1/2)[1 - (\ell_P/a_0)^2(2n+1)/24 + O(\ell_P^4/a_0^4)], with relative errors \sim 10^{-5} for a_0 = 100\ell_P.· Quantum walks and the Dirac equation. We prove that quantum walks on the 3+1 dimensional voxel lattice, using a 4-dimensional internal coin space encoding the full Dirac spinor structure with projection operators P_i^\pm = \frac{1}{2}(I_4 \pm \alpha_i), converge to the Dirac equation in the continuum limit. The coin operator encodes the mass term via \gamma^0, the conditional shift operators encode the kinetic terms via \boldsymbol{\alpha} \cdot \nabla. We eliminate fermion doubling via the Wilson term and implement exact chiral symmetry via the overlap formalism satisfying the Ginsparg-Wilson relation.· Discrete gauge theory. Gauge fields are defined on lattice edges via Wilson lines U_\mu(\mathbf{n}) = \exp(i g \ell_P A_\mu^a T^a). The discrete Yang-Mills action on plaquettes converges to the continuum action with O(\ell_P^2) error. The lattice formulation provides a manifestly gauge-invariant, finite regularization of quantum field theory with the physical cutoff \ell_P. We implement chiral fermions via the overlap operator and discuss the index theorem on the lattice.· Measurement as threshold crossing. Measurement is a physical process: when the quantum field intensity exceeds a detector threshold, energy transfers via back-reaction. The Born rule follows from a rigorous application of Rice's formula for level-crossing rates of a sinusoidal signal with Gaussian noise, combined with time-averaging over the inaccessible sub-voxel phase. No collapse postulate is required. A complete derivation is provided in the appendix.· Decoherence and the classical limit. The interaction of quantum systems with sub-Planck vacuum fluctuations of the lattice leads to a Lindblad-type master equation with decoherence rate \gamma \sim \ell_P^2 / \lambda^3. The discrete Ehrenfest theorem recovers classical equations of motion with O(\ell_P^2) corrections.· Comparison with existing approaches. A detailed comparison of the LQC with lattice QCD, causal set theory, loop quantum gravity, and quantum walks highlights the unique features: physical (not regulator) lattice, measurement as a dynamical process, and gauge-invariant formulation with chiral fermions. Why this matters: The LQC formulation bridges the discrete ontology of the canvas model with the continuous mathematics of standard quantum theory. All standard quantum mechanical results are recovered in the continuum limit, with Planck-suppressed corrections that are negligible for all currently observable phenomena. The framework honestly discusses limitations, including the current inaccessibility of experimental signatures and open problems in gravitational dynamics and spin-statistics. This paper is part of the Emergence series, building on Paper I (Unified Field Theory) and the cosmological constant derivation. It provides the rigorous mathematical foundation for quantum mechanics within the canvas model and can be evaluated independently of the broader framework. Keywords: lattice quantum mechanics, discrete spacetime, voxel lattice, quantum walks, Dirac equation, lattice gauge theory, overlap fermions, Ginsparg-Wilson relation, measurement problem, Born rule, decoherence, path integral, Trotter product formula, renormalization, holographic bound, Emergence series