Emergence II: Quantum Foundations from Wave Intersections on a Pre-Geometric Canvas

This paper builds the quantum foundations of the canvas model from first principles, deriving the core equations of quantum mechanics step by step. Starting from the six core equations of Emergence I — canvas wave equation, threshold condition, emergent spacetime lattice, quantization, back-reaction, and spatial charge gauge symmetry — we provide complete mathematical derivations for: The Klein-Gordon equation from the massive canvas wave equation, establishing the relativistic wave equation for scalar particles directly from canvas dynamics. The mass term arises naturally from the intrinsic canvas frequency ω₀ of the quantum field, with no external parameters. The Dirac equation via factorization of the Klein-Gordon operator, with full construction of the Clifford algebra and 4×4 gamma matrices. Spin emerges as a consequence of the factorization — spin-½ particles are required for the equation to be first-order, and the 720° rotation property follows from the canvas twist postulate (Emergence I). Three fermion generations from internal spatial harmonics on the 3D canvas. The Yukawa coupling to the Higgs field is proportional to the overlap integral between the fermion's standing wave mode and the localized Higgs profile. Higher harmonics are exponentially suppressed, yielding exactly three generations with non-negligible couplings. This explains why there are three — not two, not four — generations of fermions, and why the masses are exponentially hierarchical. The uncertainty principle from the Fourier transform properties of canvas fields, with no additional postulates. Quantum tunneling from the wave equation with a potential barrier, including the full transmission coefficient calculation and the exponential suppression e^{−2κa}. The Casimir effect with lattice regularization — the discrete voxel lattice provides a physical cutoff, making all sums finite without formal renormalization tricks. Feynman diagrams as physical wave intersections — propagators derived from the Green's function of the canvas wave equation, vertex factors from threshold couplings h_{ijk} ∝ 1/√T_{ijk}, loops finite due to the lattice cutoff at the Planck scale. Bell inequality violation from the shared canvas phase of entangled particles created at the same wave intersection. The hidden variable is non-local in physical space but local on the canvas, violating the CHSH inequality (S = 2√2 > 2) with no faster-than-light signaling. Quantum teleportation — the full protocol derived from the Bell state measurement mechanism, with fidelity F = 1. The Unruh effect via Bogoliubov transformation between Minkowski and Rindler modes, showing that an accelerating observer detects thermal radiation at T = ℏa/(2πck_B). The Aharonov-Bohm effect, Berry phase, quantum Zeno effect, quantum decoherence, Hong-Ou-Mandel effect, spontaneous parametric down-conversion, and Bell state measurement — all derived from the same six core equations. Why this matters: The canvas model does not assume quantum mechanics — it derives it. Every derivation starts from the canvas wave equation and threshold condition, proceeds through every algebraic step, and arrives at the standard quantum mechanical result. The measurement problem is resolved (collapse = threshold crossing), the Born rule is derived (see also the companion paper "Origin of the Born Rule"), and the effective randomness of quantum outcomes is traced to inaccessible sub-oscillation phase information. What this paper is: A complete, step-by-step derivation of quantum foundations from a deterministic pre-geometric canvas. What this paper is not: A compatibility study. This is genuine derivation, shown in full. For readers: Physicists, philosophers of physics, and anyone interested in the origins of quantum mechanics. Read this after Emergence Paper I to understand how quantum phenomena emerge from wave intersections. Keywords: canvas model, quantum foundations, Klein-Gordon equation, Dirac equation, three generations, uncertainty principle, quantum tunneling, Casimir effect, Feynman diagrams, Bell inequality, quantum teleportation, Unruh effect, Aharonov-Bohm effect, Berry phase, quantum Zeno effect, decoherence