Quantum Field Theory from the Canvas Model: A Complete Step-by-Step Derivation

We present a complete, rigorous derivation of quantum field theory from the eight primitives and four pillars of the canvas model. No steps are omitted. No results are assumed from standard QFT. Every component of the Standard Model—the gauge group, the particle content, the Lagrangian, the Feynman rules, the coupling constants, the mass spectrum, the discrete regulator, the renormalization group flow, and the structural properties of QFT—is derived from the axioms. What this paper provides: · A derivation of the gauge group U(1)×SU(2)×SU(3) from the charge primitive (P8). A field couples to an integer number q_s of spatial axes, yielding U(1) for q_s = 1, SU(2) for q_s = 2, and SU(3) for q_s = 3. Under attractor dynamics, U(q_s) reduces to SU(q_s) for q_s \geq 2, with the overall U(1) phase decoupling.· A derivation of the three generations of fermions from the threshold tensor eigenvalue equation (Pillar III). The threshold tensor is a 3 \times 3 real symmetric matrix on the internal 3D space. Its three eigenvectors correspond to three generations. Higher modes have Yukawa couplings suppressed by e^{-\beta n^2} with \beta \approx 1.868, making them unobservable.· A derivation of the coupling constants from the fundamental coupling unit \alpha_0 = 1/\ln(R_H/\ell_P) \approx 1/140. Gauge couplings at their respective thresholds satisfy \alpha_3 : \alpha_2 : \alpha_1 = 3 : 2 : 1, with \alpha_1(E_1) = \alpha_0. Renormalization group running from thresholds to low energies matches observed values.· A derivation of the fermion mass hierarchy from the overlap of the eigenvector coupling vectors with the Higgs modulation direction. The Yukawa couplings are y_{(1,1,1)} \propto e^{-3\beta}, y_{(2,1,1)} \propto 2e^{-6\beta}, y_{(2,2,1)} \propto 4e^{-9\beta}, producing the mass hierarchy m_t : m_c : m_u \approx 1 : 0.0073 : 0.000052.· A derivation of the Higgs field as the modulation of the emergent spacetime field \Phi_{\mathcal{L}} along a specific direction in the internal coupling space. The Higgs potential V(H) = -\mu^2|H|^2 + \lambda|H|^4 follows from threshold dynamics, with v = \sqrt{\mu^2/\lambda} \approx 246 GeV and m_H \approx 125 GeV.· A derivation of the Standard Model Lagrangian as the unique renormalizable, gauge-invariant Lagrangian consistent with the derived gauge group and particle content. The Lagrangian is presented in full explicit form.· Feynman rules on the discrete voxel lattice with spacing a = \ell_P. All Feynman diagrams are finite because loop integrals are over the compact Brillouin zone. The continuum limit a \to 0 recovers standard Feynman rules, with cutoff corrections suppressed by (p/M_P)^2.· A derivation of the renormalization group from Pillar IV (the feed equation). The meta-time evolution of thresholds becomes the RG flow of couplings, with beta functions \beta_i = \kappa \delta\Gamma/\delta g_i. The Standard Model one-loop beta functions follow from the particle content.· A derivation of structural properties of QFT: spin-statistics from threshold saturation, the path integral as a sum over threshold histories, confinement from voxel chain excitation, and anomaly cancellation from the particle content. Why this matters: In conventional quantum field theory, the gauge group, particle content, coupling constants, masses, Higgs potential, and regulator are empirical inputs. In the canvas model, none of these are assumed. All are derived from eight primitives and four pillars. The derivation is complete. No steps are omitted. The canvas model explains why quantum field theory has the form it does. Keywords: quantum field theory, canvas model, gauge group, Standard Model, three generations, coupling constants, fermion masses, Higgs mechanism, Feynman rules, discrete regulator, renormalization group, spin-statistics, path integral, confinement, anomaly cancellation