Modular Mathematics in the Emergence Canvas Model: Primitives, Derivations, and Results

This paper presents a consolidated treatment of modular mathematics within the Emergence Canvas Model. The central insight is that numerical factors in the Standard Model Lagrangian and Feynman rules decompose into products of contributions from eight primitive concepts. The paper formalizes the modular composition rule and demonstrates its application to the gauge sector, Higgs sector, flavor sector, strong CP, and Feynman rules. The paper distinguishes three categories of results. Category I contains results rigorously derived from the twelve postulates with no additional assumptions. Category II contains standard QFT results that operate within the Canvas framework. Category III contains interpretations that are physically motivated but not yet mathematically derived. Every derivation is shown with labeled steps. All parameter classifications (Type M, M*, S, D) are explicitly stated. The derived results include: · The d'Alembertian from the UWE and voxel lattice· \hbar = M_P \ell_P c from the fundamental amplitude scale· The gauge groups U(1), SU(2), SU(3) from the Charge primitive· The mass formula m \propto 1/T from the threshold condition· The 1/4 factor in the gauge kinetic term from P1 × P3· The gauge coupling ratios g_1^2 : g_2^2 : g_3^2 = 1 : 2/3 : 2/\pi with closed-form expressions g_3 = 5\pi/32, g_2^2 = 25\pi^3/3072, g_1^2 = 25\pi^3/2048· The CKM Wolfenstein parameter \lambda = 1/5 from the subspace sum 2+3· The CKM hierarchy |V_{us}| \sim \lambda, |V_{cb}| \sim \lambda^2, |V_{ub}| \sim \lambda^3/3· The PMNS mixing angles \theta_{12} \approx 33.6^\circ and \theta_{23} = 45^\circ, with \theta_{13} = \arcsin(1/(3\sqrt{5})) \approx 8.57^\circ as a candidate· The harmonic suppression parameter \beta = 0.167 \pm 0.005 from exact overlap integrals· The Higgs quartic coupling at the Planck scale \lambda(M_P) = 625\pi^4/301,989,888 \approx 0.0002016· The strong CP angle \theta_{\text{QCD}} = 0 from lattice topology and horizon topology· The i\epsilon prescription from causal ordering (P1)· Symmetry factors from combinatorial multiplicity (Pillar II)· The modular decomposition of propagator, vertex, and loop Feynman rules Why this matters: The Standard Model contains 19+ free parameters. The Canvas Model reduces these to zero free dimensionless parameters in the Machine, one dimensionful parameter (the Planck scale), and two cosmological boundary conditions. This paper shows how every numerical factor in the Lagrangian and Feynman rules decomposes into products of primitive contributions. The modular composition rule provides a unified framework for understanding the origin of numerical factors across all sectors of physics. Open problems are identified and honestly acknowledged, including U(1) normalization, CKM CP phase, effective Higgs direction angles, RG evolution to the electroweak scale, full seesaw diagonalization, cosmological parameter derivation, black hole entropy, and quantum foundations. These are marked as research directions, not derived results. Keywords: canvas model, modular mathematics, gauge coupling unification, CKM matrix, PMNS matrix, strong CP problem, Higgs sector, Feynman rules, Standard Model, emergence, primitive decomposition