From 2D to 4D: Dimensional Reduction and the Derivation of the Closed-Form Gauge Couplings of the Standard Model

The canvas model is formulated on a 2D pre-geometric canvas. Spacetime, gauge forces, and matter all emerge from the dynamics of fields on this canvas. But how does a 2D theory produce a 4D world? The answer is dimensional reduction: the ten structural fields on the 2D canvas contain the structure that generates four spacetime dimensions and a 3D internal coupling space. What this paper does: We derive the dimensional reduction explicitly. The 2D canvas action is mapped to the 4D effective action through the voxel lattice. The gauge couplings in 4D are expressed in terms of the structural couplings on the canvas and the geometry of the internal space. The result is a self-consistency equation that determines the gauge couplings uniquely. Combined with the attractor dynamics (Pillar IV) and the fundamental quadrant integrals, we obtain closed-form expressions for the three gauge couplings at the Planck scale: g₁² = 25π³/2048 ≈ 0.378g₂² = 25π³/3072 ≈ 0.252g₃² = 25π²/1024 ≈ 0.241 The ratios are g₁² : g₂² : g₃² = 1 : 2/3 : 2/π, matching the values required by low-energy data at the sub-percent level. Why this matters: The gauge couplings are absolute constants. They depend only on π and the integers {2,3,5} from the primitives. They do not depend on the cosmological boundary condition H₀ or the age of the universe. The normalization constant N = 1/5 is the inverse of the sum of the SU(2) and SU(3) subspace dimensions, the same integer that determines the CKM Wolfenstein parameter λ = 1/5. Both trace to the dimensional structure of the internal space. The dimensional reduction completes the gauge sector of the canvas model. Together with the attractor dynamics and the fundamental quadrant integrals, it provides a complete, parameter-free derivation of the strengths of the fundamental forces. Keywords: dimensional reduction, canvas model, gauge couplings, attractor dynamics, quadrant integrals, Standard Model, Planck scale, internal space geometry, CKM matrix