Gauge Couplings in Closed Form: Complete Derivation from the Attractor Dynamics and Internal Space Geometry

This paper derives the three gauge couplings of the Standard Model—g_1, g_2, and g_3—in closed form from the primitives of the canvas model. The derivation requires no experimental input and produces exact expressions in terms of \pi and the integers arising from the gauge subspace dimensions.

The derivation proceeds in five stages:

First, the attractor dynamics (Pillar IV) determine the structural couplings between gauge and space fields on the pre-geometric canvas: g_{\text{structural}}^2 = 16/q_s, where q_s = 1,2,3 are the gauge subspace dimensions.

Second, dimensional reduction from the 2D canvas to 4D spacetime converts these structural couplings to physical gauge couplings via a universal constant C that depends on the internal space volume. Self-consistency requires C = g_3^3/5, where 5 = 2+3 is the sum of the SU(2) and SU(3) subspace dimensions—the same integer that determines the CKM Wolfenstein parameter \lambda = 1/5.

Third, the fundamental quadrant integrals over the internal space geometry give the coupling ratios 1 : 2/3 : 2/\pi.

Fourth, the self-consistency equation g_3^2 = (32/\pi) \times (g_3^3/5) yields g_3 = 5\pi/32.

Fifth, the complete closed-form spectrum follows:

g_1^2 = \frac{25\pi^3}{2048} \approx 0.3785, \quadg_2^2 = \frac{25\pi^3}{3072} \approx 0.2524, \quadg_3^2 = \frac{25\pi^2}{1024} \approx 0.2409

The ratios are g_1^2 : g_2^2 : g_3^2 = 1 : 2/3 : 2/\pi, matching the values required by low-energy data at the sub-percent level. The absolute values match the data-required Planck-scale couplings within 0.7\%.

Why this matters:

The gauge couplings of the Standard Model are traditionally free parameters, entered by hand. This paper shows they are absolute constants, determined solely by \pi and the integers \{1,2,3,5\} from the primitives. They do not depend on the cosmological boundary condition H_0, the fundamental coupling \alpha_0, or any experimental input.

The appearance of 5 = 2+3 in both the gauge coupling formula and the CKM matrix is not a coincidence. Both derive from the same gauge subspace dimensions. The integers \{1,2,3\} are the fundamental building blocks: their sum 1+2+3 = 6 determines the Higgs proximity parameter, and their pairwise sum 2+3 = 5 determines both the CKM mixing and the gauge coupling absolute scale.

The gauge sector of the Standard Model is now fully derived. No free parameters remain.

Keywords: gauge coupling unification, canvas model, attractor dynamics, dimensional reduction, internal space geometry, Standard Model, closed-form derivation, CKM matrix, fine structure constants