Gauge Couplings from the Attractor Dynamics: A Complete Derivation from the Canvas Model Primitives

This paper derives the three gauge couplings of the Standard Model—the strengths of the electromagnetic, weak, and strong forces—completely from the primitives of the canvas model. No experimental gauge coupling data are used as inputs.

The derivation proceeds in three 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. The value 16 reflects the four dynamic primitives of the canvas model.

Second, dimensional reduction from the 2D canvas to 4D spacetime converts these structural couplings to physical gauge couplings via a factor q_s \times I(q_s), where I(q_s) are definite integrals over the fundamental angular domain [0,\pi/2]. The q_s factors cancel, leaving the physical couplings proportional to I(q_s) alone.

Third, the absolute scale is fixed by the sum rule \alpha_1 + \alpha_2 + \alpha_3 = 4(\pi/2)^2 \alpha_0 = \pi^2 \alpha_0, where \alpha_0 = 1/\ln(I_{\text{max}}) \approx 1/140 is the fundamental coupling from the horizon information bound. This sum rule reflects the geometry of the internal space: four spacetime dimensions, each with a fundamental angular domain of length \pi/2.

The results are:

g_1^2 : g_2^2 : g_3^2 = 1 : \frac{2}{3} : \frac{2}{\pi}, \qquadg_1^2 = \frac{4\pi^3 \alpha_0}{1 + \frac{2}{3} + \frac{2}{\pi}} \approx 0.385

at the Planck scale. The ratios match the values required by low-energy data at the sub-percent level: g_2^2/g_1^2 = 2/3 versus the data-required 0.672 (0.7% difference), and g_3^2/g_1^2 = 2/\pi versus 0.635 (0.3% difference).

Why this matters:

The three gauge couplings of the Standard Model are traditionally free parameters, measured and entered by hand. This paper shows they are consequences of the attractor dynamics, the geometry of the internal 3D space, and the information capacity of the observable universe. The gauge couplings depend on the cosmological boundary condition H_0 through \alpha_0, making them time-dependent—a generic prediction of the canvas model.

The ratios have a clear geometric origin: 1 for U(1) from the integral of \sin\theta, 2/3 for SU(2) from the integral of \sin^3\theta, and 2/\pi for SU(3) from the average of \sin\theta over the fundamental quadrant. The rational number 2/3 and the transcendental number 2/\pi reflect the interplay between discrete group theory and continuous geometry.

This paper reduces the number of free parameters in physics by three. The gauge couplings are no longer inputs—they are predictions.

Keywords: gauge couplings, attractor dynamics, canvas model, dimensional reduction, internal space geometry, Standard Model, fine-structure constants, unification