The Yukawa Coupling from Threshold Crossing Dynamics: A Complete Derivation from Pillar II

This paper derives the Yukawa coupling function—the origin of fermion masses—from the threshold crossing dynamics (Pillar II) of the canvas model. No experimental mass inputs are required beyond a single normalization constant computed from the top quark mass.

The Standard Model contains nine fermion masses, spanning six orders of magnitude from the electron (0.511 MeV) to the top quark (172.5 GeV). The canvas model explains mass ratios between generations through geometric overlaps and harmonic mode suppression. But the absolute scale of the Yukawa couplings—the overall normalization that sets the top quark mass—has remained an open problem.

The key insight: Closed fermion wave formation is a barrier-crossing process. The fermion and Higgs open waves must overcome an energy threshold T_{fH} to form a bound state. Gauge interactions lower this barrier by providing binding energy E_{\text{binding}}.

For Gaussian field fluctuations at the Planck scale, the barrier-crossing probability—and hence the Yukawa coupling—is:

y_f \propto \exp\!\left(-\frac{(T_0 - E_{\text{binding}}^f)^2}{2\sigma^2}\right)

where T_0 = (1+2+3)/3 = 2 is the bare threshold (the average of the gauge subspace dimensions), E_{\text{binding}}^f is the gauge binding energy computed from Casimir operators, and \sigma \sim 1 is the characteristic fluctuation amplitude.

For E_{\text{binding}} \ll T_0, this reduces to:

\boxed{y_f = y_0 \cdot \exp(T_0 E_{\text{binding}}^f) \cdot g_g^{(f)} \cdot P_g \cdot e^{-\beta\Sigma_g^2}}

The exponential form is derived, not assumed. It replaces earlier Ansätze based on linear intensity addition (I_0 + E_{\text{binding}}) or inverse threshold scaling (1/(T_0 - E_{\text{binding}})).

The binding energies are computed from the gauge Casimir operators using the closed-form gauge couplings at the Planck scale: g_1^2 = 25\pi^3/2048, g_2^2 = 25\pi^3/3072, g_3^2 = 25\pi^2/1024, with ratios g_1^2 : g_2^2 : g_3^2 = 1 : 2/3 : 2/\pi. These are absolute constants derived from the canvas model primitives.

The single normalization constant y_0 is computed by requiring that after renormalization group evolution from M_P to M_Z, the physical top quark mass is m_t = 172.5 GeV. All other fermion masses are then predicted from the ratios of geometric overlaps, harmonic suppressions, and binding energies.

Why this matters:

The derivation reveals that fermion masses are fundamentally gauge phenomena. The bare threshold T_0 = 2 is the same for all fermions. Differences arise from gauge binding energies, geometric overlaps, and harmonic mode suppression. All three factors are derived from the primitives. The single computed parameter y_0 replaces the three independent mass scale calibrations of the standard formulation.

The exponential model is the only one consistent with both the theoretical derivation (Gaussian barrier crossing) and the observed lepton-to-quark mass ratio. It predicts y_0^{(\ell)}/y_0^{(d)} \approx 0.62, within 5\% of the calibrated value—significantly better than the inverse threshold model (1.00) or the linear intensity model (0.83).

Keywords: Yukawa coupling, fermion masses, threshold crossing, Pillar II, canvas model, gauge binding energy, Casimir operators, Gaussian fluctuations, barrier crossing, top quark mass