The Yukawa Coupling from Threshold Crossing Dynamics: A Complete Derivation from First Principles

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 the mass ratios between generations through geometric overlaps and harmonic mode suppression. The absolute scale of the Yukawa couplings has been treated as a single overall normalization, computed from the top quark mass through numerical renormalization group evolution.

This paper completes the derivation. The overall Yukawa normalization y_0 is not a free parameter. It is determined by the total overlap integral of all fermion modes on the internal 3D lattice with the Gaussian Higgs profile, multiplied by the baseline threshold crossing probability \exp(-T_0^2/2) = e^{-2}. The sum over modes is fixed by the internal lattice geometry (L = 1/g_3, \sigma_H = 1/2). No experimental mass input is required.

The Yukawa coupling function is:

\boxed{y_f(M_P) = e^{-2} \left(\sum_{\mathbf{n}} \mathcal{I}_{3D}(\mathbf{n})\right) \cdot \exp(T_0 E_{\text{binding}}^f) \cdot g_g^{(f)} \cdot P_g \cdot e^{-\beta\Sigma_g^2}}

Every factor on the right-hand side is derived from the primitives of the canvas model:

· e^{-2} is the baseline threshold crossing probability, with T_0 = (1+2+3)/3 = 2· \sum_{\mathbf{n}} \mathcal{I}_{3D}(\mathbf{n}) \approx 0.1172 is the total overlap integral over all modes on the internal lattice (L = 1/g_3, \sigma_H = 1/2)· \exp(T_0 E_{\text{binding}}^f) is the gauge enhancement, with binding energies computed from the closed-form gauge couplings g_1^2 = 25\pi^3/2048, g_2^2 = 25\pi^3/3072, g_3^2 = 25\pi^2/1024· g_g^{(f)}, P_g, \beta, and \Sigma_g^2 are the geometric and harmonic factors

The top quark mass is a prediction, not an input. The framework has zero free parameters in the Yukawa sector. Together with the gauge coupling derivation, the Standard Model's gauge and Yukawa couplings are now fully derived from the eight primitives and the Four Pillars.

Why this matters:

The Yukawa couplings of the Standard Model are traditionally free parameters, entered by hand. This paper shows they are determined by the internal lattice geometry and the threshold crossing dynamics. The absolute normalization y_0 \approx 0.01586 is computed from first principles, eliminating the single calibration parameter that remained in earlier formulations. The derivation is complete. No free parameters remain.

Keywords: Yukawa coupling, fermion masses, threshold crossing, canvas model, gauge binding energy, Casimir operators, Gaussian fluctuations, barrier crossing, top quark mass, internal lattice, harmonic modes