The Absolute Yukawa Scale from First Principles: Derivation of 𝒩 in the Canvas Model

The Canvas Model derives the Standard Model from eight primitives and four dynamical pillars. All dimensionless parameters—gauge couplings, mixing angles, CP phase, and mass ratios—have been obtained from the axioms. The overall Yukawa normalization 𝒩, which sets the absolute scale of fermion masses, has remained the single calibrated parameter in the framework. What this paper does: This paper derives 𝒩 from first principles. The derivation parallels the gauge coupling result g₃ = 5π/32, proceeding through four stages: identification of the structural coupling on the 2D canvas, computation of the fermionic quadrant integral, dimensional reduction from 2D to 4D, and application of prime-structure corrections from the von Mangoldt-weighted spectral energy functional. The structural Yukawa coupling is y_struct² = 16/6 = 8/3, the analog of g_struct² = 16/q_s with the Yukawa charge being the sum of all gauge subspace dimensions Σᵢ 𝒯ᵢ = 6. The fermionic quadrant integral is I_f = 2/π², reflecting the two-fermion nature of the Yukawa vertex. The dimensional reduction factor is 1/6, the inverse of the total gauge subspace dimension sum. The S-parity energy scale provides a factor of 2. The prime zeta correction ζ(2)_prime = π²/8 arises from the spectral sum over prime squares. What this paper derives: The closed-form result is: 𝒩 = 10π / (3√3) ≈ 6.046 Applying the discrete Laplacian eigenvalue correction for the fundamental mode (1,1,1) on the internal lattice of size L = 32/(5π) ≈ 2.037 ℓ_P yields the physical Yukawa normalization 𝒩_phys ≈ 3.3, which gives y_t(M_P) ≈ 0.5 and the observed top quark mass m_t = 172.5 GeV after renormalization group evolution. Why this matters: With this result, the Canvas Model has zero free dimensionless parameters. Every parameter of the Standard Model—gauge couplings, mixing angles, CP phase, mass ratios, and the absolute Yukawa scale—is derived from the eight primitives. The framework is complete. Keywords: Yukawa normalization, Canvas Model, fermion masses, top quark, gauge coupling analogy, dimensional reduction, quadrant integral, von Mangoldt weighting, prime zeta function, Standard Model, zero free parameters