The Internal Lattice Parameter β from First Principles: Exact Overlap Integral with Framework C Gauge Couplings

The internal lattice parameter β governs the exponential suppression of Yukawa couplings for higher harmonic modes in the canvas model. It is a critical input to the fermion mass spectrum: the mass hierarchy between generations is exponentially sensitive to β through the factor e^(-2βΣ²). Previous work determined β = 0.171 by fitting the harmonic formula to the exact three-wave overlap integral over all 1,176 integer modes with Σ² ≤ 100, using the internal lattice size L = 2.74 ℓ_P derived from Framework A gauge couplings (g₃² = 0.133). The gauge couplings have since been derived in closed form (Framework C): g₃² = 25π²/1024, giving g₃ = 5π/32 and L = 1/g₃ = 32/(5π) ≈ 2.037 ℓ_P. What this paper derives: We recompute β using the correct lattice size. The exact overlap integral with the Gaussian Higgs profile is evaluated analytically in terms of the imaginary error function erfi(z). The ratio of the (4,4,4) mode to the (1,1,1) mode yields: β = 0.167 ± 0.005 The uncertainty is dominated by the ±1% theoretical uncertainty in the confinement scale L = 1/g₃. The value β = 0.171 used in the mass calculations differs from the first-principles value by 2.3%, which is absorbed by a 1.5% adjustment in the confinement length scale—well within the expected O(1) uncertainty of the strong-coupling confinement criterion. Why this matters: The derivation confirms that β is not a free parameter. It is determined by the strong coupling g₃ (derived from the attractor dynamics) and the Higgs width σ = 1/2 (derived from the spacetime threshold T_ST = 4). Both inputs are fixed by the primitives. The fermion mass hierarchy is a consequence of the same gauge structure that determines the strengths of the fundamental forces. Keywords: internal lattice parameter, Yukawa couplings, fermion masses, gauge coupling unification, canvas model, overlap integral, imaginary error function, strong coupling