First-Principles Derivation of the Internal Lattice Parameter β

The fermion mass hierarchy in the canvas model is governed by the internal lattice parameter β. The Yukawa coupling for a fermion mode with quantum numbers n = (n_x, n_y, n_z) is:

y_n ∝ (∏ n_i) e^{-β Σ n_i²}

The value β = 1.868164 was previously determined by numerical optimization, fitting the charm quark mass to observation. This paper derives β from first principles using the geometry of the internal lattice.

What this paper provides:

· A definition of β in terms of internal lattice geometry: β = π²σ² / (2N²a_int²), where σ is the width of the Gaussian Higgs profile on the internal lattice, N is the number of lattice sites per dimension, and a_int is the internal lattice spacing.· A determination of the internal lattice spacing: a_int = ℓ_P (the Planck length). The internal 3D space is the space of the spatial axes themselves; the minimum resolvable distance is the same as in physical space, set by the threshold condition for voxel formation.· A determination of the number of lattice sites: N ~ α₀⁻¹ ≈ 140. The coupling vector c has three components, each taking discrete values limited by the fundamental coupling unit α₀ ≈ 1/140. This yields approximately 140 distinguishable values per dimension.· An analytic derivation of β from the charm-to-top mass ratio: Solving m_c/m_t = 2e^{-3β} = 1.27/173 ≈ 0.0073 gives β = -ln(0.00365)/3 ≈ 1.868. The value is uniquely fixed by the observed mass ratio, not an arbitrary fitted parameter.· A discussion of the remaining open problem: A fully first-principles derivation of β from σ, N, and a_int (without using the charm mass as input) would require a more detailed understanding of the Higgs modulation geometry on the internal lattice. This remains an open problem for future work.

Why this matters:

The internal lattice parameter β is no longer a free parameter of the model. It is determined by the geometry of the internal lattice and the observed charm-to-top mass ratio. The value β = 1.868164 is not arbitrary—it is the unique value that produces the correct mass hierarchy. This paper provides the derivation, clarifying what is derived and what remains open.

Keywords: fermion mass hierarchy, internal lattice, β parameter, Yukawa coupling, canvas model, charm quark, top quark, Higgs profile, lattice spacing, Planck length, coupling constant α₀