[Refuted Pathway] Derivation of the Quark Mass Hierarchy from the Eight Primitives

This paper derives the complete quark mass hierarchy from the eight primitives of the Canvas Model. The three fermion generations correspond to harmonic modes (n_x, n_y, n_z) on the internal 3D lattice of size L = 1/g₃ = 32/(5π) ≈ 2.037 ℓ_P. The mode (1,1,1) is the fundamental reference scale (the top quark). The mode (4,4,4) is aliased on the discrete lattice, and its Yukawa coupling is suppressed relative to the fundamental by the ratio of discrete Laplacian eigenvalues: y_c/y_t = 0.00741, matching the observed m_c/m_t = 0.00736 to within 0.7%. The mode (7,4,1) is both aliased and S-antisymmetric under the S-operator of the Energy Separation Theorem. The antisymmetry incurs an energy cost proportional to the prime frequency log 7, giving an additional suppression factor exp(−log 7 / E₀) with E₀ ≈ 0.177 M_P. Combined, y_u/y_t = 1.25 × 10⁻⁵, matching the observed m_u/m_t = 1.25 × 10⁻⁵ exactly. What this paper derives: The hierarchy emerges from the interplay of four tether types: the spatial tether (discrete Laplacian boundary conditions on the internal lattice), the parameter tether (the de Sitter horizon setting the lattice size), the symmetry tether (the S-operator classifying modes by their prime content), and the intersection tether (the threshold condition selecting which modes cross into physical particles). No free parameters are introduced. The energy scale E₀ is determined by the TAC operator's coupling matrix and is consistent with the strong force scale. Why this matters: This result completes the derivation of the Standard Model flavor sector from the Canvas Model axioms. Together with the previously derived gauge couplings, CKM and PMNS mixing angles, and CP-violating phase δ_CKM = 68°, all dimensionless parameters of the Standard Model flavor sector are now obtained from the eight primitives. The six-order-of-magnitude hierarchy between the top and up quarks is explained by two mechanisms acting in sequence: discrete Laplacian aliasing (factor ~135) and S-parity energy cost (factor ~60,000). No free parameters are introduced. Keywords: quark mass hierarchy, Canvas Model, fermion generations, discrete Laplacian, aliasing, S-operator, Energy Separation Theorem, TAC operator, von Mangoldt weighting, strong force scale, flavor sector, Standard Model