The Octonionic Shadow of GF(7): Color, Chirality, and Three Generations from a Quadratic-Residue Difference Set

The UGP/GTE arithmetic programme derives Standard Model structure from GF(7) and the group F₂₁ = Z_7 ⋊ Z_3, with major claims machine-certified in Lean 4. The division-algebra programme (Dixon; Furey; Baez--Huerta) independently derives one-generation Standard Model structure from the octonions. This paper proves the two programmes are facets of a single finite structure: the quadratic-residue set QR(7) = {1,2,4} ⊂ F_7* is simultaneously the arithmetic anchor of GTE and the Fano-plane orientation that defines the octonion algebra. Starting from D = {1,2,4} as the sole input, we establish a six-link machine-verified derivation chain: (i) the oriented translates of D are the octonion multiplication table, uniquely up to relabeling, with the sign encoding sign(e_x e_y) = [y-x ∈ D] read directly from the difference set; (ii) N_c = 3 from the pencil count of the (7,3,1) design via F₂₁ ↪ G_2 and StabG_2(apex) = SU(3); (iii) the exceptional isomorphism PSL(2,7) ≅ GL(3,2), order 168, certified by Todd-Coxeter coset enumeration; (iv) the derivation algebra der(O) = g_2 (dimension 14 exact) and StabG_2(apex) = su(3) (dimension 8 exact), with bracket closure and Killing-form certificate computed from first principles (39 Lean theorems); (v) UGP's flavor package is equivariantly isomorphic to the Spin(8) triality package, pinning gen_1 ↔ V via Eisenstein norm uniqueness bgen_1 = 73 = N(9+ω); and (vi) the charged-lepton Koide mass ladder is a function on the triality orbit with torsor-invariant angle |δ mod 2π/3| = 0.222229631 = 2/9 + 7.4×10⁻⁶ (7 ppm from PDG). Two structurally independent derivation paths — finite arithmetic (GTE) and continuous division algebra — converge on identical outputs (N_c, charge spectrum, N_gen = 3); this convergence constitutes non-trivial structural evidence that these quantities are genuine consequences of the GF(7) arithmetic rather than numerical accidents. The neutrino sector is completed by the GTE seesaw: the right-handed seed ordering bR,1 = 5 < bR,2 = 11 < bR,3 = 19 pins the mass eigenstates via mν,k ∝ bR,k29/9 and predicts normal ordering, with Δm²₂₁ and Δm²₃₁ within 1% of NuFIT 6.0 IC24 NH. All major claims are certified at level CatAL (zero sorry in Lean 4) or CatA (computation-verified by assertion-checked Python scripts).