The UGP/GTE arithmetic programme derives Standard Model structure from GF (7) and the group F₂₁ = Z₇ ⋊ Z₃, 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 ⊂ F₇* 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 (eₓ eᵧ) = y-x ∈ D read directly from the difference set; (ii) Nc = 3 from the pencil count of the (7, 3, 1) design via F₂₁ ↪ G₂ and Stab₆䃒 (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) = g₂ (dimension 14 exact) and Stab₆䃒 (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 gen₁ ↔ V via Eisenstein norm uniqueness b₆₄₍䃑 = 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 (Nc, charge spectrum, Ngen = 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 bₑ, ₁ = 5 < bₑ, ₂ = 11 < bₑ, ₃ = 19 pins the mass eigenstates via m⏜, ₊ ∝ bₑ, ₊^29/9 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).
Nova Spivack (Sun,) studied this question.