What question did this study set out to answer?

The aim is to prove that two distinct arithmetic frameworks derive analogous Standard Model structures from the same finite set.

July 7, 2026Open Access

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

Key Points

The aim is to prove that two distinct arithmetic frameworks derive analogous Standard Model structures from the same finite set.
Derived structures from quadratic-residue set QR(7) via machine-verified calculations.
Used oriented translates of input set for octonion multiplication table representation.
Established a derivation chain through various algebraic and group theoretic processes.
Confirmed that two derivation paths yield the same Standard Model features: Nc = 3, charge spectrum, and three generations.
Validated the neutrino sector predictions, including mass eigenstates and normal ordering within expected ranges.
All claims certified at high levels of computational validation, supporting their structural relevance.

Abstract

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).

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