The PSL(2,7) Algebraic Structure of the Generative Triple Evolution Framework
Nova Spivack · Zenodo (CERN European Organization for Nuclear Research) · 2026
The Generative Triple Evolution (GTE) framework is built on three algebraic pillars: the Fibonacci-Möbius action on P^1(F_7), the order-21 metacyclic group F_21 = Z_7 ⋊ Z_3 (the Borel stabilizer of ∞ in PSL(2,7)), and the Eisenstein integers Z[ω] (ω = e^{2πi/3}, primitive cube root of unity). This paper proves that these three pillars are unified through PSL(2,7), the unique simple group of order 168. We establish four main results, all machine-certified in Lean 4 with zero sorry. First, F_21 is the stabilizer of ∞ ∈ P^1(F_7) in PSL(2,7) (the Borel subgroup of order 21), and together with the Fibonacci-Möbius generator, these two elements generate PGL(2,7) of order 336 (f21_is_borel_psl27, pgl27_generated_by_singer_and_borel). Second, PSL(2,7) is the unique simple group of order 168 and equals Aut(F) for the Fano plane F, and F_21 acts simply transitively on all 21 incidence flags of F (psl27_is_aut_fano, f21_regular_on_fano_flags). Third, the Eisenstein integers yield F_21 and A_4 from the same functor: 7 splits as 7Z[ω] = π π̄ (norm-7 primes), giving Z[ω]/(π) ≅ GF(7) and F_21 from the unit group; 2 is inert (2Z[ω] prime), giving Z[ω]/(2) ≅ GF(4), whose additive group is V_4 ≅ Z_2^2 and whose multiplicative group is Z_3 = μ_3, assembling to A_4 = V_4 ⋊ μ_3 (eisenstein_a4_from_inert_2). Fourth, the Klein quartic — the unique Hurwitz surface saturating 84(g−1) automorphisms for genus g — has genus equal to N_gen = 3 (the number of Standard Model generations), arising from the (2,3,7) triangle group shared with PSL(2,7) (klein_quartic_genus_eq_n_gen). These results establish a coherent group-theoretic skeleton: PSL(2,7) contains F_21 as its Borel subgroup, acts as the automorphism group of the Fano plane (with F_21 as its regular flag-transitive subgroup), and is the automorphism group of the genus-3 Klein quartic. The shared μ_3 = SU(2)_L-cyclic factor links F_21 to A_4 via the Eisenstein functor, and the manifest GTE neutrino flavor symmetries μ_3 and the μ-τ exchange Z_2 generate S_3 ≤ A_4, completing the leading-order neutrino flavor mixing group. The physical interpretation — F_21 as gauge skeleton group, A_4 as the GTE leading-order neutrino flavor symmetry yielding tri-bimaximal mixing angles, and the Klein quartic as the geometric reason for three generations — is argued at the stated confidence levels.