The PSL(2,7) Algebraic Structure of the Generative Triple Evolution Framework

The Generative Triple Evolution (GTE) framework is built on three algebraic pillars: the Fibonacci-Möbius action on P¹ (F₇), the order-21 metacyclic group F₂1 = Z₇ ⋊ Z₃ (the Borel stabilizer of ∞ in PSL (2, 7) ), and the Eisenstein integers Zω (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, F₂1 is the stabilizer of ∞ ∈ P¹ (F₇) 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 (f21ᵢsborelₚsl27, pgl27generatedbyₛingerₐndborel). Second, PSL (2, 7) is the unique simple group of order 168 and equals Aut (F) for the Fano plane F, and F₂1 acts simply transitively on all 21 incidence flags of F (psl27ᵢsₐutfano, f21ᵣegularₒnfanoflags). Third, the Eisenstein integers yield F₂1 and A₄ from the same functor: 7 splits as 7Zω = π π̄ (norm-7 primes), giving Zω/ (π) ≅ GF (7) and F₂1 from the unit group; 2 is inert (2Zω prime), giving Zω/ (2) ≅ GF (4), whose additive group is V₄ ≅ Z₂² and whose multiplicative group is Z₃ = μ₃, assembling to A₄ = V₄ ⋊ μ₃ (eisensteinₐ4fromᵢnert₂). Fourth, the Klein quartic — the unique Hurwitz surface saturating 84 (g−1) automorphisms for genus g — has genus equal to Ngen = 3 (the number of Standard Model generations), arising from the (2, 3, 7) triangle group shared with PSL (2, 7) (kleinquarticgenusₑqₙgen). These results establish a coherent group-theoretic skeleton: PSL (2, 7) contains F₂1 as its Borel subgroup, acts as the automorphism group of the Fano plane (with F₂1 as its regular flag-transitive subgroup), and is the automorphism group of the genus-3 Klein quartic. The shared μ₃ = SU (2) L-cyclic factor links F₂1 to A₄ via the Eisenstein functor, and the manifest GTE neutrino flavor symmetries μ₃ and the μ-τ exchange Z₂ generate S₃ ≤ A₄, completing the leading-order neutrino flavor mixing group. The physical interpretation — F₂1 as gauge skeleton group, A₄ 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.