Paper 25 — Fano Block Structure: The Complete Fermionic Chain and One-Loop Yukawa Corrections in S84

This paper closes three open problems in the series on zero-divisors in Cayley–Dickson algebras.

§2 (Fano block structure, OP A'.2): We prove analytically that the three Fano-block indicator vectors aᵢ lie in the kernel of the ZD adjacency matrix A, and compute the exact isotypic split of Level-3 under C₃ symmetry via GAP 4.15.1: c₆ = 8/7, c₈ = 20/7 (ratio 2:5), uniform across all three blocks. The hypothetical 0+4+4 algebraic distribution of the Level-3 irr₈ sector is excluded by a derived-negative result.

§3 (Complete fermionic chain): We collect the four-step chain ρ → irr₈ = Cl(0,7)-spinor → FS(4_s) = +1 → D|_{4_s} as an explicit corollary, attributing each link to its source paper (Papers 7, 24, 19, 23). The Clifford algebra Cl(0,7) acts on the physical sector 4_s of S₈₄.

§4 (One-loop Yukawa correction, OP 23.1): Using the Machacek–Vaughn formalism with Σ = 32 (from Paper 23), we compute the fermion-loop correction to the β-function of S₈₄: Δβ^(f) = 4(λ−4)/π², giving the complete one-loop result β^(total) = (63λ² + 4λ − 16)/π², with an infrared zero at λ_IR ≈ 0.473. The Majorana factor ½ (from FS(4_s) = +1, Paper 19) is applied throughout.

§5 (Spectral projectors, OP 24.2): Via Python computation of the spectral projectors Σ₁ = e₁+…+e₇ and Σ₂ = e₉+…+e₁₅, we show Σ₁ maps to Level-5 (μ = +4) and Σ₂ maps to Level-3 as a purely irr₈ vector (‖P₆v₂‖² = 0, ‖P₈v₂‖² = 84).

§6 (Note on related literature): A 2.5-page systematic survey distinguishes the series from 16 works in the external literature (de Marrais, Bronoff, Potton, BCDI, Gillard–Gresnigt, Wilmot, Aliferis–Leontaris–Vlachos, Okubo, Tang, Dorofeev, Masi, Dzhunushaliev, Penrose–Tod).