Paper 7 — A Clifford Representation of the Fano Zero-Divisors

We construct an explicit map ρ from the 84 active zero-divisors of the sedenion algebra A₄ into the Clifford algebra Cliff(0,7) ≅ M₈(ℝ), using the octonion left-multiplication operators L_{e_k} as generators.

The construction rests on the Distinct ε-Index Theorem: every zero-divisor pair (A,B) in ZD(n) has distinct octonion components (proved algebraically via the {0,4,8} Trichotomy and Sign-Parity Criterion). This allows defining ρ(e_p + σ e_{8+k}) := L_{e_p}, a map that depends only on the octonion index p.

Three exact properties are proved:
1. GL(3,𝔽₂)-equivariance: ρ intertwines the action of GL(3,𝔽₂) ≅ PSL(2,7) on zero-divisors with its action on the generators {L_{e_1}, ..., L_{e_7}}.

2. Anti-commutation: {ρ(A), ρ(B)} = 0 for every ZD pair — an immediate consequence of the distinct ε-index theorem and the Clifford anti-commutation relations.

3. Fano Reflection Identity: for every Fano line {p,r,t} with p⊕r⊕t = 0, L_{e_p} L_{e_r} L_{e_t} = ε_{p,r,t} · R_F where R_F = diag(±1) is the reflection fixing the 4-flat {0,p,r,t} and negating its complement. The operator Γ_{prt} is the chirality element of the quaternionic subalgebra ℍ_{prt} ⊂ 𝕆.

A remark identifies the precise gap between these results and a naïve Cliff(8) / Pauli-weight-4 formulation: Cliff(0,7) is the correct and minimal target; the triple product identity gives a Fano reflection, not a scalar; the Pauli-weight condition is not a natural invariant of the zero-divisor structure.

The Pauli decomposition of each L_{e_k} on 3 qubits is given explicitly. The X-part of each decomposition recovers the binary representation of k in 𝔽₂³, giving a canonical bijection between the 7 octonion generators and the 7 points of PG(2,𝔽₂). The triple products under single-Pauli selections realise the projective duality of PG(2,𝔽₂), connecting to the CSS stabilizer codes of the companion paper (doi:10.5281/zenodo.20431754).