Paper 8 — Fano Duality, Projective Orthogonality and the CSS Bridge

Paper 7 of this series constructed an explicit Clifford representation ρ: ZD*(4) → Cliff(0,7) and observed, via exhaustive computation, that the triple-product identity ρ̃(e_p)ρ̃(e_r)ρ̃(e_t) = φ · Q_{p,r,t} defines a bijection from the 7 Fano lines to the 7 non-zero elements of {I,Z}^⊗3, coinciding with the projective duality of PG(2,𝔽₂). That paper left open whether this duality is the X/Z duality of the Steane [[7,1,3]] code.

This paper closes the gap and proves the following.
Main Theorem (CSS Bridge, n=4). For every Fano line {p,r,t}, the Z-string Q_{p,r,t} equals the projective orthogonal complement of {p,r,t} in 𝔽₂³, and coincides with the unique Z-stabiliser of the Steane [[7,1,3]] code whose support is the complement {1,...,7} \ {p,r,t}. The correspondence is equivariant for the contragredient action of GL(3,𝔽₂): lines transform covariantly, their dual Z-stabilisers transform contravariantly via g^{-T}.

Theorem (n=5 CSS Bridge). The analogue holds for n=5: every hyperplane W of PG(3,𝔽₂) maps to the unique Z-stabiliser of the [[15,7,3]] Reed–Muller code with support equal to the complement of W. Here M = I₄ (no bit-reversal needed).

Proposition (M = I_{n-1} for all n ≥ 2). Under the MSB-first generator convention, the CSS bridge S_Z(v_W) = {1,...,2^{n-1}-1} \ W holds analytically for all n ≥ 2, with M = I_{n-1}.

As a corollary, the sign-parity constraint π_A π_B π_C = −1 (proved in the series anchor paper) acquires a unified three-way interpretation: algebraically as a sign-parity discriminant, quantum-error-correcting as the phase of a balanced ZD triple, and projective-geometrically as the orthogonality relation between a Fano line and its dual point in PG(2,𝔽₂).

A second corollary makes precise why no Clifford representation can lift the full octonion multiplication to a group algebra: the 28 non-Fano triples do not satisfy the cocycle condition, and no Z-type projective dual can be assigned to them.