Paper 5 — Active Stabilizers in Reed–Muller CSS Codes from Cayley–Dickson Zero-Divisors

We establish three results connecting the zero-divisor structure of the Cayley–Dickson algebras Aₙ to the stabilizer formalism of quantum error-correcting codes.

Step 1 (Bijection, n=4). The 7 lo-flats of ZD(4) — weight-4 subsets {a,b,c,d} ⊂ {1,...,7} with a⊕b⊕c⊕d=0 — are in explicit bijection with the 7 non-identity elements of the X-stabilizer group of the Steane [[7,1,3]] code. All 7 lo-flats carry identical sign-parity pattern: π(EE|FF)=−1 (inactive), π(EF₁)=π(EF₂)=+1 (active).

Step 2 (Operator translation). For every mixed flat F={i,j,8+k,8+l} in A₄, the algebraic identity π_A·π_B·π_C=−1 translates into the Clifford-group identity (Z_αZ_β)(X_αX_γ)(X_βX_γ) = −Y_αY_β. The global phase −1 is the precise Clifford avatar of π_Aπ_Bπ_C=−1. The operator −Y_αY_β is a detectable error with full-group syndrome weight 2^(n−2), proved for all n≥4 and verified on QPU.

Step 3 (Extension to n=5, verified on QPU). The 15 Fano planes of PG(3,F₂) are in explicit bijection with the 15 non-identity elements of the X-stabilizer group S_X ≅ (Z/2Z)⁴ of the [[15,7,3]] code (4 generators, 2⁴−1=15 non-identity elements). The phase identity and syndrome formula extend to all n≥4 (Universal Phase Lemma). Syndrome structure verified on ibm_kingston (156-qubit QPU, job d8bm67ijki0s73aq6idg, 27 May 2026).

The Type-A/Type-B criterion for Fano planes in PG(3,F₂) is resolved: a plane W is Type-B if and only if 8∈W or W={1,...,7}, where 8=(1,0,0,0)∈F₂⁴ is the bridge element between A₃ and A₄ in the Cayley–Dickson tower.