A Quartic Contact Interaction with Fano–CSS Structure

The exact identity ΔN(a,b) = N(a)N(b) − N(ab) = 4 for every zero-divisor-active pair in the sedenion algebra 𝒜₄ is promoted to a quartic contact interaction ℒ_int = λ·ΔN(φ,ψ) in a scalar field theory valued in 𝒜₄.

The interaction tensor T is computed in closed form. Its non-zero entries decompose under the XOR block structure of 𝒜₄ with a seven-fold Fano labelling indexed by PG(2,𝔽₂). The coupling constant λ is shown to be unique: the automorphism group Aut(𝒜₄) ≅ C₂ × ((C₂³).PSL(3,2)), of order 2688, acts transitively on the 168 unordered zero-divisor pairs, with stabiliser C₂ × D₈ fixing the vertex structure.

The CSS bridge of Paper 8 identifies the seven Fano blocks with the seven Z-stabilisers of the Steane [[7,1,3]] code. The vertex amplitude is computed explicitly; the angular distribution is isotropic and the amplitude coefficient equals 4, both independent of λ and of energy. ZD-active fluctuations are algebraically massless: no mass is generated by ℒ_int alone. Physical consequences are drawn for the dark-sector self-interaction structure and for the single logical qubit that survives the 𝒜₃ → 𝒜₄ threshold. SIDM constraints are discussed in the (λ, E/M) parameter plane.

QPU measurements on ibm_kingston confirm the [[7,1,3]] logical qubit structure at >94σ on both Z_L and X_L operators.

This is Paper 9 of the Cayley–Dickson zero-divisor series.