Paper 29 — The Missing Four: Fano Non-Incidence in the Sedenion Zero-Divisor Algebra

Paper 29 of the series Zero-Divisors in Cayley–Dickson Algebras.

We prove that the kernel ker(L_a) of left multiplication by any sedenion zero-divisor a ∈ ZD*(A₄) is isomorphic, as a Stab_{G₈₄}(a)-module, to the Fano non-incidence subspace V_ni(a) — the span of imaginary octonion units whose lo₃-class does not lie on the Fano line determined by a (Theorem 3.1, [Derived, GAP 4.15.1 + analytic]).

This identifies the algebraic origin of the spectral invariant S_eff = 164 = |GL(3,F₂)| − 4 in the induced gravity formula of Paper 26: the four missing modes are the Fano non-incidence directions, not a numerical coincidence. Paper 28 Observation 6.4 is promoted from [Observation] to [Derived].

Secondary results: the stabilizer Stab_{G₈₄}(a) acts irreducibly on ker(L_a) (OP 23.4 closed); the G₈₄-module structure of Level-3 is 6 ⊕ 21 ⊕ 7 ⊕ 8 over GF(5) (OP 25.1 partial); the sign vector of Σ₂ generates the 21-dimensional G₈₄-irreducible (OP 25.3 closed). The explicit vertex bipartition of each 12-vertex component of ZD*(A₄) is recorded. The question of whether the 14 massless Level-2 modes admit a Goldstone interpretation is formulated as OP 29.1.

GAP 4.15.1 scripts included. All results carry explicit epistemic labels. (on Zenodo)