Paper 41 — Fano: The First Stone

We prove that the Fano plane PG(2,F₂) is the first stone of the Cayley–Dickson zero-divisor series: a single combinatorial object that organises the series at five simultaneous structural levels — combinatorial, algebraic, group-theoretic, spectral, and physical.

The central result is a canonical bijection between the four simple F₂[GL(3,F₂)]-modules and the four physical sectors of the series:

F₂ (dim 1) = irr₁ = gravitational singlet
V₃ (dim 3) = irr₃ = gauge triplet
V₃' (dim 3) = irr₃' = fermionic shadow
St (dim 8) = irr₈ = 4_s = Majorana spinors (Steinberg module)

[Derived, GAP 4.15.1]

This identification was not stated in any of Papers 1–40. It provides a uniform representation-theoretic explanation for five properties of 4_s established independently across twenty papers: the Frobenius–Schur indicator FS(4_s)=+1 (Paper 19), Majorana isolation (Paper 32), dim(4_s)=8=dim(𝕆) (Papers 7, 24), absence from the gravitational and gauge sectors (Papers 32, 38), and the matter-gravity split (Paper 38). All follow from the fact that the Steinberg module is projective-injective for any group of Lie type in its defining characteristic.

The correspondence is stable on the full symmetry group G₈₄ of order 2688 [Derived, GAP 4.15.1]. A census of 61 distinct Fano appearances across the series is assembled and organised into five structural levels. The X-stabilizer space S_X of the Steane [[7,1,3]] CSS code and the kernel ker(L_a) share the same simple F₂[S₄]-constituents [Derived, GAP 4.15.1], providing an algebraic bridge between the quantum error correction structure of Papers 5–8 and the spectral structure of Papers 37–40. The sign-parity asymmetry of the S84-equivariant basis is resolved [Derived, computational].

Four GAP 4.15.1 scripts are deposited as supplementary files. (on Zenodo)