Paper 22b — Fano Incidence and the Level-3 Tripling
Abstract
The zero-divisor pair graph ZD*(A₄) of the sedenion algebra carries a 84×84 adjacency matrix A whose spectrum consists of five eigenvalues {-4, -2, 0, +2, +4} with multiplicities {7, 14, 42, 14, 7}. The multiplicity-42 eigenspace (Level-3) is exactly three times the multiplicity-14 eigenspace (Level-2).
This paper gives the algebraic origin of that factor of 3: the Fano plane decomposes ZD*(A₄) into seven Fano blocks B₁,…,B₇, and each block contributes one copy of the Level-2 PSL(2,7)-module to Level-3, forcing the isomorphism Level-3 ≅ 3·Level-2 as PSL(2,7)-representations.
A unified analysis of the four occurrences of the factor 3 in the series shows that all three internal occurrences (orbit structure of Aut(A₄), spinorial sector tension, Level-3 tripling) share this common Fano-incidence origin, while the Furey–Cl(6) occurrence is independent.
As a structural corollary, verified by GAP 4.15.1: the automorphism group G₈₄ of ZD*(A₄) has order 2688 and PSL(2,7) is not a subgroup of G₈₄ in its action on the 84 vertices, though it acts on the lo₃-classes via Aut(A₄).
How to Cite
@misc{levratti2026fanoincidence,
author = {Levratti, Giovanni},
title = {Fano Incidence and the Level-3 Tripling},
year = {2026},
publisher = {Zenodo},
doi = {10.5281/zenodo.20648207},
url = {https://doi.org/10.5281/zenodo.20648207},
note = {Paper 22b in the series Zero-Divisors
in Cayley--Dickson Algebras, v1}
}