Paper 24 — The Fano Dictionary: Module Structure of Vᴢᴅ and the Dark Sector Partition
Abstract
The Fano plane PG(2,𝔽₂) organises octonion multiplication in A₃ and zero-divisor incidence in A₄. This paper develops an explicit dictionary between these two roles and determines the PSL(2,7)-module structure of V_ZD = span{e₁,...,e₇, e₉,...,e₁₅} ⊂ A₄.
The main result is V_ZD ≅ 1²⊕irr₆² as PSL(2,7)-modules, established by character theory via the Cayley-Dickson extension. Comparison with Level-2 = irr₆⊕irr₈ shows that the dimensional coincidence dim(Level-2) = dim(V_ZD) = 14 is purely numerical: by Schur's lemma, no PSL(2,7)-equivariant embedding Level-2 ↪ V_ZD exists, since irr₈ has multiplicity zero in V_ZD. Open Problem PT1.6 is thus closed negatively.
The natural algebraic home of irr₈ is identified as the Clifford spinor module of Cl(0,7) via the Clifford representation of the sedenion zero-divisors, converting the negative result into a positive localization.
The Fano dictionary (7-row translation table) maps each Fano-geometric element to its algebraic meaning in A₃ and A₄. The central entry: the same Fano-line incidence that encodes product creation in A₃ encodes product annihilation in A₄. This translates the bosonic/fermionic split of Level-2 into the on-line/off-line partition of PG(2,𝔽₂): the irr₆ bosonic sector (6 modes) maps equivariantly into V_ZD; the irr₈ spinorial sector (8 algebraic / 4 Majorana modes) lies in ker(L_a) and has no equivariant image in V_ZD.
Under the IIB framework (gravity = norm N of A₄), this algebraic partition corresponds to the split between recoverable and irrecoverable dark sector content across the CCC conformal boundary.
How to Cite
@misc{levratti2026fanodict,
author = {Levratti, Giovanni},
title = {The Fano Dictionary: Module Structure of Vᴢᴅ and the Dark Sector Partition},
year = {2026},
publisher = {Zenodo},
doi = {10.5281/zenodo.20681330},
url = {https://doi.org/10.5281/zenodo.20681330},
note = {Paper 24 in the series Zero-Divisors
in Cayley--Dickson Algebras, v1}
}