Three family sectors from the three-fold cover of the Fano torus: a lattice computation

In magnetised-torus and intersecting-brane models the number of chiral families is an index: the flux felt by a fermion on a two-torus. This note computes the analogue for the nearest-neighbour honeycomb lattice and asks whether the number three can have a geometric origin on the lattice.Flux index: on the honeycomb torus, N flux quanta give exactly 2|N| zero modes, |N| per valley. The sublattice index vanishes, and all modes share one valley-weighted chirality (computed for N ≤ 6). Three chiral modes are reachable only by removing a valley, and the number is then the flux, an input. Strain produces no net index.Fano torus and its cover: the seven-hexagon honeycomb torus, whose two vertex classes are the points and lines of the Fano plane, carries no Dirac points with periodic boundary conditions. Its unique smallest cover that does has degree three [proved].A degree-three cover turns the minimal flux quantum on the base into three flux quanta on the cover.The deck translation is a symmetry exactly for lifted fluxes.The cover is the union of three copies of the base with ℤ₃ holonomies [proved].Three sectors times a valley doublet: the six zero modes on the cover factorise, to machine precision, as ℂ³ ⊗ ℂ². The doublet is the valley pair, acted on by the chiral SU(2) of the sublattice and Kekulé masses. The family factor carries the regular representation of the deck group. With a valley-selective mass, exactly three chiral modes remain, one per deck sector.S₃ structure: the inversion of the torus conjugates the deck translation, so the deck group extends to S₃ and the deck sectors carry 𝟏 ⊕ 𝟐 [proved]. This explains the 1 + 2 pattern on the literal 21-hexagon torus.Masses: masses of the chiral SU(2) cannot distinguish the sectors; a family-dependent mass must break the deck symmetry. A single-site defect is of rank one and shifts one of the six modes.The number three is the covering degree for one flux quantum on the base, the minimal non-trivial sector. That this sector is selected, that the three sectors are the Standard-Model families, and their four-dimensional chiral reading are not derived. The sectors are degenerate, and no mass hierarchy is obtained. Every statement is graded as proved, computed or argued.