Canonical Lattices and Integer Relations Associated to Rational Fans

We propose a canonical local-to-global lattice theory for rational fans. We define the $\mathit{\text{ray}}\mathit{\text{lattice}}{L}_{\mathrm{r}\mathrm{a}\mathrm{y}\mathrm{s}}(\mathrm{\Sigma })$ and the $\mathit{\text{relation}}\mathit{\text{lattice}}{L}_{\mathrm{r}\mathrm{e}\mathrm{l}}(\mathrm{\Sigma })$ as invariants functorial under fan isomorphisms. We introduce $\mathit{\text{star-local}}\mathit{\text{relation}}\mathit{\text{lattices}}$, defined via the relation lattice of the localized quotient fan, which capture the linear dependencies visible within local neighborhoods. We define a $\mathit{\text{codimension}}\mathit{\text{filtration}}$ on the global relation lattice and prove a generation theorem: the global lattice is generated by local relations supported on the stars of cones of codimension at least 1. This filtration is sensitive to the facial structure of $\mathrm{\Sigma }$; explicit examples and a conjecture suggest that subdivisions can only preserve or lower filtration depths, distinguishing fans with different combinatorial topologies.