Canonical Lattices and Integer Relations Associated to Rational Fans

Abstract: We propose a canonical local-to-global lattice theory for rational fans. We define the $\textit{ray lattice } L_{\mathrm{rays}}(Σ)$ and the $\textit{relation lattice } L_{\mathrm{rel}}(Σ)$ as invariants functorial under fan isomorphisms. We introduce $\textit{star-local relation lattices}$, defined via the relation lattice of the localized quotient fan, which capture the linear dependencies visible within local neighborhoods. We define a $\textit{codimension filtration}$ 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 $Σ$; explicit examples and a conjecture suggest that subdivisions can only preserve or lower filtration depths, distinguishing fans with different combinatorial topologies.