Mod 2 magnitude cohomology ring of real hyperplane arrangements

Let $\mathcal{A}$ be a finite central real hyperplane arrangement and let $\mathcal{G}(\mathcal{A})$ be its tope graph. Koizumi recently proved that the crossing-graded magnitude homology of $\mathcal{G}(\mathcal{A})$ is torsion-free and is freely indexed by face flags. We refine his result by proving that a fixed crossing vector and terminal chamber determine a summand of rank at most one. Over ${\mathbb{F}}_2$, we use the canonical cohomology basis to determine the crossing-graded magnitude cohomology ring of $\mathcal{G}(\mathcal{A})$.