Magnitude and motivic zeta functions of matroids

We prove that the magnitude of the tope graph of a simple oriented matroid is the specialization at $x=-1$ of the motivic zeta function of its underlying matroid. This refines the Las Vergnas–Zaslavsky theorem and defines magnitude for arbitrary matroids. For simple orientable matroids, we compute the order of the pole at $q=-1$ from chains of flats and identify it with the Varchenko–Gelfand degree of the tope parity function. Comparing pole orders, we construct a rank-six real arrangement $\mathcal{A}$ such that $\mathrm{Mag}(\mathcal{A};-t)$ has infinitely many negative coefficients, disproving Koizumi–Liu’s eventual sign alternation conjecture. We then construct a canonical multiplicative Varchenko–Gelfand filtration on the mod-$2$ magnitude cohomology of a simple oriented matroid and show that its graded dimensions recover the full motivic zeta function.