A Sharp Sampling Bound for Primitive Real Toric Arrangements

Let \(A\) be a finite spanning collection of primitive integral normals in dimension \(n\), and let \(E_A\) be the least common multiple of the largest Smith invariant factors of its nonsingular full-rank submatrices. If \(E_A>1\), every grid of denominator \(m\) with \(E_A\mid m\) and \(m\ge nE_A\) meets every ordinary stratum of the associated real toric arrangement. The multiplier \(n\) is sharp in every dimension \(n\ge2\) for every prescribed \(E_A\ge2\). For arbitrary integral normals, the successful denominators are exactly \(E_A\) times the integers at least the maximum codegree of the scaled stratum closures. The proof combines the interior-lattice-point framework of Hanlon and Painter with the standard Smith lcm period; primitivity excludes unimodular simplex closures through divisibility of facet constants. A sufficient bound for realizing all Bondal-Thomsen summands follows for smooth toric varieties with no torus factor.