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 dividing m and m >= n E_A meets every ordinary stratum of the associated real toric arrangement. The multiplier n is sharp in every dimension n >= 2 for every prescribed E_A >= 2. 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. This is a complete proof of the stated sampling theorem, not a full resolution of the broad Bondal-Thomsen polytope question associated with OWR-14299288-010 in ulamai/UnsolvedMath. The necessary denominator and exact threshold concern ordinary strata, not the coarser line-bundle labels. The preprint is AI-assisted, self-audited and unrefereed. Independent peer review, formal certification, and absolute priority are not claimed.