Real-Rooted h-Star Polynomials of Rank-Two Matroids with Parallel Classes of Size at Most Three

Let M be a rank-two matroid. This manuscript proves that if every nonloop parallel class of M has size at most three, then the Ehrhart h-star polynomial of its base polytope has only real zeros; consequently its coefficient sequence is log-concave and unimodal. The result advances the rank-two sparse-paving boundary of Ferroni, Jochemko and Schroeter from parallel-class size two to size three. A uniform analytic estimate handles every ground-set size n at least 10, while an exact Sturm certificate and finite verification settle the complete smaller connected and disconnected boundary. The manuscript does not claim real-rootedness for arbitrary rank-two matroids or all matroid base polytopes. Status: Public Beta; internally verified manuscript; external mathematical review pending.