Prism-Slice h-star Polynomials: Interlacing Through Reduced Level Five and a Level-Six Counterexample

Abstract

Let c be a positive integer capacity vector with total capacity C, let 1 ≤ k ≤ C − 1, and put m = min{k, C − k}. We prove both clauses of the prism-slice conjecture when m ≤ 5: the h-star polynomial is real-rooted, and for every legal unit split the unsplit polynomial weakly interlaces the split polynomial in the directed sense used in the paper. Hence the hypersimplex h-star polynomial is real-rooted when its rank or corank is at most five. At m = 6 and C = 12, the unit split (2,1^10) → (1^12) is an exact counterexample to universal unit-split interlacing at its first possible reduced level and minimum total capacity. Both polynomials remain real-rooted, and the general real-rootedness question remains open. The proof combines exact finite calculations, certified interval arithmetic, and analytic estimates. The deposited files include the paper, arXiv source, and a frozen reproducibility supplement with complete manifests.