Eventual negative real-rootedness for fixed-rank paving matroid base polytopes and a growing-rank root-density theorem

Let M be a paving matroid and h_M^*(x) the Ehrhart h-star polynomial of its base polytope. We prove two complementary asymptotic results. First, for every fixed rank r >= 3, all zeros of h_M^* are simple and strictly negative, uniformly over all sufficiently large n-element rank-r paving matroids; no connectivity assumption is imposed. Second, when the rank grows with the ground-set size, we give an explicit regime in which a connected paving matroid has D-o(D) distinct strictly negative zeros, where D=deg h_M^*. The second theorem assumes a mild upper bound on the sizes of effective hyperplanes and does not classify the remaining o(D) zeros. The proof combines the paving hyperplane-correction formula with uniform phase grids, endpoint scaling limits, quantitative spectral gaps, and packing estimates for the hyperplanes. The two theorems concern different asymptotic regimes and neither implies the other. The fixed-rank threshold is not made explicit. The integral proof dossier supplies the complete inequalities, endpoint certificates, and quantifiers behind summarized estimates, with its frozen 216-file supporting closure. The Adiprasito--Zhang uniform-matroid theorem and analytic methods are identified as prior work; no global-first claim or real-rootedness theorem for arbitrary matroid base polytopes is asserted. Status: internally reviewed preprint; external mathematical review and formal peer review pending. The archive includes accepted TeX/PDF artifacts, frozen Markdown, selected proof dependencies, checksums, a portable packaging-integrity checker and reproduction instructions. Historical labels in frozen dependencies are not current public review claims.