Basis density of matroids with circumference at most four

We determine the asymptotically maximum number of bases of a rank-r matroid with no U(4,5)-minor, for every fixed r >= 4. Writing r = 3q + t, where 0 <= t <= 2, the maximum basis density is r! A_r/r^r, with A_(3q) = (9/2)^q, A_(3q+1) = (128/27)(9/2)^(q-1), and A_(3q+2) = 2(9/2)^q. In particular, the rank-four density is 4/9. For ranks congruent to one modulo three, the construction uses a connected rank-four book with its common point deleted. We also determine the fixed-rank leading term with error O_r(n^(r-1)) and the connected-matroid density. Three copies of AG(3,2) disprove a proposed finite extremal bound of van der Pol, Walsh and Wigal. The proof combines published small-circumference classifications with sharp weighted basis counting and optimization over component ranks. This is a preprint, not a peer-reviewed journal publication. The accompanying reproducibility archive contains LaTeX sources, standard-library Python verification code (Python 3.10+), exact-arithmetic output, and reproduction instructions. The first public release incorporates a correction to the construction lemma using minimal dependence; its main numerical conclusions are unchanged. See revision_notes.txt for details. AI-use disclosure: OpenAI Codex assisted argument exploration, literature searches, drafting and revision, verification-code generation, and internal checks, including internal AI reviews. The author is responsible for the manuscript. No specific funding or competing interests are declared.