Exact bounds for rank-three matroid basis densities

We determine the maximum asymptotic basis density of rank-three matroids with no U_{3,6} minor: it is 64/81. A uniform gap forces every sufficiently dense simplification to be covered by two lines and one point. This gives weighted stability and the exact finite extremizers for all sufficiently large ground sets (n >= 100000). For general t, we prove both sharp parity bounds suggested by the line constructions when each component of the long-line incidence graph is either balanced or has point concurrency at most two. The proof combines ternary geometry for six-point arcs, a Gallai-Edmonds bound for saturated supports, and a penalty transform that retains all parallel-class weights. This is the first public preprint, not a peer-reviewed journal publication. The accompanying source and verification materials record revisions following an internal review; the review is not a formal journal decision. OpenAI Codex assisted mathematical exploration, proof development and inspection, literature searches, algebraic and programmatic checks, drafting and revision. The author takes full responsibility for the manuscript. No specific funding or competing interests are declared.