Distance-Eight SI-Orderings in Graphic and Cographic Matroids

We give a computer-assisted proof that two bases of a graphic or cographic matroid admit a subsequence-interchange ordering whenever their distance is at most eight. Consequently every graphic or cographic matroid of rank or corank at most eight is subsequence-interchangeably base orderable. The finite ingredient is a complete pair of directed qualifications on eight vertices: an arbitrarily prescribed edge of either tree can be made terminal, with all exchanges directed from the same source tree. Each qualification has 1,245,225 pair orbits and explicit witnesses checked on all 28 intervals. Leaf padding and two explicit degree-three lifts give the nine-vertex consequence. Common-edge contraction, parallel reduction and duality then give the basis-distance theorem. Compact certificates allow verification without rerunning the discovery searches. As a consequence, proximity with at most seven forbidden basis-sum values holds for graphic and cographic matroids in arbitrary rank and over any abelian labeling group.