Basis Proximity with Few Element Labels

Let a finite matroid be labeled in an arbitrary abelian group, and let F be a finite set of forbidden basis sums. We prove that, if an F-avoiding basis exists, one exists within exchange distance |F| of every specified basis when the matroid is represented over any field and the element labeling takes at most five distinct values. With at most six element values, the same conclusion holds when the specified basis or its ground-set complement uses at most three values. The proofs realize abstract exchange counts on actual bases; they do not require the physically selected exchanges to be bases. A source-colored Fitting decomposition handles a mixed label fiber, and a three-by-three transport argument yields constructive descent when both sides of the specified basis use at most three values. We also prove proximity for finite degree-three trees of represented 2-sum factors having at most three physical label values per factor. Representation fields may differ between factors, and the global label palette is unbounded. All statements retain group torsion. The restrictions concern element labels, independently of the size of the forbidden set.