#Color projections
For a fixed color \(c,\) take all subsets of \(E\) that occur as the \(c\)-fiber of a word in the family. Symmetric cycle exchange implies ordinary basis exchange for these subsets, and hence produces a matroid on \(E.\)
This projection construction is the first bridge to ordinary matroid theory and is already represented in the accompanying Lean development.
#Joint cell counts
One can retain more information by counting how a coloring meets subsets of positions and subsets of colors. These joint counts lead to rank-like inequalities and relaxations of the assignment polytope.
They give exact descriptions for important subclasses, including full basis-partition families and families with a determining color. They do not describe every colored-matroid assignment polytope, even after the affine hull is included.
#Intrinsic rank axioms
Finding an intrinsic rank formalism for colored matroids remains open. A successful theory must distinguish valid positive classes from the known failure of unrestricted subset-count reconstruction, and it must expose the data needed by exchange rather than assuming it implicitly.