Decomposition theorems for unmatchable pairs in groups and field extensions

A theory of matchings for finite subsets of an abelian group, introduced in connection with a conjecture of Wakeford on canonical forms for homogeneous polynomials, has since been extended to the setting of field extensions and to that of matroids. Earlier approaches have produced numerous criteria for matchability and unmatchability, but have offered little structural insight. In this paper, we develop parallel structure theorems which characterize unmatchable pairs in both abelian groups and field extensions. Our framework reveals analogous obstructions to matchability: nearly periodic decompositions of sets in the group setting correspond to decompositions of subspaces involving translates of a subfield in the linear setting. This perspective not only recovers previously known results through short proofs, but also leads to new matching criteria and guarantees the existence of nontrivial unmatchable pairs.