The Partition-Representation Degrees of the Dual Fano Matroid and a Six-Symbol Characterization of Regularity

Abstract

For a finite matroid M, let chi(M) be its set of partition-representation degrees. This preprint proves that the degree set of the dual Fano matroid is exactly the set of powers of two. It then derives two fixed-alphabet consequences for all finite matroids: a finite matroid is 6-entropic if and only if it is regular, and six is the unique alphabet size for which the entropic matroids are exactly the regular matroids. The proof propagates an M(K4) finite-group slice through the remaining circuits of the dual Fano matroid and then combines the resulting spectrum with excluded-minor theory and the existence theorem for pairs of orthogonal Latin squares. Status: Public Beta; internally verified candidate proof; external review pending.