Schur--Plucker Geometry of the MDS Locus for Principal-Ideal Codes

What the math gives to ML

The paper gives an explicit algebraic certificate for when a structured coefficient-vector code lies in the MDS locus: every maximal Pluecker coordinate factors into the constant coefficient and Schur polynomials indexed by partitions inside an (n-r) by (r-1) rectangle. This is transferable as a full-minor diversity barrier for structured neural linear operators, especially companion or polynomial state-space layers, where ordinary spectral-radius control does not prevent degenerate coordinate projections. The practical adaptation is to parameterize a companion transition polynomial and regularize its Schur coordinates, computed directly from coefficients rather than explicitly from roots. This is best tested as a targeted conditioning and observability regularizer rather than as a generic neural-network loss.