An Odd-Divisor Obstruction for Cyclically Invariant Grassmannian Initial Ideals

Over a field of characteristic zero, we prove that the Pluecker ideal of Gr(k,n) has no monomial initial ideal invariant under cyclic rotation whenever gcd(k,n) has an odd divisor greater than one. More generally, for n=mq and k=rq with q>1 odd and 1<=r<m, a permutation of cycle type q^m has trace -binomial(binomial(m,r),2) on the quadratic relations. This negative trace also excludes every invariant monomial basis of the degree-two coordinate ring. No squarefreeness, quadratic-generation or Gorenstein assumption on the putative initial ideal is required. The four-page proof uses classical standard monomial theory and the two-column Schur identity, and credits Santos--Stump--Welker's known Gr(3,6) flag-complex obstruction. Exact rational relation-matrix checks and a reproducibility script accompany the paper. This is a complete auxiliary symmetry theorem, not a solution of the broader Gorenstein-initial-ideal question in UnsolvedMath OWR-734-003; without the symmetry requirement, the Grassmannian existence case is already known. This unrefereed preprint was developed with AI assistance and is self-audited, without formal proof-assistant verification or an absolute-priority claim.