A Sharp Multiplier-Dimension Threshold for Orthogonal-Dual Product Spaces

Let A be a finite-dimensional commutative symmetric Frobenius algebra over a field k, let ₖ A=2n, and let U⊂ A be an n-dimensional subspace with orthogonal complement V=U^⊥. For a multiplier subspace L⊂ A, write $LU$ for the span of all products u and put ρU(L)=ₖ(LU). When 1∈ L, we express ρU(L) and ρV(L) through the two nontrivial flattening ranks of the restricted multiplication tensor on (L/k1)× U× V. It follows that ρU(L)=ρV(L) for every two-dimensional multiplier subspace containing $1$. We prove that this forced equality is sharp: in the purely inseparable extension ₂(s²,t²,u²,v²)⊂ ₂(s,t,u,v) there is a nonsimilar orthogonal-dual pair of eight-dimensional subspaces and a three-dimensional multiplier subspace L for which the two product dimensions are $15$ and $16$. We also describe the failure loci as determinantal strata on Grassmannians and show that multiplicative closure of a separating multiplier space can erase the defect. Thus unrestricted multiplier-subspace geometry and intermediate-field profiles sample genuinely different information.