Sharp Defect Envelopes for Orthogonal-Dual Product Spaces

Let A be a finite-dimensional commutative symmetric Frobenius algebra over a field k, with ₖ A=2n. Let U⊂ A have dimension n, put V=U^⊥, and let L⊂ A contain $1$ with ₖ L=m+1. Classical linear additive combinatorics minimizes a single product dimension for independently chosen factors; here we ask for the largest asymmetry between two product spaces constrained by Frobenius orthogonality. We prove the sharp universal bound |dimₖ(LU)-dimₖ(LV)| ≤ n-/m, 1≤ m≤ 2n-1. The upper bound follows from the orthogonal-dual flattening identity together with standard inequalities among the three mode ranks of the restricted multiplication tensor. For every admissible pair $(n,m)$ and every field k, we attain equality by a finite direct product of truncated-polynomial Frobenius algebras. Hence the exact multiplier-defect envelope in this category is n- n/m. The case $m=1$ recovers the previously proved forced equality for two-dimensional unit-containing multiplier spaces. If c is the effective multiplier mode rank of the tensor, we also prove the support-sensitive bound |ₖ(LU)-ₖ(LV)| ≤ n-max\/c,/n\},$ $ 1≤ c≤ n^2,$ and show that it is sharp for every prescribed $1≤ c≤ n$ and also at the first large-rank layer $c=n+1$ when $n≥2$.