Sharp Defect Envelopes for Orthogonal-Dual Product Spaces

Abstract

Let $A$ be a finite-dimensional commutative symmetric Frobenius algebra over a field $k$, with $\dim_k A=2n$. Let $U\subset A$ have dimension $n$, put $V=U^\perp$, and let $L\subset A$ contain $1$ with $\dim_k L=m+1$. 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 $|dim_k(LU)-dim_k(LV)|$ $\le n-\lceil\frac{n}{m}\rceil,$ $\qquad 1\le m\le 2n-1.$ 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 $n-\lceil n/m\rceil$. 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 $|\dim_k(LU)-\dim_k(LV)|$ $\le n-\max\{\lceil\frac{n}{c}\rceil,$\lceil\frac{c}{n}\right\rceil\},$ $\qquad 1\le c\le n^2,$ and show that it is sharp for every prescribed $1\le c\le n$ and also at the first large-rank layer $c=n+1$ when $n\ge2$.