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-\{c, n\}, 1 c n², and show that it is sharp for every prescribed 1 c n and also at the first large-rank layer c=n+1 when n2.