Product-Core Stratification of Large Effective Multiplier Rank in Symmetric Frobenius Algebras

Let $A$ be a $2n$-dimensional commutative symmetric Frobenius algebra over a field $k$, let $U=V^\perp$ with $\dim U=\dim V=n$, let $L\subset A$ contain the unit, and consider the Frobenius trilinear form induced by multiplication on $(L/k1)\times U\times V$. Write $(c,a,b)$ for its three mode ranks. We study the large-effective-rank regime $a=2$, $b=n$, $c>n$ through the $U$-mode kernel $K$ and its product core $R=KV$. For one-dimensional product core we determine the exact interval $n+1\le c\le 2n-4$ for every $n\ge6$, and realize every rank in the interval over every field with the sharp product-dimension defect $n-2$. More generally, if $r=\dim R$ and the secondary core vanishes, $RV=0$, then the exact interval is $n+1\le c\le 2n-2r-2,$ which is nonempty exactly when $n\ge2r+3$; again every admitted rank is realized over every field with defect $n-2$. We then analyze the first nonzero secondary-core layer $\dim R=2$, $\dim RV=1$. We prove $c\le2n-4$, improve this to $c\le2n-5$ in the rank-one branch or the branch $\lambda(s)=0$, and show that equality forces a one-dimensional nilpotent ideal with rigid orthogonal geometry. Finally, we give an explicit ten-dimensional local symmetric Frobenius algebra over an arbitrary field realizing the endpoint $(c,a,b)=(6,2,5)$ and defect $3$. Thus the rigid nilpotent endpoint branch is nonempty in the minimal half-dimension for which the endpoint $2n-4$ is strictly larger than $n$.