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^⊥ with U= V=n, let L⊂ A contain the unit, and consider the Frobenius trilinear form induced by multiplication on (L/k1)× U× 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≤ c≤ 2n-4 for every n≥6, and realize every rank in the interval over every field with the sharp product-dimension defect $n-2$. More generally, if r= R and the secondary core vanishes, $RV=0$, then the exact interval is n+1≤ c≤ 2n-2r-2, which is nonempty exactly when n≥2r+3; again every admitted rank is realized over every field with defect $n-2$. We then analyze the first nonzero secondary-core layer R=2, RV=1. We prove c≤2n-4, improve this to c≤2n-5 in the rank-one branch or the branch λ(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.