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 n6, 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 n2r+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 c2n-4, improve this to c2n-5 in the rank-one branch or the branch (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.