A twelve-term exclusion for the small Davenport constant of $E_2\times C_3^r$

Source abstract

Let $E_2$ be the extraspecial group of order $3^5$ and exponent three. For every $r\ge1$, we prove that a product-one-free sequence of length $2r+11$ over $E_2\times C_3^r$ cannot contain exactly $2r-1$ central terms. Thus the critical layer with twelve noncentral terms is excluded. The proof combines a relative moment criterion for abelian normal subgroups with a finite theorem in a symplectic four-space over $\mathbb F_3$. Under explicit subspace occupancy bounds, the family of balanced triangles that can actually be completed to a nonfull zero-sum block admits edge weights summing to one on every triangle. A dual cycle argument reduces this assertion to a potential condition on branching edges. All remaining configurations contain one of eleven minimal frames; two separately implemented exhaustive checks verify all their admissible extensions. The unrestricted check has 9544046 leaves. Complete source code and execution records are supplied. The exact value of $\mathsf{d}(E_2\times C_3^r)$ for arbitrary $r$ is not determined by this result.