Maximal Period Variation in an Explicit Eighteen-Branch G₃₄ Paired-Cover Family

We construct an explicit eighteen-branch G₃₄ paired-cover family and study the associated polarized abelian sixfolds of Weil type over K = ℚ(√−3). Using an admissible-cover degeneration with nine nodes, exact graph-incidence calculations, and a common geometric identification of the Picard–Lefschetz monodromy, we establish that the period map has generic complex rank nine, the maximum permitted by the corresponding Weil-type period domain. The proof combines finite reflection-group computations, primitive cohomology, Fox monodromy calculations, polarization identities, and an explicit nine-dimensional span of Hermitian forms. Computational authentication and geometric-bridge reviews accompany the manuscript.