An Explicit Doubled-Reflection G₃₄ Hurwitz Component with Maximal Period-Map Rank

We construct an explicit twenty-branch-point Hurwitz component associated with a specified doubled-reflection tuple in the exceptional complex reflection group G₃₄. Using a semistable degeneration with seventeen smoothing parameters, we identify the limiting weight-graded monodromy operators through graph homology, Picard–Lefschetz theory, and a fixed polarization. An exact computation over Q(√−3) produces seventeen Hermitian forms on a four-dimensional multiplicity space whose complex span has dimension sixteen. A uniform Schur-orthogonality identification relates these forms to the geometric monodromy logarithms. The multivariable nilpotent-orbit theorem then establishes that the associated period map has generic differential rank sixteen, the maximum permitted by the period-domain dimension. The result concerns the explicitly specified ordered Hurwitz component. It does not establish the Hodge conjecture or algebraicity of Weil classes. The computational verification materials include the reflection-matrix data, exact verification scripts, and reproducibility certificates. These materials are being prepared for separate public archiving, subject to provenance and redistribution checks. AI-assisted tools were used in computation, drafting, and review; responsibility for the submitted work rests with the human author.