Exceptional Symmetries, Geometric Monodromy, and Local Conductors in a Weighted Genus-Two Pencil

We classify geometric monodromy and local conductors for the genus-two pencil y^2=(z^2-a^2s)((z-1)^2-b^2s)((z-2)^2-c^2s) over algebraically closed fields of characteristic zero or at least seven. The connected geometric monodromy is Sp(4) away from three explicit elliptic-involution loci; on their union it is SL(2)×SL(2), with a factor-swapping component on the two hidden loci. The proof uses labelled Picard–Lefschetz meridians, a finite exact Lie-algebra certificate, and an alternative elliptic argument on the exceptional loci. For the rank-five primitive exterior square we determine the geometric Artin conductor and the parabolic cohomology dimension throughout the weight plane. On nonexceptional weights the formulas are 6+2m+2epsilon_tr and 2m-4+2epsilon_tr, where m counts distinct nonzero finite bad parameters and epsilon_tr detects a genuine triple collision. On the hidden locus the corresponding values are 14 and 4; on the reflection locus they are (12,4), (10,2), or (8,0), according to the weight. Explicit elliptic quotients, tame stable reduction, and equivariant specialization provide the local inertia calculations. Reproducible exact certificates are supplied separately from the geometric arguments. This integrated preprint builds on two related preprints published on 4 November 2026 (DOIs: 10.5281/zenodo.23944439 and 10.5281/zenodo.23944558). It is not peer reviewed. A separate QA archive contains reproducible finite arithmetic certificates; the geometric comparison theorems rely on mathematical arguments in the paper rather than on computation alone.