Reflection Symmetry and Symplectic Monodromy in Hyperelliptic Pencils: Moving Pairs, Critical Values, and Quartic Squares

Let A ⊂ R consist of 2g ≥ 4 distinct points, and consider the moving-pair hyperelliptic pencil y² = (∏a ∈ A(z - a)) ((z - δ)² - t²). We determine its connected geometric monodromy on first cohomology: it is Sp2g when A ≠ 2δ - A, and Sp2 g/2 × Sp2 g/2 otherwise. The symmetric case follows from two quotient pencils; in the asymmetric case, Picard–Lefschetz logarithms from isolated nodes and simultaneous two-node fibres generate the full symplectic Lie algebra. We also classify the connected monodromy on the degree-eight critical-value stratum $(2, 2, 2, 1)$ by polynomial indecomposability and establish full Sp₆ for y² = (z⁴ + z)² - t through radial vanishing cycles. The associated integral marking has a rank-three logarithmic image with cyclic quotient of order four. All equalities concern connected Zariski closures over Q₂.