Finite Symplectic Monodromy Certificates and Failure-Locus Resolution in Hyperelliptic Pencils

This paper develops a finite, reproducible certification framework for connected geometric symplectic monodromy in hyperelliptic pencils, with explicit applications in genera two and three. The approach combines primitive dyadic normalization of Picard–Lefschetz logarithms, characteristic-zero Lie-algebra dimension bounds, nonordinary braid-orbit completion, and geometric resolution of exceptional certificate loci. For genus-three degree-eight polynomial pencils with seven simple critical points and critical-value partition (2+2+2+1), we establish a full-versus-split dichotomy. Indecomposable polynomials have full connected geometric monodromy (\mathrm{Sp}_6), whereas decomposable cases admit a quadratic right factor and have connected monodromy (\mathrm{Sp}_2\times\mathrm{Sp}_4). The classification is supported by an exhaustive finite analysis of 403 marked-tree configurations, representing 179 isomorphism classes. A contrasting example is provided by the quartic-square polynomial (f(z)=(z^4+z)^2). Although its polynomial root-monodromy group is the imprimitive wreath product (S_4\wr C_2), its connected cohomological monodromy is the full group (\mathrm{Sp}_6). An exact rational Rouché–Burau certificate verifies 1,715 path segments, reconstructs four actual braid words, and establishes the full 21-dimensional symplectic Lie algebra in characteristic zero. The framework also resolves the connected-monodromy classification of an explicit genus-three moving-pair family, including six triple-root parameter values through certificates based on ordinary meridians away from the nonordinary singular fibre. The results distinguish genuine geometric reductions from failures of finite certificates caused by insufficient reduction precision, incomplete Lie rank, or omitted braid conjugation. The criteria are sufficient rather than a universal classification algorithm. Version v1.0 (6 November 2026). The accompanying supplementary archives contain the LaTeX source, exact computational certificates, independent verification programs, finite enumeration records, and geometric proof-dependency documentation.