Integral cubic torsion and Frobenius in genus two

Let \(T\) be a symplectic \(\mathbf Z_2\)-lattice of rank four and let \(L\) be its closed-surface cubic Lie quotient. For a nondegenerate symmetric monodromy pairing \(B\), we compute the two-torsion in \(\operatorname{coker}(\rho_3(\tau_B)-1)\) and its normaliser action. For primitive \(B\), the module is canonically the free cubic Lie module of the rank-two quotient by the vanishing-cycle Lagrangian, reduced modulo two. For even \(B\), we give a complete fixed-dimension formula for every graph Frobenius action, including dimensions \(2,3,4,5,6,8,10\). For the thick theta pairing we also determine the full two-primary fixed groups when integral graph Frobenius is \(\pm I\). Explicit genus-two curves over \(\mathbf Q_5\) with identical metric graphs, graph actions and ordinary component groups have different cubic invariants.