Homocyclic Jacobians of planar biconnected graphs

For every r ≥ 1, we construct a simple planar biconnected graph with 2r + 2 vertices, 4r + 1 edges, maximum degree at most four, and Jacobian (ℤ/8ℤ)^r. This disproves the bounded-multiplicity conjecture of Gaudet, Jensen, Ranganathan, Wawrykow and Weisman. We give explicit generators and compute the monodromy pairing: its matrix is B_r/8 modulo ℤ, where B_r is tridiagonal with diagonal 4, …, 4, 5 and adjacent entries −1. For every finite tree T with at least two vertices and every integer m ≥ 2, we also prove exp Jac(T[K̅_m]) = m² lcmv ∈ V(T) deg_T(v), where K̅_m is the edgeless graph on m vertices. For paths with at least three vertices, these products have vertex connectivity m and exponent 2m². They disprove the bounded-exponent finiteness conjecture attributed to Baker and Shokrieh, including its analogue for connected regular matroids.