What happens when two networks are combined into a surface? Start with one square for every pair of edges, and split each square into two triangles. Tropical geometry gives this object a way to classify configurations, called its Picard group. This research candidate describes that group using the two original networks and their independent cycles. The main difficulty is not just counting dimensions. Integer configurations can contain finite-order classes that a rational calculation misses. The proof separates edge data into cuts and cycles, constructs integer local solutions, and shows that pairs of cycles introduce no additional finite-order obstruction. It also explains how to move classes when a square's diagonal is flipped. Those moves commute. A second part treats the larger Picard group, whose continuous contributions are the Jacobian tori of the two networks. These must not be confused with the finite critical groups in the discrete calculation. The result uses connected graphs without loops, unit edge lengths, and a precise local interpretation when parallel edges occur. It proves Lazar's product formula for simple graphs and establishes its extension under that stated convention. Lazar, Cartwright and an earlier catalogue reduction are credited. The archive includes the complete written argument and exact diagnostic checks. Those finite checks test consistency; they do not replace the proof. Historical priority and unaffiliated validation remain unestablished. This is an Evidence Press unrefereed candidate, dated 8 September 2026. The full paper and evidence are linked. This voice is AI-generated, and the audio is a communication aid, not additional mathematical evidence.