Learning Verifiable Mathematical Laws for Scientific Agents via Graph-Hamiltonian World Models

Scientific agents need learned mathematical laws, not only next-state predictors. We study how an agent can learn a compact mathematical law, verify its invariants during deployment, and use it for intervention. The learned object is a graph-local Hamiltonian $H_G$: a known interaction graph decomposes the scalar energy into node and edge terms, and its gradients define dynamics through $f = J\nabla H_G$. This representation yields mathematical verification conditions for planned rollouts, including symplecticity, Hamiltonian integrability, and energy drift. For finite linear Hamiltonian bases, law learning reduces to ridge regression in the graph-local Hamiltonian dimension $P_G = |V|p_v + |E|p_e$, giving a finite-horizon planning guarantee with statistical term $\widetilde{O}(\sqrt{P_G/n})$ and matching lower bounds. Experiments on graph-coupled oscillators and nonlinear pendula validate $n/P_G$ scaling, edge dependence, geometric verification, transition-only behavior, MPC intervention gains, and a minimal residual-guided refinement loop. Graph-Hamiltonian world models therefore provide verifiable learned laws as primitives for scientific agents.