Charlot Lab · EFA · one energy, both roles — the nonlinear case
The double integrator was saturated but linear. Here the plant is genuinely nonlinear and underactuated — gravity pulls the pole down as $\sin\theta$, the torque is bounded. Yet the same claim holds: a single learned energy is the controller (descend it) and its own Lyapunov certificate (it provably contracts along that descent). The pole swings up and stays up because the entity descends one scalar.
Companion to the double-integrator demo and Energy Is the Certificate §7 · verified seed-0 (research/efa-unification)
Left is the physical pole: pivoted at the base, upright is the goal, gravity tips it over as $\sin\theta$. Right is the same state as a point (angle θ across, angular velocity θ̇ up) on the energy landscape. From 61 bounded torques the entity picks the one whose next state has lowest V — pure descent — and the pole rises and holds. The gold ring is not drawn by hand: a sound nonlinear proof (2nd-order Taylor + per-box CROWN, the state-dependent Jacobian box-ranged, the greedy controller an OR over torques) guarantees V strictly decreases every step inside it. And it is tax-free: a certificate trained with no control term still controls, and the controller certifies just as far — optimizing for either role yields both. Nano, single-seed, in simulation; the honest scope is in the paper.