Cosymplectic Reeb-Hamiltonian Layer

1. Integration point: use this as the transition block in a recurrent state-space model, latent world model, or sequence encoder. Split each hidden vector into \(q,p,t\), with \(q,p\in\mathbb{R}^{d/2}\) and one scalar \(t\); project the original hidden state into this representation once at the input and project back after the transition stack. Let an MLP \(H_\psi(q,p,t,u)\) produce the leaf Hamiltonian, where \(u\) is the current observation, action, or conditioning vector. Let a second small MLP \(v_\psi(q,p,t,u)\) produce Reeb speed, initialized near one for physical time or near a learned positive constant for abstract depth. 2. Pseudocode: compute \(g_q=\nabla_qH_\psi\) and \(g_p=\nabla_pH_\psi\) with automatic differentiation; perform the three implicit/semi-implicit updates in the displayed leapfrog formula, using one fixed-point iteration or two Newton iterations for the half-step \(p_{k+1/2}\); then set \(t_{k+1}=t_k+h\,\mathrm{softplus}(v_\psi)\) if monotonic time is required. Feed \((q_{k+1},p_{k+1},t_{k+1})\) to the next block. Add an optional residual observation update after the geometric transition, but do not let that update alter \(t\). 3. Computed from the mathematics: the canonical leaf split, Hamiltonian gradients, symplectic leapfrog map, and Reeb/transverse separation. Estimated empirically: the Hamiltonian MLP parameters, step size \(h\), the best parameterization of \(v_\psi\), and whether the approximate learned \(\beta\)-preservation survives the observation residual. Monitor the Jacobian determinant of the \((q,p)\) transition and the symplectic error \(\|D\Phi^TJ D\Phi-J\|_F\), where \(J=\begin{bmatrix}0&I\\-I&0\end{bmatrix}\). 4. First experiment: train a small latent ODE/RNN or action-conditioned world model on a synthetic harmonic oscillator and a chaotic double pendulum, comparing an equal-parameter GRU, an unconstrained MLP residual transition, and the proposed layer at equal rollout FLOPs. Evaluate one-step prediction, 100- to 1000-step rollout error, energy drift, and gradient norm stability. The target signal is lower long-horizon error and bounded energy drift at similar one-step accuracy; on a real test, use a small MuJoCo trajectory dataset and compare validation rollout error at horizons 10, 50, and 200.