Cyclic Momentum Latents

1. Integration point: attach a gate vector $m$ to the latent Hamiltonian of a world model, sequential VAE, or recurrent predictor. Use a sigmoid gate $m_i=\sigma(a_i/\tau)$ during training and anneal temperature $\tau$ so the representation becomes nearly discrete. Set a target number $r$ of cyclic coordinates, or use an $L_1$ penalty when the number is unknown. 2. Pseudocode: encode each trajectory into $(q,p)$; evaluate $H_\theta$; use autodiff to obtain every $\partial H/\partial q_i$; add $\lambda_{cyc}\sum_i m_i(\partial H/\partial q_i)^2$ to the prediction loss; propagate states with leapfrog; periodically report the variance and drift of each $p_i(t)$. Warm up with $\lambda_{cyc}=0$, then ramp it linearly after the predictor learns basic dynamics. 3. Computed from the mechanism: the cyclicity penalty follows directly from Hamilton's equation; gate values, momentum conservation, and effective latent dimension are estimated from data. The budget $r$ is a modeling choice. 4. First cheap experiment: use a synthetic oscillator with one active coordinate and one nuisance coordinate, then test a 4D latent model on pendulum trajectories under rotations or changed initial conditions. Compare a Hamiltonian model without gates, a fixed penalty on all coordinates, and the gated model. The prediction is a sharp separation in $\mathbb{E}[\delta p_i(T)]$: selected coordinates should have near-zero momentum drift, while noncyclic coordinates retain nonzero force gradients. Increasing $\lambda_{cyc}$ should produce a transition from dense to sparse coordinate usage; beyond the point where the true active coordinate is incorrectly gated, rollout error should increase sharply. Measure whether the discovered cyclic count matches $r$ and whether conservation error reaches the integrator's numerical-error floor.