Beyond Isotropy in JEPAs: Hamiltonian Geometry and Symplectic Prediction

JEPAs often regularize one-view embeddings toward an isotropic Gaussian, implicitly baking Euclidean symmetry into the representation. We show that this is not merely a benign default. For a known structured downstream geometry Hsucc0, the minimax and maximum-entropy covariance under a Hamiltonian energy budget is (c/d)H-1, and Euclidean isotropy incurs a closed-form price of isotropy. More importantly, when the downstream geometry is unknown, no geometry-independent fixed marginal target is canonical: every fixed covariance shape can be maximally misaligned for some structured geometry. We further show that even oracle one-view marginals do not identify the JEPA view-to-view predictive coupling. These results suggest that the structural bias in JEPAs should enter the cross-view coupling rather than a fixed encoder marginal. We instantiate this principle with HamJEPA, which encodes each view as a phase-space state (q,p) and predicts view-to-view transitions with a learned Hamiltonian leapfrog map, while non-isotropic scale and spectral floors prevent collapse. In a deliberately headless token protocol, HamJEPA improves over SIGReg on CIFAR-100 by +4.89 kNN@20 and +3.52 linear-probe points at 30 epochs, and by +6.45 kNN@20 and +10.64 linear-probe points at 80 epochs, while a matched MLP predictor ablation shows that the symplectic coupling is the ingredient driving the neighborhood-geometry gain. On ImageNet-100, HamJEPA-q improves by +4.82 kNN@20 and +7.52 linear-probe points at 45 epochs.