Beyond Isotropy in JEPAs: Hamiltonian Geometry and Symplectic Prediction

JEPAs often regularize embeddings toward an isotropic Gaussian, implicitly baking Euclidean symmetry into the representation.We show that when downstream tasks and token geometries carry structure, the canonical target is instead a Hamiltonian Gaussian N(0,α H⁻¹) for an SPD operator H, and enforcing Euclidean isotropy can be provably suboptimal via a closed-form price of isotropy bound.A minimax and maximum-entropy analysis under a Hamiltonian energy budget singles out N(0,α H⁻¹) as the canonical structured target.We introduce HamJEPA, which encodes views into a phase-space state $(q,p)$ and predicts via a symplectic integrator; Liouville volume preservation holds for the predictor map, and encoder collapse is prevented using a log-determinant volume floor and fixed-unit scale budgets.This result provides a practical route beyond the isotropic-sphere design principle.HamJEPA improves frozen-feature performance over SIGReg: on CIFAR-100 it increases linear-probe top-1 by $+3.75$ points (30 epochs) and ≈+10.6 points (80 epochs), and kNN@20 top-1 by $+4.89$ points (30 epochs) and ≈+6.5 points (80 epochs); on ImageNet-100 (45 epochs) it increases linear-probe top-1 by $+6.54$ points and kNN best-over-k top-1 by $+4.26$ points (best over $q/p/(q,p)$ readouts).