Marginal normality does not constrain time: a prescribed Gaussian path law for JEPA world models

SIGReg, the anti-collapse regulariser of the LeWM world model, constrains the latent marginal at each time step and says nothing about how consecutive steps relate: a latent that is constant along every trajectory attains the global minimum of both training terms, and the identifiability theory for this objective assumes exactly the cross-time structure it never enforces (Klindt et al., 2026). We close the gap with **T-SIGReg** (temporal SIGReg), which prescribes the joint law of a trajectory window, $\mathcal{N}(0,R_{\rho ,h}\otimes I_d)$, with the profile $R_{\rho ,h}$ measured from the observation stream before training, and enforces it with the characteristic-function test SIGReg already uses, pointed at a *separable sketch* $q\otimes v$ that keeps the cross-time signal a full-sphere draw averages away, at cost $O((h+1)d)$. Which measure supplies $q$ is an experimental-design variable: the temporally optimal design gives up the spatial law, and mixing the marginal directions back in restores it at a cost linear in the mixing weight. On four pixel-based control suites under a paired protocol, prescribing the measured path law raises planning success on TwoRooms from 63.7 to 100.0, improves Reacher and Cube by more than twice the largest retraining threshold we measured, and on Push-T recovers the loss an AR(1) prior incurs against the control; prescribing no cross-time structure at all falls below the control in all four suites.