Structural-Test Geometry as an Inductive Bias for Partially Observable World Models

Predictive world-model objectives need not organize latent histories according to distinctions required for later control. With privileged simulator access during training, should response-derived structural information be predicted, decoded, or imposed directly on latent geometry---and how do these choices compare with structured belief distillation? We define posterior-marginalized structural tests whose response distributions induce a pseudometric over histories, then train quotient-preserving (QP) world models by aligning shared latent Euclidean distances to that metric. Across a fixed six-task diagnostic suite, the largest differences appear under held-out structural variation: QP improves planning success by $26.1$ points $[21.5, 30.7]$ over metric decoding and $6.4$ points $[3.8, 9.0]$ over structured belief distillation. On GridWorld, metric decoding accurately recovers held-out structural distances ($\rho = .91$) while its native latent geometry remains more distorted than QP's (paired IQD difference $.22$ $[.20, .24]$), showing that relational recoverability and direct geometric alignment can diverge.