Certified World Models: Predictability Across Configuration, Horizon, and Resolution

Abstract

Scale buys interpolation; structure buys certifiable transfer. A world model's average error does not say whether a particular rollout can be trusted, or for how long. For equivariant latent world models we give a predictability certificate: a computable region spanning configuration, horizon, and resolution. Under exact equivariance, rollout error is invariant over the monoid generated by k primitive symmetries and is certified from the k generators (Theorem A); universal orbit-flatness over equivariant targets characterizes equivariance at the function level (Lemma 2), so an unconstrained architecture cannot certify the property by construction. Approximate orbit-transfer defects propagate by the finite-time Lyapunov spectrum (Theorem B): expanding channels give a logarithmic horizon $T_j(\hat{I}\mu )\sim \log (1/\hat{I}\mu )/\hat{I}{»}_j$, neutral channels accumulate recurrent defect linearly, and contracting channels accumulate a bounded nonzero floor. Exact conserved charge values are certified to all horizons only at zero defect; with one-step defect $\hat{I}\cdot$, charge-value error grows at most as $T\hat{I}\cdot$. Empirically, on a 40-dimensional learned model a ${\mathbb{Z}}_N$-equivariant network recovers the full Lyapunov spectrum ($R^2=0.98$-$0.99$) where dense and recurrent baselines fail. A cone/adapted-metric certificate reads an a-priori horizon off the model's own Jacobian, tight on uniformly hyperbolic dynamics and self-abstaining elsewhere; the resulting horizon improves a budgeted re-observation decision. For public non-equivariant world models the tangent spectrum gives a training-free candidate horizon, paired with a held-out divergence cross-check that abstains or corrects when the learned loop over-promises.

56 pages. v3: evidence hardening -- pendulum-ring mechanism doubled to n=30 seeds (Fisher p=9.5e-6), 5-task x 7-checkpoint multitask audit (0/35 cells reach the calibration band), certificate start-spread and measured episode-sensitivity analyses; prose pass; conclusions unchanged. Code: https://github.com/TimothyWang418/se3-ejepa