Scale Buys Interpolation, Structure Buys a Horizon: Certified Predictability for Equivariant World Models

Abstract:Scale buys interpolation; structure buys a certified horizon. A world model's average error says nothing about whether a particular prediction can be trusted, or for how long. For equivariant latent world models we give a computable, multi-step certificate of the predictable horizon: $T$-step rollout error is provably constant over each symmetry orbit (Theorem A) and stratified channel-by-channel by the predictor's Lyapunov spectrum, $T_j(\epsilon)\sim\log(1/\epsilon)/\lambda_j$. The horizon is two-sided -- a matching lower bound makes approximate equivariance provably horizon-limited -- and the certificate is exclusive to structure: orbit-constant error characterizes equivariance, so no non-equivariant model has it at any scale. Empirically, on 40-D Lorenz-96 only a $\mathbb{Z}_N$-equivariant network recovers the full Lyapunov spectrum ($R^2{=}0.98$); dense and recurrent baselines fail. Because the spectrum is faithful, the certificate acts, a priori: under a fixed sensing budget a $c\times$-inflated certificate provably needs $c\times$ the budget, and the equivariant certificate meets a budget its inflated dense counterpart cannot -- with zero calibration data. The same read-out, unchanged, audits public pretrained world models training-free: TD-MPC2 checkpoints land on the certificate's own scope taxonomy -- calibrated where strongly expansive (ratio 0.94-1.02), optimistic where weakly expansive, correctly abstaining where contracting -- a map a deployed monitor replicates cell-by-cell, out-of-sample. Across the official 1M-317M multitask ladder, calibration does not improve with parameters. On V-JEPA 2-AC (1B, real robot data) the measured cross-check correctly overrides an over-promising tangent spectrum -- the cross-validated audit, not the raw number, is the deployable object. Scale buys interpolation, not a calibrated horizon.

23 pages (9 main + appendices). Code: this https URL

Machine Learning (cs.LG); Robotics (cs.RO); Dynamical Systems (math.DS)

arXiv:2606.13092 [cs.LG]

(or arXiv:2606.13092v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2606.13092 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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