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, $T_j(ε)\sim\log(1/ε)/λ_j$. 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.

• The paper’s main contribution is demonstrating that structural equivariance enables certified prediction horizons in learned world models.
• It introduces a computable certificate based on the Lyapunov exponents of latent dynamics, establishing tight bounds on multi-step errors.
• Equivariant models empirically outperform scale-only approaches, ensuring reliable planning, auditing, and budgeted sensing in complex systems.

Certified Predictability in Equivariant World Models: Structure Versus Scale

Overview and Contributions

This paper addresses the crucial distinction between average prediction error and actionable predictability in learned world models for planning and control. The central thesis is that model scale alone enables interpolation, whereas structural equivariance enables a certified, actionable prediction horizon, quantifiable in terms of the Lyapunov spectrum of the latent dynamics. The authors formalize and prove tight certification guarantees for equivariant latent world models: multi-step rollout error is orbit-constant and ties directly to the eigenstructure of the predictive Jacobian. The horizon certificate stratifies channels according to their Lyapunov exponents, yielding a computable guarantee of trustworthiness over prediction horizon for each symmetry orbit.

The paper provides:

• A precise multi-step predictability certificate for equivariant latent world models (Theorem A, Theorem B, Proposition 6).
• Tight lower and upper bounds on prediction horizon under approximate equivariance, showing horizon degradation is inevitable when equivariance is not exact.
• Characterization of structure-exclusive guarantees: no non-equivariant model, regardless of scale, possesses this certificate (Lemma 2).
• Identification of symmetry-conserved/invariant subspaces that can achieve unbounded certified horizons (Proposition 4, Proposition 5).
• A practical audit protocol for public world models, training-free, demonstrating actionable deployment and budgeted sensing.
• Rigorous empirical validation across chaotic, high-dimensional, compositional, and real-dynamics tasks.

  Figure 1: Certified region comparison: equivariant models certify the entire generated monoid $\langle S \rangle$ from $k$ generator checks up to a horizon determined by the predictor spectrum, while non-equivariant models certify only interpolation tubes.

Theoretical Framework

The central theoretical result is that in models with latent equivariance, rollout errors are constant over group orbits; this property is both necessary and sufficient for equivariance (Theorem A, Lemma 2). The certified prediction horizon $T_j(\epsilon)$ for channel $j$ is governed by its Lyapunov exponent $\lambda_j$ via the law $T_j(\epsilon) \sim \log(1/\epsilon)/\lambda_j$.

Approximate equivariance yields an explicit horizon limitation: a residual equivariance defect is exponentially amplified by channel expansion, matching upper and lower bounds (Proposition 6). Only exact equivariance or conservation ($\lambda_j \le 0$) yields horizon-unbounded guarantees.

Figure 2: Numerical confirmation of the tight lower bound: orbit-error variation matches the analytic $\epsilon e^{\lambda T}$ with exceptional precision; channels with $\lambda \le 0$ remain bounded, confirming infinite certified horizon.

The certification procedure is computable from model-side checks on symmetry generators (Lemma 1), local spectral estimation at query latents, and escalation for conserved channels (Algorithm 1). The prefactor in degradation is the splitting conditioning, computable from the spectral projector angles.

Structural Exclusivity and Empirical Validation

The certificate is structurally exclusive: orbit-constant error is equivalent to equivariance, and cannot be attained in non-equivariant architectures regardless of parameter count or augmentation (Lemma 2, E1, E3, E4). Empirical validation includes:

• Recovery of the full Lyapunov spectrum for chaotic systems only in equivariant models (e.g., $40$-D Lorenz-96, $k$0) versus dense/recurrent baselines failing spectra estimation (E2).
• Graceful horizon limitation in augmentation-only baselines, with a measured floor at $k$1, never attaining orbit-exactness (E3).

  Figure 3: Empirical sweep: (a) $k$2-D Lorenz-96 spectrum, only $k$3-equivariant model recovers true exponents; (b) Equivariant certificate meets sensing budget, dense certificate over-allocates; (c) Audit of official TD-MPC2 checkpoints stratified by expansion/contraction regimes.

Conservation Laws and Horizon Stratification

Channels identified by conservation laws are certified for unbounded horizons (Proposition 4, Proposition 5). Representation-theoretic placement ensures that conserved charges are uniquely associated with isotypic blocks. The multi-step horizon law applies generically, but the structure enables hard guarantees for conserved quantities.

Practical Implications: Model Auditing and Budgeted Sensing

The certified horizon is actionable for planning and monitoring: under fixed sensing budgets, structure-faithful certificates achieve budget-minimal calibration, whereas mis-estimated horizons from dense certificates cost proportional budget inflation (Proposition 9, E12). Public world models (TD-MPC2, LeWM, V-JEPA 2-AC) are audited training-free, with cross-validated horizon stratification identifying calibrated, optimistic, abstaining, and bias-driven deployment regimes (E13–E16).

Figure 4: Scale does not buy a calibrated horizon: official multitask ladder sweep shows no monotonic improvement in calibration with parameter count; horizon is a property of model structure, not scale.

Empirical Sweep: Chaotic and High-Dimensional Systems

Experiments validate the horizon staircase law across synthetic chaotic spectra, learned models of real chaotic systems (Lorenz, Hénon, Rössler), and high-dimensional ($k$4) Lorenz-96, confirming per-channel horizon recovery only in equivariant models. Dense and recurrent baselines fail the spectrum estimation and thus cannot be trusted for horizon calibration.

Figure 5: Certified horizon law across chaotic models: measured horizon slope matches textbook Lyapunov exponents to within $k$5–$k$6.

Figure 6: High-dimensional spectral horizon: recovered versus true Lyapunov exponents across $k$7 channels, equivariant model lies on $k$8, dense model fails completely.

Decision Value and Deployment Scope

The certificate's deployment value concentrates where decision predicates align with certified quantities; misalignment incurs irreducible penalties (Proposition 11). For monitoring and budgeted re-observation, the certificate delivers actionable cadence with zero new calibration data. Control returns, however, are diluted by map-level task tolerances, and the certificate's value is maximal when the certified predicate is task-aligned.

Limitations and Future Directions

The certificate is exact only for genuine dynamical symmetries and is gracefully approximate elsewhere, with two-sided degradation measured and theoretically bounded. The prefactor's worst-case tail is disclosed, and emergence of the Noether hinge in learned dynamics is open. Scaling and modality transfer exact flatness but do not solve architecture-agnostic pixel prediction. Safety guarantees were inconclusive at the tested scale.

Future directions include the study of symmetry emergence in learned dynamics, task value alignment for certificate deployment, and architecture-independent stabilization for pixel predictors. Extensions to non-uniformly hyperbolic and non-compact group actions will be crucial for further generalizing certified predictability.

Conclusion

The paper establishes a sharp division between interpolation capacity attained by scaling and actionable predictability horizon delivered exclusively by structure. Equivariant world models certify, without retraining or calibration, which situations they can handle and for how long, with a tight bound dictated by the predictor's spectrum. The certified region is structurally exclusive; trust in a rollout is a property of the loop's dynamics and symmetry, not its parameter count. The audit protocol, horizon certificate, and empirical validations demonstrate practical deployment value, particularly for monitoring and budgeted sensing in high-stakes and high-dimensional environments. The actionable horizon is enabled only by structure, not scale.

Figure 7: Certified horizon versus demanded resolution across controlled spectrum: the measured staircase matches the predicted law $k$9, stratifying channels by their dynamical expansion.