提出可验证的世界模型预测能力，能给出可靠预测时长与精度的数学保证。
Certified World Models: Predictability Across Configuration, Horizon, and Resolution
- 基于对称性构建潜在世界模型，用生成元计算可预测范围。
- 实测40维模型中对称网络准确恢复李雅普诺夫谱（R²=0.98-0.99）。
- 无需训练即可通过雅可比矩阵预估预测时长，适合高可靠性场景。
世界模型的平均误差无法判断特定轨迹是否可信或能持续多久。针对等变潜在世界模型，本文提出一个可计算的可预测性证书：覆盖配置、预测时长和分辨率的区域。在精确等变条件下，轨迹误差在由k个基本对称性生成的幺半群上不变，且可从生成元处认证（定理A）；对等变目标的全局轨道平坦性刻画了函数层面的等变性（引理2），因此非约束架构无法构造性地认证该性质。近似轨道转移缺陷随有限时间李雅普诺夫谱传播（定理B）：发散通道给出对数时长 $T_j(ε) ilde{\log}(1/ε)/λ_j$，中性通道线性累积缺陷，收缩通道累积有界非零底限。精确守恒量值仅在零缺陷下可跨所有时长认证；一步缺陷为η时，电荷值误差最多以Tη增长。实验上，在一个40维学习模型中，$\b{Z}_N$-等变网络恢复了完整的李雅普诺夫谱（R²=0.98–0.99），而密集与循环基线模型失败。锥/自适应度量证书可从模型自身雅可比矩阵读取先验时长，在均匀双曲动力学下紧致，在其他情况自动回避；由此得到的时长改进了预算内重观测决策。对于公开的非等变世界模型，切向谱提供无训练候选时长，并搭配保留发散交叉检验，当模型过度承诺时主动回避或修正。
原文摘要 · Abstract (English)
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 $T_j(ε)\sim\log(1/ε)/λ_j$, 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 $η$, charge-value error grows at most as $Tη$. 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.
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