Exact equivariance, kept through training, buys zero-shot generalisation across the symmetry group

Abstract

A latent world model built from an equivariant encoder $E$ and an equivariant predictor $f$ inherits a provable symmetry of its training loss: when the world's dynamics genuinely carries a group $G$ acting on latents by an orthogonal representation $ρ(g)$, the one-step prediction relMSE is exactly invariant across the whole group, so fitting the dynamics on a restricted slice of orientations mathematically determines it on the entire orbit (jǔ yī fǎn sān). We verify this end-to-end at laptop scale (CPU/MPS, fully seeded). [A] The symmetry survives a real Muon/AdamW + EMA + VICReg run -- composed encode-then-predict residual $\sim 10^{-6}$ after optimisation, not just at initialisation, and under any optimiser. [B] One-step error is flat to five digits across the group, while a same-hypothesis-class non-equivariant baseline fits the slice but breaks out-of-distribution (VN $\times 1.00$ vs baseline $\times 13.8$ in 2D, $\times 17.2$ in 3D, $\times 157$ over the full $\mathrm{SE}(3)$ ladder), with the equivariant model $4.5$-$7.4\times$ smaller. [C] The same isometry argument lifts to closed loop: under a matching equivariant planner the control trajectory at orientation $g$ is exactly $ρ(g)$ applied to the seen one, so closed-loop error is invariant across the group -- float-floor-exact in 2D/$\mathrm{SO}(2)$ on real PushT and statistically flat in 3D/$\mathrm{SE}(3)$ (disjoint 95% CIs). We stress-test the prior against Sutton's Bitter Lesson: augmentation, brute-force scale, and soft-equivariance each close at most the across-group task metric, never the float-floor exactness. Because equivariance is closed under composition, the $H$-fold rollout stays flat ($\times 1.00$, $\le 2\times 10^{-7}$) at every horizon, while the baseline's residual compounds with $H$. Out of scope: task-success sweeps, planner-free invariance, and scaling.