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

arXiv:2606.03003v1 Announce Type: cross 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 rho(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 (ju yi fan san). 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 imes 1.00 vs baseline imes 13.8 in 2D, imes 17.2 in 3D, imes 157 over the full SE(3) ladder), with the equivariant model 4.5-7.4imes smaller. [C] The same isometry argument lifts to closed loop: under a matching equivariant planner the control trajectory at orientation g is exactly rho(g) applied to the seen one, so closed-loop error is invariant across the group -- float-floor-exact in 2D/SO(2) on real PushT and statistically flat in 3D/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 (imes 1.00, le 2imes 10^{-7}) at every horizon, while the baseline's residual compounds with H. Out of scope: task-success sweeps, planner-free invariance, and scaling.

Source: arXiv cs.AI | 2026-06-03