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

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 (j\v{u} y\=i f\v{a}n s\=an). 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.