Noether World Models: An Exactly Conserved Family of Latent Invariants

World models predict by rolling a learned latent transition forward over many steps. Current methods pursue conservation through a soft penalty, a symplectic integrator or an implicit projection, so each exact guarantee covers a small set of scalars fixed before training and paid for with an implicit update. We propose the Noether World Model (NWM), which assembles the generator as a graph Laplacian over object subspaces, $\mathbf{H}=\mathbf{L}\otimes I_m$, and advances the latent with a Cayley transform. The resulting unitary operator conserves a $2m^2$-dimensional family of quadratic invariants exactly at every value of its weights, and that family is exactly the set of quadratic forms with that property. Experiments on three physics datasets and three planning environments show that NWM holds the entire family at double precision whereas no baseline holds more than the norm, with $32\times$ fewer parameters than the strongest physics-structured baseline. Decoded quality in distribution separates the nine designs by only 1.3 dB, whereas under a zero-shot change of force law NWM leads the unconstrained transitions by 1.9 dB, so the family pays where the data leaves the training distribution. Exactness has a price, since the operator fixes every member of the family and most goals in a latent it did not shape lie off the reachable set. The same family can nonetheless enter the training objective of a world model whose predictor we do not replace, and it improves planning only where that predictor already drifts, a condition measurable before retraining.