A learned operator leaks the invariant.

A group action preserves an invariant: a phase for U(1), a norm for SU(2), a symplectic form for Sp(2n), the invariant mass for the Poincaré group and for a Lorentz boost, the quadratic Casimir for SU(3). Each point is one state. Its distance from the ring is how far its invariant moved when the group acted, on a compressed scale marked at 1%, 100%, 100 times and 10,000 times. A point on the ring kept its invariant. Blue is ARCHON's exact action. Amber is an operator trained to reproduce the same action from examples. The figure steps each group from the training range out to pure noise.

The blue points stay on the ring at every distance. On U(1), SU(2) and Sp(2n) the learned operator carries a violation of 5% to 18% that training does not remove. On the Poincaré group its violation grows from 7 in the training range to 13,896 on noise. Under extreme shift the learned violation is 650,000 to 2.1 million times ARCHON's on those four groups, and 60 to 70 times on a Lorentz boost. On SU(3) the learned operator loses 87% to 98% of the Casimir. The points drawn are one trained operator on 128 states, and the readouts are the figures of the full runs.