Operating-point locality of soft constraints in flow-matching generative models

Few-step flow-matching generators such as MeanFlow expose two operating modes—a one-function-evaluation (1-NFE) map and a multi-step Euler sampler— and neither respects, by construction, the continuous symmetries of the physical point clouds they are increasingly used to generate. We study soft constraints (a symmetry-encouraging equivariance penalty, and a physics sum-rule residual) added to the training loss, and ask whether a constraint enforced during training governs the samples. It does, but only locally: a soft constraint lowers the tar- get violation only at the sampling operating point—the (t,r) pair, and even the step count K—at which it was enforced, and relaxes elsewhere. Constraining the 1-NFE endpoint makes one-step samples equivariant (16×on SO(3) molecules, ∼600×on SO(2) jets) while multi-step samples stay at the unconstrained level; constraining the sampler’s velocity step does the reverse. A per-step audit locates the cause in a state mismatch, and a diffusion control isolates it. Enforcing the constraint at both operating points makes the generator equivariant at both (81×at 1-step and 3.1×at 10-step on molecules; 96×and 2.6×on jets), an equal-budget ablation shows this is coverage rather than regularization strength, a relocation control moves the effect by moving the training step count, and marginalizing over K yields one model equivariant at every sampling budget. The same law governs a physics constraint, and we measure its boundary: under a non-compact Lorentz boost the fix holds at one step only. This is a mechanism paper; it makes no sample-fidelity claim.