Gauge Freedom as a Numerical Design Variable in Generative Dynamics

Continuous-time generative objectives may determine a probability path without determining the dynamics that realise it. Here we show that this non-identifiability can become a numerical design variable once the dynamics are discretised. For a fixed path and solver, we optimise only over generators that are exactly equivalent in continuous time, and propose a projection result that characterises the leading solver error along this exact fibre. For a two-dimensional Gaussian contraction, a closed-form path-preserving rotation cancels the leading covariance defect of an order-$p$ Runge--Kutta method and raises its marginal order from $p$ to $p+1$, allowing a step-dependent correction to make every grid-point marginal exact. For an anisotropic Gaussian flow-matching path, a one-parameter exact gauge reduces eight-step Euler endpoint error from $0.3835$ to $7.90\times10^{-6}$ in squared Wasserstein distance, even though it strictly increases the population matching loss. Thus the representative preferred by training need not be the one preferred by the sampler.