The Gauge Freedom of Rotational Drift in Diffusion Models

The forward process of a diffusion model is a design choice, yet most standard constructions use a purely dissipative drift $f=-\beta (t)x$ — the gradient component of the Helmholtz decomposition. We ask what the missing divergence-free half contributes within the exactly solvable family $dx=(-\gamma I+r(\tau )J_0)xd\tau +\sqrt{2}d{W}_{\tau }$, where $J_0^{\top }=-J_0$ is fixed. It leaves the generated distribution unchanged: an explicit time-dependent change of coordinates — rotation, rescaling, and time reparametrization — reduces the Fokker–Planck equation to the standard heat equation for every skew profile $J(\tau )$, so rotating and non-rotating processes differ only by the explicit map $R(\tau )$, which is the identity at the generation endpoint. Rotation is not inert, though — at constant rotation strength the process carries a stationary probability current with entropy production rate exactly $\sigma ={\rho }^2\Vert J_0{\Vert }_F^2=2\lfloor d/2\rfloor {\rho }^2$, breaking detailed balance underneath an endpoint law that is untouched. The rotational content resides in the path measure and is absent from the marginal score in the co-moving frame. The two conditions delimiting this gauge are not symmetric in kind. The reduction requires conformal dissipation, but the gauge then holds throughout the entire conformal class, anisotropic noise included, so the first condition is a design degree of freedom rather than a boundary. Realized through the co-moving frame of the rotational field, the gauge persists exactly for Killing fields; a differential rotation, though divergence-free and norm-preserving, already escapes it. Under model misspecification, rotational mismatch becomes visible through a single-peaked visibility window whose position is set by the trade-off between rotation and the dissipative decay of non-radial structure, weakly dependent on rotation strength and shifted later by multimode geometry, while its depth grows with rotation strength and its shape separates covariance-driven from multimode-driven visibility. We verify these claims analytically and through closed-form evaluations in 2D and at $d=256$ on MNIST, with trained-model checks in the local heat-ball framework. Source code is available at https://github.com/Anonymous-510/anonymous-submission.