Beyond Gradient Flow: Identifiability and Recovery from Distribution Snapshots

Inferring dynamics from snapshots of evolving distributions is fundamentally underdetermined: the Fokker-Planck equation constrains the drift $F$ only through its score-weighted divergence $\nabla\cdot F+F\cdot\nabla\log\rho$, leaving a $\rho$-solenoidal gauge invisible to any single-time constraint. Time-indexed transport formulations cannot resolve this ambiguity: every admissible marginal path admits a curl-free explanation, minimum-action reconstruction selects it, and marginal fit alone cannot distinguish dynamically inequivalent explanations. Requiring one autonomous field to explain several marginals instead makes part of the hidden circulation visible as $\nabla\log\rho$ changes across marginals. Separating instantaneous Fokker-Planck source constraints from the snapshot experiment, we show that the source constraints identify the field modulo the kernel of a stacked score-weighted divergence operator. For generic Gaussian shape variation, source constraints at $K\ge m$ time points in intrinsic dimension $m$ eliminate every polynomial gauge direction, whereas finitely many density snapshots alone admit aliasing; we give the obstruction explicitly. At a Gaussian anchor, for Sobolev smoothness $s$ and $n$ samples per time point, we derive a conditional lower rate $(nK)^{-2s/(2s+m+1)}$ for the tangent snapshot experiment, with a matching upper rate in a degreewise benchmark. Strong-form fitting is non-orthogonal to score error and cannot be repaired by spectral filtering. Instead, we estimate using smooth test functions while retaining the known diffusion term, and derive a finite-sample bound that separates sampling error from fixed-grid quadrature bias. Planted-circulation experiments confirm the predicted gauge contraction and expose a design tension between cross-slice information and covariance-aware whitening.