Time-resolved operator archetypes characterize dynamical sensitivity during cell-state transitions

During development, cells traverse gene expression states where their local dynamical sensitivity changes sharply, yet existing computational methods provide limited access to when and where this sensitivity peaks along a trajectory. Here we introduce scJDO (single-cell Jacobian Differential Operators), a framework that characterizes how local dynamical sensitivity evolves during cell fate transitions, together with an explicit account of what that representation can and cannot recover from snapshot data. scJDO treats time-indexed Jacobians as explicit analytical objects, projecting the temporal sequence of operators into a shared subspace and decomposing it into recurrent operator archetypes with interpretable temporal activation profiles. Unlike methods that derive Jacobians from splicing-kinetic vector fields, scJDO learns a neural drift field directly from cell-state geometry via diffusion score matching, enabling Jacobian analysis on trajectory-resolved scRNA-seq datasets regardless of splicing-data availability. Applied to a dense time-course of induced pluripotent stem cell (iPSC) reprogramming, scJDO resolves a quantitative operator-level signature that distinguishes diverted from productive fate: the productive trajectory executes a sequential handoff from an early MEF-exit operator regime to a late pluripotency-associated regime, whereas the diverted trajectory maintains the early regime and instead activates a distinct stress-associated archetype. We validate scJDO across four settings: synthetic benchmarks with analytically known ground truth, branching hematopoiesis, dense real time-course reprogramming, and a perturbational setting using Schrodinger bridges in K562 CRISPRi Perturb-seq. We compare against the two most widely used single-cell Jacobian methods on a dataset where all three are runnable, finding that scJDO shares significantly more gene-level and directional operator structure with Dynamo than expected by chance while providing operator-level analysis on datasets without splicing kinetics. We further characterize the boundary of the representation directly. At a fate-decision saddle, eight mathematically distinct readouts of the same learned drift field are consistent with a single explanation: a drift field fit to snapshot density reproduces density-dominant separation between committed branches rather than the low-variance transverse instability that defines the decision. Together, scJDO provides an operator-level view of single-cell dynamics and an explicit characterization of its own identifiability boundary.