Label-free flow operators and stability operators from single-cell time courses: what local dynamical operators can and cannot identify

Local Jacobians of single-cell dynamics are used to locate unstable cells along trajectories, but what they identify from measured data has rarely been tested against independent truth. We present scJDO2, which separates two per-cell operators by the time information they require: a score operator, the Hessian of a score-matched log density, available from any snapshot; and a flow operator, the Jacobian of a drift fitted to a measured velocity, here estimated label-free from a sampled time course by entropic optimal transport. Both are summarized in a shared operator space of magnitude-free descriptors, with the invariance of each stated, and evaluated with pre-registered tests. On synthetic data calibrated to six real datasets, the flow operator from the time course alone separated fate-bifurcation cells from committed cells on held-out seeds (AUC 0.89-1.00) with a-priori settings; tuning on development seeds overfit. On LARRY lineage-traced hematopoiesis, its fitted drift predicted the clonal displacement of barcoded cells beyond population movement (Spearman 0.56, permutation p < 0.001; 0.50 after removing the mean shift) and a held-out sampling day better than mean-displacement and no-movement baselines (energy distance 0.69, 95 % CI 0.66-0.73, vs 0.87 and 0.88); lineage thus validates the velocity, not λ₁. Mini-batch transport reproduced dense transport (velocity cosine 0.9998) and scaled to all 130,887 cells (transport 5 min and 2.5 GB; drift fit and Jacobians 21 min and 1.3 GB). On an independent simulator it detected two of three bifurcations and two of three cycles and never reported instability on linear trajectories. Several readings failed. Neither operator recovers the values of the true Jacobian; the flow operators leading eigenvalue ranks branching cells empirically. The flow operator depends on the transport metric and inverts on Factor Analysis. Operator archetypes isolated bifurcation cells but did not outperform expression clustering and were unstable across fit seeds. The score operator did not predict CRISPRi knockdown responses or sister-cell fate coherence. A confirmatory test of transition timing could not be constructed, and diffusion is not identifiable at typical sampling intervals. The released software reports these diagnostics.