Embryo lineages don't take the least-action route: after removing the centroid measurement floor, worm paths still run about twice as costly

How far developmental trajectories depart from the routes inferred dynamics consider natural now has a controlled measurement: C. elegans lineage-centroid paths carry about 3.79 times the action of the minimum-action path between the same endpoints. Earlier readings of development as least-action or variational free-energy minimization assumed an optimality principle rather than computing the optimal path through learned dynamics and comparing it, with controls, to the paths embryos actually take. Across 179 C. elegans lineage-branch paths the median observed-to-minimum action ratio is 3.79; over 17 model configurations the grand median is 3.79 with a nested-bootstrap 95% CI of [3.56, 4.04], and every configuration median exceeds 3.1.

That ratio cannot be read directly as suboptimal development, because taking the centroid of a spreading, bifurcating ensemble carries a generic action cost unrelated to the route any cell takes. Without calibration the same pipeline would report development as roughly 4–9 times suboptimal in both species, mostly artifact in one and entirely artifact in the other. Forward-simulating 2,000 particles per branch under the learned field gives a centroid ratio of 2.07 (IQR 1.61–2.39); observed paths sit significantly above this floor (median per-path fold 1.96, Wilcoxon p = 4×10⁻³¹) and significantly below random-partition centroids at 4.77 (p = 3×10⁻⁵).

The roughly two-fold excess that survives calibration in the worm comes from lineages moving faster than the least-action route, not from detours or fighting the flow. A geometry-versus-timing decomposition assigns 87% of the log-excess to the geometric factor, but set-level route comparisons cannot separate a different route from the same route at a different pace, so speed had to be tested directly. Observed paths traverse state space 43% faster than minimum-action paths between the same endpoints (Wilcoxon p = 1×10⁻²⁷), pass through slightly weaker-field regions (median |f| 1.80 vs 1.92, p = 8×10⁻⁹), and have a positive signed field projection indistinguishable from minimum-action paths (p = 0.67); straight-line routes move against the field in 93% of steps.

Zebrafish replication yields a boundary rather than the same conclusion: post-epiboly observed ratios sit at the model's own ensemble floor, so the excess there is attributed to centroid geometry. The worm has an invariant, highly deterministic lineage, whereas the zebrafish atlas has stage-level time resolution and defines paths from URD branches rather than invariant lineages, so raw magnitudes are not directly comparable. For 25 zebrafish post-epiboly branches the observed ratio is 7.95 against a forward-simulated ensemble floor of 10.03 (p = 5×10⁻³) and is indistinguishable from random partitions; on a uniform 1-hour grid the worm gives 2.30 and zebrafish 2.41.

The demonstrated scope is 179 lineage-branch centroid paths from 87,702 C. elegans cells over 150–750 min in a 50-dimensional PCA latent space, plus 25 URD branch paths from 38,731 zebrafish cells over the post-epiboly 4.7–12 h window; the action is defined relative to the learned autonomous dynamics and is not a measure of fitness or energetic cost. The authors propose applying the framework to frog and mouse atlases next, provided comparisons use matched resolution against calibrated floors.

The authors state that the calibrated claim is a measurement, not a mechanism: it establishes a path-specific excess and localizes its immediate correlate to speed, while whether that excess is selected or tolerated, and its single-cell-resolution form, remain open. The ratio grows monotonically with latent dimensionality (about 3.2 at 20-D, 3.79 at 50-D, 5.0–5.5 at 100-D), and the 50-D choice rests on action-independent forecasting and lineage-coherence criteria. The energetic reading is an order-of-magnitude calibration: transcriptional remodeling coupled to the action rate is about 0.1% of the embryonic ATP budget, but translation-coupled costs reach 10–50% of turnover, so energetics cannot be excluded.