Residual Layers and Flow Calls Along a Common Path

Residual layers and time-conditioned flow calls both build transformations through local updates, but more updates can encode a richer process or merely resolve the same process more finely. We compare them by the full path of intermediate transformations. For compactly supported, time-Lipschitz fields in fixed dimension d and spatial regularity s∈1,2, we prove a sharp worst-case all-prefix information rate $\tilde{\Theta }({\epsilon }^{-\max \{d/s,(1+d/s)/2\}})$ on sufficiently resolved Euler grids. When d≥ s, the path and endpoint have the same leading information power, while full-field reconstruction costs more. A finite time-conditioned network attains the path rate with its implementation costs counted; its encoded path can also be traversed with O(ε^-1/2) aligned calls and linear interpolation readout on the original grid. Conditional execution bounds and controlled examples show how call placement affects path error. The distinction separates information stored about a process from the resolution used to execute it.