Davie discrete sewing lemma ($a_{s,t}\le C\omega (s,t{)}^{\theta },\theta >1$)

For a finite time grid, suppose a nonnegative error is zero on neighboring points and obeys $a_{s,t}\le a_{s,u}+a_{u,t}+L\omega (s,t{)}^{1/p}{a}_{s,u}+A\omega (s,t{)}^{\theta }$. For a control function for rough paths and $\theta >1$, sufficiently small control intervals have the displayed bound with constants independent of the grid. Splitting near half the control size makes the higher-order errors contract, and handles a crossing single grid interval by its zero error. Applying this to local Euler composition errors yields compactness of approximate pathlevel solutions.