A fixed RATTLE residual budget allows an O(epsilon) drift envelope

Fix compact phase tubes ${\mathcal{C}}_0\subset \mathrm{relint}({\mathcal{C}}_1)$ in a convex phase-coordinate chart with closure inside a regular chart, and $T>0$. Put $h=T\mathrm{/}N$ and suppose the exact-solve RATTLE iterates from the common initial state stay in ${\mathcal{C}}_0$ for the grids considered. Suppose every finite RATTLE solve succeeds, its branch iterates stay in the selected local balls over ${\mathcal{C}}_1$, and its scaled residuals are bounded by a fixed sufficiently small $\epsilon >0$. After each output apply the consistency retraction from Holonomic phase residuals control distance to the consistent phase manifold before the next step. For sufficiently small $h$, the restored trajectory stays in ${\mathcal{C}}_1$ and has an $O(T\epsilon )$ global error envelope relative to exact-solve RATTLE; refinement alone does not force this upper bound to vanish. If both scaled residual tolerances are instead $O(h^2)$, the global solver contribution is $O(h^2)$, while the unprojected solver outputs have position and tangency residuals $O(h^4)$ and $O(h^3)$. These are upper bounds; they do not assert a matching lower bound or that residual tolerances alone certify forward error.