Reset and smooth-mode flow propagate a transverse event-time error

Let $\tau$ be an exact event time and $\widehat{\tau }$ its reported time, with $\mathrm{\Delta }=\mathrm{\mid }\widehat{\tau }-\tau \mathrm{\mid }$. On the interval $J$ between these times, suppose the incoming exact mode extension $y^{-}$ and dense reconstruction $U^{-}$ satisfy $\mathrm{\parallel }{U}^{-}(t)-y^{-}(t){\mathrm{\parallel }}_2\le \eta$, and $\mathrm{\parallel }{f}^{-}(t,y^{-}(t)){\mathrm{\parallel }}_2\le M^{-}$. Assume $\eta ,M^{-},M^{+},L_R,L_{R,t},L^{+}\ge 0$. Suppose the reset is defined on a neighborhood containing $(\tau ,y^{-}(\tau ))$, $(\widehat{\tau },U^{-}(\widehat{\tau }))$, and the coordinate segments joining them, with Lipschitz bounds $L_R$ in state and $L_{R,t}$ in time there. Let $y^{+}$ start from $R(\tau ,y^{-}(\tau ))$ at time $\tau$, and let ${\tilde{y}}^{+}$ start from $R(\widehat{\tau },U^{-}(\widehat{\tau }))$ at time $\widehat{\tau }$. Assume both outgoing solution pieces through $t_0=\max \{\tau ,\widehat{\tau }\}$ lie in a common convex tube where the outgoing field is bounded by $M^{+}$ and is $L^{+}$-Lipschitz in state. For $s\ge t_0$ while both outgoing solutions remain in a common convex tube on which the field is $L^{+}$-Lipschitz, we have $\mathrm{\parallel }{\tilde{y}}^{+}(t_0)-y^{+}(t_0){\mathrm{\parallel }}_2\le L_R\eta +(L_R{M}^{-}+L_{R,t}+M^{+})\mathrm{\Delta }\mathrm{.}$ For $s\ge t_0$ while both outgoing solutions remain in the tube, $\mathrm{\parallel }{\tilde{y}}^{+}(s)-y^{+}(s){\mathrm{\parallel }}_2\le e^{L^{+}(s-t_0)}[L_R\eta +(L_R{M}^{-}+L_{R,t}+M^{+})\mathrm{\Delta }]\mathrm{.}$