Finite separated transverse events give an event-aligned hybrid trajectory error bound

Consider exact and reported hybrid trajectories with the same initial data and a common finite sequence of transitions at exact event times ${\tau }_1<\cdots <{\tau }_N$ and reported times ${\widehat{\tau }}_1<\cdots <{\widehat{\tau }}_N$. Assume the events are separated by $\delta >0$, are transverse with margins ${\nu }_j>0$, and the reported event times are upper endpoints of certified root boxes, so ${\tau }_j\le {\widehat{\tau }}_j$. After each event, both trajectories are compared at the reported event time, when both are in the outgoing mode. Let $E_j$ bound their state discrepancy at this aligned mode-segment start, with $E_0=0$.

For mode segment $j$, let $U_j$ be the accumulated piecewise dense reconstruction from the reported start, with a certified local error bound ${\eta }_j^{\mathrm{l}\mathrm{o}\mathrm{c}}$ relative to the exact mode solution from that same start. Let $L_j$ be a state-Lipschitz bound, $H_j$ the segment duration, and $L_{g,j},{\epsilon }_j$ the guard state-Lipschitz and reported guard-residual bounds. Let $D_j$ be a certified total error bound between $U_j$ and the exact reference incoming-mode continuation on the prefix. It may be supplied directly; a sufficient ledger calculation is $D_j:={\eta }_j^{\mathrm{l}\mathrm{o}\mathrm{c}}+e^{L_j{H}_j}{E}_j,$ when the two exact mode solutions share the aligned start time, their initial discrepancy is at most $E_j$, and both remain in a common convex tube on which the field is $L_j$-Lipschitz. Then $D_j$ bounds the reference error on the incoming-mode prefix and ${\mathrm{\Delta }}_{j+1}:=\frac{L_{g,j}{D}_j+{\epsilon }_j}{{\nu }_{j+1}}$ bounds ${\widehat{\tau }}_{j+1}-{\tau }_{j+1}$. Assume ${\mathrm{\Delta }}_{j+1}<\delta \mathrm{/}3$ for each event; the event separation then ensures no later exact event occurs before the reported reset time. At each event assume the reset has state and time Lipschitz constants $L_{R,j+1},L_{R,t,j+1}$ on a domain containing both compared reset inputs and the coordinate segments joining them, the incoming field is bounded by $M_j^{-}$ over the event-time bracket, and both outgoing solutions remain through their alignment time in a common tube where the outgoing field is bounded by $M_{j+1}^{+}$ and is $L_{j+1}^{+}$-Lipschitz in state. If the reported reset value is approximate, let ${\epsilon }_{R,j+1}\ge 0$ be a certified bound on its evaluation error, with ${\epsilon }_{R,j+1}=0$ for exact evaluation. Then $E_{j+1}:=L_{R,j+1}{D}_j+(L_{R,j+1}{M}_j^{-}+L_{R,t,j+1}+M_{j+1}^{+}){\mathrm{\Delta }}_{j+1}+{\epsilon }_{R,j+1}$ bounds the state discrepancy at the next aligned mode-segment start. The terminal segment has its supplied total bound $D_N$; when the sufficient common-tube ledger calculation applies, one may take $D_N={\eta }_N^{\mathrm{l}\mathrm{o}\mathrm{c}}+e^{L_N{H}_N}{E}_N$. These finite recurrences bound state errors on matched smooth-mode segments and reset states at aligned times; they make no same-clock state claim between an exact jump and its later reported jump.