Analytic pendulum: checking the complex bounds and compact-iterate hypothesis

Analytic-data check. Both functions extend entirely. On $D(1)$ around $\bar{U}$, $\mathrm{\mid }p\mathrm{\mid }<5$ and $\mathrm{\mid }\mathrm{Im}q\mathrm{\mid }<1$. Since $\mathrm{\mid }\sin (x+iy){\mathrm{\mid }}^2={\sin }^{2}x+{\sinh }^{2}y\le {\cosh }^2(1)$, $\mathrm{\parallel }{X}_A\mathrm{\parallel }\le \cosh (1)$ and $\mathrm{\parallel }{X}_B\mathrm{\parallel }<5$, both bounded by $M$. The sets are star-shaped about zero, $K$ is compact in $U$, and its real boundary margin is $2$. The analytic labels are potential first, kinetic second, so the composition is kick-drift-kick.

Consequence and perturbations. Choose the theorem's sufficiently small $h_0$ also with $h_0\le 1\mathrm{/}(2M)$, using the computed margin $\delta =2$. Its mechanical-stage argument bounds every full internal segment by $2Mh\le 1$ from its retained starting point in $K$, so all stages lie in $B_3(0)\subset U$ and the analytic splitting equals velocity Verlet. All analytic-data hypotheses now hold, so [F1] gives the stated energy conclusion if numerical retention holds. If computed points $x_j\in K$ additionally have per-step state errors at most $\eta$, and every comparison segment from $x_{j+1}$ to ${\mathrm{\Psi }}_h(x_j)$ and from ${\mathrm{\Psi }}_h(x_j)$ to the modified-flow endpoint lies in $D(1\mathrm{/}8)$, the estimate instead reads $\mathrm{\mid }H(x_n)-H(x_0)\mathrm{\mid }\le 2C_E{h}^2+Cnh{e}^{-c\mathrm{/}h}+C_{G}n\eta$. Neither conclusion certifies retention or predicts empirical machine error.