The finite modified Hamiltonian has a bounded one-step defect

Since an instance has $z\in K^{\mathrm{\prime }}$, assume $K^{\mathrm{\prime }}\mathrm{\ne }\mathrm{\varnothing }$; otherwise the estimate is vacuous. Set $K=K^{\mathrm{\prime }}$. The local method construction in A finite formal logarithm has Hamiltonian coefficients on a star-shaped domain gives smooth step branches on a neighborhood of $K$ for uniformly small $\mathrm{\mid }h\mathrm{\mid }$: cover $K$ by finitely many smaller convex balls whose enlarged closed balls lie in $U$, and take the minimum of their positive step bounds. For midpoint, the construction identifies every admitted ball with the same local branch by its common Picard sequence. The enlarged balls control all method stages and spatial Taylor segments. For the modified flow, apply [F3] to the smooth augmented field $(w,s)\mapsto (X_{H_s^{[N]}}(w),0)$ on $U\times \mathbb{R}$. Its open flow domain contains $\{0\}\times K\times [-1,1]$; compactness gives a common interval $[-\delta ,\delta ]$ of existence, with $\delta >0$. The projected flow image of this compact time-state-parameter product is a compact subset of $U$. Together with the finitely many enlarged balls it gives a compact tube $K^{\mathrm{\prime }\mathrm{\prime }}⋐U$ containing the trajectories and all Taylor segments; the latter are taken within individual convex balls, without assuming $K^{\mathrm{\prime }}$ convex. After decreasing the common step bound to at most $\min (\delta ,1)$, [F1] applies with initial set $K$ and control tube $K^{\mathrm{\prime }\mathrm{\prime }}$, giving one $C_N$ independent of $z\in K^{\mathrm{\prime }}$ and $h$. The stipulated endpoint segment in [A1] remains in the original $K^{\mathrm{\prime }}$ for the scalar energy comparison.

By [F3], the modified Hamiltonian vector field has a local flow from $z$. By [F2], that flow preserves $H_h^{[N]}$, so $H_h^{[N]}({\phi }_h^{[N]}(z))=H_h^{[N]}(z)$.

Parameterize the segment in [A1] by $\gamma (t)=H_h^{[N]}({\phi }_h^{[N]}(z)+t({\mathrm{\Phi }}_h(z)-{\phi }_h^{[N]}(z)))$. By [F4] and [F6], $\mathrm{\mid }{\gamma }^{\mathrm{\prime }}(t)\mathrm{\mid }\le G_N\mathrm{\parallel }{\mathrm{\Phi }}_h(z)-{\phi }_h^{[N]}(z)\mathrm{\parallel }$; [F5] then bounds the endpoint difference by the same quantity. Thus $\mathrm{\mid }{H}_h^{[N]}({\mathrm{\Phi }}_h(z))-H_h^{[N]}({\phi }_h^{[N]}(z))\mathrm{\mid }\le G_N\mathrm{\parallel }{\mathrm{\Phi }}_h(z)-{\phi }_h^{[N]}(z)\mathrm{\parallel }\le G_N{C}_N{h}^{N+2}$ by [F1]. For $h\le 1$ this is at most $G_N{C}_N{h}^{N+1}$. Substituting the equality from step 1.2 proves the statement with $C_1=G_N{C}_N$.