Block variation bounds for a frozen-slow Euler step

Given: The dimensions, macro and micro steps, field bound, exact solution, and admissible microstates in the Statement.

View the autonomous field as $\widehat{F}(t,u):=F(u)$ on $\mathbb{R}\times \mathrm{\Omega }$. An exact IVP solution then satisfies its Volterra equation on every subinterval, with the value at the left endpoint as initial data (First-order systems, initial value problems, and solutions on intervals, A first-order initial value problem is equivalent to its Volterra integral equation).

For an integrable vector field on an increasing interval, $\mathrm{\parallel }{\int }_a^{b}V(t)dt{\mathrm{\parallel }}_2\le {\int }_a^b\mathrm{\parallel }V(t){\mathrm{\parallel }}_2$ (For $a\le b$ and $f:[a,b]\to {\mathbb{R}}^m$ integrable when $a<b$, $\parallel {\int }_a^{b}f{\parallel }_2\le {\int }_a^b\parallel f{\parallel }_2$; for $a<b$, $\parallel f{\parallel }_2$ is integrable).

At each fast-loop iteration whose pre-update state $(x_0,z_j)$ lies in $D$, the procedure sets $z_{j+1}=z_j+hg(x_0,z_j)$, with the same frozen $x_0$ at every microstep (Frozen-slow explicit Euler with fast microsteps). If all $m$ pre-update states lie in $D$, all loop updates are performed even when the final state $(x^{+},z_m)$ is outside $D$ and the procedure consequently returns domain-exit.

If an integrable real-valued function $w$ satisfies $0\le w\le C$ on $[a,b]$, then ${\int }_a^{b}w\le C(b-a)$ (If $f\le g$ on $[a,b]$ and both are integrable then ${\int }_a^{b}f\le {\int }_a^{b}g$; and $m(b-a)\le {\int }_a^{b}f\le M(b-a)$, claim 3).

$U([0,H])\subseteq S$ and every microstate $(x_0,z_j)$ belongs to $S$, where $F$ is bounded by $M$.