Almost Sure Tracking on the Synthetic Copy: Jump Times of the Deterministic-Count Clocks, Almost Sure Conflict-Freeness of Every Record, Almost Sure Null Mass of the Untracked Records, and Trimming an Event to the Tracked Set

Adopt the setting and notation of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record (and hence of The Record-Driven Causal Intensity of the Open-Loop Aggregate Solution: Joint Measurability of the Record-Frozen Control, the Regularised Recursion Path, Non-Anticipation, and Measurability of the Likelihood and Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution): the natural numbers $N\ge1$, $l\ge2$, $m\ge1$, $\tilde{l}\ge1$, the real numbers $B\ge0$, $T>0$ and $R>0$ with $R\ge NBT$, the probability space $(\Omega,\mathcal{F},P)$ carrying the independent family of driving variables $K^{c}$, $V^{c}_i$, $U^{c,j}_i$, the cells $I_{c,j}=(b^{c}_{j-1},b^{c}_j]\subseteq(0,R]$ (with $0=b^{c}_0<b^{c}_1<\dots<b^{c}_{J_c}=R$ and natural numbers $J_c\ge1$) indexed by the finite set $\mathsf{L}$ of pairs $(c,j)$ with $c$ a transition label and $1\le j\le J_c$, the $\sigma$-algebra $\mathcal{U}$ generated by all $U^{c,j}_i$, the event $\Omega^{U}_0$, the set $\mathbb{N}_0^{\mathsf{L}}$ with its vectors $e_{c,j}$, the cell-count vector $\mathsf{K}=(\mathsf{K}_{c,j})_{(c,j)\in\mathsf{L}}$ with values in $\mathbb{N}_0^{\mathsf{L}}$, the deterministic-count clocks $\mathsf{P}^{(y)}=(\mathsf{P}^{(y),c})_c$ ($y\in\mathbb{N}_0^{\mathsf{L}}$) and the copy clocks $\mathsf{P}^{\sharp}$, the observation record space $(\mathbf{R},\mathcal{R},\rho)$ with horizon $T$ and $\tilde{l}$ channels (with $\rho$ $\sigma$-finite), the record-frozen control paths $a^{r}$ ($r\in\mathbf{R}$) of the policy $h$, the aggregate lattice $\mathbb{G}_N$ and the point $x_0\in\mathbb{G}_N$, and the conflict-free sets $\mathsf{G}^{(y)}\subseteq\mathbf{R}\times\Omega$ and $\mathsf{G}^{\sharp}\subseteq\mathbf{R}\times\Omega$ of claim 3 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record: writing $\mathsf{P}^{(y)}(\omega)$ for the clock family $(u\mapsto\mathsf{P}^{(y),c}_u(\omega))_c$ and $\mathsf{P}^{(y),c}(\omega)$ for its $c$-component, a pair $(r,\omega)$ lies in $\mathsf{G}^{(y)}$ if and only if the data $(\mathsf{P}^{(y)}(\omega),a^{r},x_0)$ are conflict-free in the sense of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution, and similarly for $\mathsf{G}^{\sharp}$ with the clock family $\mathsf{P}^{\sharp}(\omega)$. Jump times and $k$-th jump times $\tau_k(\cdot)$ of counting paths are those of Counting Path and Its Jump Times. Let $\mathsf{m}\ge1$ be a natural number (the move size), and for $\omega\in\Omega$ let the set of tracked records $\mathsf{T}_\omega$ be the set of all $r\in\mathbf{R}$ with $(r,\omega)\in\mathsf{G}^{\sharp}$ and $(r,\omega)\in\mathsf{G}^{(\mathsf{K}(\omega)-\mathsf{m}e_{c,j})}$ for every $(c,j)\in\mathsf{L}$ with $\mathsf{K}_{c,j}(\omega)\ge\mathsf{m}$ (the tracked records of Uniform Pair-Exponent Bound on the Synthetic Copy: Removed-Clock Intensities, the Insertion Response on the Clock-Good Event, and Bounds on the Pair Covariances).

1. (Jump times of the deterministic-count clocks) Let $\omega\in\Omega^{U}_0$, $y\in\mathbb{N}_0^{\mathsf{L}}$ and $c$ a transition label. A real number $u>0$ is a jump time of the counting path $u'\mapsto\mathsf{P}^{(y),c}_{u'}(\omega)$ if and only if $u=U^{c,j}_i(\omega)$ for some $j\in\{1,\dots,J_c\}$ and some $i\in\{1,\dots,y_{c,j}\}$; and for every natural number $n$ with $\tau_n(\mathsf{P}^{(y),c}(\omega))<+\infty$, the number $\tau_n(\mathsf{P}^{(y),c}(\omega))$ is a jump time of that path.

2. (Every record is almost surely conflict-free) For every $y\in\mathbb{N}_0^{\mathsf{L}}$ and every $r\in\mathbf{R}$, the set $\Omega^{r,y}=\{\omega\in\Omega:\ (r,\omega)\notin\mathsf{G}^{(y)}\}$ belongs to $\mathcal{U}$ and $P(\Omega^{r,y})=0$.

3. (Almost surely, almost every record is tracked) For $y\in\mathbb{N}_0^{\mathsf{L}}$ and $\omega\in\Omega$ the set $B^{(y)}_\omega=\{r\in\mathbf{R}:\ (r,\omega)\notin\mathsf{G}^{(y)}\}$ belongs to $\mathcal{R}$; the set $\Omega_y=\{\omega\in\Omega:\ \rho(B^{(y)}_\omega)=0\}$ belongs to $\mathcal{U}$ and $P(\Omega_y)=1$; the set $\Omega'=\bigcap_{y\in\mathbb{N}_0^{\mathsf{L}}}\Omega_y$ belongs to $\mathcal{U}$ and $P(\Omega')=1$; and for every $\omega\in\Omega'$ one has $\mathbf{R}\setminus\mathsf{T}_\omega\in\mathcal{R}$ and $\rho(\mathbf{R}\setminus\mathsf{T}_\omega)=0$.

4. (Trimming an event to the tracked set) For every event $G_0\in\mathcal{F}$ the event $G=G_0\cap\Omega'$ belongs to $\mathcal{F}$, satisfies $\rho(\mathbf{R}\setminus\mathsf{T}_\omega)=0$ for every $\omega\in G$, and $P(\Omega\setminus G)=P(\Omega\setminus G_0)$.