The Copy Clocks Are Independent Poisson Clocks with Horizon R, and Every Record Is Almost Surely Conflict-Free for Them

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, as recalled in The Copy Clocks Have Independent Poisson Increments on the Clock Interval: Law Identity with the Uniform Poisson Path and Applicability of the Window Discrepancy Bound: 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 natural numbers $J_c\ge1$ and the finite index set $\mathsf{L}$ of pairs $(c,j)$ with $c$ a transition label and $1\le j\le J_c$, the countable set $\mathbb{N}_0^{\mathsf{L}}$ of cell-count vectors, the probability space $(\Omega,\mathcal{F},P)$ carrying the independent family of driving variables $K^{c}$, $V^{c}_i$, $U^{c,j}_i$ ($c$ ranging over the transition labels, $i\in\mathbb{N}$, $(c,j)\in\mathsf{L}$), the cells $I_{c,j}\subseteq(0,R]$, the event $\Omega^{U}_0$, the cell-count vector $\mathsf{K}$ with coordinates $\mathsf{K}_{c,j}$, the deterministic-count clocks $\mathsf{P}^{(y)}=(\mathsf{P}^{(y),c})_c$ ($y\in\mathbb{N}_0^{\mathsf{L}}$) with $\mathsf{P}^{(y),c}_u=\mathbf{1}_{\Omega^{U}_0}\sum_{j=1}^{J_c}\sum_{i=1}^{y_{c,j}}\mathbf{1}\{U^{c,j}_i\le u\}$, and the copy clocks $\mathsf{P}^{\sharp}=(\mathsf{P}^{\sharp,c})_c$, $\mathsf{P}^{\sharp,c}_u(\omega)=\mathsf{P}^{(\mathsf{K}(\omega)),c}_u(\omega)$, that is, $\mathsf{P}^{\sharp,c}_u=\mathbf{1}_{\Omega^{U}_0}\sum_{j=1}^{J_c}\sum_{i=1}^{\mathsf{K}_{c,j}}\mathbf{1}\{U^{c,j}_i\le u\}$ for $u\ge0$; together with the observation-record objects of that setting, namely the record space $\mathbf{R}$, the record-frozen control paths $a^{r}$ ($r\in\mathbf{R}$), the initial point $x_0$ in the aggregate lattice $\mathbb{G}_N$, and the conflict-free sets $\mathsf{G}^{(y)}$ and $\mathsf{G}^{\sharp}$ 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, recalled in 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 (a pair $(r,\omega)$ lies in $\mathsf{G}^{\sharp}$ if and only if the data $(\mathsf{P}^{\sharp}(\omega),a^{r},x_0)$ are conflict-free in the sense of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution, and similarly for $\mathsf{G}^{(y)}$ with $\mathsf{P}^{(y)}(\omega)$). Then:

1. (Counting paths, constant beyond $R$.) For every label $c$ and every $\omega\in\Omega$ the path $u\mapsto\mathsf{P}^{\sharp,c}_u(\omega)$ is a counting path on $[0,\infty)$ which is constant on $[R,\infty)$.

2. (Poisson clocks with horizon $R$.) For every label $c$, the process $\mathsf{P}^{\sharp,c}=(\mathsf{P}^{\sharp,c}_u)_{u\ge0}$ is a Poisson clock with horizon $R$ on $(\Omega,\mathcal{F},P)$.

3. (Independence across labels.) The family of $\sigma$-algebras $\sigma(\mathsf{P}^{\sharp,c}_u:u\ge0)$, indexed by the transition labels $c$, is independent.

4. (Every record is almost surely conflict-free for the copy clocks.) For every $r\in\mathbf{R}$ the set $\Omega^{r,\sharp}=\{\omega\in\Omega:(r,\omega)\notin\mathsf{G}^{\sharp}\}$ is an event with $P(\Omega^{r,\sharp})=0$.