Proof of Probability of the Good Event on the Synthetic Copy: Poisson Tail for the Cell Counts, the Window Discrepancy Bound for the Clock-Good Event, and the Bound on the Complement of the Good Event

We use claims 2 and 4 of Basic Properties of a Measure (monotonicity and countable subadditivity of $P$, a finite union being a sequence padded with the empty set) and claim 3 of that lemma in the form $P(\Omega\setminus A)=1-P(A)$.

Step 1 (claim 1). Let $q=(c,j)\in\mathsf{L}$. By The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record, for the label $c$ the variables $K^{c}$, $(V^{c}_i)_i$, $(U^{c,j'}_i)_{j',i}$ with the cells $I_{c,1},\dots,I_{c,J_c}$ form the data of Uniform Representation of the Rate-One Poisson Counting Path on a Bounded Interval: Distinct Points, Poisson Increments, Conditional Law Given the Cell Counts, and Point Insertion, and $\mathsf{K}_{c,j}$ is the cell count $C_j$ of that lemma; by its claim 2, $C_j$ has the Poisson distribution with parameter $|I_j|=|I_{c,j}|=\mu_q$. Now let $\mathsf{m}<\mu_q$ and put $x=\mu_q-\mathsf{m}>0$. Claim 3 of Series Formula, Exponential Moments, and Chernoff Tail Bounds for the Poisson Distribution, applied to the Poisson variable $\mathsf{K}_q$ with parameter $\mu_q$, gives $P(\mathsf{K}_q\le\mu_q-x)\le\exp(-\varpi_{\mu_q}(x))$, and $\mu_q-x=\mathsf{m}$. Since $\{\mathsf{K}_q<\mathsf{m}\}\subseteq\{\mathsf{K}_q\le\mathsf{m}\}$, monotonicity gives $P(\mathsf{K}_q<\mathsf{m})\le\exp(-\varpi_{\mu_q}(\mu_q-\mathsf{m}))$.

Step 2 (claim 2). Assume $R\in\mathbb{N}$, $M\in\mathbb{N}$ with $L\le M$, and $D>2$; put $x=D-2>0$. For each label $c\in\mathcal{L}$ we construct an event $G_c$ as follows. By claim 3 of 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 with $n=R$ (a natural number with $n\le R$), the process $\mathsf{P}^{\sharp,c}$ satisfies the hypotheses of Uniform Window Discrepancy Bound for a Counting Process with Poisson Increments on the Integer Grid with $n=R$; apply that lemma with its window length taken to be $M$ and with this $x$. It yields an event $G_c\in\mathcal{F}$ (it lies in the $\sigma$-algebra generated by $\mathsf{P}^{\sharp,c}_0,\dots,\mathsf{P}^{\sharp,c}_R$, a sub-$\sigma$-algebra of $\mathcal{F}$) with $P(\Omega\setminus G_c)\le2\,(R+1)(M+3)\exp\bigl(-\varpi_{M+2}(D-2)\bigr)$ such that for every $\omega\in G_c$ and all real $0\le u\le u'\le R$ with $u'-u\le M$, $\bigl|\mathsf{P}^{\sharp,c}_{u'}(\omega)-\mathsf{P}^{\sharp,c}_{u}(\omega)-(u'-u)\bigr|\le x+2=D .$ Now let $\omega\in\Omega^{U}_0\cap\bigcap_{c\in\mathcal{L}}G_c$. For every label $c$ and all real $0\le u\le u'\le R$ with $u'-u\le L$ we have $u'-u\le M$, so the displayed bound holds; by the definition of the clock-good event in 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 (the set of $\omega\in\Omega^{U}_0$ satisfying exactly these bounds), $\omega\in G_{L,D}$. Hence $\Omega\setminus G_{L,D}\subseteq(\Omega\setminus\Omega^{U}_0)\cup\bigcup_{c\in\mathcal{L}}(\Omega\setminus G_c).$ The set $G_{L,D}$ belongs to $\mathcal{F}$ by claim 1 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, and $P(\Omega\setminus\Omega^{U}_0)=1-P(\Omega^{U}_0)=0$ by claim 1 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record. Monotonicity and subadditivity over the $l(l-1)$ labels give $P(\Omega\setminus G_{L,D})\le0+\sum_{c\in\mathcal{L}}P(\Omega\setminus G_c)\le2\,l(l-1)\,(R+1)(M+3)\exp\bigl(-\varpi_{M+2}(D-2)\bigr).$

Step 3 (claim 3). Each $\mathsf{K}_q$ is a random variable (claim 1 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record), so $\{\mathsf{K}_q\ge\mathsf{m}\}\in\mathcal{F}$, and $G^{\mathsf{m}}=G_{L,D}\cap\bigcap_{q\in\mathsf{L}}\{\mathsf{K}_q\ge\mathsf{m}\}$ is a finite intersection of members of $\mathcal{F}$, hence in $\mathcal{F}$. Its complement is $\Omega\setminus G^{\mathsf{m}}=(\Omega\setminus G_{L,D})\cup\bigcup_{q\in\mathsf{L}}\{\mathsf{K}_q<\mathsf{m}\},$ so by subadditivity, claim 1 (applicable since $\mathsf{m}<\mu_q$ for every $q$) and claim 2, $P(\Omega\setminus G^{\mathsf{m}})\le P(\Omega\setminus G_{L,D})+\sum_{q\in\mathsf{L}}P(\mathsf{K}_q<\mathsf{m})\le2\,l(l-1)\,(R+1)(M+3)\exp\bigl(-\varpi_{M+2}(D-2)\bigr)+\sum_{q\in\mathsf{L}}\exp\bigl(-\varpi_{\mu_q}(\mu_q-\mathsf{m})\bigr).$