Proof of Clock-Reading Bound for the Aggregate Recursion: the Recursion up to a Time Depends Only on the Clocks Below the Consumed Levels

Throughout, "the recursion" for a clock family means the aggregate recursion of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution for the data $(p,a,x_0)$ with the fixed $a$ and $x_0$, and claim 1 of that theorem is used freely: each $t\mapsto\mathsf{C}^{c,(k)}_t$ is nondecreasing and continuous on $[0,T]$ with $\mathsf{C}^{c,(k)}_{\theta_k}=\kappa^{c}_k$, $\theta_{k+1}>\theta_k$, and $\kappa^{c}_{k+1}=\mathsf{C}^{c,(k)}_{\theta_{k+1}}$. Consequently the recursion consumed time $t\mapsto\mathsf{C}^{\mathrm{rec},c}_t$ is nondecreasing on $[0,T]$: it is nondecreasing on each interval $[\theta_k,\theta_{k+1}]$ ($k<K$) and constant on $[\theta_K,T]$, and consecutive pieces share their endpoint values.

Two facts about counting paths. Let $q,q'$ be counting paths agreeing on $[0,w]$ for some $w\ge0$, and let $j\ge1$ be a natural number. (F1) If $\tau_j(q)\le w$ then $\tau_j(q')=\tau_j(q)$: write $S=\{u\ge0:q(u)\ge j\}$ and $S'=\{u\ge0:q'(u)\ge j\}$, which have the same intersection with $[0,w]$; $S$ contains its greatest lower bound $\tau_j(q)$ by right-continuity and monotonicity of $q$ (Counting Path and Its Jump Times), so $\tau_j(q)\in S'$ and $\tau_j(q')\le\tau_j(q)$; and $S'\cap[0,\tau_j(q))=S\cap[0,\tau_j(q))=\emptyset$, so $\tau_j(q')\ge\tau_j(q)$. (F2) If $\tau_j(q)>w$ then $\tau_j(q')>w$: $q'(w)=q(w)<j$, and $q'$ is right-continuous and integer-valued, so $q'<j$ on some interval $[w,w+\varepsilon)$ and therefore $\tau_j(q')\ge w+\varepsilon>w$.

Claim 1. Let $p,p',\mathbf{w},r$ be as in the claim. We prove by induction on $k$ the statement $S(k)$: if $k\le K$ and $\theta_k\le r$, then $k\le K'$, $\theta'_k=\theta_k$, $x'^{(k)}=x^{(k)}$, and $\kappa'^{c}_k=\kappa^{c}_k$ for every $c$. $S(0)$ holds since both recursions start from $\theta_0=0$, $x_0$, and $\kappa^{c}_0=0$. Assume $S(k)$ and let $k<K$ with $\theta_{k+1}\le r$; we prove $S(k+1)$. Since $k<K$, step $k$ is performed by the $p$-recursion, so $\theta_k<T$ and $x^{(k)}\in\mathbb{G}_N$; as $\theta_k<\theta_{k+1}\le r$, $S(k)$ applies, the same holds for the primed data, and step $k$ is performed by the $p'$-recursion too. Both steps use the same functions $\mathsf{C}^{c,(k)}$ (which depend only on $\theta_k$, $x^{(k)}$, $\kappa^{c}_k$ and $a$). For every label $c$, $\kappa^{c}_k=\mathsf{C}^{\mathrm{rec},c}_{\theta_k}\le\mathsf{C}^{\mathrm{rec},c}_r\le w_c$ by monotonicity, so $p^{c}(\kappa^{c}_k)=p'^{c}(\kappa^{c}_k)$ and the index $j=p^{c}(\kappa^{c}_k)+1$ is the same for both; call the label $c$ small if $\lambda^{c}_k=\tau_j(p^{c})\le w_c$ and large otherwise. By (F1), $\lambda'^{c}_k=\lambda^{c}_k$ for small labels, and by (F2), $\lambda'^{c}_k>w_c$ for large labels. Moreover, for every $c$, $\mathsf{C}^{c,(k)}_{\theta_{k+1}}=\kappa^{c}_{k+1}=\mathsf{C}^{\mathrm{rec},c}_{\theta_{k+1}}\le\mathsf{C}^{\mathrm{rec},c}_r\le w_c$, so for a large label and every $t\in[\theta_k,\theta_{k+1}]$ one has $\mathsf{C}^{c,(k)}_t\le w_c<\min(\lambda^{c}_k,\lambda'^{c}_k)$; since the set $\{t\in[\theta_k,T]:\mathsf{C}^{c,(k)}_t\ge\lambda\}$ is closed (continuity) and so contains its greatest lower bound when nonempty, this gives $h^{c}_k>\theta_{k+1}$ and $h'^{c}_k>\theta_{k+1}$ for large labels (with the convention that $+\infty$ exceeds every real number). For small labels $h'^{c}_k=h^{c}_k$, the defining sets being identical. Now $\theta_{k+1}=\min(T,\min_ch^{c}_k)$, and the large labels do not attain this minimum, so $\theta_{k+1}=\min(T,\min_{c\ \mathrm{small}}h^{c}_k)$ (read as $T$ if there is no small label); hence

$$\theta'_{k+1}=\min\Bigl(\min\bigl(T,\min_{c\ \mathrm{small}}h'^{c}_k\bigr),\ \min_{c\ \mathrm{large}}h'^{c}_k\Bigr)=\min\Bigl(\theta_{k+1},\ \min_{c\ \mathrm{large}}h'^{c}_k\Bigr)=\theta_{k+1}.$$

The firing sets agree: a large label lies in neither $\mathcal{J}_k$ nor $\mathcal{J}'_k$, because $\mathsf{C}^{c,(k)}_{\theta_{k+1}}\le w_c$ is below both $\lambda^{c}_k$ and $\lambda'^{c}_k$, while for a small label the membership conditions coincide. Therefore $x'^{(k+1)}=x^{(k+1)}$ and $\kappa'^{c}_{k+1}=\mathsf{C}^{c,(k)}_{\theta_{k+1}}=\kappa^{c}_{k+1}$, and $k+1\le K'$ because the $p'$-recursion performed step $k$. This proves $S(k+1)$.

Stopping. If $\theta_K\le r$, then $S(K)$ gives $K\le K'$ with identical data at index $K$; the $p$-recursion stops at $K$ because $\theta_K=T$ or $x^{(K)}\notin\mathbb{G}_N$, and the same condition holds for the primed data, so the $p'$-recursion stops at $K$ as well (it cannot have stopped earlier, since $K\le K'$): $K'=K$.

Agreement on $[0,r]$. Let $k$ be the largest index with $k\le K$ and $\theta_k\le r$. If $k=K$, then $K'=K$ and all step data agree, so $\Sigma^{\mathrm{rec}}$ and $\Sigma'^{\mathrm{rec}}$ agree on $[0,T]$ (both are $x^{(i)}$ on $[\theta_i,\theta_{i+1})$ for $i<K$ and $x^{(K)}$ on $[\theta_K,T]$), and likewise the recursion consumed times agree on $[0,T]$. If $k<K$, then $\theta_{k+1}>r$, so on $[\theta_k,r]$ the $p$-recursion path equals $x^{(k)}$ and its consumed times equal $\mathsf{C}^{c,(k)}$; we show $\theta'_{k+1}>r$, after which the same description holds for the primed recursion on $[\theta_k,r]$ (and on $[0,\theta_k)$ the paths and consumed times agree by $S(0),\dots,S(k)$). Indeed $r<T$ (as $\theta_{k+1}\le T$), and step $k$ is performed by both recursions with the same $\mathsf{C}^{c,(k)}$ and the same small/large classification as above; for a small label $h'^{c}_k=h^{c}_k\ge\theta_{k+1}>r$; for a large label, every $t\in[\theta_k,r]$ has $\mathsf{C}^{c,(k)}_t\le\mathsf{C}^{c,(k)}_r=\mathsf{C}^{\mathrm{rec},c}_r\le w_c<\lambda'^{c}_k$, so $h'^{c}_k>r$ by the closedness argument; hence $\theta'_{k+1}=\min(T,\min_ch'^{c}_k)>r$. Finally, for $s\in[0,r]$, $\mathsf{C}^{\mathrm{rec},c}_s\le\mathsf{C}^{\mathrm{rec},c}_r\le w_c$, so $\mathsf{N}^{\mathrm{rec},c}_s=p^{c}(\mathsf{C}^{\mathrm{rec},c}_s)=p'^{c}(\mathsf{C}^{\mathrm{rec},c}_s)=\mathsf{N}'^{\mathrm{rec},c}_s$. The final assertion ($\mathsf{C}'^{\mathrm{rec},c}_r\le w_c$) is the case $s=r$ of the agreement of consumed times.

Claim 2. Fix $r$ and $\mathbf{w}$. Define the stopped clocks $\bar{\mathsf{P}}^{c}_u=\mathsf{P}^{c}_{\min(u,w_c)}$ ($u\ge0$). Each path of $\bar{\mathsf{P}}^{c}$ is a counting path (it agrees with the counting path of $\mathsf{P}^{c}$ on $[0,w_c]$ and is constant afterwards, which preserves the four properties of Counting Path and Its Jump Times), each $\bar{\mathsf{P}}^{c}_u$ is $\mathcal{K}_{\mathbf{w}}$-measurable, and $\bar{\mathsf{P}}^{c}(\omega)$ agrees with $\mathsf{P}^{c}(\omega)$ on $[0,w_c]$ for every $\omega$. Run the recursion for the data $(\bar{\mathsf{P}}(\omega),a,x_0)$ and mark its quantities with a bar. By claim 4 of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution, applied to the family $\bar{\mathsf{P}}$ with a one-point parameter space, each $\bar{\Sigma}^{\gamma}_s$, $\bar{\mathsf{C}}^{c}_s$ and $\bar{\mathsf{N}}^{c}_s$ is measurable with respect to the $\sigma$-algebra generated by the variables $\bar{\mathsf{P}}^{c}_u$, $u\in[0,NBs]$, hence with respect to $\mathcal{K}_{\mathbf{w}}$.

(a) By claim 1 applied at each $\omega$ with $p=\mathsf{P}(\omega)$ and $p'=\bar{\mathsf{P}}(\omega)$, on $C_{\mathbf{w}}$ one has $\bar{\mathsf{C}}^{c}_r=\mathsf{C}^{c}_r\le w_c$ for every $c$, so $C_{\mathbf{w}}\subseteq\bar{C}_{\mathbf{w}}:=\{\bar{\mathsf{C}}^{c}_r\le w_c\ \text{for every }c\}$; and by claim 1 with the roles of the two clock families exchanged (its hypothesis is symmetric), $\bar{C}_{\mathbf{w}}\subseteq C_{\mathbf{w}}$. Hence $C_{\mathbf{w}}=\bar{C}_{\mathbf{w}}$, which lies in $\mathcal{K}_{\mathbf{w}}$ by the measurability just recorded.

(b) Let $\mathcal{D}$ be the class of all $F\in\mathcal{F}$ for which there is an $H\in\mathcal{K}_{\mathbf{w}}$ with $F\cap C_{\mathbf{w}}=H\cap C_{\mathbf{w}}$. Then $\mathcal{D}$ is a $\sigma$-algebra: $\Omega\in\mathcal{D}$ (take $H=\Omega$); if $F\cap C_{\mathbf{w}}=H\cap C_{\mathbf{w}}$ then $(\Omega\setminus F)\cap C_{\mathbf{w}}=(\Omega\setminus H)\cap C_{\mathbf{w}}$; and countable unions are handled by taking the union of the corresponding $H$'s. It contains every generator of $\mathfrak{F}_r$: for $Z$ one of $\Sigma^{\gamma}_s$, $\mathsf{C}^{c}_s$, $\mathsf{N}^{c}_s$ with $s\in[0,r]$, $\bar{Z}$ the corresponding barred quantity and $B'$ a Borel set, claim 1 gives $Z=\bar{Z}$ on $C_{\mathbf{w}}$, so $\{Z\in B'\}\cap C_{\mathbf{w}}=\{\bar{Z}\in B'\}\cap C_{\mathbf{w}}$ with $\{\bar{Z}\in B'\}\in\mathcal{K}_{\mathbf{w}}$. Hence $\mathfrak{F}_r\subseteq\mathcal{D}$, which is (b).

The consequence. $\mathsf{C}^{c}_r$ is $\mathfrak{F}_r$-measurable by definition of $\mathfrak{F}_r$, and $0\le\mathsf{C}^{c}_r\le NBr$ by claim 1 of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution ($\mathsf{C}^{c,(k)}_t\le NBt$ and $\kappa^{c}_K\le NB\theta_K$). With $\mathcal{I}=\{\emptyset,\Omega\}$, the $\sigma$-algebra generated by $\mathcal{I}$ and the variables $\mathsf{P}^{c}_u$, $u\le w_c$, is $\mathcal{K}_{\mathbf{w}}$, and (a), (b) are the two requirements of the clock-reading bound of Fresh-Start Property for Independent Poisson Clocks Read at Levels Satisfying a Clock-Reading Bound (there only up to null events; here they hold exactly).