Disagreement Ledger and Cascade Containment for One-Agent-Moved Reconstructions

Adopt the setting, notation, and hypotheses of Shared-Clock Coupling of One-Agent-Moved Reconstructions: the controlled $N$-agent dynamics with transition-rate family $\beta$ on $l$ states with control set $\mathcal{A}$, rate bound $B$, and Lipschitz constant $K_\beta$, the horizon $T>0$, the driving system $(\Omega,\mathcal{F},P)$ with transition clocks $Y^{i,\sigma\gamma}$, the policy $h$, the moved agent $i_0$ and the moved initial states of claim 1 of One-Agent-Move Ratio of the Record Density Kernel (which agree with the original initial states off $i_0$), the observation record space $(\mathbf{R},\mathcal{R})$, the two fixed reconstruction data sets with reconstructed state paths $\sigma^{r,i}$, $\sigma'^{r,i}$ and domains $G$, $G'$, the record-frozen control paths $a^r$, and the reconstructed transition consumed times $A^{r,i,\sigma\gamma}$, $A'^{r,i,\sigma\gamma}$. Assume additionally the transition-rate floor: a real $\beta_{\min}>0$ with $\beta(\sigma,\gamma,\Sigma,a)\ge\beta_{\min}$ for all states $\sigma\neq\gamma$, all $a\in\mathcal{A}$, and all $\Sigma$ in the probability simplex $\Delta^{l}$. Fix $r\in\mathbf{R}$ and write $\hat{\Omega}:=\Omega^r\cap\Omega'^r$ for the intersection of the probability-one events of claim (d) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records, at every point of which $(r,\omega)\in G\cap G'$ and the two reconstructed collections are solutions of the controlled dynamics for the record-frozen policy $\hat{h}^r$, on the original and on the moved driving system respectively --- two driving systems on the same probability space sharing the same clocks and differing only in the initial state of agent $i_0$.

Fix an agent $i\neq i_0$. For a transition label $c=(i,\sigma\gamma)$ of agent $i$ (an ordered pair of distinct states), write $A^{r,c}:=A^{r,i,\sigma\gamma}$, $A'^{r,c}:=A'^{r,i,\sigma\gamma}$, $Y^{c}:=Y^{i,\sigma\gamma}$, and define the reconstructed counters $N^{r,c}_u:=Y^{c}_{A^{r,c}_u}$ and $N'^{r,c}_u:=Y^{c}_{A'^{r,c}_u}$, the evaluations of the counting path of the clock at the consumed levels. Define pathwise, for $t\in[0,T]$: the frontier consumed time $A^{\vee,c}_t:=\max(A^{r,c}_t,A'^{r,c}_t)$; the frontier count $\mathcal{Y}^{\vee,c}_t:=Y^{c}_{A^{\vee,c}_t}$; and the gap $\Gamma_c(t):=|A'^{r,c}_t-A^{r,c}_t|$. Say that the label $c$ has a frontier crossing at $u\in(0,T]$ if $\mathcal{Y}^{\vee,c}_u>\mathcal{Y}^{\vee,c}_v$ for every $v\in[0,u)$. The dependence of all of these objects on $r$ and on $\omega\in\hat{\Omega}$ is suppressed throughout.

Fix $\omega\in\hat{\Omega}$ and consider the disagreement set $\{u\in[0,T]:\sigma^{r,i}_u\neq\sigma'^{r,i}_u\}$; by claim 1 below it is the union of a finite, possibly empty, family of maximal nonempty subintervals (its connected components), each containing its left endpoint, called its onset, and the two state paths agree, and are constant, on a left neighbourhood of every onset. The length of an interval is the difference of its supremum and its infimum. Call such a maximal interval clean if exactly one of the two state paths jumps at its onset $u$ (both paths being constant on a left neighbourhood of $u$ and piecewise constant, jumping at $u$ means differing at $u$ from the value on a left neighbourhood); for a clean interval, with $\bar{\sigma}$ the common value of both paths on a left neighbourhood of $u$, the jumping solution is called leading and the other trailing, the leading transition at $u$ is $\bar{\sigma}\to\gamma$ for some state $\gamma\neq\bar{\sigma}$, the label $c=(i,\bar{\sigma}\gamma)$ is the onset clock, and the onset level $L$ is the value of the leading $c$-consumed time at $u$. Call a clean interval ordered if the trailing $c$-consumed time at $u$ is strictly less than $L$. Let $J$ denote the set of times $v$ with $u<v\le\sup I$, where $u$ is the onset and $\sup I$ the supremum of the interval $I$, at which some transition counter of agent $i$ jumps in at least one of the two solutions (a finite set, by claim 1); note that $J$ includes the time at which the interval ends by a counter jump, when it so ends. Call the interval bad if one of the following holds: (i) it is not clean; (ii) it is clean but not ordered; (iii) it is clean and ordered, $J\neq\varnothing$, and it is not the case that at $\min J$ (a minimum, by claim 1) exactly one counter jump occurs among the two solutions, namely a jump of the trailing solution's onset-clock counter. The witness of a bad interval is its onset in cases (i) and (ii), and $\min J$ in case (iii), case (iii) being understood to apply only when (i) and (ii) do not.

Then at every $\omega\in\hat{\Omega}$ the following hold.

1. (Structure) Each transition counter of agent $i$ in either solution is nondecreasing, right-continuous, integer-valued, and has finitely many jumps in $[0,T]$, each of unit size, and in each solution at most one transition counter of agent $i$ jumps at any given time; both state paths of agent $i$ are piecewise constant with finitely many jumps. The disagreement set of agent $i$ is the union of a finite, possibly empty, family of maximal nonempty intervals; distinct maximal intervals are separated, each contains its left endpoint, every onset is strictly positive, and the two state paths agree, and are constant, on a left neighbourhood of every onset. Each frontier count $\mathcal{Y}^{\vee,c}$ is nondecreasing and right-continuous with jumps of unit size, and the label $c$ has a frontier crossing at $u$ exactly when $\mathcal{Y}^{\vee,c}$ jumps at $u$. Finally, the number $\mathcal{O}^{(i)}$ of maximal intervals satisfies $\mathcal{O}^{(i)}\le2\,\mathfrak{m}_i$, where $\mathfrak{m}_i:=\sum_{(\sigma,\gamma)}Y^{i,\sigma\gamma}_{BT}$, the sum over ordered pairs of distinct states.

2. (Resolution) Every maximal interval that is not bad is clean and ordered and, with onset $u$ and onset clock $c$, has length at most $\Gamma_{c}(u)/\beta_{\min}$. Every clean ordered interval has a frontier crossing of its onset clock at its onset; and for every clean ordered interval with onset $u$ and onset clock $c$, and every time $v$ in the interval such that no transition counter of agent $i$ jumps in either solution at any time of $(u,v]$, one has $\beta_{\min}\,(v-u)\le\Gamma_{c}(u)$.

3. (Cascade containment) If some bad interval of agent $i$ has witness at most $x\in[0,T]$, then there are times $s\le s'\le x$ and transition labels $c$, $c'$ of agent $i$ such that $c$ has a frontier crossing at $s$, $c'$ has a frontier crossing at $s'$, $(s,c)\neq(s',c')$, and $s'-s\le\Gamma_{c}(s)/\beta_{\min}$.