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

Fix $r\in\mathbf{R}$, an agent $i\neq i_0$, and $\omega\in\hat{\Omega}$; all assertions are pathwise at this $\omega$, where by claim (d) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records the two reconstructed collections are solutions of the controlled dynamics for the record-frozen policy on the original and the moved driving system, which share their clocks, with the properties of conditions 1--6 of Solution of the Controlled N-Agent Dynamics; the reconstructed transition consumed times have the integral representation over $[0,u]$ with integrand $\eta^{r,i,\sigma}_x\,\beta(\sigma,\gamma,\Sigma^r_x,a^r(x))$ for the label $(i,\sigma\gamma)$, and likewise with primes; integrals over subintervals are combined by additivity and bounded by monotonicity, single points being null on the restricted measure spaces. We use three facts, valid in each solution. (F1) A jump of the $(\sigma,\gamma)$-counter of agent $i$ at time $u$ forces the agent to change state from $\sigma$ to $\gamma$ at $u$: at most one counter jumps at any time (shown in claim 1 below), so the state identity of condition 6 moves exactly the two indicators, and a decrease of an indicator is possible only from the value $1$. (F2) A state change of agent $i$ from $\sigma$ to $\gamma$ at $u$ forces its $(\sigma,\gamma)$-counter to jump at $u$: by condition 6 the drop of $\eta^{i,\sigma}$ forces some counter with source $\sigma$ to jump and the rise of $\eta^{i,\gamma}$ forces some counter with target $\gamma$ to jump, and at most one counter jumps at $u$, so it is the $(\sigma,\gamma)$-counter. (F3) Each consumed time $A$ is continuous and nondecreasing with $A_0=0$ (claim 1 of Cumulative-Rate Time Change: Regularity, Substitution, and Crossing Times), and the counter $N_u=Y_{A_u}$ jumps at $u>0$ exactly when $A$ first reaches, at $u$, a jump time of the clock path, having been strictly below it before $u$: either $A$ is constant on a left neighbourhood of $u$, and then $N_{u^-}=Y_{A_u}=N_u$ and there is no jump and no first reach; or $A_v<A_u$ for all $v<u$, and then $N_{u^-}=\lim_{v\uparrow u}Y_{A_v}=Y_{A_u^-}$ by continuity of $A$ and monotonicity of $Y$, so $N$ jumps at $u$ exactly when the counting path jumps at the level $A_u$, that is, when $A_u$ is a jump time first reached at $u$. In particular jump times of clock paths are strictly positive (counting paths start at $0$), and all consumed times are bounded by $BT$, the integrands being at most $B$ by the rate bound.

Claim 1. Counters. Each counter is nondecreasing and right-continuous (composition of the continuous nondecreasing consumed time with the right-continuous nondecreasing clock path), integer-valued, and bounded by $Y^{c}_{BT}$, hence has finitely many jumps. Each jump has unit size: at a jump time $\tau$, by the dichotomy of (F3), $N_\tau-N_{\tau^-}=Y_{A_\tau}-Y_{A_\tau^-}$, the jump of the counting path at the single level $A_\tau$, which is $1$: clause 4 of Counting Path and Its Jump Times bounds jumps by $1$ and integrality with a genuine jump forces equality. At most one transition counter of agent $i$ jumps at a given time in each solution: by condition 3 of Solution of the Controlled N-Agent Dynamics the grand total of all counters of the solution agrees with the restriction of a counting path, whose jumps have size at most $1$, while each counter is nondecreasing and integer-valued, so two counters jumping at one time would give the grand total a jump of size at least $2$. States and intervals. By the state identity of condition 6 each of the two state paths of agent $i$ is constant between the finitely many counter-jump times of its solution, hence piecewise constant with finitely many jumps; so $[0,T]$ splits into a common refinement of finitely many intervals of constancy of both paths, and the disagreement set is a finite, possibly empty, union of maximal nonempty intervals. Each maximal interval contains its left endpoint $u$: the paths are right-continuous (counters are) and constant on $[u,u+\varepsilon)$ for small $\varepsilon>0$, so disagreement just after $u$ forces disagreement at $u$. Every onset is strictly positive because the states agree at $0$: by claim (c) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records the reconstructed initial states are the driving systems' initial states, and the moved initial states of One-Agent-Move Ratio of the Record Density Kernel agree with the original ones off the moved agent. Distinct maximal intervals are separated: if the closures of two of them met at a point $u$, with $I_1$ ending and $I_2$ beginning at $u$, then $u\in I_2$ (maximal intervals contain their left endpoints) and $I_1\cup I_2$ would be an interval contained in the disagreement set, contradicting the maximality of $I_1$; the finitely many maximal intervals therefore have pairwise disjoint closures. Hence for every onset $u>0$ there is $\varepsilon>0$ with $[u-\varepsilon,u)$ disjoint from the disagreement set and inside a common constancy interval of both paths: the paths agree and are constant on a left neighbourhood of every onset. Frontier counts. $A^{\vee,c}$ is continuous and nondecreasing (maximum of two such), so $\mathcal{Y}^{\vee,c}=Y^{c}\circ A^{\vee,c}$ is nondecreasing and right-continuous, and by the dichotomy of (F3) applied verbatim to $A^{\vee,c}$ its jumps have unit size; the label $c$ has a frontier crossing at $u$ exactly when $\sup_{v<u}\mathcal{Y}^{\vee,c}_v=\mathcal{Y}^{\vee,c}_{u^-}<\mathcal{Y}^{\vee,c}_u$, that is, exactly when $\mathcal{Y}^{\vee,c}$ jumps at $u$. Count of intervals. At each onset $u>0$ the paths agree on a left neighbourhood of $u$ and differ at $u$, so at least one path jumps at $u$, whence by (F2) at least one counter of agent $i$ jumps at $u$ in one of the two solutions; onsets of distinct maximal intervals are distinct times; and the total number of counter jumps of agent $i$ over both solutions is at most $\sum_{(\sigma,\gamma)}(N^{r,(i,\sigma\gamma)}_T+N'^{r,(i,\sigma\gamma)}_T)\le2\,\mathfrak{m}_i$, so $\mathcal{O}^{(i)}\le2\,\mathfrak{m}_i$.

Claim 2. A not-bad interval is clean and ordered by the definition of bad. Let its onset be $u$, the paths equal to $\bar{\sigma}$ on a left neighbourhood of $u$ (claim 1). Exactly one path jumps at $u$; by (F2) and (F1) the jumping (leading) solution executes $\bar{\sigma}\to\gamma$ via a jump of the counter of $c=(i,\bar{\sigma}\gamma)$, which by (F3) occurs when the leading $c$-consumed time first reaches, at $u$, a jump time $L$ of $Y^{c}$, having been strictly below it before $u$; so the onset data are as described. By orderedness the trailing $c$-consumed value at $u$ is strictly below $L$, and since consumed times are nondecreasing it is strictly below $L$ at all earlier times as well; thus $\Gamma_c(u)=L-(\text{trailing value at }u)>0$.

Frontier crossing at the onset (for every clean ordered interval). The leading $c$-consumed time is strictly below $L$ before $u$ and equals $L$ at $u$, the trailing one is strictly below $L$ at $u$ and before; so $A^{\vee,c}_v<L$ for $v<u$ and $A^{\vee,c}_u=L$, and, $L$ being a jump time of the counting path, $\mathcal{Y}^{\vee,c}_v=Y^{c}_{A^{\vee,c}_v}\le Y^{c}_L-1<Y^{c}_L=\mathcal{Y}^{\vee,c}_u$ for $v<u$: a frontier crossing at $u$.

Slow-resolution display. Let the interval be clean and ordered with onset $u$ and data $(c,\bar{\sigma},\gamma,L)$, and let $v$ be a time in the interval such that no transition counter of agent $i$ jumps in either solution at any time of $(u,v]$. Then by (F2) neither agent-$i$ path changes state on $(u,v]$: the trailing solution's agent occupies $\bar{\sigma}$ on $[u,v]$, so on this interval the trailing $c$-consumed time has integrand $\eta'^{r,i,\bar{\sigma}}_x\,\beta(\bar{\sigma},\gamma,\Sigma'^r_x,a^r(x))=\beta(\bar{\sigma},\gamma,\Sigma'^r_x,a^r(x))\ge\beta_{\min}$ (unprimed when the trailing solution is the unprimed one), by the transition-rate floor, the reconstructed empirical state measure lying in the simplex by claim (c) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records. Moreover the trailing $c$-consumed time remains at most $L$ on $[u,v]$: it starts strictly below $L$ at $u$, it is continuous, and its first reach of any jump time of $Y^{c}$ in the interval $(\text{trailing value at }u,\,L]$ --- of which $L$ is one --- would by (F3) be a counter jump at a time of $(u,v]$, which is excluded; so it never crosses the smallest such jump time on $(u,v]$, and in particular stays at most $L$ there. Hence $\beta_{\min}\,(v-u)\le(\text{trailing value at }v)-(\text{trailing value at }u)\le L-(\text{trailing value at }u)=\Gamma_c(u)$, which is the displayed inequality of claim 2.

Length bound. Let the interval be not bad, and let $J$ be as in the statement, a subset of the finitely many counter-jump times (claim 1). If $J=\varnothing$, every time $v$ of the interval other than its onset satisfies the hypothesis of the slow-resolution display, so the length is at most $\Gamma_c(u)/\beta_{\min}$. If $J\neq\varnothing$, let $w:=\min J$: since the interval is not bad, at $w$ exactly the trailing solution's $c$-counter jumps, its consumed time first reaching, at $w$, a jump time $\hat{L}$ of $Y^{c}$; every $v$ in the interval with $v<w$ satisfies the slow-resolution hypothesis, and, the trailing consumed time being at most $L$ before $w$ by that argument and continuous, $\hat{L}=(\text{trailing value at }w)\le L$. By (F1) the trailing agent moves $\bar{\sigma}\to\gamma$ at $w$, so both paths equal $\gamma$ at $w$ and the interval is contained in $[u,w)$; and $\beta_{\min}\,(w-u)\le\hat{L}-(\text{trailing value at }u)\le\Gamma_c(u)$ by the same integrand bound on $(u,w)$ and continuity. So the length is at most $\Gamma_c(u)/\beta_{\min}$.

Claim 3. Suppose some bad interval has witness at most $x$, and let $w^{*}$ be the smallest witness of a bad interval of agent $i$ (the intervals being finitely many), so $0<w^{*}\le x$. Write $N^{c}$, $N'^{c}$ for the two counters of a label $c$. We first prove the following invariant, for every $t\in[0,w^{*})$: (i) if the two paths agree at $t$, then $N^{c}_t=N'^{c}_t$ for every label $c$ of agent $i$; (ii) if $t$ lies in a maximal interval, then that interval is clean and ordered, with onset $u\le t$ and data $(c,\bar{\sigma},\gamma,L)$; on $[u,t]$ the leading solution's agent occupies $\gamma$ and the trailing solution's agent occupies $\bar{\sigma}$; the leading $c$-consumed time equals $L$ and the trailing $c$-consumed time is strictly below $L$ on $[u,t]$; the leading $c$-counter exceeds the trailing $c$-counter by exactly one, equivalently exactly one jump time of $Y^{c}$ lies in the interval from the trailing $c$-consumed value at $t$ (exclusive) to $L$ (inclusive), namely $L$ itself; and the two solutions' counters agree on every label other than $c$.

Induction. Let $0<\tau_1<\dots<\tau_K<w^{*}$ be the times in $(0,w^{*})$ at which some counter of agent $i$ jumps in at least one solution (finitely many by claim 1). On $[0,\tau_1)$ (on all of $[0,w^{*})$ if $K=0$) no counter has jumped, so by condition 6 the states equal the initial states, which agree off $i_0$ (as in claim 1), and all counters vanish: (i) holds, and no maximal interval meets $[0,\tau_1)$ (every onset carries a counter jump, by claim 1). Between consecutive listed times, and after $\tau_K$ up to $w^{*}$, counters and states are constant, and in case (ii) the trailing $c$-consumed time rises continuously without reaching any jump time of $Y^{c}$: by (ii) the only one available in the interval from its current value to $L$ is $L$ itself, and reaching a jump time is a counter jump by (F3), hence one of the listed times; so whichever of (i), (ii) held just after the preceding listed time persists, the strict bound below $L$ included. It remains to treat an event time $\tau=\tau_k$, assuming the invariant on a left neighbourhood of $\tau$.

Events during agreement. Suppose the paths agree on a left neighbourhood of $\tau$, in a common state $\bar{\sigma}$, so by (i) all counters agree there. If both solutions jump the same label $\tilde{c}$ at $\tau$: by (F1) both agents move from the source of $\tilde{c}$, which is then $\bar{\sigma}$, to its target, so the paths still agree at $\tau$, and both $\tilde{c}$-counters increase by one (claim 1), preserving (i). If exactly one counter jumps overall at $\tau$, say the label $c$ in one solution: by (F1) that solution's agent moves $\bar{\sigma}\to\gamma$ with $c=(i,\bar{\sigma}\gamma)$, the other path does not move (F2), and a maximal interval opens at $\tau$ with clean onset and data $(c,\bar{\sigma},\gamma,L)$, where $L$ is the level first reached at $\tau$ by the jumping solution's $c$-consumed time (F3). It is ordered: if the trailing $c$-consumed time had reached $L$ strictly before $\tau$, then by monotonicity it stays at least $L$, and its counter counts the jump at $L$ from that reach on, while the leading $c$-counter counts only jumps strictly below $L$ before $\tau$; so $N'^{c}>N^{c}$ on a left neighbourhood of $\tau$, contradicting (i); if it reached $L$ first at $\tau$, its counter would jump at $\tau$ as well (F3), contradicting one-sidedness; and by continuity it cannot exceed $L$ at $\tau$ without having reached it at or before $\tau$. So the trailing value at $\tau$ is strictly below $L$, and (ii) is established at $\tau$, the counter gap being one on $c$ and zero elsewhere. Otherwise jumps occur at $\tau$ in both solutions on distinct labels $c_1\neq c_2$ (at most one per solution, claim 1): by (F1) both sources equal $\bar{\sigma}$ and the targets differ (equal targets would force $c_1=c_2$), so the paths differ at $\tau$ and a maximal interval opens at $\tau$ whose onset is not clean --- a bad interval with witness $\tau<w^{*}$, contradicting the minimality of $w^{*}$; this case cannot occur.

Events inside an interval. Suppose a maximal interval is open on a left neighbourhood of $\tau$, whether or not $\tau$ itself lies in it, with data $(c,\bar{\sigma},\gamma,L)$ and onset $u<\tau$, so (ii) holds just before $\tau$. By the induction no counter jumped at any time of $(u,\tau)$, and $\tau\le\sup I^{}$ for this interval $I$ (which is open just before $\tau$), so $\tau=\min J$ for it. If the jump configuration at $\tau$ were anything other than exactly one jump, of the trailing solution's $c$-counter, the interval would be bad with witness $\tau<w^{*}$, contradicting minimality. So the trailing $c$-counter jumps alone; by (ii) the only jump time of $Y^{c}$ it can first reach is $L$, so it crosses $L$; the two $c$-counters equalize, by (F1) the trailing agent moves $\bar{\sigma}\to\gamma$, the paths agree at $\tau$ (both $\gamma$), the interval ends at $\tau$, and (i) is restored, all other counters untouched. This completes the induction.

Extraction. Consider the time $w^{*}$, the witness of a bad interval $I^{*}$; the invariant holds on a left neighbourhood of $w^{*}$. Case (a): the onset of $I^{*}$ is $w^{*}$ and is not clean. Just before $w^{*}$ the paths agree in a common state $\bar{\sigma}$ and (i) holds. Both paths jump at $w^{*}$ (an onset with no jumping path is impossible, and one jumping path is cleanness), so by (F2) and claim 1 each solution jumps exactly one counter, with labels $c_1$ and $c_2$; both sources equal $\bar{\sigma}$ (F1), and $c_1\neq c_2$, since equal labels would give equal targets and agreement at $w^{*}$, contradicting the onset. For $c_1$: the jumping solution's $c_1$-consumed time first reaches a jump time $L_1$ of $Y^{c_1}$ at $w^{*}$ (F3); the other solution's $c_1$-consumed value is strictly below $L_1$ at $w^{*}$: reaching $L_1$ strictly earlier would, by the monotonicity argument above, make its $c_1$-counter strictly larger on a left neighbourhood of $w^{*}$, contradicting (i); reaching it first at $w^{*}$ would make its $c_1$-counter jump at $w^{*}$, contradicting claim 1 for that solution, whose one jump at $w^{*}$ is on $c_2$; and continuity excludes exceeding $L_1$ without reaching it. Hence $A^{\vee,c_1}<L_1$ strictly before $w^{*}$ and $=L_1$ at $w^{*}$, and as in claim 2 the label $c_1$ has a frontier crossing at $w^{*}$; symmetrically so does $c_2$. Taking $s=s'=w^{*}\le x$ and the labels $c_1\neq c_2$ gives $(s,c_1)\neq(s',c_2)$ and $s'-s=0\le\Gamma_{c_1}(s)/\beta_{\min}$, proving the claim in this case. A clean onset at $w^{*}$ is automatically ordered: cleanness means exactly one path jumps, hence by (F2) and claim 1 exactly one counter jumps overall (a counter jump of the other solution would move its path, by (F1)), and the orderedness argument of the events-during-agreement case applies verbatim at $\tau=w^{*}$, using only the invariant on a left neighbourhood; so no witness arises from case (ii) of badness at $w^{*}$. Case (b): $I^{*}$ is clean and ordered with onset $u^{*}<w^{*}$ and $w^{*}=\min J$. By claim 2 the onset clock $c$ has a frontier crossing at $u^{*}$; and (ii) holds just before $w^{*}$. The configuration at $w^{*}$ is not the lone trailing $c$-jump, $w^{*}$ being a witness. The leading solution cannot jump $c$ at $w^{*}$: its $c$-consumed time is constant, equal to $L$, on $[u^{*},w^{*}]$ (its agent occupies $\gamma\neq\bar{\sigma}$ on $[u^{*},w^{*})$, so the integrand vanishes there, the endpoint being null), and a constant consumed time has no first reach (F3). The trailing solution jumps at most one counter (claim 1). Hence some solution jumps a label $c''\neq c$ at $w^{*}$: were all jumps at $w^{*}$ on $c$, they could only be trailing $c$-jumps, that is, the excluded lone configuration. Fix such a $c''$, jumped by one or both solutions. The $c''$-counters agree just before $w^{*}$ (invariant (ii)). If exactly one solution jumps $c''$: its $c''$-consumed time first reaches a jump time $L''$ at $w^{*}$, and the other solution's $c''$-consumed value is strictly below $L''$ at $w^{*}$ (reaching $L''$ strictly earlier contradicts, by the monotonicity argument, the equality of the $c''$-counters just before $w^{*}$; reaching it first at $w^{*}$ is excluded by the case assumption, being a second $c''$-jump; exceeding without reaching is excluded by continuity); so, as in claim 2, $c''$ has a frontier crossing at $w^{*}$. If both solutions jump $c''$ at $w^{*}$, with levels $L''_1$, $L''_2$ first reached at $w^{*}$: both $c''$-consumed times are strictly below their respective levels before $w^{*}$, so $A^{\vee,c''}<\max(L''_1,L''_2)$ strictly before $w^{*}$ and $=\max(L''_1,L''_2)$ at $w^{*}$, and again $c''$ has a frontier crossing at $w^{*}$, the maximum being a jump time. Finally, the window: no counter of agent $i$ jumps in either solution at any time of $(u^{*},v]$ for $v<w^{*}$ in the interval, so the slow-resolution display of claim 2 gives $\beta_{\min}\,(v-u^{*})\le\Gamma_{c}(u^{*})$ for all such $v$, whence $\beta_{\min}\,(w^{*}-u^{*})\le\Gamma_{c}(u^{*})$ in the limit. Taking $s=u^{*}$ with the crossing of $c$ and $s'=w^{*}\le x$ with the crossing of $c''$ gives $(s,c)\neq(s',c'')$ and $s'-s\le\Gamma_{c}(s)/\beta_{\min}$, as required.