Proof of Shared-Clock Coupling of One-Agent-Moved Reconstructions

Throughout fix $(r,\omega)\in G\cap G'$ and abbreviate $\sigma^i_u=\sigma^{r,i}_u(\omega)$, $\sigma'^i_u=\sigma'^{r,i}_u(\omega)$, $\eta^{i,\delta}_u=\eta^{r,i,\delta}_u(\omega)$, $\eta'^{i,\delta}_u=\eta'^{r,i,\delta}_u(\omega)$, $\Sigma_u=\Sigma^r_u(\omega)$, $\Sigma'_u=\Sigma'^r_u(\omega)$, and $A_u=A^{r,i,\sigma\gamma}_u(\omega)$, $A'_u=A'^{r,i,\sigma\gamma}_u(\omega)$ when the indices are clear. By claim (c) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records, each path $u\mapsto\sigma^i_u$ and $u\mapsto\sigma'^i_u$ is piecewise constant and right-continuous in the sense of condition 1 of Solution of the Controlled N-Agent Dynamics, with $\sigma^i_0=\varsigma^i_0(\omega)$ and $\sigma'^i_0=\varsigma'^i_0(\omega)$, and $\eta^{i,\delta}_u$ is the indicator that $\sigma^i_u=\delta$.

Claim 1. By piecewise constancy, for each $i\neq i_0$ the set of $u$ at which the two agent-$i$ paths differ is a finite union of intervals each containing its left endpoint (on each interval of a common refinement of the two partitions into intervals of constancy, both paths are constant, so the difference set is a union of members of that refinement, each of which contains its left endpoint). Hence $D^r(\omega)$, a finite union of such sets over $i\neq i_0$, has a least element when nonempty, namely the least of the left endpoints of the participating intervals. This least element is not $0$: for $i\neq i_0$ the initial values agree, since the moved initial states agree with the original ones off agent $i_0$.

Claim 2. Let $u<\zeta^r(\omega)$. By minimality no $i\neq i_0$ differs at $u$. For every $\delta$, $N\Sigma'^{\delta}_u-N\Sigma^{\delta}_u=\sum_{i=1}^N\bigl(\eta'^{i,\delta}_u-\eta^{i,\delta}_u\bigr)=\eta'^{i_0,\delta}_u-\eta^{i_0,\delta}_u,$ using the exactly-one-state property of claim (c) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records. The right side is the $\delta$-coordinate of $e_{\sigma'^{i_0}_u}-e_{\sigma^{i_0}_u}$, proving the display, and the coordinate absolute sum of a difference of two standard basis vectors is at most $2$. For the left limits at $u\le\zeta^r(\omega)$ with $u>0$: the paths are piecewise constant, so all left limits exist and are attained on an interval $[u-\varepsilon,u)$ with $\varepsilon>0$; since the asserted identities hold at every point of $[u-\varepsilon,u)\subseteq[0,\zeta^r(\omega))$, they hold for the left limits.

Claim 3. Let $u\le\zeta^r(\omega)\wedge T$, $i\neq i_0$, and $(\sigma,\gamma)$. The difference of the two defining integrals is the integral of the difference by linearity, both integrands being bounded by the rate bound $B$ and measurable. For $x<\zeta^r(\omega)$ the indicators $\eta^{i,\sigma}_x$ and $\eta'^{i,\sigma}_x$ agree by claim 2, so the integrand difference there equals $\eta^{i,\sigma}_x(\beta(\sigma,\gamma,\Sigma'_x,a^r(x))-\beta(\sigma,\gamma,\Sigma_x,a^r(x)))$, of absolute value at most $K_\beta\sum_\delta|\Sigma'^\delta_x-\Sigma^\delta_x|\le 2K_\beta/N$ by the Lipschitz hypothesis and claim 2; the single point $x=\zeta^r(\omega)$, when it belongs to $[0,u]$, does not affect the integrals, finitely many points being immaterial by Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval. Monotonicity then gives $|A'_u-A_u|\le(2K_\beta/N)u$. The observation bound is identical, with $K_{\tilde{\beta}}$, the observation rates, and the consumed observation times of the two reconstruction data sets, whose integral representation is claim (c) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records.

Claim 4. Let $\omega\in\Omega^r\cap\Omega'^r$ with $(r,\omega)\in G\cap G'$ and $\zeta=\zeta^r(\omega)\le T$. By claim (d) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records, at $\omega$ the reconstructed processes at $r$ satisfy conditions 1--6 of Solution of the Controlled N-Agent Dynamics for the record-frozen policy, for each of the two systems on the same clocks; in particular the transition counters are the compositions of the clock paths with the consumed times ($Y^{i,\sigma\gamma}$ evaluated along $A^{r,i,\sigma\gamma}$, and along $A'^{r,i,\sigma\gamma}$ for the moved system), the state identity of condition 6 holds, and by condition 3 the grand total of each system's counters coincides with the restriction of a counting path, so that in each system at most one counter jumps at any one time, jumps being by exactly one. We use three facts, valid in each of the two per-record solutions. (F1) A jump of the $(\sigma,\gamma)$-counter of agent $i$ at a time $u$ forces the agent's path to change state from $\sigma$ to $\gamma$ at $u$: by the at-most-one-jump property the jumping counter is the only counter changing at $u$, so by the state identity $\eta^{i,\gamma}$ increases by one at $u$ and $\eta^{i,\sigma}$ decreases by one, and the decrease is possible only from the value $1$, the occupation indicators taking values in $\{0,1\}$. (F2) Conversely, a state change of agent $i$ from $\sigma$ to $\gamma$ at a time $u$ forces the $(\sigma,\gamma)$-counter of agent $i$ to jump at $u$: by the state identity some counter draining state $\sigma$ of agent $i$ and some counter filling state $\gamma$ of agent $i$ must jump at $u$, and at most one counter jumps at $u$, so they are the same counter, the $(\sigma,\gamma)$ one. (F3) If a consumed time is continuous and nondecreasing (claim 1 of Cumulative-Rate Time Change: Regularity, Substitution, and Crossing Times), first reaches a level $L$ at a time $u$ (that is, its value at $u$ is $L$ and its values before $u$ are strictly below $L$), and $L$ is a jump time of the clock path, then the corresponding counter jumps at $u$; conversely, a jump of the counter at $u$ forces the consumed time to be strictly below its value at $u$ before $u$, and that value to be a jump time of the clock path.

By claim 1 and minimality, some $i\neq i_0$ has $\sigma^i_\zeta\neq\sigma'^i_\zeta$, while the left limits at $\zeta$ agree by claim 2. If neither system's agent-$i$ state changed at $\zeta$, the two values would equal the common left limit, a contradiction; so at least one system's agent-$i$ path jumps at $\zeta$, and by fact (F2) some transition counter of agent $i$ jumps at $\zeta$ in that system. Moreover there is a pair $(\sigma,\gamma)$ for which exactly one of the two systems' agent-$i$ counters jumps at $\zeta$: otherwise, for every pair, the two counters jump simultaneously at $\zeta$ or not at all; each system executes at most one transition at $\zeta$, so by facts (F1) and (F2) agent $i$ would execute the same transition (or none) in both systems, and with equal left limits the values at $\zeta$ would be equal, a contradiction. Fix such $(i,\sigma,\gamma)$, and suppose the unprimed counter jumps at $\zeta$ and the primed one does not (the other case is symmetric). By fact (F3), $L_0:=A_\zeta$ is a jump time of the path of $Y^{i,\sigma\gamma}$ and $A_u<L_0$ for $u<\zeta$; in particular $L_0>0$, jump times of counting paths being strictly positive.

Suppose, for contradiction, that $A'_\zeta=A_\zeta=L_0$. We construct a strictly decreasing sequence $L_0>L_1>L_2>\dots$ of jump times of the path of $Y^{i,\sigma\gamma}$, all in $(0,L_0]$, together with strictly decreasing times $\zeta>u_0>u_1>\dots$; this is impossible, since the number of jump times of a counting path in $(0,L_0]$ is at most its value at $L_0$, a finite integer, and the contradiction gives $A'_\zeta\neq A_\zeta$.

Base step. Since $A'$ is continuous and nondecreasing with $A'_0=0<L_0=A'_\zeta$, there is a least time $u_0$ with $A'_{u_0}=L_0$, and $A'_u<L_0$ for $u<u_0$, with $u_0>0$. By fact (F3) the primed counter jumps at $u_0$. Since the primed counter does not jump at $\zeta$, $u_0\neq\zeta$, so $u_0<\zeta$. Also, since $A_u<L_0$ for $u<\zeta$, the level $L_1:=A_{u_0}$ satisfies $L_1<L_0$.

Inductive step. Suppose given $u_k<\zeta$ at which the primed counter jumps. By fact (F1) the primed agent $i$ changes state from $\sigma$ to $\gamma$ at $u_k$. Since $u_k<\zeta$, the two agent-$i$ paths agree at $u_k$ and their left limits at $u_k$ agree (claim 2), so the unprimed agent $i$ also changes state from $\sigma$ to $\gamma$ at $u_k$, and by fact (F2) the unprimed counter jumps at $u_k$. By fact (F3), $L_{k+1}:=A_{u_k}$ is a jump time of the clock path, $A_u<L_{k+1}$ for $u<u_k$, and $L_{k+1}>0$. Now $L_{k+1}<L_k$: for $k=0$ this was noted in the base step; for $k\ge1$, the unprimed counter jumps at $u_{k-1}$ with level $L_k=A_{u_{k-1}}$ and $A_u<L_k$ for $u<u_{k-1}$, and $u_k<u_{k-1}$, so $A_{u_k}<L_k$. Since $A'$ is continuous and nondecreasing with $A'_0=0<L_{k+1}<A'_{u_k}$ (for $k=0$ one has $A'_{u_0}=L_0>L_1$, and for $k\ge1$ one has $A'_{u_k}=L_k>L_{k+1}$), there is a least time $u_{k+1}$ with $A'_{u_{k+1}}=L_{k+1}$, and $A'_u<L_{k+1}$ for $u<u_{k+1}$, with $u_{k+1}>0$; monotonicity of $A'$ and $A'_{u_{k+1}}=L_{k+1}<A'_{u_k}$ give $u_{k+1}<u_k$. By fact (F3) the primed counter jumps at $u_{k+1}$, completing the induction.

The closed interval with endpoints $A_\zeta$ and $A'_\zeta$ contains the jump time $A_\zeta$ (or, in the symmetric case, $A'_\zeta$), and $|A_\zeta-A'_\zeta|\le 2K_\beta\zeta/N$ by claim 3. This proves claim 4.

Claim 5. Let $c$ be real. For $c<0$ the set where $\zeta^r\le c$ is empty. For $c\in[0,T]$, the set of $(r,\omega)\in G\cap G'$ with $\zeta^r(\omega)\le c$ equals the union, over $i\neq i_0$ and over $q$ ranging over the fixed countable set consisting of the rationals of $[0,c]$ together with $c$ itself, of the sets where $\sigma^{r,i}_q(\omega)\neq\sigma'^{r,i}_q(\omega)$. Indeed, suppose $\zeta^r(\omega)=u\le c$: the paths differ at $u$ and, by right-continuity and piecewise constancy, on $[u,u+\varepsilon)$ for some $\varepsilon>0$ depending on $(r,\omega)$; if $u=c$ they differ at the index point $c$, and if $u<c$ they differ at every rational of the nonempty interval with endpoints $u$ and the smaller of $u+\varepsilon$ and $c$, which contains a rational of $[0,c]$. Conversely a difference at some index point $q\le c$ gives $\zeta^r(\omega)\le q\le c$. Each set where $\sigma^{r,i}_q\neq\sigma'^{r,i}_q$ is a member of $\mathcal{R}\otimes\mathcal{T}$ by claim (a) of Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records for the two data sets (the primed fields are measurable with respect to $\mathcal{R}\otimes\mathcal{T}'\subseteq\mathcal{R}\otimes\mathcal{T}$, by claim 1 of One-Agent-Move Ratio of the Record Density Kernel), and $G\cap G'\in\mathcal{R}\otimes\mathcal{T}$. For $c\in[T,T+1)$ the set where $\zeta^r\le c$ is the one just described with $c$ replaced by $T$; for $c\ge T+1$ it is all of $\mathbf{R}\times\Omega$. Hence the sets where $\zeta^r\le c$ are measurable for every real $c$, proving claim 5.