Proof of The Closeness, Fourth-Moment and Discrepancy Hypotheses of the Estimand Assembly Lemma Supplied by an N-Agent Solution under the Cost Bound: Control-Lipschitz Bound on the Record Discrepancy, Close Records from the Path-Closeness Event, and the Explicit Mean-Square Bound

Throughout, a composition $g\circ f$ of measurable maps is measurable, since $(g\circ f)^{-1}(E)=f^{-1}(g^{-1}(E))$; a map into a product of two measurable spaces whose two coordinates are measurable is measurable with respect to the product $\sigma$-algebra, by claim 2 of Generator Criterion for Measurability applied to the generating rectangles, whose preimages are intersections of the two coordinate preimages (the rectangle argument); and for $x\in\Delta^{l}$ one has $|x|\le1$, because $x^{\gamma}\ge0$, $\sum_{\gamma}x^{\gamma}=1$ and $|x|^{2}=\sum_{\gamma}(x^{\gamma})^{2}\le(\sum_{\gamma}x^{\gamma})^{2}=1$ (claim 1 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square, the cross terms being nonnegative), so that $|x-y|\le2$ for $x,y\in\Delta^{l}$ by claims 5 and 6 there (this is also recorded in claim 4 of The Observation-Centred Fluctuation at an Intermediate Time: Restriction of the Mean-Field Flow and of the Realized Control to a Shorter Horizon, Representation through the Aggregate State and the Record Prefix, and Transport of Its Joint Law with the Record to the Synthetic Copy). Monotonicity, positive homogeneity and additivity of integrals of nonnegative measurable functions are those of Linearity and Monotonicity of the Lebesgue Integral, and $\int\mathbf{1}_{A}\,d\nu=\nu(A)$ is The Integral of an Indicator Function is the Measure of the Set. We use repeatedly that for real numbers $0\le a\le b$ one has $a^{2}\le b^{2}$, $a^{4}\le b^{4}$, $a^{1/2}\le b^{1/2}$ and $a^{1/4}\le b^{1/4}$, by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field applied once or twice in either direction.

Claim 1. (a) The setting of Law Identity between the N-Agent Aggregate Path with Its Observation Record and the Synthetic Copy's Regularised Path with Its Record with horizon $s$ requires the setting of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record with the horizon $s$ (adopted on the copy side, with its probability space $(\Omega,\mathcal{F},P)$ in the role of $(\Omega^{\flat},\mathcal{F}^{\flat},P^{\flat})$ and $\mathsf{x}_{0}$ in the role of $x_{0}$), the strict inequality $R>NBs$ (hypothesis (L)), an $N$-agent driving system with $l$ states and $\tilde{l}$ channels (here $(\Omega^{\mathrm{ag}},\mathcal{F}^{\mathrm{ag}},P^{\mathrm{ag}})$), a solution of the controlled $N$-agent dynamics on $[0,s]$ for the policy of the copy side and for the same data $m$, $\mathcal{A}$, $\beta$, $B$, $\tilde{\beta}$, $\tilde{B}$ and horizon (here the restricted solution, which is a solution on $[0,s]$ for $h^{(s)}$ on this driving system by the setting of The Observation-Centred Fluctuation at an Intermediate Time: Restriction of the Mean-Field Flow and of the Realized Control to a Shorter Horizon, Representation through the Aggregate State and the Record Prefix, and Transport of Its Joint Law with the Record to the Synthetic Copy, the data being the same by (L)), and reconstruction data for this driving system and $h^{(s)}$ with horizon $s$ (fixed in the adopted setting). Thus that setting is instantiated. The additional hypotheses of claim 5 of The Observation-Centred Fluctuation at an Intermediate Time: Restriction of the Mean-Field Flow and of the Realized Control to a Shorter Horizon, Representation through the Aggregate State and the Record Prefix, and Transport of Its Joint Law with the Record to the Synthetic Copy are exactly this instantiation together with $P^{\mathrm{ag}}(\Sigma_{0}=\mathsf{x}_{0})=1$, which is part of (L); hence its claims 1--5 are available. For $r\in\mathbf{R}_{s}$ and $t\in[0,s]$, the record-frozen flow of (AF) is $\Phi^{r}_{t}=\mathsf{S}^{(s)}_{t}(z_{0},a^{r})$, the mean-field flow with horizon $s$ of claim 2 of Stability of the Mean-Field Flow under Perturbation of the Initial State and Weak Convergence of Controls evaluated at the base point and at the class of the record-frozen control path of the copy side's policy at $r$; by (L) the base point is $S_{0}$ and that path is $a^{(s),r}$, so $\Phi^{r}_{t}=\mathsf{S}^{(s)}_{t}(S_{0},a^{(s),r})=\Phi^{(s),r}_{t}$ (the dense sequence entering neither flow). Consequently $X''(\omega,r)=\sqrt{N}(\bar\Sigma^{\sharp,r}_{s}(\omega)-\Phi^{r}_{s})=\sqrt{N}(\bar\Sigma^{\sharp,r}_{s}(\omega)-\Phi^{(s),r}_{s})=X''_{s}(\omega,\theta,r)$ for every $(\omega,\theta,r)\in\Omega^{\sharp}$, by the defining formulas.

(b) By claim 1 of The Observation-Centred Fluctuation at an Intermediate Time: Restriction of the Mean-Field Flow and of the Realized Control to a Shorter Horizon, Representation through the Aggregate State and the Record Prefix, and Transport of Its Joint Law with the Record to the Synthetic Copy, the Data paragraph of The Realized Control as the Record-Frozen Control at the Observation Record, and Measurability of the Path-and-Record Closeness Set is instantiated with horizon $s$ by the $N$-agent data, the policy $h^{(s)}$, the restricted solution and the fixed dense sequence. For the comparison data: each component of $t\mapsto S_{t}$ is continuous on $[0,T]$, hence its restriction to $[0,s]$ is continuous on $[0,s]$ (given $t_{0}\in[0,s]$ and $\varepsilon>0$, a $\delta>0$ serving for all $t\in[0,T]$ with $|t-t_{0}|<\delta$ serves in particular for all such $t\in[0,s]$, which is continuity relative to $[0,s]$); $|S_{t}|\le1=K^{*}$ because $S_{t}\in\Delta^{l}$; and $t\mapsto A_{t}$ ($t\in[0,s]$) takes values in $\mathcal{A}$ with components continuous on $[0,s]$ (by the same restriction argument), hence measurable with respect to $\mathcal{B}_{[0,s]}$ by claim 4 of Measurability of Countable Suprema, Bounded Pointwise Limits, Monotone Functions, and Continuous Functions. For the path data: $\mathbb{G}_{N}$ is a nonempty finite subset of $\mathbb{R}^{l}$, since it contains $\Sigma_{0}(\omega)$ for any $\omega$ (claim 4 of The Observation-Centred Fluctuation at an Intermediate Time: Restriction of the Mean-Field Flow and of the Realized Control to a Shorter Horizon, Representation through the Aggregate State and the Record Prefix, and Transport of Its Joint Law with the Record to the Synthetic Copy) and, by its definition in Open-Loop Aggregate Solution Driven by Aggregate Transition Clocks ($Nx^{\gamma}$ a nonnegative integer for every $\gamma$, and $0\le x^{\gamma}\le1$ on $\Delta^{l}$), is contained in the finite set of points $x$ with $Nx^{\gamma}\in\{0,1,\dots,N\}$ for all $\gamma$. Thus the whole setting of The Realized Control as the Record-Frozen Control at the Observation Record, and Measurability of the Path-and-Record Closeness Set is instantiated with horizon $s$, and its claims 2, 3 and 4 furnish $\mathsf{d}$, $\mathsf{s}$ and $\mathsf{E}(\varepsilon,\varepsilon')\in\mathcal{C}\otimes\mathcal{R}_{s}$ (the $\sigma$-algebra written $\mathcal{C}_{s}$ there being $\mathcal{C}$, both generated by the sets $\{p:p(t)=y\}$), with $\mathsf{s}(p)=\sup_{t\in[0,s]}|p(t)-S_{t}|$ and $\mathsf{d}(r)=\int_{[0,s]}|a^{(s),r}(t)-A_{t}|\,dt$ as displayed. Finally, for $\omega\in\Omega^{\mathrm{ag}}_{0}$, claim 2 of that lemma applied to the restricted solution, whose record is $W^{(s)}$ and whose realized control is $\hat{\alpha}^{(s)}$, gives $\mathsf{d}(W^{(s)}(\omega))=\int_{[0,s]}|\hat{\alpha}^{(s)}(t,\omega)-A_{t}|\,dt$, and $\hat{\alpha}^{(s)}(t,\omega)=\hat{\alpha}(t,\omega)$ for $t\in[0,s]$ by claim 3 of The Observation-Centred Fluctuation at an Intermediate Time: Restriction of the Mean-Field Flow and of the Realized Control to a Shorter Horizon, Representation through the Aggregate State and the Record Prefix, and Transport of Its Joint Law with the Record to the Synthetic Copy.

Claim 2. Fix $r\in\mathbf{R}_{s}$, $t\in[0,s]$ and a label $c=(\sigma,\gamma_{0})\in\mathcal{L}$, and write $a=a^{(s),r}(t)$ and $a'=A_{t}$; both lie in $\mathcal{A}$ (claim 1 of The Record-Frozen Control as a Measurable Map into the Weakly Metrized Control Set, and the Record-Frozen Mean-Field Flow: Joint Measurability, Flow Equation, and Measurability in the Record with horizon $s$, available by claim 1 of The Observation-Centred Fluctuation at an Intermediate Time: Restriction of the Mean-Field Flow and of the Realized Control to a Shorter Horizon, Representation through the Aggregate State and the Record Prefix, and Transport of Its Joint Law with the Record to the Synthetic Copy; and $A$ takes its values in $\mathcal{A}$ because $(S,A)$ is a mean-field trajectory pair for $\beta$, as recorded in the adopted $N$-agent setting), and $S_{t}\in\Delta^{l}$. Let $(U,W_{\beta},\bar\beta)$ be the twice continuously differentiable extension of (X), with $\Delta^{l}\subset U$ and $\mathcal{A}\subseteq W_{\beta}$, both open and convex, so that $U\times W_{\beta}$ is an open subset of $\mathbb{R}^{l+m}$ (condition 2 of that definition). The segment $\{(S_{t},a+\tau(a'-a)):\tau\in[0,1]\}$ between the points $x=(S_{t},a)$ and $y=(S_{t},a')$ of $\mathbb{R}^{l+m}$ lies in $\Delta^{l}\times\mathcal{A}\subseteq U\times W_{\beta}$, because $\mathcal{A}$ is convex; and $|y-x|=|a'-a|$, the first $l$ coordinates of $y-x$ vanishing (claim 1 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square). By claim 1 of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect (in the instance of its setting fixed by the adopted copy setting through Closeness to a Mean-Field Pair on the Synthetic Copy: Clock Discrepancy, Gronwall Comparison of the Weighted Response with the Profile Response, and the Observation-Information Bound with Mean-Field Data), $\psi_{c}$ is of class $C^{2}$ on $U\times W_{\beta}$, hence of class $C^{1}$ there (clause 2 of C^k Maps on a Euclidean Open Set), and at every point of $\Delta^{l}\times\mathcal{A}$, in particular at every point of the segment, $|\partial_{i}\psi_{c}|\le B+K$ and $|\partial_{j}\partial_{i}\psi_{c}|\le3K$ for all $i,j\in\{1,\dots,l+m\}$. Part (i) of Multivariate Taylor Expansion with Uniform Second-Order Remainder, applied with $n=l+m$, $W=U\times W_{\beta}$, $f=\psi_{c}$ and $M_{1}=B+K$, gives

$$|\psi_{c}(S_{t},a')-\psi_{c}(S_{t},a)|\le\sqrt{l+m}\,(B+K)\,|a'-a|=\Lambda_{1}|a'-a| .$$

For $\gamma\in\{1,\dots,l\}$ the function $\partial_{\gamma}\psi_{c}$ is of class $C^{1}$ on $U\times W_{\beta}$ (clause 2 of C^k Maps on a Euclidean Open Set, $\psi_{c}$ being of class $C^{2}$), and its partial derivatives $\partial_{j}\partial_{\gamma}\psi_{c}$ are bounded by $3K$ on the segment; part (i) of the Taylor lemma with $f=\partial_{\gamma}\psi_{c}$ and $M_{1}=3K$ gives $|\partial_{\gamma}\psi_{c}(S_{t},a')-\partial_{\gamma}\psi_{c}(S_{t},a)|\le\sqrt{l+m}\,3K\,|a'-a|$. Since $g^{c}=(\partial_{1}\psi_{c},\dots,\partial_{l}\psi_{c})$, the vector $g^{c}(S_{t},a')-g^{c}(S_{t},a)$ has $l$ components each of absolute value at most $\sqrt{l+m}\,3K|a'-a|$, so its squared norm is at most $l\,(l+m)\,9K^{2}|a'-a|^{2}$ (claim 1 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square) and

$$|g^{c}(S_{t},a')-g^{c}(S_{t},a)|\le\sqrt{l}\sqrt{l+m}\,3K\,|a'-a|=\Lambda_{3}|a'-a| .$$

Summing the two bounds over the $l(l-1)$ labels gives the displayed inequality of claim 2 with $C_{\mathrm{Lip}}=l(l-1)(\Lambda_{1}+\Lambda_{3})$. The integrand of $\mathsf{D}^{r}_{\mathrm{ctl}}$ (the left-hand side as a function of $t$, with the comparison pair $(S,A)$ on $[0,s]$ by (L)) is bounded and measurable by claim 1 of Closeness to a Mean-Field Pair on the Synthetic Copy: Clock Discrepancy, Gronwall Comparison of the Weighted Response with the Profile Response, and the Observation-Information Bound with Mean-Field Data, and $t\mapsto|a^{(s),r}(t)-A_{t}|$ is measurable by claim 2 of The Realized Control as the Record-Frozen Control at the Observation Record, and Measurability of the Path-and-Record Closeness Set; integrating the pointwise inequality over $[0,s]$ (monotonicity and positive homogeneity) gives $\mathsf{D}^{r}_{\mathrm{ctl}}\le C_{\mathrm{Lip}}\int_{[0,s]}|a^{(s),r}(t)-A_{t}|\,dt=C_{\mathrm{Lip}}\,\mathsf{d}(r)$.

Claim 3. The tolerances. Both tolerances are nonnegative real numbers: $\varepsilon_{S}(N)=(C_{\mathrm{flw}}+1)N^{-1/4}$ and $\varepsilon_{\mathrm{ctl}}(N)=(TC_{\mathrm{ctl}})^{1/2}N^{-1/4}$ are nonnegative, because $N^{-1/4}>0$, $C_{\mathrm{ctl}}=2(\mathcal{Z}^{\sharp}+1)>0$ with $\mathcal{Z}^{\sharp}\ge0$ (claims 3 and 4 of Closeness of the Realized Control and the Realized Mean-Field Flow on a High-Probability Event under the Cost Bound), $(TC_{\mathrm{ctl}})^{1/2}$ is a nonnegative square root, and $C_{\mathrm{flw}}=C_{S}C_{\mathrm{ctl}}^{1/2}$ with $C_{S}=e^{\Lambda_{b}T}\sqrt{l}\,K_{2}\sqrt{T}$ (claim 4 there) is nonnegative, $K_{2}=2\sqrt{l}(l-1)K_{1}$ being nonnegative since $K_{1}\ge0$ by claim 1 of The Transition-Rate Family and Aggregate Drift of Affine-Controlled Data; and $C_{\mathrm{Lip}}=l(l-1)(\Lambda_{1}+\Lambda_{3})\ge0$ since $B,K\ge0$. This is the first clause of (CL), so that, from here on, (CP) holds in full with $\mathsf{w}^{\mathrm{clk}}=N(\Lambda_{1}s\,\varepsilon_{S}+\varepsilon_{\mathrm{ctl}})\ge0$, and claims 1 and 2 of Mean-Square Assembly of the Estimand Linearisation on the Synthetic Copy: Approximation of the Recentred Copy Endpoint by an Affine Function of the Parameter are available by its scope statement.

Joint measurability of the regularised paths. Write $\bar\Pi(\omega,r)=\bar\Sigma^{\sharp,r}(\omega)\in\mathsf{Path}$ for $(\omega,r)\in\Omega\times\mathbf{R}_{s}$; this is $\Pi^{\sharp}(\omega,0,r)$ for the map $\Pi^{\sharp}$ of Law Identity between the N-Agent Aggregate Path with Its Observation Record and the Synthetic Copy's Regularised Path with Its Record, which takes its values in $\mathsf{Path}$ and is measurable with respect to $(\mathcal{F}\otimes\mathcal{B}(\mathbb{R}^{d}))\otimes\mathcal{R}_{s}$ and $\mathcal{C}$ by its claim 1 (available by claim 1(a)). By claim 4 of Sections of Product-Measurable Sets and Maps Are Measurable, and Insertion Maps into Products Are Measurable (fixing the middle coordinate $0\in\mathbb{R}^{d}$), $\bar\Pi$ is measurable with respect to $\mathcal{F}\otimes\mathcal{R}_{s}$ and $\mathcal{C}$. By the rectangle argument the map $(\omega,r)\mapsto(\bar\Pi(\omega,r),r)$ is measurable with respect to $\mathcal{F}\otimes\mathcal{R}_{s}$ and $\mathcal{C}\otimes\mathcal{R}_{s}$ (its second coordinate being the projection, measurable by claim 5 of Sections of Product-Measurable Sets and Maps Are Measurable, and Insertion Maps into Products Are Measurable), so, writing $\mathsf{E}=\mathsf{E}(\varepsilon_{S}(N),\varepsilon_{\mathrm{ctl}}(N))\in\mathcal{C}\otimes\mathcal{R}_{s}$ (claim 1(b)), the set

$$\mathsf{B}=\bigl\{(\omega,r)\in\Omega\times\mathbf{R}_{s}:\ (\bar\Pi(\omega,r),r)\notin\mathsf{E}\bigr\}$$

belongs to $\mathcal{F}\otimes\mathcal{R}_{s}$, and its sections $\mathsf{B}_{\omega}=\{r:(\omega,r)\in\mathsf{B}\}$ belong to $\mathcal{R}_{s}$ for every $\omega$ (claim 1 of Sections of Product-Measurable Sets and Maps Are Measurable, and Insertion Maps into Products Are Measurable). For $\omega\in G\cap\Omega'$ one has $\mathsf{R}^{\mathrm{cl}}_{\omega}=\mathsf{T}_{\omega}\cap(\mathbf{R}_{s}\setminus\mathsf{B}_{\omega})$.

Membership and inclusion. For $\omega\notin G\cap\Omega'$, $\mathsf{R}^{\mathrm{cl}}_{\omega}=\emptyset\in\mathcal{R}_{s}$ and $\emptyset\subseteq\mathsf{T}_{\omega}$. For $\omega\in G\cap\Omega'$, $\mathbf{R}_{s}\setminus\mathsf{T}_{\omega}\in\mathcal{R}_{s}$ by claim 3 of Almost Sure Tracking on the Synthetic Copy: Jump Times of the Deterministic-Count Clocks, Almost Sure Conflict-Freeness of Every Record, Almost Sure Null Mass of the Untracked Records, and Trimming an Event to the Tracked Set (its setting being that of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record with the move size $\mathsf{m}$, adopted), so $\mathsf{T}_{\omega}\in\mathcal{R}_{s}$, hence $\mathsf{R}^{\mathrm{cl}}_{\omega}\in\mathcal{R}_{s}$ as an intersection of two members, and $\mathsf{R}^{\mathrm{cl}}_{\omega}\subseteq\mathsf{T}_{\omega}$ by construction.

Closeness on the close records. Let $\omega\in G$ and $r\in\mathsf{R}^{\mathrm{cl}}_{\omega}$; then $\omega\in G\cap\Omega'$ (otherwise $\mathsf{R}^{\mathrm{cl}}_{\omega}$ is empty) and $(\bar\Sigma^{\sharp,r}(\omega),r)\in\mathsf{E}$, that is, $\mathsf{s}(\bar\Sigma^{\sharp,r}(\omega))\le\varepsilon_{S}(N)$ and $\mathsf{d}(r)\le\varepsilon_{\mathrm{ctl}}(N)$. The first gives $|\bar\Sigma^{\sharp,r}_{t}(\omega)-S_{t}|\le\mathsf{s}(\bar\Sigma^{\sharp,r}(\omega))\le\varepsilon_{S}(N)=\varepsilon_{S}$ for every $t\in[0,s]$, the path deviation being the least upper bound of these distances; the second gives, by claim 2, $\mathsf{D}^{r}_{\mathrm{ctl}}\le C_{\mathrm{Lip}}\,\mathsf{d}(r)\le C_{\mathrm{Lip}}\,\varepsilon_{\mathrm{ctl}}(N)=\varepsilon_{\mathrm{ctl}}$.

Integrals over $\rho$-null sets vanish. Let $\omega\in\Omega$ and let $\mathsf{N}\in\mathcal{R}_{s}$ with $\rho(\mathsf{N})=0$. The map $r\mapsto\ell^{\sharp,\omega}(r)$ is $\mathcal{R}_{s}$-measurable with values in $[0,\infty)$ (a section of the $\mathcal{R}_{s}\otimes\mathcal{F}$-measurable map of claim 3 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record, by claim 3 of Sections of Product-Measurable Sets and Maps Are Measurable, and Insertion Maps into Products Are Measurable). For natural numbers $n$ put $f_{n}=\min(\ell^{\sharp,\omega},n)\,\mathbf{1}_{\mathsf{N}}$, measurable by claims 1, 3 and 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, nonnegative, with $f_{n}\le f_{n+1}$ and $\sup_{n}f_{n}=\ell^{\sharp,\omega}\mathbf{1}_{\mathsf{N}}$ pointwise (the value $\ell^{\sharp,\omega}(r)$ being a real number, exceeded by no $n$ larger than it). Since $f_{n}\le n\mathbf{1}_{\mathsf{N}}$, monotonicity and positive homogeneity give $\int f_{n}\,d\rho\le n\rho(\mathsf{N})=0$, and Monotone Convergence Theorem gives $\int_{\mathbf{R}_{s}}\ell^{\sharp,\omega}\mathbf{1}_{\mathsf{N}}\,d\rho=\sup_{n}\int f_{n}\,d\rho=0$.

The non-close mass. Since $\ell^{\sharp,\omega}\ge0$ integrates to $1$ against $\rho$ for every $\omega$ (claim 3 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record) and $0\le\ell^{\sharp,\omega}\mathbf{1}_{\mathbf{R}_{s}\setminus\mathsf{R}^{\mathrm{cl}}_{\omega}}\le\ell^{\sharp,\omega}$, the integral $\pi^{\mathrm{nc}}_{\omega}$ is defined (the indicator of $\mathbf{R}_{s}\setminus\mathsf{R}^{\mathrm{cl}}_{\omega}\in\mathcal{R}_{s}$ being measurable) and lies in $[0,1]$ by monotonicity. For $\omega\notin G\cap\Omega'$, $\pi^{\mathrm{nc}}_{\omega}=\int\ell^{\sharp,\omega}\,d\rho=1$. For $\omega\in G\cap\Omega'$, the complement of $\mathsf{R}^{\mathrm{cl}}_{\omega}=\mathsf{T}_{\omega}\cap(\mathbf{R}_{s}\setminus\mathsf{B}_{\omega})$ is the disjoint union of $\mathsf{B}_{\omega}$ and $(\mathbf{R}_{s}\setminus\mathsf{T}_{\omega})\setminus\mathsf{B}_{\omega}$, so by additivity

$$\pi^{\mathrm{nc}}_{\omega}=\int_{\mathbf{R}_{s}}\ell^{\sharp,\omega}\mathbf{1}_{\mathsf{B}_{\omega}}\,d\rho+\int_{\mathbf{R}_{s}}\ell^{\sharp,\omega}\mathbf{1}_{(\mathbf{R}_{s}\setminus\mathsf{T}_{\omega})\setminus\mathsf{B}_{\omega}}\,d\rho=I(\omega),\qquad I(\omega)=\int_{\mathbf{R}_{s}}\ell^{\sharp,\omega}(r)\,\mathbf{1}_{\mathsf{B}}(\omega,r)\,\rho(dr),$$

the second integral vanishing by the previous paragraph, because $(\mathbf{R}_{s}\setminus\mathsf{T}_{\omega})\setminus\mathsf{B}_{\omega}$ is a member of $\mathcal{R}_{s}$ contained in $\mathbf{R}_{s}\setminus\mathsf{T}_{\omega}$, which for $\omega\in\Omega'$ belongs to $\mathcal{R}_{s}$ and has $\rho$-measure $0$ by claim 3 of Almost Sure Tracking on the Synthetic Copy: Jump Times of the Deterministic-Count Clocks, Almost Sure Conflict-Freeness of Every Record, Almost Sure Null Mass of the Untracked Records, and Trimming an Event to the Tracked Set, so that the smaller set has $\rho$-measure $0$ by monotonicity of the measure $\rho$. Now $\mathbf{1}_{\mathsf{B}}$ is an $\mathcal{F}\otimes\mathcal{R}_{s}$-measurable map $\Omega\times\mathbf{R}_{s}\to[0,\infty)$, so by claim 2 of Mean-Square Assembly of the Estimand Linearisation on the Synthetic Copy: Approximation of the Recentred Copy Endpoint by an Affine Function of the Parameter (which, by the scope statement preceding its claim 1, holds in the present instance although (CL), (FM) and (DM) have not yet been established) the map $\omega\mapsto I(\omega)$ is $\mathcal{F}$-measurable and

$$\mathbb{E}[I]=\int_{\Omega^{\sharp}}\mathbf{1}_{\mathsf{B}}\,d\mu^{\sharp},$$

$\mathbf{1}_{\mathsf{B}}$ being regarded as a function on $\Omega^{\sharp}$ through the coordinate maps. Hence $\pi^{\mathrm{nc}}=\mathbf{1}_{\Omega\setminus(G\cap\Omega')}+\mathbf{1}_{G\cap\Omega'}\,I$ is $\mathcal{F}$-measurable (claims 1, 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions; $G\in\mathcal{F}$ by (G), and $\Omega'\in\mathcal{F}$ because claim 3 of Almost Sure Tracking on the Synthetic Copy: Jump Times of the Deterministic-Count Clocks, Almost Sure Conflict-Freeness of Every Record, Almost Sure Null Mass of the Untracked Records, and Trimming an Event to the Tracked Set places it in the $\sigma$-algebra generated by the driving variables $U^{c,j}_{i}$, which is contained in $\mathcal{F}$). This establishes (CL).

The bound on $\bar\pi^{\mathrm{nc}}$. Pointwise, $\mathbf{1}_{G}\pi^{\mathrm{nc}}=\mathbf{1}_{G\setminus\Omega'}+\mathbf{1}_{G\cap\Omega'}I\le\mathbf{1}_{\Omega\setminus\Omega'}+I$, and $\mathbb{E}[\mathbf{1}_{\Omega\setminus\Omega'}]=P(\Omega\setminus\Omega')=0$ since $P(\Omega')=1$ (claim 3 of Almost Sure Tracking on the Synthetic Copy: Jump Times of the Deterministic-Count Clocks, Almost Sure Conflict-Freeness of Every Record, Almost Sure Null Mass of the Untracked Records, and Trimming an Event to the Tracked Set); by monotonicity and additivity, $\bar\pi^{\mathrm{nc}}\le\mathbb{E}[I]$. On $\Omega^{\sharp}$, $\mathbf{1}_{\mathsf{B}}(\omega,r)=\mathbf{1}\{(\Pi^{\sharp},\mathsf{D})\notin\mathsf{E}\}(\omega,\theta,r)=F'(\Pi^{\sharp},\mathsf{D})(\omega,\theta,r)$ with $F'=\mathbf{1}_{(\mathsf{Path}\times\mathbf{R}_{s})\setminus\mathsf{E}}$, a $\mathcal{C}\otimes\mathcal{R}_{s}$-measurable map into $[0,\infty]$ (claim 1 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions), because $\Pi^{\sharp}(\omega,\theta,r)=\bar\Pi(\omega,r)$ and $\mathsf{D}(\omega,\theta,r)=r$. Claim 2 of Law Identity between the N-Agent Aggregate Path with Its Observation Record and the Synthetic Copy's Regularised Path with Its Record with $P^{\mathrm{ag}}(D_{0})=1$, where $D_{0}=\{\Sigma_{0}=\mathsf{x}_{0}\}$, gives

$$\mathbb{E}[I]=\int_{\Omega^{\sharp}}F'(\Pi^{\sharp},\mathsf{D})\,d\mu^{\sharp}=\mathbb{E}^{\mathrm{ag}}\bigl[\mathbf{1}_{D_{0}}F'(\Pi,W^{(s)})\bigr]\le\mathbb{E}^{\mathrm{ag}}\bigl[F'(\Pi,W^{(s)})\bigr]=P^{\mathrm{ag}}\bigl(\{(\Pi,W^{(s)})\notin\mathsf{E}\}\bigr),$$

the record of the restricted solution being $W^{(s)}$, and the set $\{(\Pi,W^{(s)})\notin\mathsf{E}\}$ belonging to $\mathcal{F}^{\mathrm{ag}}$ because $\Pi$ and $W^{(s)}$ are measurable (claim 1 of that lemma) and the pair is measurable by the rectangle argument. Let $\omega\in\Omega^{\mathrm{cl}}_{N}$. Then $\omega\in\Omega^{\mathrm{ag}}_{0}$ by the definition of $\Omega^{\mathrm{cl}}_{N}$ in claim 2 of The Path-Closeness Event under the Cost Bound: Closeness of the Empirical State Measure to the Mean-Field Trajectory and of the Record-Frozen Control to the Mean-Field Control on an Event of Probability $1-O(N^{-1/2})$, so $\Pi(\omega)$ is the path $t\mapsto\Sigma_{t}(\omega)$ on $[0,s]$ (definition of $\Pi$ in Law Identity between the N-Agent Aggregate Path with Its Observation Record and the Synthetic Copy's Regularised Path with Its Record); by claim 2 of The Path-Closeness Event under the Cost Bound: Closeness of the Empirical State Measure to the Mean-Field Trajectory and of the Record-Frozen Control to the Mean-Field Control on an Event of Probability $1-O(N^{-1/2})$, $|\Sigma_{t}(\omega)-S_{t}|\le\varepsilon_{S}(N)$ for every $t\in[0,T]$, so $\varepsilon_{S}(N)$ is an upper bound of the set whose least upper bound is $\mathsf{s}(\Pi(\omega))$, whence $\mathsf{s}(\Pi(\omega))\le\varepsilon_{S}(N)$; and by the same claim with $t=s$ and by claim 1(b), $\mathsf{d}(W^{(s)}(\omega))=\int_{[0,s]}|\hat{\alpha}(t,\omega)-A_{t}|\,dt\le\varepsilon_{\mathrm{ctl}}(N)$. Hence $(\Pi(\omega),W^{(s)}(\omega))\in\mathsf{E}$, that is, $\Omega^{\mathrm{cl}}_{N}\subseteq\{(\Pi,W^{(s)})\in\mathsf{E}\}$, and by monotonicity of $P^{\mathrm{ag}}$ and claim 2 of The Path-Closeness Event under the Cost Bound: Closeness of the Empirical State Measure to the Mean-Field Trajectory and of the Record-Frozen Control to the Mean-Field Control on an Event of Probability $1-O(N^{-1/2})$,

$$\bar\pi^{\mathrm{nc}}\le P^{\mathrm{ag}}\bigl(\{(\Pi,W^{(s)})\notin\mathsf{E}\}\bigr)\le P^{\mathrm{ag}}\bigl(\Omega^{\mathrm{ag}}\setminus\Omega^{\mathrm{cl}}_{N}\bigr)\le N^{-1/2}+c_{Q}\kappa^{\sharp}N^{-1}.$$

Claim 4. Claims 1 and 2 of Mean-Square Assembly of the Estimand Linearisation on the Synthetic Copy: Approximation of the Recentred Copy Endpoint by an Affine Function of the Parameter are used below in the present instance, which is legitimate by the scope statement preceding its claim 1, although (FM) and (DM) have not yet been established. Let $\mathsf{Q}_{s}=(\mathbb{Q}\cap[0,s])\cup\{s\}$, the countable set of claim 2 of that lemma (countable by The Integers and the Rational Numbers are Countable and claims 3 and 6 of Basic Properties of Countable Sets, and nonempty), so that it is the set of terms of a sequence $(t_{n})_{n\in\mathbb{N}}$ by Countable Set; by that claim, $\bar{\mathsf{e}}(\omega,r)=\sup_{n}\sqrt{N}\,|\bar\Sigma^{\sharp,r}_{t_{n}}(\omega)-\Phi^{r}_{t_{n}}|$. Define $F:\mathsf{Path}\times\mathbf{R}_{s}\to\mathbb{R}$ by $F(p,r)=\sup_{n}f_{n}(p,r)$ with $f_{n}(p,r)=N^{2}|p(t_{n})-\Phi^{r}_{t_{n}}|^{4}$. Each $f_{n}$ is measurable with respect to $\mathcal{C}\otimes\mathcal{R}_{s}$: $(p,r)\mapsto p(t_{n})^{\gamma}$ is the composition of the projection onto $\mathsf{Path}$ with the $\mathcal{C}$-measurable map $p\mapsto\phi(p(t_{n}))$, $\phi(y)=y^{\gamma}$, of claim 1 of The Space of Piecewise Constant Paths in a Finite Set: Measurability of Evaluations, Restrictions, Left Limits, and Occupation Integrals; $(p,r)\mapsto\Phi^{r,\gamma}_{t_{n}}$ is the composition of the projection onto $\mathbf{R}_{s}$ with the $\mathcal{R}_{s}$-measurable map of claim 1 of Mean-Square Assembly of the Estimand Linearisation on the Synthetic Copy: Approximation of the Recentred Copy Endpoint by an Affine Function of the Parameter; and $f_{n}$ is obtained from these by differences, squares, sums and multiplication by $N^{2}$ (claims 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, $|z|^{4}=(\sum_{\gamma}(z^{\gamma})^{2})^{2}$). Since $p(t_{n})$ and $\Phi^{r}_{t_{n}}$ lie in $\Delta^{l}$, $0\le f_{n}\le16N^{2}$, so $F$ is real-valued and $\mathcal{C}\otimes\mathcal{R}_{s}$-measurable by claim 1 of Measurability of Countable Suprema, Bounded Pointwise Limits, Monotone Functions, and Continuous Functions. For a bounded family $(a_{n})$ of nonnegative reals, $(\sup_{n}a_{n})^{4}=\sup_{n}a_{n}^{4}$: each $a_{n}^{4}\le(\sup_{k}a_{k})^{4}$, and conversely each $a_{n}\le(\sup_{k}a_{k}^{4})^{1/4}$, so $\sup_{n}a_{n}\le(\sup_{k}a_{k}^{4})^{1/4}$ and $(\sup_{n}a_{n})^{4}\le\sup_{k}a_{k}^{4}$ (monotonicity of fourth powers and fourth roots). Hence $F(\bar\Pi(\omega,r),r)=\bar{\mathsf{e}}(\omega,r)^{4}$, and on $\Omega^{\sharp}$, $\bar{\mathsf{e}}^{4}=F(\Pi^{\sharp},\mathsf{D})$. By claim 2 of Law Identity between the N-Agent Aggregate Path with Its Observation Record and the Synthetic Copy's Regularised Path with Its Record with $P^{\mathrm{ag}}(D_{0})=1$,

$$\int_{\Omega^{\sharp}}\bar{\mathsf{e}}^{4}\,d\mu^{\sharp}=\int_{\Omega^{\sharp}}F(\Pi^{\sharp},\mathsf{D})\,d\mu^{\sharp}=\mathbb{E}^{\mathrm{ag}}\bigl[\mathbf{1}_{D_{0}}F(\Pi,W^{(s)})\bigr]\le\mathbb{E}^{\mathrm{ag}}\bigl[F(\Pi,W^{(s)})\bigr].$$

For $\omega\in\Omega^{\mathrm{ag}}_{0}$: $\Pi(\omega)(t)=\Sigma_{t}(\omega)$, and $\Phi^{W^{(s)}(\omega)}_{t}=\Phi^{(s),W^{(s)}(\omega)}_{t}=\Phi_{t}(\omega)$ for $t\in[0,s]$ by claim 1(a) and claim 3 of The Observation-Centred Fluctuation at an Intermediate Time: Restriction of the Mean-Field Flow and of the Realized Control to a Shorter Horizon, Representation through the Aggregate State and the Record Prefix, and Transport of Its Joint Law with the Record to the Synthetic Copy, so $F(\Pi(\omega),W^{(s)}(\omega))=\sup_{n}N^{2}|\Sigma_{t_{n}}(\omega)-\Phi_{t_{n}}(\omega)|^{4}\le N^{2}Q(\omega)^{4}$, because $|\Sigma_{t}(\omega)-\Phi_{t}(\omega)|\le Q(\omega)$ for every $t\in[0,T]$ by The Empirical State Measure Deviates from the Realized Mean-Field Flow by at Most the Noise Majorant, applied in the instance of the setting of Pre-Stopping-Time Envelope and Restricted Moment Bounds for the State Fluctuation Process fixed in the Data of The Path-Closeness Event under the Cost Bound: Closeness of the Empirical State Measure to the Mean-Field Trajectory and of the Record-Frozen Control to the Mean-Field Control on an Event of Probability $1-O(N^{-1/2})$, in which $Q$ is the noise majorant and the realized mean-field flow is $\Phi$ (formed with the base point $S_{0}$, the initial point of the trajectory of the stationary triple). For $\omega\notin\Omega^{\mathrm{ag}}_{0}$, $F\le16N^{2}$. Hence $F(\Pi,W^{(s)})\le N^{2}Q^{4}+16N^{2}\mathbf{1}_{\Omega^{\mathrm{ag}}\setminus\Omega^{\mathrm{ag}}_{0}}$ pointwise, and by monotonicity, additivity and positive homogeneity, using $P^{\mathrm{ag}}(\Omega^{\mathrm{ag}}\setminus\Omega^{\mathrm{ag}}_{0})=0$ ($\Omega^{\mathrm{ag}}_{0}$ having probability one by Solution of the Controlled N-Agent Dynamics) and claim 1 of The Path-Closeness Event under the Cost Bound: Closeness of the Empirical State Measure to the Mean-Field Trajectory and of the Record-Frozen Control to the Mean-Field Control on an Event of Probability $1-O(N^{-1/2})$,

$$\int_{\Omega^{\sharp}}\bar{\mathsf{e}}^{4}\,d\mu^{\sharp}\le N^{2}\,\mathbb{E}^{\mathrm{ag}}[Q^{4}]\le N^{2}\cdot c_{Q}\kappa^{\sharp}N^{-2}=c_{Q}\kappa^{\sharp}.$$

Thus (FM) holds with $\mathsf{c}_{4}=c_{Q}\kappa^{\sharp}$, a real number which is nonnegative because it dominates $N^{2}\mathbb{E}^{\mathrm{ag}}[Q^{4}]\ge0$.

Claim 5. The concluding statements of claims 2 and 3 of Moment Toolkit for the Synthetic Copy: Square-Integrable Majorants of the Window Discrepancies of the Copy Clocks and the Fourth Moment of a Weighted Centred Cell-Count Sum refer to the setting and hypotheses of Mean-Square Assembly of the Estimand Linearisation on the Synthetic Copy: Approximation of the Recentred Copy Endpoint by an Affine Function of the Parameter; the setting and the hypotheses (OC), (X), (W), (G), (G$'$), (CL), (AF), (CP), (FM) and (DM) of Mean-Square Assembly of the Estimand Linearisation on the Synthetic Copy: Approximation of the Recentred Copy Endpoint by an Affine Function of the Parameter are those of Mean-Square Assembly of the Estimand Linearisation on the Synthetic Copy: Approximation of the Recentred Copy Endpoint by an Affine Function of the Parameter verbatim (the later version adds only the scope statement on its claims 1 and 2, a parenthetical remark inside its claim 1, and the explicit naming of $\mathcal{F}^{\sharp}$ and $\lambda_{d}$, these being the objects already fixed by The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record, so that the setting and hypotheses, and hence the instances, are the same), so those concluding statements apply to the present instance; the concluding statement of claim 3 is what is used below, that of claim 2 being replaced by a direct verification. The setting of Moment Toolkit for the Synthetic Copy: Square-Integrable Majorants of the Window Discrepancies of the Copy Clocks and the Fourth Moment of a Weighted Centred Cell-Count Sum is that of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record, adopted on the copy side, and its claims 1 and 2 require $R$ to be a natural number, which holds by (L); so for the data $w_{1}=\mathsf{w}^{\mathrm{clk}}$, $w_{2}=\mu_{\max}$, $x_{1}$, $x_{2}$ its claim 2 furnishes a family $(\Xi^{c})_{c\in\mathcal{L}}$ of $\mathcal{F}$-measurable square-integrable maps $\Omega\to[0,\infty)$ with $\mathrm{Disc}_{\mathsf{w}^{\mathrm{clk}}}(\mathsf{P}^{\sharp,c}(\omega))+\mathrm{Disc}_{\mu_{\max}}(\mathsf{P}^{\sharp,c}(\omega))\le\Xi^{c}(\omega)$ for every $\omega\in\Omega$, in particular for every $\omega\in G$, and with the displayed bound on $\lVert\Xi^{c}\rVert_{2}=(\mathbb{E}[(\Xi^{c})^{2}])^{1/2}$; here $\mathsf{w}^{\mathrm{clk}}=N(\Lambda_{1}s\,\varepsilon_{S}+\varepsilon_{\mathrm{ctl}})$ is formed, as in Mean-Square Assembly of the Estimand Linearisation on the Synthetic Copy: Approximation of the Recentred Copy Endpoint by an Affine Function of the Parameter with horizon $s$, from the tolerances of claim 3. This is hypothesis (DM) for this family. The present instance of the setting of Mean-Square Assembly of the Estimand Linearisation on the Synthetic Copy: Approximation of the Recentred Copy Endpoint by an Affine Function of the Parameter (horizon $s$) has the hypotheses (OC), (X), (W), (G), (G$'$), (AF) and (CP) by assumption, (CL) by claim 3, (FM) by claim 4 and (DM) as just shown, and its probability space, copy clocks, clock horizon, cells, cell-count vector and bijection are those of Moment Toolkit for the Synthetic Copy: Square-Integrable Majorants of the Window Discrepancies of the Copy Clocks and the Fourth Moment of a Weighted Centred Cell-Count Sum; hence the concluding statement of claim 3 of that lemma gives $\mathsf{k}_{4}^{4}\le3(\sum_{q}\alpha_{q}^{2}\mu_{q})^{2}+\sum_{q}\alpha_{q}^{4}\mu_{q}$ for the cell coefficients $\alpha$ of the adopted setting. All hypotheses of Mean-Square Assembly of the Estimand Linearisation on the Synthetic Copy: Approximation of the Recentred Copy Endpoint by an Affine Function of the Parameter being satisfied, its claim 4 gives $\lVert\alpha\cdot\Theta+\varsigma(\mathsf{D})-\mathbf{c}\cdot X''\rVert_{2}\le\mathsf{e}_{1}+\dots+\mathsf{e}_{5}$ with its constants formed from the present data: $\mathsf{e}_{1}=|\mathbf{c}|\bar\Phi^{2}\sqrt{N}|\mathsf{x}_{0}-z_{0}|$ with $z_{0}=S_{0}$; $\mathsf{e}_{2}$ with the horizon $s$, $\mathsf{c}_{4}=c_{Q}\kappa^{\sharp}$, $\varepsilon_{S}=\varepsilon_{S}(N)$ and $\varepsilon_{\mathrm{ctl}}=C_{\mathrm{Lip}}\varepsilon_{\mathrm{ctl}}(N)$, which is the displayed expression; $\mathsf{e}_{3}$ with $(\mathbb{E}[(\Xi^{c})^{2}])^{1/2}=\lVert\Xi^{c}\rVert_{2}$; $\mathsf{e}_{4}=\sqrt{\eta}|\alpha|$; and $\mathsf{e}_{5}=\sqrt{2}(\mathsf{g}^{1/4}+(\bar\pi^{\mathrm{nc}})^{1/4})(|\mathbf{c}|\mathsf{c}_{4}^{1/4}+\mathsf{k}_{4}/\sqrt{N})$, which is at most $\hat{\mathsf{e}}_{5}$ because $0\le\bar\pi^{\mathrm{nc}}\le N^{-1/2}+c_{Q}\kappa^{\sharp}N^{-1}$ by claim 3, fourth roots are monotone, and the second factor is nonnegative; this gives the second inequality of the display. Finally, $\mathbf{c}\cdot X''$ is the function $(\omega,\theta,r)\mapsto\mathbf{c}\cdot X''(\omega,r)=\mathbf{c}\cdot X''_{s}(\omega,\theta,r)$ on $\Omega^{\sharp}$ by claim 1(a), which identifies the left-hand side as stated.