Proof 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

Throughout, measurable for real-valued maps is with respect to the named $\sigma$-algebra and the Borel $\sigma$-algebra of the real line; sums, scalar multiples, products, absolute values and maxima of measurable real-valued maps are measurable by claims 2, 3 and 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, constants and indicators of measurable sets by claim 1 there, and a composition of measurable maps is measurable directly from Measurable Function and Real-Valued Measurable Function (a preimage under the composition is the preimage under the first map of a preimage under the second). Integrals of nonnegative measurable functions are those of Lebesgue Integral of a Nonnegative Measurable Function, monotone and additive by Linearity and Monotonicity of the Lebesgue Integral. Whenever a map $f=(f^{1},\dots,f^{k})$ into a Euclidean space has measurable components, its Euclidean norm $|f|$ and every dot product $v\cdot f$ with a fixed vector $v$ are measurable, by Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable applied to the sequentially continuous maps $z\mapsto|z|$ (sequentially continuous since $\bigl||z|-|z'|\bigr|\le|z-z'|$, which follows from claim 6 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ applied twice, to $z=(z-z')+z'$ and to $z'=(z'-z)+z$; we refer to this as the reverse triangle inequality) and $z\mapsto v\cdot z$ (sequentially continuous since $|v\cdot z-v\cdot z'|\le|v|\,|z-z'|$ by Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$ and Cauchy-Schwarz Inequality for the Euclidean Dot Product); likewise the square root of a nonnegative measurable function is measurable, by the same lemma applied to $t\mapsto\sqrt{t}$ on $E=[0,\infty)$, which is sequentially continuous because $|\sqrt{a}-\sqrt{b}|\le\sqrt{|a-b|}$ for $a,b\ge0$ (square both sides). We also use that $1/n\to0$: given $\varepsilon>0$, claim 3 of The Archimedean Property of the Real Numbers provides $n_0$ with $1/n_0<\varepsilon$, and then $1/n\le1/n_0<\varepsilon$ for all $n\ge n_0$. We use the elementary inequalities $(a+b)^{2}\le2a^{2}+2b^{2}$ for real $a,b$ (recorded in Square-Integrable Random Variables and the Mean-Square Inner Product) and $\sqrt{x+y}\le\sqrt{x}+\sqrt{y}$ for real $x,y\ge0$ (square both sides). Finally, for $x,y\in\Delta^l$ one has $|x-y|\le\sqrt{2}$: by claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, $|x-y|^{2}=\sum_{\gamma}(x^{\gamma}-y^{\gamma})^{2}\le\sum_{\gamma}\bigl((x^{\gamma})^{2}+(y^{\gamma})^{2}\bigr)\le\sum_{\gamma}(x^{\gamma}+y^{\gamma})=2$, since $x^{\gamma}y^{\gamma}\ge0$ and $0\le x^{\gamma}\le1$ give $(x^{\gamma})^{2}\le x^{\gamma}$.

Product-space bookkeeping. Write $\varpi_{13}:\Omega^{\sharp}\to\Omega\times\mathbf{R}$, $\varpi_{13}((\omega,\theta),r)=(\omega,r)$, $\varpi_{1}:\Omega^{\sharp}\to\Omega$, $\varpi_{1}((\omega,\theta),r)=\omega$, and $\varpi_{12}:\Omega^{\sharp}\to\Omega\times\mathbb{R}^d$, $\varpi_{12}((\omega,\theta),r)=(\omega,\theta)$. By Product Sigma-Algebra, $\mathcal{F}\otimes\mathcal{R}$ is generated by the rectangles $A\times C$ ($A\in\mathcal{F}$, $C\in\mathcal{R}$), and $\varpi_{13}^{-1}(A\times C)=(A\times\mathbb{R}^d)\times C$ is a measurable rectangle of $(\mathcal{F}\otimes\mathcal{B}(\mathbb{R}^d))\otimes\mathcal{R}=\mathcal{F}^{\sharp}$, because $A\times\mathbb{R}^d$ is a measurable rectangle of $\mathcal{F}\otimes\mathcal{B}(\mathbb{R}^d)$; hence $\varpi_{13}$ is measurable by claim 2 of Generator Criterion for Measurability. The maps $\varpi_{12}$ and $\varpi_{1}$ are measurable by claim 5 of Sections of Product-Measurable Sets and Maps Are Measurable, and Insertion Maps into Products Are Measurable (a coordinate projection, and the composition of two coordinate projections). Thus a function of $(\omega,r)$ measurable with respect to $\mathcal{F}\otimes\mathcal{R}$, or a function of $\omega$ measurable with respect to $\mathcal{F}$, is, regarded as a function on $\Omega^{\sharp}$ through $\varpi_{13}$ or $\varpi_{1}$, measurable with respect to $\mathcal{F}^{\sharp}$; likewise the coordinates $\Theta_q$ of $\Theta$ and the coordinates $\mathsf{K}_q\circ\varpi_{1}$ are $\mathcal{F}^{\sharp}$-measurable ($\Theta$ is measurable by claim 4 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record and its coordinates are obtained by the projections of claim 1 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets, whose $\mathcal{B}_d$ is $\mathcal{B}(\mathbb{R}^d)$ by claim 5 there). The exchange of coordinates $(r,\omega)\mapsto(\omega,r)$ is measurable from $\mathcal{R}\otimes\mathcal{F}$ to $\mathcal{F}\otimes\mathcal{R}$ by claim 5 of Sections of Product-Measurable Sets and Maps Are Measurable, and Insertion Maps into Products Are Measurable, which justifies the transport of measurability announced in the statement.

The copy integral of a function of $(\omega,r)$. Let $F:\Omega\times\mathbf{R}\to[0,\infty)$ be $\mathcal{F}\otimes\mathcal{R}$-measurable. By claim 3 of Image Measures, Measures with Densities, and Change of Variables ($\mu^{\sharp}$ being the measure with density $\mathsf{q}^{\sharp}$ with respect to $(P\otimes\lambda_d)\otimes\rho$, and $\mathsf{q}^{\sharp}$ being $\mathcal{F}^{\sharp}$-measurable with values in $[0,\infty)$ by claim 4 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record),

The three measures $P$, $\lambda_d$ and $\rho$ are $\sigma$-finite ($P$ is finite, $\lambda_d$ is $\sigma$-finite as a product of the $\sigma$-finite Lebesgue measure of claim 5 of Existence of Lebesgue Measure on the Real Line by Finite Products of Lebesgue Measure and Coordinate Integration on $\mathbb{R}^l$, and $\rho$ is $\sigma$-finite as recorded in The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record), and hence so is $P\otimes\lambda_d$ (a product of $\sigma$-finite measures, Existence and Uniqueness of the Product Measure), so Tonelli and Fubini Theorems applies to $(\Omega\times\mathbb{R}^d,\mathcal{F}\otimes\mathcal{B}(\mathbb{R}^d),P\otimes\lambda_d)$ and $(\mathbf{R},\mathcal{R},\rho)$, and then to $(\Omega,\mathcal{F},P)$ and $(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d),\lambda_d)$: integrating first over $r$, then over $\theta$, then over $\omega$, the right-hand side equals

where the inner integral $\mathsf{I}_F(\omega)=\int_{\mathbf{R}}F(\omega,r)\ell^{\sharp,\omega}(r)\rho(dr)\in[0,\infty]$ is an $\mathcal{F}$-measurable function of $\omega$ by the Tonelli theorem applied on $\Omega\times\mathbf{R}$ (the map $(\omega,r)\mapsto F(\omega,r)\ell^{\sharp,\omega}(r)$ being $\mathcal{F}\otimes\mathcal{R}$-measurable, since $(r,\omega)\mapsto\ell^{\sharp,\omega}(r)$ is $\mathcal{R}\otimes\mathcal{F}$-measurable by 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 where the factor $\varphi_\eta(\theta-\mathsf{K}(\omega)/\sqrt{N})$, a finite nonnegative constant with respect to $r$, was taken out of the $r$-integral by the linearity of the integral (Linearity and Monotonicity of the Lebesgue Integral). Now fix $\omega$. If $\mathsf{I}_F(\omega)<\infty$, the constant $\mathsf{I}_F(\omega)$ may be taken out of the $\theta$-integral by the same linearity, and since $\int_{\mathbb{R}^d}\varphi_\eta(\theta-a)\,d\lambda_d(\theta)=1$ for every $a\in\mathbb{R}^d$ (claim 1 of The Gaussian Smoothing Weight: Normalization, Derivatives, Exponential Tilting, Moments, and First-Order Remainder), the $\theta$-integral equals $\mathsf{I}_F(\omega)$; if $\mathsf{I}_F(\omega)=\infty$, then, as $\varphi_\eta>0$ everywhere (claim 1 of that lemma), the $\theta$-integrand is identically $+\infty$, in particular at least $1$ everywhere, so by monotonicity (Linearity and Monotonicity of the Lebesgue Integral) and The Integral of an Indicator Function is the Measure of the Set the $\theta$-integral is at least $\lambda_d(\mathbb{R}^d)=\infty$ (Lebesgue measure has infinite total mass: it is a product of copies of the Lebesgue measure of the line, Finite Products of Lebesgue Measure and Coordinate Integration on $\mathbb{R}^l$, which gives mass $n$ to $[0,n]$ for every $n$ by claim 4 of Existence of Lebesgue Measure on the Real Line), hence equals $+\infty=\mathsf{I}_F(\omega)$ as well. In both cases the $\theta$-integral equals $\mathsf{I}_F(\omega)$, and we obtain

If $F$ depends on $\omega$ alone, then $\mathsf{I}_F(\omega)=F(\omega)\int_{\mathbf{R}}\ell^{\sharp,\omega}\,d\rho=F(\omega)$, because $\int_{\mathbf{R}}\ell^{\sharp,\omega}\,d\rho=1$ 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); so $\int_{\Omega^{\sharp}}F\circ\varpi_{1}\,d\mu^{\sharp}=\mathbb{E}[F]$. We also record that, for every fixed $\omega$, the measure $\rho^{\omega}$ with density $\ell^{\sharp,\omega}$ with respect to $\rho$ (claim 3 of Image Measures, Measures with Densities, and Change of Variables; the section $r\mapsto\ell^{\sharp,\omega}(r)$ is $\mathcal{R}$-measurable by claim 3 of Sections of Product-Measurable Sets and Maps Are Measurable, and Insertion Maps into Products Are Measurable) has total mass $\rho^{\omega}(\mathbf{R})=\int_{\mathbf{R}}\ell^{\sharp,\omega}\,d\rho=1$, so $(\mathbf{R},\mathcal{R},\rho^{\omega})$ is a probability space, on which $\int_{\mathbf{R}}f\,d\rho^{\omega}=\int_{\mathbf{R}}f\,\ell^{\sharp,\omega}\,d\rho$ for every $\mathcal{R}$-measurable $f\ge0$; the Cauchy--Schwarz inequality of claim 1 of Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm is therefore available on it, in the form $\int_{\mathbf{R}}XY\ell^{\sharp,\omega}\,d\rho\le\bigl(\int_{\mathbf{R}}X^{2}\ell^{\sharp,\omega}\,d\rho\bigr)^{1/2}\bigl(\int_{\mathbf{R}}Y^{2}\ell^{\sharp,\omega}\,d\rho\bigr)^{1/2}$ for nonnegative $\mathcal{R}$-measurable $X,Y$ with finite second moments. In the same way we use claim 1 of Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm on $(\Omega,\mathcal{F},P)$ and on the copy.

Claim 1. The flow lemma. The data 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 are: an affine-controlled transition-rate family on $l$ states with nonempty convex compact control set and a Lipschitz constant, supplied by (AF) together with the transition-rate family $\beta$ and the constants of The Transition-Rate Family and Aggregate Drift of Affine-Controlled Data and of Stability of the Mean-Field Flow under Perturbation of the Initial State and Weak Convergence of Controls, all formed from $(\beta_0,\beta_1)$ with the rate bound $B=\sup\beta$ identified in (AF); the horizon $T>0$; a fixed dense sequence in $L^{2}([0,T];\mathbb{R}^{m})$, supplied by $(\mathsf{v}_n)$; the natural number $\tilde{l}\ge1$ and the record space $(\mathbf{R},\mathcal{R})$ with horizon $T$ and $\tilde{l}$ channels, which are those of the adopted setting; an $\mathcal{A}$-valued observation-driven control policy $h$ with horizon $T$, control dimension $m$ and $\tilde{l}$ channels, supplied by the adopted $h$ (with values in $\mathcal{A}$, and $\mathcal{A}$-valued by (AF)); and a point of $\Delta^l$, supplied by $z_0$. Its record-frozen control paths are the adopted $a^{r}$ (the same policy and the same definition The Record-Frozen Control Path and Record-Frozen Policy). Hence its claims are available with $z_0$ in the role of its $x_0$. Claim 4 there gives, for every $r\in\mathbf{R}$: $\Phi^{r}_0=z_0$; $|\Phi^{r}_t-\Phi^{r}_u|\le K_b|t-u|$ with $K_b=2\sqrt{l}\,(l-1)B$; the components of $\Phi^{r}$ are continuous on $[0,T]$, hence measurable with respect to $\mathcal{B}_{[0,T]}$ by claim 4 of Measurability of Countable Suprema, Bounded Pointwise Limits, Monotone Functions, and Continuous Functions, so that $\Phi^{r}$ is a measurable map from $[0,T]$ to $\Delta^l$; and $\Phi^{r}_t=z_0+\int_{[0,t]}b(\Phi^{r}_u,a^{r}(u))\,du$ for every $t$, where $b$ is the aggregate state drift of $\beta$ (claim 3 of The Transition-Rate Family and Aggregate Drift of Affine-Controlled Data, the drift used in Stability of the Mean-Field Flow under Perturbation of the Initial State and Weak Convergence of Controls being that of The Transition-Rate Family and Aggregate Drift of Affine-Controlled Data). Claim 5 there gives the joint measurability of $(t,r)\mapsto\Phi^{r,\gamma}_t$ with respect to $\mathcal{B}_{[0,T]}\otimes\mathcal{R}$ and the $\mathcal{R}$-measurability of $r\mapsto\Phi^{r,\gamma}_t$ for fixed $t$.

The linearisation lemma. The setting of Pathwise Estimand Linearisation of an Open-Loop Aggregate Solution Along a Comparison Pair: Cell-Count Coefficients from the Fundamental Solution and the Three Error Terms consists of the setting of Linearisation of a Perturbed Controlled Aggregate Flow Along a Comparison Pair: Exact Variation-of-Constants Identity and Residual Bound together with the cells-and-clocks data. The former requires: $l\ge2$, $m\ge1$, a nonempty $\mathcal{A}\subseteq\mathbb{R}^m$, real $B\ge0$ and $T>0$, a transition-rate family $\beta$ on $l$ states with control set $\mathcal{A}$ and rate bound $B$, a twice continuously differentiable extension $(U,V,\bar\beta)$ of $\beta$ with derivative bound $K$ (supplied by $(U,W_\beta,\bar\beta)$ of (X), whose second component is the set written $V$ in Twice Continuously Differentiable Extension of a Transition-Rate Family), the labels $c$ with $v_c$, the label rates $\psi_c$, state gradients $g^{c}$, drift Jacobian $\mathcal{E}$ and the constant $\Lambda_2$ of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect (the very objects adopted 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 and Weighted Response and Pair-Exponent Quadratic Form on the Synthetic Copy: the Linearised Injection Equation, Comparison with the Profile Injection, and the Observation-Information Bound, which adopt them from Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect), and a comparison pair $(S,\mathsf{A})$ of measurable maps into $\Delta^l$ and $\mathcal{A}$ with $u\mapsto\mathcal{E}(S_u,\mathsf{A}_u)$ entrywise continuous, together with the fundamental solution $\Phi^{\mathcal{E}}$ and a bound $\bar\Phi$; the comparison pair of the adopted setting has measurable components with values in $\Delta^l$ and $\mathcal{A}$ (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 the continuity, $\Phi^{\mathcal{E}}$ and $\bar\Phi$ are supplied by (CP). The cells-and-clocks data require: the natural number $N\ge1$ with the lattice $\mathbb{G}_N$, the notions of clock family, control path and open-loop aggregate solution of Open-Loop Aggregate Solution Driven by Aggregate Transition Clocks (formed from the same $N$, $l$, $m$, $\mathcal{A}$, $B$, $T$, $\beta$); a real $R>0$ with $R\ge NBT$, natural numbers $J_c\ge1$ and boundaries $0=\mathsf{b}^{c}_0<\dots<\mathsf{b}^{c}_{J_c}=R$, all supplied by the adopted clock horizon and cells; the mean-field label rates $\phi_c(t)=\psi_c(S_t,\mathsf{A}_t)$, which are those of the adopted setting; and an estimand direction, supplied by $\mathbf{c}$. The objects $H^{c}_u=\Phi^{\mathcal{E}}(T,u)\mathcal{E}^{\star}_uv_c$, $\lVert H^{c}\rVert_1$ and the cell coefficients are defined from these data alone. It remains to see that the entry times exist, i.e. that for $q=(c,j)$ with $\bar{\mathsf{C}}^{c}_T\ge\mathsf{b}^{c}_j$ the set $\mathsf{S}_q=\{u\in[0,T]:\bar{\mathsf{C}}^{c}_u\ge\mathsf{b}^{c}_j\}$ has a least element. By claim 2 of Injection Weights from a Bounded Profile Along Mean-Field Clocks: Label-Rate Form of the Fluctuation Covariance, Weight Bounds, the Prior Quadratic Form, and the Step-Function Injection (whose objects the adopted setting forms from $\phi_c$), $u\mapsto\bar{\mathsf{C}}^{c}_u$ is nondecreasing; moreover, for $0\le u<t\le T$, $\bar{\mathsf{C}}^{c}_t-\bar{\mathsf{C}}^{c}_u=N\int_{[0,T]}\mathbf{1}_{(u,t]}\phi_c\,d\lambda_{[0,T]}\le NB\,\lambda_{[0,T]}((u,t])=NB(t-u)$, by linearity and monotonicity of the integral, $0\le\phi_c\le B$ (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), The Integral of an Indicator Function is the Measure of the Set and claim 4 of Existence of Lebesgue Measure on the Real Line. Now $\mathsf{S}_q$ is nonempty ($T\in\mathsf{S}_q$) and bounded below by $0$, so $\tau=-\sup\{-u:u\in\mathsf{S}_q\}$ exists by Least Upper Bound Property of the Real Numbers and is the greatest lower bound of $\mathsf{S}_q$; for every natural number $n$ there is $u_n\in\mathsf{S}_q$ with $u_n<\tau+1/n$ (otherwise $\tau+1/n$ would be a lower bound exceeding $\tau$), whence $\bar{\mathsf{C}}^{c}_\tau\ge\bar{\mathsf{C}}^{c}_{u_n}-NB(u_n-\tau)\ge\mathsf{b}^{c}_j-NB/n$ for every $n$. If $B=0$ this reads $\bar{\mathsf{C}}^{c}_\tau\ge\mathsf{b}^{c}_j$ directly; if $B>0$ and we had $\bar{\mathsf{C}}^{c}_\tau<\mathsf{b}^{c}_j$, claim 3 of The Archimedean Property of the Real Numbers with $\varepsilon=(\mathsf{b}^{c}_j-\bar{\mathsf{C}}^{c}_\tau)/(NB)$ would give an $n$ with $NB/n<\mathsf{b}^{c}_j-\bar{\mathsf{C}}^{c}_\tau$, a contradiction. Hence $\bar{\mathsf{C}}^{c}_\tau\ge\mathsf{b}^{c}_j$, so $\tau\in\mathsf{S}_q$ is the least element $\bar\tau_q$. Thus $\alpha_q$ is defined for every $q$, and the vector $\alpha$ is defined. (When $\bar\Sigma^{\sharp,r}(\omega)$ is an open-loop aggregate solution, as in claim 3, the entry times of Pathwise Estimand Linearisation of an Open-Loop Aggregate Solution Along a Comparison Pair: Cell-Count Coefficients from the Fundamental Solution and the Three Error Terms for that instance are the least elements of the same sets $\mathsf{S}_q$, so its cell coefficients are the $\alpha_q$ just defined.) This proves claim 1.

Claim 2. Countable reduction of the supremum. Fix $\omega\in\Omega$ and $r\in\mathbf{R}$ and put $f(t)=\sqrt{N}\,|\bar\Sigma^{\sharp,r}_t(\omega)-\Phi^{r}_t|$ for $t\in[0,T]$. Since $\bar\Sigma^{\sharp,r}_t(\omega)\in\mathbb{G}_N\subseteq\Delta^l$ (claim 2(a) of The Record-Driven Causal Intensity of the Open-Loop Aggregate Solution: Joint Measurability of the Record-Frozen Control, the Regularised Recursion Path, Non-Anticipation, and Measurability of the Likelihood, which applies to the copy clocks by 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 $\Phi^{r}_t\in\Delta^l$, one has $0\le f(t)\le\sqrt{2N}$, so $\sup_{t\in[0,T]}f(t)$ and $\sup_{t\in\mathsf{Q}_T}f(t)$ exist by Least Upper Bound Property of the Real Numbers and lie in $[0,\sqrt{2N}]$, with $\sup_{\mathsf{Q}_T}f\le\sup_{[0,T]}f$ since $\mathsf{Q}_T\subseteq[0,T]$. For the reverse inequality let $t\in[0,T]$; if $t=T$ then $t\in\mathsf{Q}_T$. Let $t<T$. By claim 2(a) of The Record-Driven Causal Intensity of the Open-Loop Aggregate Solution: Joint Measurability of the Record-Frozen Control, the Regularised Recursion Path, Non-Anticipation, and Measurability of the Likelihood, with $\vartheta_0<\dots<\vartheta_{K'}$ the recursion times of the data $(\mathsf{P}^{\sharp}(\omega),a^{r},x_0)$, the path $s\mapsto\bar\Sigma^{\sharp,r}_s(\omega)$ is constant on each $[\vartheta_k,\vartheta_{k+1})$ ($k<K'$) and on $[\vartheta_{K'},T]$; so there is a real $\delta>0$ with $t+\delta\le T$ and $\bar\Sigma^{\sharp,r}_s(\omega)=\bar\Sigma^{\sharp,r}_t(\omega)$ for all $s\in[t,t+\delta)$ (take $\delta=\vartheta_{k+1}-t$ if $t\in[\vartheta_k,\vartheta_{k+1})$, and $\delta=T-t$ if $t\in[\vartheta_{K'},T)$). By claim 3 of The Archimedean Property of the Real Numbers pick a natural number $n_0$ with $1/n_0<\delta$, and for each natural number $n\ge n_0$ let $m_n$ be the least natural number with $m_n>nt$ (the set of such numbers is nonempty by claim 1 of that theorem and has a least element by The Natural Numbers Are Well Ordered); then $m_n-1\le nt<m_n$, so $t_n=m_n/n$ is a rational number with $t<t_n\le t+1/n<t+\delta\le T$, i.e. $t_n\in\mathsf{Q}_T\cap[t,t+\delta)$. Hence $\bar\Sigma^{\sharp,r}_{t_n}(\omega)=\bar\Sigma^{\sharp,r}_t(\omega)$ and, by the reverse triangle inequality and the Lipschitz bound of claim 1,

Reindexing by $k\mapsto n=k+n_0-1$ so that the sequence starts at $k=1$, we have $|f(t)-f(t_{k+n_0-1})|\le\sqrt{N}K_b/(k+n_0-1)\le\sqrt{N}K_b/k$, and $\sqrt{N}K_b/k\to0$ (a scalar multiple of $1/k\to0$, preliminaries); so claim 3 of Order Properties of Limits of Real Sequences gives $f(t_n)\to f(t)$ along the reindexed sequence, and $f(t)\le\sup_{\mathsf{Q}_T}f$ by claim 1 of that theorem, as $f(t_n)\le\sup_{\mathsf{Q}_T}f$ for all $n$. Taking the supremum over $t$ gives $\bar{\mathsf{e}}(\omega,r)=\sup_{t\in\mathsf{Q}_T}f(t)\in[0,\sqrt{2N}]$.

Measurability. For fixed $t\in[0,T]$ and $\gamma\in\{1,\dots,l\}$, the map $(r,\omega)\mapsto\bar\Sigma^{\sharp,r,\gamma}_t(\omega)$ is $\mathcal{R}\otimes\mathcal{F}$-measurable, being the section at $t$ (claim 3 of Sections of Product-Measurable Sets and Maps Are Measurable, and Insertion Maps into Products Are Measurable) of the $\mathcal{B}_{[0,T]}\otimes(\mathcal{R}\otimes\mathcal{F})$-measurable map of claim 2(d) of The Record-Driven Causal Intensity of the Open-Loop Aggregate Solution: Joint Measurability of the Record-Frozen Control, the Regularised Recursion Path, Non-Anticipation, and Measurability of the Likelihood; composing with the exchange of coordinates, $(\omega,r)\mapsto\bar\Sigma^{\sharp,r,\gamma}_t(\omega)$ is $\mathcal{F}\otimes\mathcal{R}$-measurable. The map $(\omega,r)\mapsto\Phi^{r,\gamma}_t$ is $\mathcal{F}\otimes\mathcal{R}$-measurable, being the composition of the coordinate projection $(\omega,r)\mapsto r$ (claim 5 of Sections of Product-Measurable Sets and Maps Are Measurable, and Insertion Maps into Products Are Measurable) with the map $r\mapsto\Phi^{r,\gamma}_t$, which is $\mathcal{R}$-measurable for fixed $t$ by claim 5 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 (available by claim 1). Hence $(\omega,r)\mapsto\sqrt{N}|\bar\Sigma^{\sharp,r}_t(\omega)-\Phi^{r}_t|$ is $\mathcal{F}\otimes\mathcal{R}$-measurable for each fixed $t$ (componentwise differences, then the norm), with values in $[0,\sqrt{2N}]$. The set $\mathsf{Q}_T$ is countable (by The Integers and the Rational Numbers are Countable and claims 3 and 6 of Basic Properties of Countable Sets) and nonempty, so by Countable Set it is the set of terms of a sequence $(t_n)_{n\in\mathbb{N}}$, and $\bar{\mathsf{e}}=\sup_n f_n$ with $f_n(\omega,r)=\sqrt{N}|\bar\Sigma^{\sharp,r}_{t_n}(\omega)-\Phi^{r}_{t_n}|$; claim 1 of Measurability of Countable Suprema, Bounded Pointwise Limits, Monotone Functions, and Continuous Functions (uniform bound $\sqrt{2N}$) shows that $\bar{\mathsf{e}}$ is $\mathcal{F}\otimes\mathcal{R}$-measurable. The components of $X''(\omega,r)=\sqrt{N}(\bar\Sigma^{\sharp,r}_T(\omega)-\Phi^{r}_T)$ are measurable by the case $t=T$ above, and $|\mathbf{c}\cdot X''|\le|\mathbf{c}|\,|X''|\le|\mathbf{c}|\,\bar{\mathsf{e}}$ by Cauchy-Schwarz Inequality for the Euclidean Dot Product and the definition of $\bar{\mathsf{e}}$ (the value at $t=T$ is at most the supremum). By the bookkeeping paragraph, $\bar{\mathsf{e}}$ and $\mathbf{c}\cdot X''$ are bounded random variables on the copy, and the displayed integral formula is $(\ast)$.

Poisson law and $\mathsf{k}_4$. For $q=(c,j)$, the cell count $\mathsf{K}_{c,j}$ is the cell count $C_j$ of Uniform Representation of the Rate-One Poisson Counting Path on a Bounded Interval: Distinct Points, Poisson Increments, Conditional Law Given the Cell Counts, and Point Insertion for the data of the label $c$ (so identified in The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record), which by claim 2 of that lemma has the Poisson distribution with parameter $|I_{c,j}|=\mu_q$. By claim (b) of Factorial Moments and Moments of Every Order of the Poisson Distribution, $\mathsf{K}_q^{4}$ is integrable with $\mathbb{E}[\mathsf{K}_q^{4}]\le8^{4}+16\mu_q^{4}$, and by Moments of the Poisson Distribution $\mathsf{K}_q$ is square-integrable. Now $|\alpha\cdot(\mathsf{K}-\mu)|\le\sum_{q}|\alpha_q|(\mathsf{K}_q+\mu_q)$, and for nonnegative reals $a_1,\dots,a_d$ one has $(\sum_qa_q)^{2}\le d\sum_qa_q^{2}$ (Cauchy--Schwarz, Cauchy-Schwarz Inequality for the Euclidean Dot Product, against the all-ones vector), hence $(\sum_qa_q)^{4}\le d^{2}(\sum_qa_q^{2})^{2}\le d^{3}\sum_qa_q^{4}$; with $(\mathsf{K}_q+\mu_q)^{4}\le8(\mathsf{K}_q^{4}+\mu_q^{4})$ (apply $(a+b)^{2}\le2a^{2}+2b^{2}$ twice) this gives

whose expectation is finite; so $(\alpha\cdot(\mathsf{K}-\mu))^{4}$ is integrable (it is measurable, as a polynomial in the $\mathsf{K}_q$) and $\mathsf{k}_4$ is a finite nonnegative real number. In the same way $(\alpha\cdot(\mathsf{K}-\mu))^{2}$ is integrable, so $W=|\alpha\cdot(\mathsf{K}-\mu)|/\sqrt{N}$ is a square-integrable random variable on $(\Omega,\mathcal{F},P)$, and $W^{2}$ is square-integrable too, with $\mathbb{E}[W^{4}]=\mathsf{k}_4^{4}/N^{2}$.

The Gaussian term. Put $\xi=\Theta-\mathsf{K}\circ\varpi_1/\sqrt{N}$, a map $\Omega^{\sharp}\to\mathbb{R}^d$ with $\mathcal{F}^{\sharp}$-measurable coordinates, and let $g(z)=(\alpha\cdot z)^{2}$, a sequentially continuous map on $\mathbb{R}^d$; then $(\alpha\cdot\xi)^{2}=g\circ\xi$ is $\mathcal{F}^{\sharp}$-measurable, nonnegative, and it depends on $(\omega,\theta)$ only. Exactly as in the derivation of $(\ast)$ (density, then Tonelli integrating first over $r$, with $\int_{\mathbf{R}}\ell^{\sharp,\omega}d\rho=1$, then over $\theta$ and $\omega$),

For fixed $\omega$, claim 2 of Translation and Reflection Invariance of Lebesgue Measure on $\mathbb{R}^n$ applied to the measurable function $z\mapsto(\alpha\cdot z)^{2}\varphi_\eta(z)$ and the translation by $-x_\omega$ shows that the inner integral equals $\int_{\mathbb{R}^d}(\alpha\cdot z)^{2}\varphi_\eta(z)\,d\lambda_d(z)$. With $Z_\alpha(z)=\alpha\cdot z/\eta$ and $\kappa_\alpha=|\alpha|^{2}/\eta$ as in claim 3 of The Gaussian Smoothing Weight: Normalization, Derivatives, Exponential Tilting, Moments, and First-Order Remainder, $(\alpha\cdot z)^{2}\varphi_\eta(z)=\eta^{2}Z_\alpha(z)^{2}\varphi_\eta(z)$, and claim 4 there gives $\int_{\mathbb{R}^d}Z_\alpha^{2}\varphi_\eta\,d\lambda_d=\kappa_\alpha$ (the integrand being integrable, its integral as a nonnegative function is this value); hence the inner integral equals $\eta^{2}\kappa_\alpha=\eta|\alpha|^{2}$ for every $\omega$, and $\int_{\Omega^{\sharp}}(\alpha\cdot\xi)^{2}d\mu^{\sharp}=\eta|\alpha|^{2}$. The same computation with $\alpha$ replaced by the basis vector $e_q$ gives $\int_{\Omega^{\sharp}}\xi_q^{2}\,d\mu^{\sharp}=\eta$, so $\xi_q$ is square-integrable on the copy; $\mathsf{K}_q\circ\varpi_1/\sqrt{N}$ is square-integrable on the copy because $\int_{\Omega^{\sharp}}(\mathsf{K}_q\circ\varpi_1)^{2}d\mu^{\sharp}=\mathbb{E}[\mathsf{K}_q^{2}]<\infty$ by $(\ast)$; hence $\Theta_q=\xi_q+\mathsf{K}_q\circ\varpi_1/\sqrt{N}$ is square-integrable (closure of square-integrability under sums, Square-Integrable Random Variables and the Mean-Square Inner Product), and $\lVert\alpha\cdot\Theta-\alpha\cdot\mathsf{K}\circ\varpi_1/\sqrt{N}\rVert_2=\lVert\alpha\cdot\xi\rVert_2=\sqrt{\eta}\,|\alpha|$. This proves claim 2.

Claim 3. Let $\omega\in G$ and $r\in\mathsf{R}^{\mathrm{cl}}_\omega$. By (G), $G\subseteq G_{L,D}$, and $G_{L,D}\subseteq\Omega^{U}_0$ by its definition in Uniform Pair-Exponent Bound on the Synthetic Copy: Removed-Clock Intensities, the Insertion Response on the Clock-Good Event, and Bounds on the Pair Covariances; by (G$'$), $\omega\in G^{\mathsf{m}}$; and $r\in\mathsf{T}_\omega$ by (CL), so $(r,\omega)\in\mathsf{G}^{\sharp}$ by the definition of the tracked records in Uniform Pair-Exponent Bound on the Synthetic Copy: Removed-Clock Intensities, the Insertion Response on the Clock-Good Event, and Bounds on the Pair Covariances.

The solution and its consumed clocks. By claim 2(c) of The Record-Driven Causal Intensity of the Open-Loop Aggregate Solution: Joint Measurability of the Record-Frozen Control, the Regularised Recursion Path, Non-Anticipation, and Measurability of the Likelihood (applied to the copy clocks), $\bar\Sigma^{\sharp,r}(\omega)$ is the unique open-loop aggregate solution on $[0,T]$ for the data $(\mathsf{P}^{\sharp}(\omega),a^{r},x_0)$; here $\mathsf{P}^{\sharp}(\omega)$ is a clock family (every path of $\mathsf{P}^{\sharp,c}$ is a counting path, claim 2 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record) and $a^{r}$ is a control path in the sense of Open-Loop Aggregate Solution Driven by Aggregate Transition Clocks: its components are the sections at fixed $r$ of the jointly measurable map of claim 1 of The Record-Driven Causal Intensity of the Open-Loop Aggregate Solution: Joint Measurability of the Record-Frozen Control, the Regularised Recursion Path, Non-Anticipation, and Measurability of the Likelihood, measurable by claim 3 of Sections of Product-Measurable Sets and Maps Are Measurable, and Insertion Maps into Products Are Measurable, and its values lie in $\mathcal{A}$ by 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. The consumed clock times $\mathsf{C}^{\sharp,c,r}_t(\omega)$ of the adopted setting are, by their definition in Weighted Response and Pair-Exponent Quadratic Form on the Synthetic Copy: the Linearised Injection Equation, Comparison with the Profile Injection, and the Observation-Information Bound (for $\omega\in G^{\mathsf{m}}$ and $r\in\mathsf{T}_\omega$), the consumed clock times of this very solution in the sense of Open-Loop Aggregate Solution Driven by Aggregate Transition Clocks.

Cell counts. The cell counts of Pathwise Estimand Linearisation of an Open-Loop Aggregate Solution Along a Comparison Pair: Cell-Count Coefficients from the Fundamental Solution and the Three Error Terms for the clock family $\mathsf{P}^{\sharp}(\omega)$ are $\mathsf{P}^{\sharp,c}_{\mathsf{b}^{c}_j}(\omega)-\mathsf{P}^{\sharp,c}_{\mathsf{b}^{c}_{j-1}}(\omega)$. Since $\omega\in\Omega^{U}_0$, the definition of the clocks in The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record gives, with $y=\mathsf{K}(\omega)$,

because on $\Omega^{U}_0$ every $U^{c,j'}_i(\omega)$ lies in $I_{c,j'}=(\mathsf{b}^{c}_{j'-1},\mathsf{b}^{c}_{j'}]$ and the cells of the clock $c$ are pairwise disjoint, so the indicator equals $1$ exactly when $j'=j$.

The flow. By claim 1, $\Phi^{r}$ is a measurable map from $[0,T]$ to $\Delta^l$ satisfying $\Phi^{r}_t=\Phi^{r}_0+\int_{[0,t]}b(\Phi^{r}_u,a^{r}_u)\,du$ for every $t$, with $\Phi^{r}_0=z_0$, for the same control path $a^{r}$ and with $b$ the drift of Linearisation of a Perturbed Controlled Aggregate Flow Along a Comparison Pair: Exact Variation-of-Constants Identity and Residual Bound (claim 1 there identifies that drift with the aggregate state drift of $\beta$). So $\Phi^{r}$ is admissible as the flow $y$ of Pathwise Estimand Linearisation of an Open-Loop Aggregate Solution Along a Comparison Pair: Cell-Count Coefficients from the Fundamental Solution and the Three Error Terms, with $y_0=z_0$.

The clock-discrepancy hypothesis. By (CL), $|\bar\Sigma^{\sharp,r}_t(\omega)-S_t|\le\varepsilon_S$ for every $t$ and $\mathsf{D}^{r}_{\mathrm{ctl}}\le\varepsilon_{\mathrm{ctl}}$. Since $\omega\in G^{\mathsf{m}}$ and $r\in\mathsf{T}_\omega$, claim 2 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 yields $|\mathsf{C}^{\sharp,c,r}_u(\omega)-\bar{\mathsf{C}}^{c}_u|\le N(\Lambda_1T\varepsilon_S+\varepsilon_{\mathrm{ctl}})=\mathsf{w}^{\mathrm{clk}}$ for every label $c$ and every $u\in[0,T]$, the mean-field clocks $\bar{\mathsf{C}}^{c}$ of that lemma and of Pathwise Estimand Linearisation of an Open-Loop Aggregate Solution Along a Comparison Pair: Cell-Count Coefficients from the Fundamental Solution and the Three Error Terms being both those of Injection Weights from a Bounded Profile Along Mean-Field Clocks: Label-Rate Form of the Fluctuation Covariance, Weight Bounds, the Prior Quadratic Form, and the Step-Function Injection formed from $\phi_c$. This is hypothesis (CD) with $w_1=\mathsf{w}^{\mathrm{clk}}\ge0$.

The linearisation bound. All data of Pathwise Estimand Linearisation of an Open-Loop Aggregate Solution Along a Comparison Pair: Cell-Count Coefficients from the Fundamental Solution and the Three Error Terms are now in place for the instance $p=\mathsf{P}^{\sharp}(\omega)$, $a=a^{r}$, $x_0$, $\Sigma=\bar\Sigma^{\sharp,r}(\omega)$, $y=\Phi^{r}$, $\mathbf{c}$ and $w_1=\mathsf{w}^{\mathrm{clk}}$, with cell counts $\mathsf{K}_q(\omega)$ and cell coefficients $\alpha_q$ (claim 1). Its claim 3, with $e_t=\bar\Sigma^{\sharp,r}_t(\omega)-\Phi^{r}_t$ and $\bar e=\sup_{u\in[0,T]}\sqrt{N}|e_u|=\bar{\mathsf{e}}(\omega,r)=:\bar{\mathsf{e}}$, gives

since $\sqrt{N}\,\mathbf{c}\cdot(\Sigma_T-y_T)=\mathbf{c}\cdot X''(\omega,r)$ and $\frac{1}{\sqrt{N}}\sum_q\alpha_q(\mathsf{K}_q(\omega)-\mu_q)=\alpha\cdot(\mathsf{K}(\omega)-\mu)/\sqrt{N}$, where

with $\mathsf{d}_u=\sum_{c}|g^{c}(S_u,a^{r}_u)-g^{c}(S_u,\mathsf{A}_u)|$ the control-gradient discrepancy of Linearisation of a Perturbed Controlled Aggregate Flow Along a Comparison Pair: Exact Variation-of-Constants Identity and Residual Bound, and

(the window discrepancies being those of Cell-Count Form of a Compensated Counting-Path Functional Along a Time Change: Window-Discrepancy Bounds for the Time-Change and Partial-Cell Errors formed with $R$, exactly as in (DM)). We bound the two integrals. For every $u\in[0,T]$, by claim 6 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, (CL) and the definition of $\bar{\mathsf{e}}$,

so by monotonicity of the integral, The Integral of an Indicator Function is the Measure of the Set and claim 1 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval, $\int_{[0,T]}|\Phi^{r}_u-S_u|\,du\le T(\varepsilon_S+\bar{\mathsf{e}}/\sqrt{N})$. Next, $\mathsf{d}_u$ is at most the integrand of $\mathsf{D}^{r}_{\mathrm{ctl}}$ at $u$ (which adds the nonnegative terms $|\psi_c(S_u,a^{r}_u)-\psi_c(S_u,\mathsf{A}_u)|$; both integrands are bounded measurable by claim 1 of Linearisation of a Perturbed Controlled Aggregate Flow Along a Comparison Pair: Exact Variation-of-Constants Identity and Residual Bound and 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), so $\int_{[0,T]}\mathsf{d}_u\,du\le\mathsf{D}^{r}_{\mathrm{ctl}}\le\varepsilon_{\mathrm{ctl}}$. Finally, by (DM) and $\omega\in G$, $\mathsf{E}_{\mathrm{cell}}\le\frac{|\mathbf{c}|}{\sqrt{N}}\sum_{c}(\sqrt{2}+\lVert H^{c}\rVert_1)\Xi^{c}(\omega)$. Inserting these three bounds gives exactly $|\alpha\cdot(\mathsf{K}(\omega)-\mu)/\sqrt{N}-\mathbf{c}\cdot X''(\omega,r)|\le\tilde{Z}(\omega,r)$ with $\tilde{Z}$ as displayed in the statement.