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.

Scope of claims 1 and 2. The preliminaries, the product-space bookkeeping, the derivation of $(\ast)$ and the proofs of claims 1 and 2 below use only the adopted setting with hypotheses (OC), (X), (W), (G), (G$'$), (AF) and (CP), and of (CL) at most the fact that $\varepsilon_S$ and $\varepsilon_{\mathrm{ctl}}$ are real numbers (in fact they do not mention them): they invoke The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record, 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, The Gaussian Smoothing Weight: Normalization, Derivatives, Exponential Tilting, Moments, and First-Order Remainder, 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, Factorial Moments and Moments of Every Order of the Poisson Distribution, Moments of the Poisson Distribution, the flow lemmas named in (AF), the linearisation setting named in (CP), the objects $\psi_c$, $g^{c}$, $\mathcal{E}$ and $\Lambda_2$ of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect as adopted through 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, the comparison data and claim 1 only 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 (which are free of $\varepsilon_S$ and $\varepsilon_{\mathrm{ctl}}$; its claims 2--4 are used only in the proof of claim 3), the mean-field clocks 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 and general measure-theoretic tools, and they mention none of $\mathsf{w}^{\mathrm{clk}}$, $\mathsf{R}^{\mathrm{cl}}_\omega$, $\pi^{\mathrm{nc}}$, $\bar\pi^{\mathrm{nc}}$, $\mathsf{c}_4$ or $\Xi^{c}$. Hypotheses (CL) (beyond its first clause), (FM) and (DM) enter only in the proofs of claims 3 and 4, where each use is marked. This establishes the scope statement of claims 1 and 2 made in the statement, the measurability of $\bar{\mathsf{e}}$ invoked in (FM) being the measurability assertion of claim 2.

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 $\bar{\mathsf{e}}$,