Proof of The Van Trees Data of the Trimmed Synthetic Copy Supplied by an N-Agent Solution: Transport to the Record Support, Regularity of the Smoothed Joint Density, the Information Bound in the Injection Direction, and the Pairing of the Cell Coefficients with the Profile Response

Throughout, the composition of measurable maps is measurable (preimages compose), and a map into a product of two measurable spaces is measurable as soon as the preimage of every measurable rectangle is measurable, by claim 2 of Generator Criterion for Measurability, the rectangles generating the product $\sigma$-algebra; we use both facts without further comment. Write $\mathsf{N}=\Omega^{\sharp}\setminus\Omega^{\mathrm{tr}}$. By claim 5 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record, the data $(d,\eta,\mathsf{p},f)$ on $(\mathbf{R}_{s},\mathcal{R}_{s},\rho)$ satisfy all hypotheses of Gaussian Smoothing of a Kernel-Weighted Discrete Family with Countable Support: Joint Density, van Trees Regularity, and the Symmetrised Directional Score Bound, and, as recorded in the setting of Pathwise Reduction of the Symmetrised Move Information on the Synthetic Copy: Removed-Clock Likelihoods, the Shifted Family, and the Averaging Bound, claims 1--3 of Gaussian Smoothing of a Kernel-Weighted Discrete Family with Countable Support: Joint Density, van Trees Regularity, and the Symmetrised Directional Score Bound are available for the $n=d$ moves $a_{q}$ and the weights $w$ (indexed by $\mathsf{L}$ through the fixed bijection), $\mathsf{J}^{\mathrm{sym}}$ being its symmetrised kernel-weighted move information for these moves and weights; $\bar{f}$, $\mathsf{R}_{+}$, $g$, $g_{q}(\theta,r)=g(\theta-a_{q},r)$ and $\bar{g}=\tfrac12g+\tfrac{1}{2d}\sum_{q}g_{q}$ are the objects of that lemma. We refer to this as the kernel instance. 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, $\mu^{\sharp}$ is a probability measure and $\Theta$ and $\mathsf{D}$ are measurable with respect to $\mathcal{F}^{\sharp}$ and $\mathcal{B}(\mathbb{R}^{d})$, respectively $\mathcal{R}_{s}$; hence the pair $(\Theta,\mathsf{D})$ is measurable with respect to $\mathcal{F}^{\sharp}$ and $\mathcal{B}(\mathbb{R}^{d})\otimes\mathcal{R}_{s}$, and claim 5 of that lemma gives $\int_{\Omega^{\sharp}}F(\Theta,\mathsf{D})\,d\mu^{\sharp}=\int F\,g\,d(\lambda_{d}\otimes\rho)$ for every $\mathcal{B}(\mathbb{R}^{d})\otimes\mathcal{R}_{s}$-measurable $F\ge0$ (the density identity). Finally, by claim 1(a) 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, claims 1--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 available and $X''(\omega,r)=X''_{s}(\omega,\theta,r)$ on $\Omega^{\sharp}$.

Claim 1. The record support. By claim 1 of the kernel instance, $\bar{f}$ is $\mathcal{R}_{s}$-measurable, $\mathsf{R}_{+}\in\mathcal{R}_{s}$, $g$ is finite and measurable with respect to $\mathcal{B}(\mathbb{R}^{d})\otimes\mathcal{R}_{s}$, and $g(\theta,r)>0$ if and only if $r\in\mathsf{R}_{+}$. Hence $\Omega^{\mathrm{tr}}=\mathsf{D}^{-1}(\mathsf{R}_{+})\in\mathcal{F}^{\sharp}$ and $\mathsf{N}=\mathsf{D}^{-1}(\mathbf{R}_{s}\setminus\mathsf{R}_{+})=(\Theta,\mathsf{D})^{-1}(\mathbb{R}^{d}\times(\mathbf{R}_{s}\setminus\mathsf{R}_{+}))$. Let $F$ be the indicator of the rectangle $\mathbb{R}^{d}\times(\mathbf{R}_{s}\setminus\mathsf{R}_{+})$, which is measurable (claim 1 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions). Then $\mathbf{1}_{\mathsf{N}}=F(\Theta,\mathsf{D})$, and $F(\theta,r)g(\theta,r)=0$ for every $(\theta,r)$, because $g(\theta,r)=0$ when $r\notin\mathsf{R}_{+}$. By The Integral of an Indicator Function is the Measure of the Set and the density identity, $\mu^{\sharp}(\mathsf{N})=\int_{\Omega^{\sharp}}F(\Theta,\mathsf{D})\,d\mu^{\sharp}=\int Fg\,d(\lambda_{d}\otimes\rho)=0$, the integral of the zero function being $0$ (the only nonnegative simple minorant of $0$ is $0$, whose integral is $0$). By additivity of the measure $\mu^{\sharp}$, $\mu^{\sharp}(\Omega^{\mathrm{tr}})=\mu^{\sharp}(\Omega^{\sharp})-\mu^{\sharp}(\mathsf{N})=1$.

The trimmed copy. By claim 1 of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions applied to $(\Omega^{\sharp},\mathcal{F}^{\sharp},\mu^{\sharp})$ and $X_{0}=\Omega^{\mathrm{tr}}$, the restriction $(\Omega^{\mathrm{tr}},\mathcal{F}^{\mathrm{tr}},\mu^{\mathrm{tr}})$ with $\mathcal{F}^{\mathrm{tr}}=\{A\in\mathcal{F}^{\sharp}:A\subseteq\Omega^{\mathrm{tr}}\}$ and $\mu^{\mathrm{tr}}(A)=\mu^{\sharp}(A)$ is a measure space, and $\mu^{\mathrm{tr}}(\Omega^{\mathrm{tr}})=1$, so it is a probability space. Let $Z:\Omega^{\sharp}\to[0,\infty]$ be $\mathcal{F}^{\sharp}$-measurable. The zero extension of $Z^{\mathrm{tr}}$ is $Z\mathbf{1}_{\Omega^{\mathrm{tr}}}$, which is measurable: $\{Z\mathbf{1}_{\Omega^{\mathrm{tr}}}>a\}$ equals $\{Z>a\}\cap\Omega^{\mathrm{tr}}$ for $a\ge0$ and $\Omega^{\sharp}$ for $a<0$. By the same claim, $Z^{\mathrm{tr}}$ is $\mathcal{F}^{\mathrm{tr}}$-measurable and $\int_{\Omega^{\mathrm{tr}}}Z^{\mathrm{tr}}\,d\mu^{\mathrm{tr}}=\int_{\Omega^{\sharp}}Z\mathbf{1}_{\Omega^{\mathrm{tr}}}\,d\mu^{\sharp}$. Since $Z=Z\mathbf{1}_{\Omega^{\mathrm{tr}}}+Z\mathbf{1}_{\mathsf{N}}$ pointwise, additivity of the integral (Linearity and Monotonicity of the Lebesgue Integral) gives $\int Z\,d\mu^{\sharp}=\int Z\mathbf{1}_{\Omega^{\mathrm{tr}}}\,d\mu^{\sharp}+\int Z\mathbf{1}_{\mathsf{N}}\,d\mu^{\sharp}$, and it remains to see that the last integral is $0$. For $n\in\mathbb{N}$ put $Z_{n}=\min(Z,n)\mathbf{1}_{\mathsf{N}}$, measurable because $\{Z_{n}>a\}$ is $\{Z>a\}\cap\mathsf{N}$ for $0\le a<n$, empty for $a\ge n$, and $\Omega^{\sharp}$ for $a<0$. Then $0\le Z_{n}\le n\mathbf{1}_{\mathsf{N}}$, so by monotonicity and positive homogeneity and The Integral of an Indicator Function is the Measure of the Set, $\int Z_{n}\,d\mu^{\sharp}\le n\,\mu^{\sharp}(\mathsf{N})=0$. The sequence $(Z_{n})$ is nondecreasing with pointwise limit $Z\mathbf{1}_{\mathsf{N}}$ (at a point of $\mathsf{N}$ with $Z<\infty$ the terms equal $Z$ from some index on; where $Z=\infty$ they equal $n$; off $\mathsf{N}$ all vanish), so the monotone convergence theorem gives $\int Z\mathbf{1}_{\mathsf{N}}\,d\mu^{\sharp}=\sup_{n}\int Z_{n}\,d\mu^{\sharp}=0$. This proves the integral identity.

Laws and mean-square norms. Let $(\mathsf{Z},\mathcal{G})$ be a measurable space and $V:\Omega^{\sharp}\to\mathsf{Z}$ measurable. For $A\in\mathcal{G}$, $(V^{\mathrm{tr}})^{-1}(A)=V^{-1}(A)\cap\Omega^{\mathrm{tr}}$ belongs to $\mathcal{F}^{\sharp}$ and is contained in $\Omega^{\mathrm{tr}}$, hence belongs to $\mathcal{F}^{\mathrm{tr}}$; so $V^{\mathrm{tr}}$ is measurable. Moreover, by The Integral of an Indicator Function is the Measure of the Set and the integral identity applied to $Z=\mathbf{1}_{V^{-1}(A)}$ (whose restriction is the indicator of $(V^{\mathrm{tr}})^{-1}(A)$ in $\Omega^{\mathrm{tr}}$), $\mu^{\mathrm{tr}}((V^{\mathrm{tr}})^{-1}(A))=\int_{\Omega^{\sharp}}\mathbf{1}_{V^{-1}(A)}\,d\mu^{\sharp}=\mu^{\sharp}(V^{-1}(A))$; that is, the two image measures agree on $\mathcal{G}$. If $Z$ is a random variable on the copy, then $Z^{\mathrm{tr}}$ is a random variable on the trimmed copy (the case $\mathsf{Z}=\mathbb{R}$), $(Z^{\mathrm{tr}})^{2}=(Z^{2})^{\mathrm{tr}}$, and the integral identity applied to $Z^{2}$ gives $\mathbb{E}^{\mathrm{tr}}[(Z^{\mathrm{tr}})^{2}]=\int_{\Omega^{\sharp}}Z^{2}\,d\mu^{\sharp}$; by Square-Integrable Random Variables and the Mean-Square Inner Product, $Z$ is square-integrable if and only if $Z^{\mathrm{tr}}$ is, and then the mean-square norms, the square roots of these two equal quantities, agree.

Consequences. (a) By 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 (with the estimand direction $\mathbf{c}$), the pair $V=(\mathbf{c}\cdot X''_{s},\mathsf{D})$ is measurable with respect to $\mathcal{F}^{\sharp}$ and $\mathcal{B}(\mathbb{R})\otimes\mathcal{R}_{s}$, $\mathbf{c}\cdot X''_{s}$ is square-integrable on the copy, and the image measure of $\mu^{\sharp}$ under $V$ equals the image measure of $P^{\mathrm{ag}}$ under $(\mathbf{c}\cdot X'_{s},W^{(s)})$. Since $V^{\mathrm{tr}}=(\mathbf{c}\cdot X^{\mathrm{tr}},\mathsf{D}^{\mathrm{tr}})$ and $(\mathbf{c}\cdot X''_{s})^{\mathrm{tr}}=\mathbf{c}\cdot X^{\mathrm{tr}}$, the preceding paragraph gives (a). (b) Let $m^{\mathrm{est}}:\mathbf{R}_{s}\to\mathbb{R}$ be measurable with respect to $\mathcal{R}_{s}$ and the Borel $\sigma$-algebra. For a Borel set $E'$, the preimage of $E'$ under $m^{\mathrm{est}}|_{\mathsf{R}_{+}}$ is $(m^{\mathrm{est}})^{-1}(E')\cap\mathsf{R}_{+}$, which belongs to $\mathcal{R}_{s}$ and is contained in $\mathsf{R}_{+}$, hence belongs to $\mathcal{R}_{+}$; so $m^{\mathrm{est}}|_{\mathsf{R}_{+}}$ is $\mathcal{R}_{+}$-measurable. Since $\mathsf{D}^{\mathrm{tr}}$ takes its values in $\mathsf{R}_{+}$ and is the restriction of $\mathsf{D}$, $m^{\mathrm{est}}|_{\mathsf{R}_{+}}\circ\mathsf{D}^{\mathrm{tr}}$ is the restriction $(m^{\mathrm{est}}\circ\mathsf{D})^{\mathrm{tr}}$ of the random variable $m^{\mathrm{est}}(\mathsf{D})$ on the copy ($m^{\mathrm{est}}\circ\mathsf{D}$ is measurable as a composition), and the preceding paragraph gives the equivalence of square-integrability and the equality of norms. The constant map $\varsigma$ is $\mathcal{R}_{s}$-measurable (its preimages are $\emptyset$ or $\mathbf{R}_{s}$). Now let $y_{1},y_{2}>0$ and $(\Xi^{c})_{c}$ be as in claim 5 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, with $y_{1}$, $y_{2}$ in the roles of its $x_{1}$, $x_{2}$. That claim gives $\lVert Z\rVert_{2}\le\mathsf{e}_{1}+\mathsf{e}_{2}+\mathsf{e}_{3}+\mathsf{e}_{4}+\hat{\mathsf{e}}_{5}$ on the copy for $Z=\alpha\cdot\Theta+\varsigma(\mathsf{D})-\mathbf{c}\cdot X''_{s}$ (its final sentence identifying the left-hand side there with the norm of this $Z$), a random variable which is square-integrable because its three summands are (claim 4 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, all of whose hypotheses hold by that claim 5, together with claim 1(a) 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 and the closure of square-integrability under sums and scalar multiples recorded in Square-Integrable Random Variables and the Mean-Square Inner Product), and $Z^{\mathrm{tr}}=\alpha\cdot\Theta^{\mathrm{tr}}+\varsigma(\mathsf{D}^{\mathrm{tr}})-\mathbf{c}\cdot X^{\mathrm{tr}}$; the preceding paragraph transfers the bound to the trimmed copy.

Claim 2. The restricted record space. By claim 1 of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions, $(\mathsf{R}_{+},\mathcal{R}_{+},\rho_{+})$ with $\mathcal{R}_{+}=\{A\in\mathcal{R}_{s}:A\subseteq\mathsf{R}_{+}\}$ and $\rho_{+}(A)=\rho(A)$ is a measure space. The measure $\rho$ is $\sigma$-finite (as recorded in 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): there is a sequence $(A_{n})$ in $\mathcal{R}_{s}$ with union $\mathbf{R}_{s}$ and $\rho(A_{n})<\infty$; then $(A_{n}\cap\mathsf{R}_{+})$ is a sequence in $\mathcal{R}_{+}$ with union $\mathsf{R}_{+}$ and $\rho_{+}(A_{n}\cap\mathsf{R}_{+})\le\rho(A_{n})<\infty$ by monotonicity of $\rho$, so $\rho_{+}$ is $\sigma$-finite.

The restricted product $\sigma$-algebra. Let $(\mathsf{Y},\mathcal{Y})$ be a measurable space and put $\mathsf{H}_{\mathsf{Y}}=\mathsf{Y}\times\mathsf{R}_{+}$, a measurable rectangle, so $\mathsf{H}_{\mathsf{Y}}\in\mathcal{Y}\otimes\mathcal{R}_{s}$. Write $\mathcal{H}_{1}=\{E\in\mathcal{Y}\otimes\mathcal{R}_{s}:E\subseteq\mathsf{H}_{\mathsf{Y}}\}$ for the restriction of $\mathcal{Y}\otimes\mathcal{R}_{s}$ to $\mathsf{H}_{\mathsf{Y}}$, a $\sigma$-algebra on $\mathsf{H}_{\mathsf{Y}}$ by claim 1 of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions, and $\mathcal{H}_{2}=\mathcal{Y}\otimes\mathcal{R}_{+}$, the $\sigma$-algebra on $\mathsf{H}_{\mathsf{Y}}$ generated by the rectangles $A\times B$ with $A\in\mathcal{Y}$ and $B\in\mathcal{R}_{+}$. Every such rectangle belongs to $\mathcal{Y}\otimes\mathcal{R}_{s}$ (as $B\in\mathcal{R}_{s}$) and is contained in $\mathsf{H}_{\mathsf{Y}}$, hence belongs to $\mathcal{H}_{1}$; since $\mathcal{H}_{2}$ is the smallest $\sigma$-algebra on $\mathsf{H}_{\mathsf{Y}}$ containing these rectangles, $\mathcal{H}_{2}\subseteq\mathcal{H}_{1}$. Conversely let $\mathcal{D}$ be the family of all $E\in\mathcal{Y}\otimes\mathcal{R}_{s}$ with $E\cap\mathsf{H}_{\mathsf{Y}}\in\mathcal{H}_{2}$. It contains $\mathsf{Y}\times\mathbf{R}_{s}$ (whose intersection with $\mathsf{H}_{\mathsf{Y}}$ is the rectangle $\mathsf{H}_{\mathsf{Y}}$), is closed under complements (the intersection of the complement of $E$ with $\mathsf{H}_{\mathsf{Y}}$ is $\mathsf{H}_{\mathsf{Y}}\setminus(E\cap\mathsf{H}_{\mathsf{Y}})$) and under countable unions (intersection distributes over unions), and contains every rectangle $A\times B$ with $A\in\mathcal{Y}$, $B\in\mathcal{R}_{s}$, because $(A\times B)\cap\mathsf{H}_{\mathsf{Y}}=A\times(B\cap\mathsf{R}_{+})$ with $B\cap\mathsf{R}_{+}\in\mathcal{R}_{+}$. Thus $\mathcal{D}$ is a $\sigma$-algebra containing the generators of $\mathcal{Y}\otimes\mathcal{R}_{s}$, so $\mathcal{D}=\mathcal{Y}\otimes\mathcal{R}_{s}$; for $E\in\mathcal{H}_{1}$ this gives $E=E\cap\mathsf{H}_{\mathsf{Y}}\in\mathcal{H}_{2}$. Hence $\mathcal{H}_{1}=\mathcal{H}_{2}$. From now on we use this with $(\mathsf{Y},\mathcal{Y})=(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$, so that $\mathsf{H}_{\mathsf{Y}}=\mathsf{H}$ and $\mathcal{H}_{1}=\mathcal{H}_{2}=\mathcal{B}(\mathbb{R}^{d})\otimes\mathcal{R}_{+}$.

The restricted product measure. By Lebesgue Measure on $\mathbb{R}^n$, $\mathcal{B}(\mathbb{R}^{d})$ is the $\sigma$-algebra $\mathcal{B}_{d}$ of Finite Products of Lebesgue Measure and Coordinate Integration on $\mathbb{R}^l$ and $\lambda_{d}$ is the product measure $\lambda_{d}$ of that lemma, which is $\sigma$-finite by its claim 1. The restriction $(\lambda_{d}\otimes\rho)|_{\mathsf{H}}$ of the product measure $\lambda_{d}\otimes\rho$ to $\mathsf{H}$ is a measure on $\mathcal{H}_{1}=\mathcal{H}_{2}$ (claim 1 of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions) with $(\lambda_{d}\otimes\rho)|_{\mathsf{H}}(A\times B)=(\lambda_{d}\otimes\rho)(A\times B)=\lambda_{d}(A)\rho(B)=\lambda_{d}(A)\rho_{+}(B)$ for $A\in\mathcal{B}(\mathbb{R}^{d})$ and $B\in\mathcal{R}_{+}$. Since $\lambda_{d}$ and $\rho_{+}$ are $\sigma$-finite, Existence and Uniqueness of the Product Measure applied to $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\lambda_{d})$ and $(\mathsf{R}_{+},\mathcal{R}_{+},\rho_{+})$ shows that exactly one measure on $\mathcal{H}_{2}$ has these rectangle values, namely $\lambda_{d}\otimes\rho_{+}$; hence $(\lambda_{d}\otimes\rho)|_{\mathsf{H}}=\lambda_{d}\otimes\rho_{+}$. Consequently, for every $\mathcal{H}_{2}$-measurable $\phi:\mathsf{H}\to[0,\infty]$ with zero extension $\tilde\phi$ to $\mathbb{R}^{d}\times\mathbf{R}_{s}$, claim 1 of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions gives

$$\int_{\mathsf{H}}\phi\,d(\lambda_{d}\otimes\rho_{+})=\int_{\mathbb{R}^{d}\times\mathbf{R}_{s}}\tilde\phi\,d(\lambda_{d}\otimes\rho)\qquad(\ast)$$

(the extension identity); and a function on $\mathsf{H}$ is $\mathcal{H}_{2}$-measurable if and only if its zero extension is $\mathcal{B}(\mathbb{R}^{d})\otimes\mathcal{R}_{s}$-measurable. In particular, if $\psi$ is a real-valued $\mathcal{B}(\mathbb{R}^{d})\otimes\mathcal{R}_{s}$-measurable function on $\mathbb{R}^{d}\times\mathbf{R}_{s}$, its restriction $\psi|_{\mathsf{H}}$ is $\mathcal{H}_{2}$-measurable: the preimage of a Borel set $E'$ under $\psi|_{\mathsf{H}}$ is $\psi^{-1}(E')\cap\mathsf{H}\in\mathcal{H}_{1}=\mathcal{H}_{2}$.

Random variables and record. Each $\Theta_{q}$ is square-integrable on the copy 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 (available by its scope statement, as recorded in the adopted setting), so $\Theta^{\mathrm{tr}}_{q}$ is a square-integrable random variable on the trimmed copy by claim 1. For $B\in\mathcal{R}_{+}$, $(\mathsf{D}^{\mathrm{tr}})^{-1}(B)=\mathsf{D}^{-1}(B)$ (as $B\subseteq\mathsf{R}_{+}$), which belongs to $\mathcal{F}^{\sharp}$ and is contained in $\Omega^{\mathrm{tr}}$, hence to $\mathcal{F}^{\mathrm{tr}}$: $\mathsf{D}^{\mathrm{tr}}$ is measurable with respect to $\mathcal{F}^{\mathrm{tr}}$ and $\mathcal{R}_{+}$. By claim 1, $\Theta^{\mathrm{tr}}$ is measurable with respect to $\mathcal{F}^{\mathrm{tr}}$ and $\mathcal{B}(\mathbb{R}^{d})$, so the pair $(\Theta^{\mathrm{tr}},\mathsf{D}^{\mathrm{tr}}):\Omega^{\mathrm{tr}}\to\mathsf{H}$ is measurable with respect to $\mathcal{F}^{\mathrm{tr}}$ and $\mathcal{H}_{2}$; let $\nu^{\mathrm{tr}}$ be its image measure under $\mu^{\mathrm{tr}}$, the joint law.

The density; hypothesis (i). $g^{\mathrm{tr}}$ is $\mathcal{H}_{2}$-measurable (restriction of the measurable $g$), finite-valued, and strictly positive on $\mathsf{H}$ by claim 1 of the kernel instance. Let $E\in\mathcal{H}_{2}$. Then $E\in\mathcal{B}(\mathbb{R}^{d})\otimes\mathcal{R}_{s}$ and $E\subseteq\mathsf{H}$, so $(\Theta,\mathsf{D})^{-1}(E)\subseteq\mathsf{D}^{-1}(\mathsf{R}_{+})=\Omega^{\mathrm{tr}}$ and therefore $(\Theta^{\mathrm{tr}},\mathsf{D}^{\mathrm{tr}})^{-1}(E)=(\Theta,\mathsf{D})^{-1}(E)$. By claim 1 (image measures, with $V=(\Theta,\mathsf{D})$), The Integral of an Indicator Function is the Measure of the Set, the density identity with $F=\mathbf{1}_{E}$, and $(\ast)$ applied to $\phi=\mathbf{1}_{E}g^{\mathrm{tr}}$ (whose zero extension is $\mathbf{1}_{E}g$, because $E\subseteq\mathsf{H}$),

$$\nu^{\mathrm{tr}}(E)=\mu^{\sharp}\bigl((\Theta,\mathsf{D})^{-1}(E)\bigr)=\int\mathbf{1}_{E}\,g\,d(\lambda_{d}\otimes\rho)=\int_{\mathsf{H}}\mathbf{1}_{E}\,g^{\mathrm{tr}}\,d(\lambda_{d}\otimes\rho_{+}) .$$

Thus $\nu^{\mathrm{tr}}$ is the measure with density $g^{\mathrm{tr}}$ with respect to $\lambda_{d}\otimes\rho_{+}$, which is hypothesis (i) of The Multivariate van Trees Inequality for the data named in the claim (the $\sigma$-algebra $\mathcal{B}_{d}$ there being $\mathcal{B}(\mathbb{R}^{d})$ and its $\lambda_{d}$ being Lebesgue measure, as noted above).

Hypothesis (ii). Let $r\in\mathsf{R}_{+}$. The function $\theta\mapsto g^{\mathrm{tr}}(\theta,r)=g(\theta,r)$ on $\mathbb{R}^{d}$ is strictly positive and of class $C^{1}$ by claim 2 of the kernel instance; its partial derivatives are $\partial_{q}g^{\mathrm{tr}}(\theta,r)=\partial_{q}g(\theta,r)$, so that $\partial_{q}g^{\mathrm{tr}}=(\partial_{q}g)|_{\mathsf{H}}$.

Hypothesis (iii). By claim 2 of the kernel instance, $\partial_{q}g$ is $\mathcal{B}(\mathbb{R}^{d})\otimes\mathcal{R}_{s}$-measurable, so $\partial_{q}g^{\mathrm{tr}}$ is $\mathcal{H}_{2}$-measurable. The function $\phi(\theta,r)=(1+\sum_{k}|\theta_{k}|)|\partial_{q}g^{\mathrm{tr}}(\theta,r)|$ on $\mathsf{H}$ has zero extension $\mathbf{1}_{\mathsf{H}}(1+\sum_{k}|\theta_{k}|)|\partial_{q}g|\le(1+\sum_{k}|\theta_{k}|)|\partial_{q}g|$, so by $(\ast)$, monotonicity of the integral and claim 2 of the kernel instance, $\int_{\mathsf{H}}\phi\,d(\lambda_{d}\otimes\rho_{+})\le\int(1+\sum_{k}|\theta_{k}|)|\partial_{q}g|\,d(\lambda_{d}\otimes\rho)<\infty$.

Hypothesis (iv$'$). Hypotheses (i)--(iii) being verified, $\mathsf{sc}_{q}$ is a random variable on the trimmed copy, as noted in hypothesis (iv) of The Multivariate van Trees Inequality. Put $\phi=(\partial_{q}g^{\mathrm{tr}}/g^{\mathrm{tr}})^{2}$, a nonnegative $\mathcal{H}_{2}$-measurable function on $\mathsf{H}$: the quotient $\partial_{q}g^{\mathrm{tr}}/g^{\mathrm{tr}}$ is measurable by Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable, applied to the continuous map $(u_{1},u_{2})\mapsto u_{1}/u_{2}$ on $\mathbb{R}\times(0,\infty)$ composed with the measurable pair $(\partial_{q}g^{\mathrm{tr}},g^{\mathrm{tr}})$, exactly as noted in hypothesis (iv) of The Multivariate van Trees Inequality ($g^{\mathrm{tr}}>0$), and its square is measurable by claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. Then $\mathsf{sc}_{q}^{2}=\phi\circ(\Theta^{\mathrm{tr}},\mathsf{D}^{\mathrm{tr}})$, and by claims 2 and 3 of Image Measures, Measures with Densities, and Change of Variables,

$$\mathbb{E}^{\mathrm{tr}}[\mathsf{sc}_{q}^{2}]=\int_{\mathsf{H}}\phi\,d\nu^{\mathrm{tr}}=\int_{\mathsf{H}}\phi\,g^{\mathrm{tr}}\,d(\lambda_{d}\otimes\rho_{+})=\int_{\mathsf{H}}\frac{(\partial_{q}g^{\mathrm{tr}})^{2}}{g^{\mathrm{tr}}}\,d(\lambda_{d}\otimes\rho_{+})=\int\mathbf{1}_{\mathsf{H}}\frac{(\partial_{q}g)^{2}}{g}\,d(\lambda_{d}\otimes\rho)\le\frac1\eta ,$$

the third equality pointwise on $\mathsf{H}$, the fourth by $(\ast)$ (the zero extension of $(\partial_{q}g^{\mathrm{tr}})^{2}/g^{\mathrm{tr}}$ being the integrand of claim 2 of the kernel instance, read as $0$ off $\mathsf{H}$), and the inequality by that claim. Hence $\mathsf{sc}_{q}$ is square-integrable: hypothesis (iv$'$) of Score Identities and the Mixture-Weight Directional van Trees Inequality holds, and that lemma applies on the trimmed copy with $l:=d$, $(Y,\mathcal{G},\mu):=(\mathsf{R}_{+},\mathcal{R}_{+},\rho_{+})$, $\Theta:=\Theta^{\mathrm{tr}}$, $D:=\mathsf{D}^{\mathrm{tr}}$, $p:=g^{\mathrm{tr}}$, and (in its claim 2) $n=d$ and the shifts $a_{q}$.

The mixture-weight information. The mixture weight of claim 2 of Score Identities and the Mixture-Weight Directional van Trees Inequality for the shifts $a_{q}$ is $\bar{p}(\theta,r)=\tfrac12g^{\mathrm{tr}}(\theta,r)+\tfrac{1}{2d}\sum_{q}g^{\mathrm{tr}}(\theta-a_{q},r)$ for $(\theta,r)\in\mathsf{H}$, which is $\bar{g}^{\mathrm{tr}}$ by definition and equals $\bar{g}(\theta,r)$, the mixture weight of the kernel instance, since $g^{\mathrm{tr}}(\theta-a_{q},r)=g(\theta-a_{q},r)=g_{q}(\theta,r)$; by claim 1 of the kernel instance (applied to $\bar{g}$), $\bar{g}>0$ on $\mathsf{H}$, so $\bar{g}^{\mathrm{tr}}$ is strictly positive. For $\mathsf{u}\in\mathbb{R}^{d}$, $\partial_{\mathsf{u}}g^{\mathrm{tr}}=\sum_{q}\mathsf{u}_{q}\partial_{q}g^{\mathrm{tr}}=(\partial_{\mathsf{u}}g)|_{\mathsf{H}}$, so $\mathcal{I}_{\mathsf{u}}$ as defined in the statement is the mixture-weight information $\int_{\mathsf{H}}(\partial_{\mathsf{u}}g^{\mathrm{tr}})^{2}/\bar{p}\,d(\lambda_{d}\otimes\rho_{+})$ of that lemma (its integrand measurable by that claim); the zero extension of the integrand is $\mathbf{1}_{\mathsf{H}}(\partial_{\mathsf{u}}g)^{2}/\bar{g}$ read as $0$ off $\mathsf{H}$, and $(\ast)$ gives $\mathcal{I}_{\mathsf{u}}=\int\mathbf{1}_{\mathsf{H}}(\partial_{\mathsf{u}}g)^{2}/\bar{g}\,d(\lambda_{d}\otimes\rho)$, finite by claim 2 of Score Identities and the Mixture-Weight Directional van Trees Inequality. Finally $\partial_{t\mathsf{u}}g^{\mathrm{tr}}=\sum_{q}t\mathsf{u}_{q}\partial_{q}g^{\mathrm{tr}}=t\,\partial_{\mathsf{u}}g^{\mathrm{tr}}$, so the integrand defining $\mathcal{I}_{t\mathsf{u}}$ is $t^{2}$ times that defining $\mathcal{I}_{\mathsf{u}}$, and $\mathcal{I}_{t\mathsf{u}}=t^{2}\mathcal{I}_{\mathsf{u}}$ by positive homogeneity of the integral (for $t=0$ both sides are $0$).

Claim 3. Kernel bound. By claim 2, $\mathcal{I}_{\mathsf{u}}^{1/2}$ is the left-hand side of the inequality of claim 3 of the kernel instance for the direction $\mathsf{u}=\sum_{q}w_{q}a_{q}$ (which is the $u$ of that lemma for the weights $w$), and $\mathsf{u}=\sum_{q}w_{q}\mathsf{m}e_{q}/\sqrt{N}=(\mathsf{m}/\sqrt{N})w$ by the scalar multiplication and addition of $\mathbb{R}^{d}$. That claim gives $\mathcal{I}_{\mathsf{u}}^{1/2}\le(\mathsf{J}^{\mathrm{sym}})^{1/2}+\sqrt{2}\sum_{q}|w_{q}|(\exp(\kappa_{q})-1-\kappa_{q})^{1/2}$ with $\kappa_{q}=\lVert a_{q}\rVert^{2}/\eta$; here $\lVert a_{q}\rVert=\mathsf{m}\lVert e_{q}\rVert/\sqrt{N}=\mathsf{m}/\sqrt{N}$ (claim 5 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square; $\lVert e_{q}\rVert=1$ by its claim 1), so $\kappa_{q}=\mathsf{m}^{2}/(N\eta)=\kappa_{\mathrm{mv}}$ for every $q$, and $\sum_{q}|w_{q}|=\lVert w\rVert_{1}$. This is the first display.

The mean-field information bound. The setting of Information Bound on the Synthetic Copy with Mean-Field Data: Prior Energy of the Profile, Observation Information Along the Profile Response, and the Non-Close Record Mass consists of the setting of Assembly of the Symmetrised Move Information on the Synthetic Copy: Prior Part, Record Part, and the Bad Part with Exponentially Small Under-Likelihood Mass with its hypotheses (OC), (X), (W), (G), (P), of the setting and hypotheses 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 for the weights $w$ (the comparison pair, the profile $\varpi$ with bound $\Lambda$, the profile response $\bar\psi$ with bound $\mathsf{M}$), of (G$'$), of (CL) and of the real number $\zeta>0$. All of these except (P) and (CL) are part of the setting adopted from 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 (which adopts them through Mean-Square Assembly of the Estimand Linearisation on the Synthetic Copy: Approximation of the Recentred Copy Endpoint by an Affine Function of the Parameter); (P) is assumed in the statement; and (CL) holds for the tolerances $\varepsilon_{S}$, $\varepsilon_{\mathrm{ctl}}$ and the family $(\mathsf{R}^{\mathrm{cl}}_{\omega})_{\omega}$ by claim 3 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, whose bound $\bar\pi^{\mathrm{nc}}\le N^{-1/2}+c_{Q}\kappa^{\sharp}N^{-1}$ we also use. Hence claim 4 of Information Bound on the Synthetic Copy with Mean-Field Data: Prior Energy of the Profile, Observation Information Along the Profile Response, and the Non-Close Record Mass applies:

$$\mathsf{J}^{\mathrm{sym}}\le(1+\delta)\Bigl[\kappa_{0}^{\mathrm{mf}}N\mathcal{P}+\mathsf{Q}^{\mathrm{cl}}+\lVert w\rVert_{1}^{2}\bigl(\mathsf{e}^{\star}_{N}+\bar{E}_{N}\bar\pi^{\mathrm{nc}}+\mathsf{c}_{N}\mathsf{j}^{\star}\bigr)\Bigr]+2d\lVert w\rVert_{1}^{2}\mathsf{B}.$$

Since $\bar{E}_{N}=\tilde{l}\,s\,\Gamma^{2}(A^{\mathrm{ins}})^{2}/(N\underline{b})\ge0$ (claim 3 of 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, in the horizon-$s$ instance; the numerator is a product of nonnegative reals and $N\underline{b}>0$) and $1+\delta>0$, replacing $\bar\pi^{\mathrm{nc}}$ by its upper bound gives $\mathsf{J}^{\mathrm{sym}}\le\mathsf{J}^{\mathrm{mf}}$, a real number. Inserting this into the first display (the square root being nondecreasing) and squaring gives the bound on $\mathcal{I}_{\mathsf{u}}$, and $\mathcal{I}_{\mathsf{u}}=\mathcal{I}_{\sqrt{N}\mathsf{z}}=N\mathcal{I}_{\mathsf{z}}$ by the homogeneity of claim 2.

Claim 4. The Jacobian bound. We instantiate Linearisation of a Perturbed Controlled Aggregate Flow Along a Comparison Pair: Exact Variation-of-Constants Identity and Residual Bound with horizon $s$: its data $l$, $m$, $\mathcal{A}$, $B$, $\beta$, the extension $(U,W_{\beta},\bar\beta)$ with derivative bound $K$ (hypothesis (X)), the comparison pair $(S,A)|_{[0,s]}$ with $\mathcal{E}^{\star}$ continuous (hypothesis (CP)), $\Phi^{\mathcal{E}}$ and $\bar\Phi$ are adopted. Its drift $b(\Sigma,\alpha')=\sum_{c}v_{c}\psi_{c}(\Sigma,\alpha')$ is the aggregate state drift of $\beta$: for $(\Sigma,\alpha')\in\Delta^{l}\times\mathcal{A}$ one has $\psi_{c}(\Sigma,\alpha')=\Sigma^{\sigma}\bar\beta(\sigma,\gamma,\Sigma,\alpha')=\Sigma^{\sigma}\beta(\sigma,\gamma,\Sigma,\alpha')$ for $c=(\sigma,\gamma)$, by the definition of the label rates in Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect and clause 1 of Twice Continuously Differentiable Extension of a Transition-Rate Family, and the $\gamma'$-th component of $\sum_{c}v_{c}\psi_{c}$ is $\sum_{\sigma\neq\gamma'}\Sigma^{\sigma}\beta(\sigma,\gamma',\Sigma,\alpha')-\sum_{\gamma\neq\gamma'}\Sigma^{\gamma'}\beta(\gamma',\gamma,\Sigma,\alpha')$, since $v_{c}^{\gamma'}=1$ if $\gamma=\gamma'$, $-1$ if $\sigma=\gamma'$ and $0$ otherwise. For the remaining data take the empty record $r_{\emptyset}\in\mathbf{R}_{s}$ (the unique member of the cell $C_{\emptyset}$ of the observation record space), the control path $a=a^{(s),r_{\emptyset}}$, whose components are measurable by The Record-Frozen Control Path and Record-Frozen Policy and whose 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 (available in the horizon-$s$ instance by 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), the paths $x=y=\Phi^{r_{\emptyset}}$, measurable with values in $\Delta^{l}$ and satisfying $\Phi^{r_{\emptyset}}_{t}=z_{0}+\int_{[0,t]}b(\Phi^{r_{\emptyset}}_{u},a_{u})\,du$ for every $t$ by the same claim 1, and the perturbation $\mathfrak{m}=0$, so that both integral equations of that lemma hold. Its claim 1 then gives $|\mathcal{E}(\Sigma,\alpha')z|\le\Lambda_{\mathcal{E}}|z|$ for all $(\Sigma,\alpha')\in\Delta^{l}\times\mathcal{A}$ and $z\in\mathbb{R}^{l}$, in particular $|\mathcal{E}^{\star}_{u}v_{c}|\le\Lambda_{\mathcal{E}}|v_{c}|=\sqrt{2}\,\Lambda_{\mathcal{E}}$, where $|v_{c}|=\sqrt{2}$ because $v_{c}=\delta_{\gamma}-\delta_{\sigma}$ with $\sigma\neq\gamma$ has exactly two nonzero coordinates, equal to $1$ and $-1$ (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). With the bound of (CP), $|H^{c}_{u}|=|\Phi^{\mathcal{E}}(s,u)(\mathcal{E}^{\star}_{u}v_{c})|\le\bar\Phi^{2}|\mathcal{E}^{\star}_{u}v_{c}|\le\sqrt{2}\,\bar\Phi^{2}\Lambda_{\mathcal{E}}=:\bar{H}$ for all $u\in[0,s]$.

Variation of constants for the profile response. We apply Variation of Constants with Bounded Measurable Forcing and the Two-Parameter Fundamental Solution with horizon $s$, $k=l$ and $A(u)=\mathcal{E}^{\star}_{u}$, whose entries are continuous by (CP); its two-parameter fundamental solution $\Phi^{\mathcal{E}}(t,u)=\Phi^{\mathcal{E},1}(t)\Psi^{\mathcal{E},1}(u)$ is the $\Phi^{\mathcal{E}}$ of (CP), by the definition there, and a bound $\bar{A}$ for $A$ and a row-sum bound for $\Phi^{\mathcal{E},1}$ and $\Psi^{\mathcal{E},1}$ (a number in the role written $\bar\Phi$ in that lemma, in general different from the $\bar\Phi$ of (CP)) exist as noted there. Let $\mathsf{f}(u)=\Theta^{\mathrm{fl}}(S_{u},A_{u})\varpi_{u}$ ($u\in[0,s]$), bounded with measurable components 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. The profile response $\bar\psi$ has measurable components, is bounded by $\mathsf{M}$ and satisfies $\bar\psi_{t}=\int_{[0,t]}(\mathcal{E}^{\star}_{u}\bar\psi_{u}+\mathsf{f}(u))\,du$ for every $t\in[0,s]$ (the profile response equation), which is the equation of claim 3 of Variation of Constants with Bounded Measurable Forcing and the Two-Parameter Fundamental Solution with $\xi=0_{\mathbb{R}^{l}}$ and $g:=\mathsf{f}$. By its claims 2 and 3, $\bar\psi_{t}=\Phi^{\mathcal{E},1}(t)0_{\mathbb{R}^{l}}+\int_{[0,t]}\Phi^{\mathcal{E}}(t,u)\mathsf{f}(u)\,du=\int_{[0,t]}\Phi^{\mathcal{E}}(t,u)\mathsf{f}(u)\,du$ (claim 3 of Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum with the scalar $0$), the integrand being bounded measurable. Next, claim 1 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 (applied to $\mathcal{A}$, $\beta$ and its rate bound $B$, with $\Theta^{\mathrm{fl}}$ the aggregate fluctuation covariance of $\beta$) gives, for $\Sigma=S_{u}\in\Delta^{l}$, $\alpha'=A_{u}\in\mathcal{A}$ and $y=\varpi_{u}$, $\Theta^{\mathrm{fl}}(S_{u},A_{u})\varpi_{u}=\sum_{c=(\sigma,\gamma)}v_{c}(v_{c}\cdot\varpi_{u})S^{\sigma}_{u}\beta(\sigma,\gamma,S_{u},A_{u})$, and $S^{\sigma}_{u}\beta(\sigma,\gamma,S_{u},A_{u})=\psi_{c}(S_{u},A_{u})=\phi_{c}(u)$ as shown in the preceding paragraph. This proves the two displayed identities of the claim. Write $\mathsf{f}_{c}(u)=(v_{c}\cdot\varpi_{u})\phi_{c}(u)$, so that $\mathsf{f}=\sum_{c}\mathsf{f}_{c}v_{c}$ (finite sum of scalar multiples in $\mathbb{R}^{l}$); each $\mathsf{f}_{c}$ is bounded and measurable (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, it being the integrand defining $w_{c,j}$), and $|\mathsf{f}_{c}(u)|\le|v_{c}||\varpi_{u}|\phi_{c}(u)\le\sqrt{2}\,\Lambda\,\phi_{c}(u)$ by The Cauchy-Schwarz Inequality on $\mathbb{R}^n$, the profile bound and $\phi_{c}\ge0$.

The coefficient functions. For $c\in\mathcal{L}$ and $u\in[0,s]$ put $\mathsf{a}_{c}(u)=\mathbf{c}\cdot(\Phi^{\mathcal{E}}(s,u)v_{c})$. By claim 4 of Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum, $\mathsf{a}_{c}(u)=\sum_{\gamma,\gamma'}\Phi^{\mathcal{E}}(s,u)_{\gamma\gamma'}\mathbf{c}^{\gamma}v_{c}^{\gamma'}$ is a finite linear combination of the entries of $u\mapsto\Phi^{\mathcal{E}}(s,u)$, which are continuous on $[0,s]$ by claim 1 of Variation of Constants with Bounded Measurable Forcing and the Two-Parameter Fundamental Solution; hence $\mathsf{a}_{c}$ is continuous (Continuity of Sums and Products of Real-Valued Functions on a Metric Space), therefore measurable (claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval), and $|\mathsf{a}_{c}(u)|\le|\mathbf{c}|\,|\Phi^{\mathcal{E}}(s,u)v_{c}|\le\sqrt{2}\,|\mathbf{c}|\bar\Phi^{2}$ (The Cauchy-Schwarz Inequality on $\mathbb{R}^n$ and (CP)). By claim 3 of Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum (applied inductively over the finite sum), $\Phi^{\mathcal{E}}(s,u)\mathsf{f}(u)=\sum_{c}\mathsf{f}_{c}(u)\Phi^{\mathcal{E}}(s,u)v_{c}$, so by linearity of the componentwise integral and of the dot product,

$$\mathbf{c}\cdot\bar\psi_{s}=\sum_{c\in\mathcal{L}}\int_{[0,s]}\mathsf{a}_{c}(u)\mathsf{f}_{c}(u)\,du .$$

The representation of the coefficient functions. For $\tau\in[0,s]$ let $\chi_{\tau}$ be the indicator of $[\tau,s]\cap[0,s]$, a member of $\mathcal{B}_{[0,s]}$. We show

$$\mathsf{a}_{c}(\tau)=\mathbf{c}\cdot v_{c}+\int_{[0,s]}\chi_{\tau}(u)\,\mathbf{c}\cdot H^{c}_{u}\,du\qquad(\tau\in[0,s]).\qquad(\ast\ast)$$

The map $u\mapsto\mathcal{E}^{\star}_{u}v_{c}$ is bounded measurable (its components are finite sums of products of the continuous, hence bounded and measurable, entries of $\mathcal{E}^{\star}$ with constants; Continuous Real-Valued Functions on a Compact Interval are Bounded, claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval, claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions), so by claim 2 of Variation of Constants with Bounded Measurable Forcing and the Two-Parameter Fundamental Solution the map $u\mapsto H^{c}_{u}=\Phi^{\mathcal{E}}(s,u)\mathcal{E}^{\star}_{u}v_{c}$ is bounded measurable, with $|H^{c}_{u}|\le\bar{H}$. Claim 1 of that lemma with $t=s$ and $v=v_{c}$ gives $\Phi^{\mathcal{E}}(s,\tau)v_{c}-v_{c}=\int_{[\tau,s]}H^{c}_{u}\,du$, the right-hand side being $0$ when $\tau=s$ by the convention of that lemma. Fix a component $\gamma$ and write $h=H^{c,\gamma}$. If $\tau<s$: let $\tilde{h}_{\tau}$ be the function on $\mathbb{R}$ equal to $h$ on $[\tau,s]$ and to $0$ elsewhere. The restriction $h|_{[\tau,s]}$ is $\mathcal{B}_{[\tau,s]}$-measurable (a preimage $S\cap[0,s]$ of a Borel set under $h$, with $S$ Borel, has trace $S\cap[\tau,s]$ on $[\tau,s]$) and bounded, hence integrable with respect to $\lambda_{[\tau,s]}$ (bounded by $\bar{H}$, whose integral is $\bar{H}(s-\tau)$ by The Integral of an Indicator Function is the Measure of the Set, positive homogeneity and claim 1 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval); by claim 2 of that toolkit on $[\tau,s]$, $\int_{[\tau,s]}h|_{[\tau,s]}\,d\lambda_{[\tau,s]}=\int_{\mathbb{R}}\tilde{h}_{\tau}\,d\lambda$. The same $\tilde{h}_{\tau}$ is the zero extension of the bounded measurable function $\chi_{\tau}h$ on $[0,s]$, so the same claim on $[0,s]$ gives $\int_{[0,s]}\chi_{\tau}h\,d\lambda_{[0,s]}=\int_{\mathbb{R}}\tilde{h}_{\tau}\,d\lambda$. Hence $\int_{[\tau,s]}h|_{[\tau,s]}\,d\lambda_{[\tau,s]}=\int_{[0,s]}\chi_{\tau}h\,d\lambda_{[0,s]}$. If $\tau=s$: $\chi_{s}$ is the indicator of $\{s\}$, and $\lambda_{[0,s]}(\{s\})=\lambda(\{s\})\le\lambda([s-\epsilon,s])=\lambda_{[s-\epsilon,s]}([s-\epsilon,s])=\epsilon$ for every $\epsilon\in(0,s]$ (monotonicity of $\lambda$ and claim 1 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval on $[s-\epsilon,s]$), so $\lambda_{[0,s]}(\{s\})=0$ and $|\int_{[0,s]}\chi_{s}h\,d\lambda_{[0,s]}|\le\int_{[0,s]}\chi_{s}\bar{H}\,d\lambda_{[0,s]}=\bar{H}\lambda_{[0,s]}(\{s\})=0$, in agreement with the convention. In both cases, summing over $\gamma$ with the coefficients $\mathbf{c}^{\gamma}$ and using linearity of the integral, $\mathsf{a}_{c}(\tau)-\mathbf{c}\cdot v_{c}=\mathbf{c}\cdot(\Phi^{\mathcal{E}}(s,\tau)v_{c}-v_{c})=\int_{[0,s]}\chi_{\tau}(u)\,\mathbf{c}\cdot H^{c}_{u}\,du$, which is $(\ast\ast)$. Consequently, for $\tau,\tau'\in[0,s]$, writing $\Delta(\tau,\tau')$ for the set of $u\in[0,s]$ with $\chi_{\tau}(u)\neq\chi_{\tau'}(u)$, which is $[\min(\tau,\tau'),\max(\tau,\tau'))$, one has $|\chi_{\tau}-\chi_{\tau'}|=\mathbf{1}_{\Delta(\tau,\tau')}$ and

$$|\mathsf{a}_{c}(\tau)-\mathsf{a}_{c}(\tau')|\le\int_{[0,s]}\mathbf{1}_{\Delta(\tau,\tau')}(u)\,|\mathbf{c}|\,|H^{c}_{u}|\,du\le|\mathbf{c}|\,\bar{H}\,\lambda_{[0,s]}\bigl(\Delta(\tau,\tau')\bigr)$$

by linearity, monotonicity, The Cauchy-Schwarz Inequality on $\mathbb{R}^n$ and The Integral of an Indicator Function is the Measure of the Set.

Time cells. Fix $c\in\mathcal{L}$. 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 (horizon $s$): $\bar{\mathsf{C}}^{c}$ is nondecreasing; the time cells $\bar{J}_{c,1},\dots,\bar{J}_{c,J_{c}}$ belong to $\mathcal{B}_{[0,s]}$, are pairwise disjoint with union $[0,s]$, so that $\sum_{j}\mathbf{1}_{\bar{J}_{c,j}}=1$ on $[0,s]$ and $\sum_{j}\lambda_{[0,s]}(\bar{J}_{c,j})=\lambda_{[0,s]}([0,s])=s$ (additivity of $\lambda_{[0,s]}$ and claim 1 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval); and $N\int_{[0,s]}\mathbf{1}_{\bar{J}_{c,j}}\phi_{c}\,du\le\mu_{c,j}\le\mu_{\max}$, where by the convention of that lemma $\int_{J}\cdot\,du$ denotes the integral over $[0,s]$ of the integrand multiplied by $\mathbf{1}_{J}$. By the definition of the injection weights, $\frac{\mathsf{m}}{N}w_{c,j}=\int_{[0,s]}\mathbf{1}_{\bar{J}_{c,j}}\mathsf{f}_{c}\,du$. Call $j$ full if $\bar{\mathsf{C}}^{c}_{s}\ge\mathsf{b}^{c}_{j}$ and partial otherwise; by (CP) and 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, for full $j$ the set $\{u\in[0,s]:\bar{\mathsf{C}}^{c}_{u}\ge\mathsf{b}^{c}_{j}\}$ has the least element $\bar\tau_{c,j}$ and $\alpha_{c,j}=\mathsf{a}_{c}(\bar\tau_{c,j})$, while $\alpha_{c,j}=0$ for partial $j$. Two observations. (T1) Let $j$ be full and $u\in\bar{J}_{c,j}$; then $\Delta(\bar\tau_{c,j},u)\subseteq\bar{J}_{c,j}$. Indeed, write $\bar\tau=\bar\tau_{c,j}$; by minimality and monotonicity, $\bar{\mathsf{C}}^{c}_{u'}\ge\mathsf{b}^{c}_{j}$ holds exactly when $u'\ge\bar\tau$; and $u\in\bar{J}_{c,j}$ means $\bar{\mathsf{C}}^{c}_{u}\le\mathsf{b}^{c}_{j}$ and, when $j\ge2$, $\bar{\mathsf{C}}^{c}_{u}>\mathsf{b}^{c}_{j-1}$. If $u<\bar\tau$ and $u'\in[u,\bar\tau)$, then $\bar{\mathsf{C}}^{c}_{u'}<\mathsf{b}^{c}_{j}$ and $\bar{\mathsf{C}}^{c}_{u'}\ge\bar{\mathsf{C}}^{c}_{u}$, so $u'\in\bar{J}_{c,j}$. If $u\ge\bar\tau$ and $u'\in[\bar\tau,u)$, then $\mathsf{b}^{c}_{j}\le\bar{\mathsf{C}}^{c}_{u'}\le\bar{\mathsf{C}}^{c}_{u}\le\mathsf{b}^{c}_{j}$, so $\bar{\mathsf{C}}^{c}_{u'}=\mathsf{b}^{c}_{j}\in I_{c,j}$ and $u'\in\bar{J}_{c,j}$ (for $j=1$, $\bar{J}_{c,1}=\{\bar{\mathsf{C}}^{c}\le\mathsf{b}^{c}_{1}\}$ contains $u'$ as well). (T2) At most one partial $j$ has $\bar{J}_{c,j}\neq\emptyset$: if $j<j'$ are partial and $u\in\bar{J}_{c,j'}$, then $j'\ge2$ and $\bar{\mathsf{C}}^{c}_{s}\ge\bar{\mathsf{C}}^{c}_{u}>\mathsf{b}^{c}_{j'-1}\ge\mathsf{b}^{c}_{j}$, contradicting the partiality of $j$.

The pairing bound. By the definition of $\mathsf{z}$ and the dot product, $\alpha\cdot\mathsf{z}=\sum_{q}\alpha_{q}\mathsf{z}_{q}=\frac{\mathsf{m}}{N}\sum_{q}w_{q}\alpha_{q}=\sum_{c}\sum_{j}\alpha_{c,j}\int_{[0,s]}\mathbf{1}_{\bar{J}_{c,j}}\mathsf{f}_{c}\,du$, while by the display for $\mathbf{c}\cdot\bar\psi_{s}$ and $\sum_{j}\mathbf{1}_{\bar{J}_{c,j}}=1$, $\mathbf{c}\cdot\bar\psi_{s}=\sum_{c}\sum_{j}\int_{[0,s]}\mathbf{1}_{\bar{J}_{c,j}}\mathsf{a}_{c}\mathsf{f}_{c}\,du$. Hence, by linearity and monotonicity of the integral (all integrands being bounded measurable),

$$|\alpha\cdot\mathsf{z}-\mathbf{c}\cdot\bar\psi_{s}|\le\sum_{c\in\mathcal{L}}\sum_{j=1}^{J_{c}}\int_{[0,s]}\mathbf{1}_{\bar{J}_{c,j}}(u)\,|\alpha_{c,j}-\mathsf{a}_{c}(u)|\,|\mathsf{f}_{c}(u)|\,du .$$

Fix $c$. For full $j$ and $u\in\bar{J}_{c,j}$, (T1) and the bound following $(\ast\ast)$ give $|\alpha_{c,j}-\mathsf{a}_{c}(u)|=|\mathsf{a}_{c}(\bar\tau_{c,j})-\mathsf{a}_{c}(u)|\le|\mathbf{c}|\bar{H}\lambda_{[0,s]}(\bar{J}_{c,j})$ (monotonicity of $\lambda_{[0,s]}$), so the $j$-th term is at most $|\mathbf{c}|\bar{H}\lambda_{[0,s]}(\bar{J}_{c,j})\cdot\sqrt{2}\Lambda\int_{[0,s]}\mathbf{1}_{\bar{J}_{c,j}}\phi_{c}\,du\le|\mathbf{c}|\bar{H}\sqrt{2}\Lambda\frac{\mu_{\max}}{N}\lambda_{[0,s]}(\bar{J}_{c,j})$; summing over the full $j$ and using $\sum_{j}\lambda_{[0,s]}(\bar{J}_{c,j})=s$ bounds their total by $\sqrt{2}|\mathbf{c}|\bar{H}\Lambda s\,\mu_{\max}/N=2|\mathbf{c}|\bar\Phi^{2}\Lambda_{\mathcal{E}}\Lambda s\,\mu_{\max}/N$. For partial $j$ the term is $\int_{[0,s]}\mathbf{1}_{\bar{J}_{c,j}}|\mathsf{a}_{c}||\mathsf{f}_{c}|\,du\le\sqrt{2}|\mathbf{c}|\bar\Phi^{2}\cdot\sqrt{2}\Lambda\,\mu_{\max}/N$, and it vanishes when $\bar{J}_{c,j}=\emptyset$; by (T2) the partial terms total at most $2|\mathbf{c}|\bar\Phi^{2}\Lambda\,\mu_{\max}/N$. Summing over the $l(l-1)$ labels gives $|\alpha\cdot\mathsf{z}-\mathbf{c}\cdot\bar\psi_{s}|\le\mathsf{e}_{\mathrm{inj}}$.

The remaining bounds. For $q\in\mathsf{L}$, $\alpha\cdot a_{q}=(\mathsf{m}/\sqrt{N})\,\alpha\cdot e_{q}=(\mathsf{m}/\sqrt{N})\,\alpha_{q}$, and $|\alpha_{q}|\le\sqrt{2}|\mathbf{c}|\bar\Phi^{2}$ (it is $\mathsf{a}_{c}(\bar\tau_{q})$ or $0$), which gives the bound on $\max_{q}|\alpha\cdot a_{q}|$. Finally $\lVert\alpha\rVert^{2}=\sum_{q}\alpha_{q}^{2}\le d\cdot2|\mathbf{c}|^{2}\bar\Phi^{4}$ (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 taking nonnegative square roots, $\lVert\alpha\rVert\le\sqrt{2d}\,|\mathbf{c}|\bar\Phi^{2}$. This completes the proof.