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

Setting. Adopt the setting, notation, hypothesis (L) and conventions 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 hence of The Path-Closeness Event under the Cost Bound: Closeness of the Empirical State Measure to the Mean-Field Trajectory and of the Record-Frozen Control to the Mean-Field Control on an Event of Probability $1-O(N^{-1/2})$, 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, Mean-Square Assembly of the Estimand Linearisation on the Synthetic Copy: Approximation of the Recentred Copy Endpoint by an Affine Function of the Parameter, 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, Assembly of the Symmetrised Move Information on the Synthetic Copy: Prior Part, Record Part, and the Bad Part with Exponentially Small Under-Likelihood Mass, 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, 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, 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, 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, Pathwise Reduction of the Symmetrised Move Information on the Synthetic Copy: Removed-Clock Likelihoods, the Shifted Family, and the Averaging Bound and The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record), including its $N$-agent side (the probability space $(\Omega^{\mathrm{ag}},\mathcal{F}^{\mathrm{ag}},P^{\mathrm{ag}})$ carrying the $N$-th solution for the fixed natural number $N\ge N_{\mathrm{cl}}$, the stationary mean-field triple $(S,A,P)$ with initial point $S_{0}$, the empirical state measure $\Sigma$, the observation-centred fluctuation $X'_{t}$, the constants $c_{Q}$ and $\kappa^{\sharp}$), its intermediate time $s\in(0,T]$ with the record prefix $W^{(s)}$, the record space $(\mathbf{R}_{s},\mathcal{R}_{s},\rho)$ with horizon $s$ and $\tilde{l}$ channels, and the recentred copy endpoint $X''_{s}$, and its copy side: the probability space $(\Omega,\mathcal{F},P)$ with expectation $\mathbb{E}$, the natural numbers $l\ge2$, $m\ge1$, $\tilde{l}\ge1$, the real numbers $B\ge0$, $K\ge0$, $\tilde{B}\ge0$, $\tilde{K}\ge0$, $\underline{b}>0$, the set $\mathcal{L}$ of transition labels $c=(\sigma,\gamma)$ with vectors $v_{c}=\delta_{\gamma}-\delta_{\sigma}$, the label rates $\psi_{c}$, state gradients $g^{c}$ and drift Jacobian $\mathcal{E}$, the index set $\mathsf{L}$ of the cells with $d$ elements, identified with $\{1,\dots,d\}$ by the fixed bijection, the cell lengths $\mu_{q}$ and $\mu_{\max}$, the cell boundaries $\mathsf{b}^{c}_{j}$, the cell-count vector $\mathsf{K}$, the move size $\mathsf{m}$, the standard basis vectors $e_{q}$ ($q\in\mathsf{L}$) of Euclidean space $\mathbb{R}^{d}$, the smoothing parameter $\eta\in(0,1]$ with the Gaussian smoothing weight $\varphi_{\eta}$, the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^{d})$ and Lebesgue measure $\lambda_{d}$ on $\mathbb{R}^{d}$, the synthetic copy $(\Omega^{\sharp},\mathcal{F}^{\sharp},\mu^{\sharp})$ with $\Omega^{\sharp}=\Omega\times\mathbb{R}^{d}\times\mathbf{R}_{s}$ and $\mathcal{F}^{\sharp}=(\mathcal{F}\otimes\mathcal{B}(\mathbb{R}^{d}))\otimes\mathcal{R}_{s}$, its parameter $\Theta=(\Theta_{q})_{q\in\mathsf{L}}$ and record $\mathsf{D}$, the likelihoods $\ell^{\sharp,\omega}$, the event $G$ with $\mathsf{g}=P(\Omega\setminus G)$, the comparison pair $(S^{\mathrm{cp}},\mathsf{A})$, equal to $(S,A)$ on $[0,s]$ by (L), with the mean-field label rates $\phi_{c}(t)=\psi_{c}(S_{t},A_{t})$ and mean-field clocks $\bar{\mathsf{C}}^{c}_{t}=N\int_{[0,t]}\phi_{c}(u)\,du$, the matrices $\mathcal{E}^{\star}_{t}=\mathcal{E}(S_{t},A_{t})$, the two-parameter fundamental solution $\Phi^{\mathcal{E}}(t,u)$ ($t,u\in[0,s]$) with bound $\bar\Phi$, the estimand direction $\mathbf{c}\in\mathbb{R}^{l}$, the maps $H^{c}_{u}=\Phi^{\mathcal{E}}(s,u)\mathcal{E}^{\star}_{u}v_{c}$, the entry times $\bar\tau_{q}$ and the cell coefficients $\alpha=(\alpha_{q})_{q\in\mathsf{L}}\in\mathbb{R}^{d}$ of (CP) (so that $\alpha_{q}=\mathbf{c}\cdot(\Phi^{\mathcal{E}}(s,\bar\tau_{q})v_{c})$ when $\bar{\mathsf{C}}^{c}_{s}\ge\mathsf{b}^{c}_{j}$ for $q=(c,j)$, and $\alpha_{q}=0$ otherwise), the constant map $\varsigma$, the constants $\Lambda_{\mathcal{E}}=\sqrt{2}\,l(l-1)\sqrt{l}\,(B+K)$ and $C_{\mathrm{Lip}}$, the tolerances $\varepsilon_{S}(N)$ and $\varepsilon_{\mathrm{ctl}}(N)$ and, from claims 3 and 5 of that lemma, the tolerances $\varepsilon_{S}$, $\varepsilon_{\mathrm{ctl}}$ of (CL), the close records $\mathsf{R}^{\mathrm{cl}}_{\omega}$, the non-close masses $\pi^{\mathrm{nc}}_{\omega}$ and $\bar\pi^{\mathrm{nc}}$, and, for each choice of the data of its claim 5 (two positive reals in the roles written $x_{1}$, $x_{2}$ there, and a family $(\Xi^{c})_{c\in\mathcal{L}}$), the constants $\mathsf{e}_{1},\dots,\mathsf{e}_{4}$ and $\hat{\mathsf{e}}_{5}$ of that claim. Adopt moreover the following objects of the adopted copy setting, which that lemma leaves unused: from The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record the parameter lattice $\mathsf{S}=\{y/\sqrt{N}:y\in\mathbb{N}_{0}^{\mathsf{L}}\}$, the count mass function $\mathsf{p}$, the record kernel $f$ and the smoothed joint density $g$ of its claim 5; from Pathwise Reduction of the Symmetrised Move Information on the Synthetic Copy: Removed-Clock Likelihoods, the Shifted Family, and the Averaging Bound the moves $a_{q}=\mathsf{m}e_{q}/\sqrt{N}$ ($q\in\mathsf{L}$) and the symmetrised kernel-weighted move information $\mathsf{J}^{\mathrm{sym}}\in[0,\infty]$ formed with these moves and the injection weights $w$ introduced next; from 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 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 $\varpi$ with bound $\Lambda\ge0$, the time cells $\bar{J}_{c,j}$, the injection weights $w=(w_{q})_{q\in\mathsf{L}}$ formed from $\phi_{c}$ and $\varpi$, with $\lVert w\rVert_{1}=\sum_{q}|w_{q}|$, the profile energy $\mathcal{P}$, the profile response $\bar\psi$ with bound $\mathsf{M}$ and the observation information matrix $\tilde{D}(x)$, the aggregate fluctuation covariance of $\beta$, written $\Theta$ in those two lemmas and $\Theta^{\mathrm{fl}}$ here, and the constants $\mathsf{e}_{F}$, $\epsilon_{\psi}$, $\kappa$ of its claims 2--4 formed with $\varepsilon_{S}$, $\varepsilon_{\mathrm{ctl}}$; from 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 the real number $\zeta>0$ and the constants $\mathsf{E}^{\star}_{N}$, $\mathsf{e}^{\star}_{N}$, $\kappa_{0}$ (written $\kappa_{0}^{\mathrm{mf}}$ here), $\bar{\mathsf{j}}$, $\mathsf{j}^{\star}$ and $\mathsf{Q}^{\mathrm{cl}}$; and from 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 and Assembly of the Symmetrised Move Information on the Synthetic Copy: Prior Part, Record Part, and the Bad Part with Exponentially Small Under-Likelihood Mass the constants $\Gamma$, $A^{\mathrm{ins}}$ (the constant written $A_{0}$ there and in Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect, renamed as in 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), $\bar{E}_{N}$, $\mathsf{c}_{N}=\exp(\bar{E}_{N})-1$ and $\varepsilon_{0}=\Gamma A^{\mathrm{ins}}/(N\underline{b})$. Assume in addition hypothesis (P) 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: a real number $\delta$ with $0<\delta\le1$, the inequality $\varepsilon_{0}\le\tfrac12$ (a restriction on the adopted data, satisfied for all sufficiently large $N$ once the remaining data are fixed), an integer $\theta_{\mathrm{P}}\ge1$ (written $\theta$ there) with $2\theta_{\mathrm{P}}\varepsilon_{0}\le1$, and real numbers $x^{\mathrm{P}}_{q}>0$ ($q\in\mathsf{L}$; written $x_{q}$ there) with $\mathsf{m}(x^{\mathrm{P}}_{q}+\mathsf{m})\le\delta\mu_{q}/2$; adopt the constants $\mathsf{E}^{\mathrm{ch}}_{\theta_{\mathrm{P}}}$, $\bar\Pi$ and $\mathsf{B}$ of that theorem formed from them (the letters $\bar\Pi$ and $\mathsf{B}$, left free by 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, are thus bound here to these constants).

The record support and the trimmed copy. Put

$$\bar{f}(r)=\sum_{x\in\mathsf{S}}\mathsf{p}(x)f(x,r)\in[0,\infty],\qquad \mathsf{R}_{+}=\{r\in\mathbf{R}_{s}:\bar{f}(r)>0\},\qquad \Omega^{\mathrm{tr}}=\Omega\times\mathbb{R}^{d}\times\mathsf{R}_{+}=\mathsf{D}^{-1}(\mathsf{R}_{+})$$

(the sum over the countable set $\mathsf{S}$ in the sense of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions, as in Gaussian Smoothing of a Kernel-Weighted Discrete Family with Countable Support: Joint Density, van Trees Regularity, and the Symmetrised Directional Score Bound); the set $\mathsf{R}_{+}$ is the record support. Let $(\Omega^{\mathrm{tr}},\mathcal{F}^{\mathrm{tr}},\mu^{\mathrm{tr}})$ be the restriction of $(\Omega^{\sharp},\mathcal{F}^{\sharp},\mu^{\sharp})$ to $\Omega^{\mathrm{tr}}$ (defined by claim 1), the trimmed copy, with expectation $\mathbb{E}^{\mathrm{tr}}$; for a map $V$ on $\Omega^{\sharp}$ write $V^{\mathrm{tr}}$ for its restriction to $\Omega^{\mathrm{tr}}$, in particular $\Theta^{\mathrm{tr}}=(\Theta^{\mathrm{tr}}_{q})_{q\in\mathsf{L}}$, $\mathsf{D}^{\mathrm{tr}}$ and $X^{\mathrm{tr}}=(X''_{s})^{\mathrm{tr}}$; the map $\mathsf{D}^{\mathrm{tr}}$ takes its values in $\mathsf{R}_{+}$ and is regarded as $\mathbf{R}_{s}$-valued in claim 1 (except where it is composed with a map defined on $\mathsf{R}_{+}$) and as $\mathsf{R}_{+}$-valued in claim 2. Let $(\mathsf{R}_{+},\mathcal{R}_{+},\rho_{+})$ be the restriction of $(\mathbf{R}_{s},\mathcal{R}_{s},\rho)$ to $\mathsf{R}_{+}$ (so that $\mathcal{R}_{+}=\{A\in\mathcal{R}_{s}:A\subseteq\mathsf{R}_{+}\}$; defined by claim 1, which gives $\mathsf{R}_{+}\in\mathcal{R}_{s}$), put $\mathsf{H}=\mathbb{R}^{d}\times\mathsf{R}_{+}$, let $g^{\mathrm{tr}}=g|_{\mathsf{H}}$ be the restriction of $g$ to $\mathsf{H}$, and let $\bar{g}^{\mathrm{tr}}:\mathsf{H}\to[0,\infty)$ be given by $\bar{g}^{\mathrm{tr}}(\theta,r)=\tfrac12g^{\mathrm{tr}}(\theta,r)+\tfrac{1}{2d}\sum_{q\in\mathsf{L}}g^{\mathrm{tr}}(\theta-a_{q},r)$. Write $\partial_{q}$ for the partial derivative with respect to the coordinate $q$ of $\mathbb{R}^{d}$ and $\partial_{\mathsf{u}}=\sum_{q\in\mathsf{L}}\mathsf{u}_{q}\partial_{q}$ for $\mathsf{u}\in\mathbb{R}^{d}$. For $\mathsf{u}\in\mathbb{R}^{d}$ put

$$\mathcal{I}_{\mathsf{u}}=\int_{\mathsf{H}}\frac{(\partial_{\mathsf{u}}g^{\mathrm{tr}})^{2}}{\bar{g}^{\mathrm{tr}}}\,d(\lambda_{d}\otimes\rho_{+})\in[0,\infty]$$

(well defined by claim 2, where it is identified with the mixture-weight information in the direction $\mathsf{u}$ for the shifts $a_{q}$, $q\in\mathsf{L}$, $n=d$). Put

$$\mathsf{u}=\sum_{q\in\mathsf{L}}w_{q}a_{q}=\frac{\mathsf{m}}{\sqrt{N}}\,w,\qquad \mathsf{z}=\frac{\mathsf{u}}{\sqrt{N}}=\frac{\mathsf{m}}{N}\,w,\qquad \kappa_{\mathrm{mv}}=\frac{\mathsf{m}^{2}}{N\eta},$$ $$\mathsf{J}^{\mathrm{mf}}=(1+\delta)\Bigl[\kappa_{0}^{\mathrm{mf}}N\mathcal{P}+\mathsf{Q}^{\mathrm{cl}}+\lVert w\rVert_{1}^{2}\Bigl(\mathsf{e}^{\star}_{N}+\bar{E}_{N}\bigl(N^{-1/2}+c_{Q}\kappa^{\sharp}N^{-1}\bigr)+\mathsf{c}_{N}\mathsf{j}^{\star}\Bigr)\Bigr]+2d\,\lVert w\rVert_{1}^{2}\,\mathsf{B},\qquad \mathsf{e}_{\mathrm{inj}}=2\,l(l-1)\,|\mathbf{c}|\,\Lambda\,\bar\Phi^{2}\,(\Lambda_{\mathcal{E}}s+1)\,\frac{\mu_{\max}}{N}.$$

Conventions. Write $\lVert\cdot\rVert$ for the Euclidean norm on $\mathbb{R}^{d}$ (so that $\lVert\alpha\rVert$ is the quantity written $|\alpha|$ in the constant $\mathsf{e}_{4}=\sqrt{\eta}|\alpha|$ of the adopted setting) and $|\cdot|$ for that on $\mathbb{R}^{l}$ and for the absolute value; $x\cdot y$ for the dot product; $\exp$ for the exponential function; $t^{1/2}=\sqrt{t}$ for the nonnegative square root; $\lambda_{[0,s]}$ for the restricted Lebesgue measure on $[0,s]$ and $\int_{[0,s]}\cdot\,dt$ for the integral with respect to it; measurability of maps between measurable spaces, measurability and integrals of $[0,\infty]$-valued functions, and image measures and measures with densities as in those items; $\lVert\cdot\rVert_{2}$ for the mean-square norm on the copy or on the trimmed copy, as indicated; and $\bar{g}$ for the mixture weight of Gaussian Smoothing of a Kernel-Weighted Discrete Family with Countable Support: Joint Density, van Trees Regularity, and the Symmetrised Directional Score Bound for the moves $a_{q}$. Notational cautions: $\Theta$ is the parameter of the copy and $\Theta^{\mathrm{fl}}$ the aggregate fluctuation covariance; $\mathsf{S}$ is the parameter lattice, the mean-field flow $\mathsf{S}(\cdot,\cdot)$ of the adopted setting not being used; the shifts $a_{q}$, indexed by $\mathsf{L}$, are points of $\mathbb{R}^{d}$, unrelated to the record-frozen control paths $a^{(s),r}$; the letter $\mathsf{u}$ denotes a direction in $\mathbb{R}^{d}$ and $u$ a time; the scores of The Multivariate van Trees Inequality (written $S_{i}$ there) are written $\mathsf{sc}_{q}$; $\kappa_{\mathrm{mv}}$ is the quantity written $\kappa_{j}$ in Gaussian Smoothing of a Kernel-Weighted Discrete Family with Countable Support: Joint Density, van Trees Regularity, and the Symmetrised Directional Score Bound, unrelated to the constant $\kappa$ of Closeness to a Mean-Field Pair on the Synthetic Copy: Clock Discrepancy, Gronwall Comparison of the Weighted Response with the Profile Response, and the Observation-Information Bound with Mean-Field Data and to $\kappa^{\sharp}$; $\mathsf{H}$ is a subset of $\mathbb{R}^{d}\times\mathbf{R}_{s}$, unrelated to the maps $H^{c}$; $\bar{f}$ is the marginal of the record kernel, unrelated to the profile injection $\bar{F}_{t}$; $\theta$ denotes the second coordinate of a point of $\Omega^{\sharp}$; $\delta$ is the real number of (P), the standard basis vectors of $\mathbb{R}^{l}$ being written $\delta_{1},\dots,\delta_{l}$ only inside $v_{c}=\delta_{\gamma}-\delta_{\sigma}$; the reals $x^{\mathrm{P}}_{q}$ of (P) are unrelated to the two positive reals of 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, which are written $y_{1}$, $y_{2}$ in claim 1(b) below; $\mathsf{B}$ is the bad part 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 and $B$ the rate bound; the one-parameter fundamental solution of $u\mapsto\mathcal{E}^{\star}_{u}$ and its inverse, written $\Phi$ and $\Psi$ in Variation of Constants with Bounded Measurable Forcing and the Two-Parameter Fundamental Solution, are written $\Phi^{\mathcal{E},1}$ and $\Psi^{\mathcal{E},1}$ where they occur, the letters $\Phi$ (realized mean-field flow) and $\Psi_{t}$ (the maps 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) being taken; $\mathbb{N}_{0}$ is the set of natural numbers together with $0$, as in The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record; $(\mathsf{Z},\mathcal{G})$ in claim 1 is a generic measurable space, unrelated to the event $G$, to the drift Jacobian $\mathcal{E}$, to the closeness sets $\mathsf{E}(\varepsilon,\varepsilon')$ of the adopted conventions and to the constants $\mathsf{E}^{\star}_{N}$, $\mathsf{E}^{\mathrm{ch}}_{\theta_{\mathrm{P}}}$; $m^{\mathrm{est}}$ in claim 1(b) is a map on records, unrelated to the control dimension $m$ and to the move size $\mathsf{m}$; $\varpi$ denotes the profile of the adopted copy setting, while the Chernoff exponent of Series Formula, Exponential Moments, and Chernoff Tail Bounds for the Poisson Distribution, written $\varpi_{\mu}$ in Assembly of the Symmetrised Move Information on the Synthetic Copy: Prior Part, Record Part, and the Bad Part with Exponentially Small Under-Likelihood Mass and $\varpi_{k}$ 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, occurs here only inside the adopted constants $\bar\Pi$ and $\lVert\Xi^{c}\rVert_{2}$; and $\lambda$ without subscript is Lebesgue measure on the real line, as in Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval, the intensities $\lambda^{\sharp,\omega}$ of the adopted setting always carrying superscripts.

Then the following hold.

1. (The record support and the trimmed copy.) $\mathsf{R}_{+}\in\mathcal{R}_{s}$, $\Omega^{\mathrm{tr}}\in\mathcal{F}^{\sharp}$ and $\mu^{\sharp}(\Omega^{\mathrm{tr}})=1$; the trimmed copy is a probability space; for every $\mathcal{F}^{\sharp}$-measurable $Z:\Omega^{\sharp}\to[0,\infty]$ the restriction $Z^{\mathrm{tr}}$ is $\mathcal{F}^{\mathrm{tr}}$-measurable and $\int_{\Omega^{\mathrm{tr}}}Z^{\mathrm{tr}}\,d\mu^{\mathrm{tr}}=\int_{\Omega^{\sharp}}Z\,d\mu^{\sharp}$; for every measurable space $(\mathsf{Z},\mathcal{G})$ and every map $V:\Omega^{\sharp}\to\mathsf{Z}$ measurable with respect to $\mathcal{F}^{\sharp}$ and $\mathcal{G}$, the restriction $V^{\mathrm{tr}}$ is measurable with respect to $\mathcal{F}^{\mathrm{tr}}$ and $\mathcal{G}$ and its image measure under $\mu^{\mathrm{tr}}$ equals the image measure of $V$ under $\mu^{\sharp}$; and a random variable $Z$ on the copy is square-integrable if and only if $Z^{\mathrm{tr}}$ is square-integrable on the trimmed copy, in which case the two mean-square norms agree. Consequently: (a) the pair $(\mathbf{c}\cdot X^{\mathrm{tr}},\mathsf{D}^{\mathrm{tr}})$ is measurable with respect to $\mathcal{F}^{\mathrm{tr}}$ and $\mathcal{B}(\mathbb{R})\otimes\mathcal{R}_{s}$, $\mathbf{c}\cdot X^{\mathrm{tr}}$ is square-integrable on the trimmed copy, and the image measure of $\mu^{\mathrm{tr}}$ under $(\mathbf{c}\cdot X^{\mathrm{tr}},\mathsf{D}^{\mathrm{tr}})$ on $\mathcal{B}(\mathbb{R})\otimes\mathcal{R}_{s}$ equals the image measure of $P^{\mathrm{ag}}$ under $(\mathbf{c}\cdot X'_{s},W^{(s)})$; (b) for every map $m^{\mathrm{est}}:\mathbf{R}_{s}\to\mathbb{R}$ measurable with respect to $\mathcal{R}_{s}$ and the Borel $\sigma$-algebra (an estimator; such maps play the role of the map written $m$ in Score Identities and the Mixture-Weight Directional van Trees Inequality), the restriction $m^{\mathrm{est}}|_{\mathsf{R}_{+}}$ is measurable with respect to $\mathcal{R}_{+}$, $m^{\mathrm{est}}|_{\mathsf{R}_{+}}\circ\mathsf{D}^{\mathrm{tr}}$ is the restriction of $m^{\mathrm{est}}(\mathsf{D})=m^{\mathrm{est}}\circ\mathsf{D}$ to $\Omega^{\mathrm{tr}}$, written $m^{\mathrm{est}}(\mathsf{D}^{\mathrm{tr}})$, and $m^{\mathrm{est}}(\mathsf{D})$ is square-integrable on the copy if and only if $m^{\mathrm{est}}(\mathsf{D}^{\mathrm{tr}})$ is square-integrable on the trimmed copy, with equal mean-square norms; in particular this applies to the constant map $\varsigma$. Moreover, for every choice of real numbers $y_{1}>0$, $y_{2}>0$ and of a family $(\Xi^{c})_{c\in\mathcal{L}}$ 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 the reals written $x_{1}$, $x_{2}$ there),

$$\bigl\lVert\alpha\cdot\Theta^{\mathrm{tr}}+\varsigma(\mathsf{D}^{\mathrm{tr}})-\mathbf{c}\cdot X^{\mathrm{tr}}\bigr\rVert_{2}\ \le\ \mathsf{e}_{1}+\mathsf{e}_{2}+\mathsf{e}_{3}+\mathsf{e}_{4}+\hat{\mathsf{e}}_{5},$$

the norm being that of the trimmed copy and the constants those of that claim for these data.

2. (Van Trees regularity of the trimmed copy.) $(\mathsf{R}_{+},\mathcal{R}_{+},\rho_{+})$ is a measure space whose measure is $\sigma$-finite; for every measurable space $(\mathsf{Y},\mathcal{Y})$ the set $\mathsf{Y}\times\mathsf{R}_{+}$ belongs to $\mathcal{Y}\otimes\mathcal{R}_{s}$ and the restriction of the $\sigma$-algebra $\mathcal{Y}\otimes\mathcal{R}_{s}$ to $\mathsf{Y}\times\mathsf{R}_{+}$ is the product $\sigma$-algebra $\mathcal{Y}\otimes\mathcal{R}_{+}$; in particular the restriction of $\mathcal{B}(\mathbb{R}^{d})\otimes\mathcal{R}_{s}$ to $\mathsf{H}$ is $\mathcal{B}(\mathbb{R}^{d})\otimes\mathcal{R}_{+}$, and the restriction of the product measure $\lambda_{d}\otimes\rho$ to $\mathsf{H}$ is the product measure $\lambda_{d}\otimes\rho_{+}$. Each $\Theta^{\mathrm{tr}}_{q}$ is a square-integrable random variable on the trimmed copy, and $\mathsf{D}^{\mathrm{tr}}$ is measurable with respect to $\mathcal{F}^{\mathrm{tr}}$ and $\mathcal{R}_{+}$. The function $g^{\mathrm{tr}}:\mathsf{H}\to[0,\infty)$ is measurable with respect to $\mathcal{B}(\mathbb{R}^{d})\otimes\mathcal{R}_{+}$, strictly positive, and is a density of the joint law of $(\Theta^{\mathrm{tr}},\mathsf{D}^{\mathrm{tr}})$ under $\mu^{\mathrm{tr}}$ with respect to $\lambda_{d}\otimes\rho_{+}$; and the probability space $(\Omega^{\mathrm{tr}},\mathcal{F}^{\mathrm{tr}},\mu^{\mathrm{tr}})$, the natural number $d$ (in the role of $l$), the measurable space $(\mathsf{R}_{+},\mathcal{R}_{+})$ with the measure $\rho_{+}$, the random variables $\Theta^{\mathrm{tr}}_{q}$, the map $\mathsf{D}^{\mathrm{tr}}$ and the density $g^{\mathrm{tr}}$ satisfy hypotheses (i), (ii) and (iii) of The Multivariate van Trees Inequality and hypothesis (iv$'$) of Score Identities and the Mixture-Weight Directional van Trees Inequality, the scores $\mathsf{sc}_{q}=\partial_{q}g^{\mathrm{tr}}(\Theta^{\mathrm{tr}},\mathsf{D}^{\mathrm{tr}})/g^{\mathrm{tr}}(\Theta^{\mathrm{tr}},\mathsf{D}^{\mathrm{tr}})$ satisfying $\mathbb{E}^{\mathrm{tr}}[\mathsf{sc}_{q}^{2}]\le1/\eta$ for every $q\in\mathsf{L}$. Consequently claims 1 and 2 of Score Identities and the Mixture-Weight Directional van Trees Inequality are available on the trimmed copy for these data and the shifts $a_{q}$; $\bar{g}^{\mathrm{tr}}$ is the mixture weight of that lemma, strictly positive on $\mathsf{H}$, and $\mathcal{I}_{\mathsf{u}}$ is its mixture-weight information in the direction $\mathsf{u}$; and for every $\mathsf{u}\in\mathbb{R}^{d}$,

$$\mathcal{I}_{\mathsf{u}}=\int_{\mathbb{R}^{d}\times\mathbf{R}_{s}}\mathbf{1}_{\mathsf{H}}\,\frac{(\partial_{\mathsf{u}}g)^{2}}{\bar{g}}\,d(\lambda_{d}\otimes\rho)<\infty,$$

the integrand being read as $0$ off $\mathsf{H}$, as in claim 3 of Gaussian Smoothing of a Kernel-Weighted Discrete Family with Countable Support: Joint Density, van Trees Regularity, and the Symmetrised Directional Score Bound, while $\mathcal{I}_{t\mathsf{u}}=t^{2}\,\mathcal{I}_{\mathsf{u}}$ for every real $t$.

3. (Information in the injection direction.)

$$\mathcal{I}_{\mathsf{u}}^{1/2}\ \le\ (\mathsf{J}^{\mathrm{sym}})^{1/2}+\sqrt{2}\,\lVert w\rVert_{1}\bigl(\exp(\kappa_{\mathrm{mv}})-1-\kappa_{\mathrm{mv}}\bigr)^{1/2}$$

(the right-hand side being $\infty$ if $\mathsf{J}^{\mathrm{sym}}=\infty$). Moreover all hypotheses 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 hold for the adopted data, its hypothesis (CL) holding with the tolerances $\varepsilon_{S}$, $\varepsilon_{\mathrm{ctl}}$ and the close records $\mathsf{R}^{\mathrm{cl}}_{\omega}$ of 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; consequently $\mathsf{J}^{\mathrm{sym}}\le\mathsf{J}^{\mathrm{mf}}<\infty$, and

$$N\,\mathcal{I}_{\mathsf{z}}=\mathcal{I}_{\mathsf{u}}\ \le\ \Bigl((\mathsf{J}^{\mathrm{mf}})^{1/2}+\sqrt{2}\,\lVert w\rVert_{1}\bigl(\exp(\kappa_{\mathrm{mv}})-1-\kappa_{\mathrm{mv}}\bigr)^{1/2}\Bigr)^{2}.$$

4. (Pairing of the cell coefficients with the profile response.) For every $c\in\mathcal{L}$ and $u\in[0,s]$, $|H^{c}_{u}|\le\sqrt{2}\,\bar\Phi^{2}\Lambda_{\mathcal{E}}$, and for all $t,u\in[0,s]$

$$\bar\psi_{t}=\int_{[0,t]}\Phi^{\mathcal{E}}(t,u')\,\Theta^{\mathrm{fl}}(S_{u'},A_{u'})\,\varpi_{u'}\,du',\qquad \Theta^{\mathrm{fl}}(S_{u},A_{u})\,\varpi_{u}=\sum_{c\in\mathcal{L}}v_{c}\,(v_{c}\cdot\varpi_{u})\,\phi_{c}(u).$$

Moreover

$$\alpha\cdot\mathsf{z}=\frac{\mathsf{m}}{N}\sum_{q\in\mathsf{L}}w_{q}\alpha_{q},\qquad \bigl|\alpha\cdot\mathsf{z}-\mathbf{c}\cdot\bar\psi_{s}\bigr|\ \le\ \mathsf{e}_{\mathrm{inj}},\qquad \max_{q\in\mathsf{L}}|\alpha\cdot a_{q}|\ \le\ \frac{\sqrt{2}\,|\mathbf{c}|\,\bar\Phi^{2}\,\mathsf{m}}{\sqrt{N}},\qquad \lVert\alpha\rVert\ \le\ \sqrt{2d}\,|\mathbf{c}|\,\bar\Phi^{2}.$$