Instantiation of the Trimmed-Copy Van Trees Data from an N-Agent Solution at the Power Scales: Construction of the Copy Side, Verification of the Copy Hypotheses, and Domination of Its Constants by the Scale-Set Majorants

The $N$-agent side and the intermediate time. Adopt the two paragraphs The $N$-agent side and The intermediate time of the setting 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 the setting, notation and standing hypotheses 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 $1-O(N^{-1/2})$ and Asymptotic Lower Bound for the Recentred N-Agent Cost without Uniform Control-Moment Hypotheses, with the renamings made there), but neither its copy side nor its hypothesis (L), which are constructed, respectively verified, below. Thus we have: the natural numbers $l\ge2$, $m\ge1$, $\tilde{l}\ge1$; the affine-controlled transition-rate family $(\beta_{0},\beta_{1})$ on $l$ states with nonempty convex compact control set $\mathcal{A}\subseteq\mathbb{R}^{m}$ and Lipschitz constant $\Lambda^{\mathrm{aff}}$, its transition-rate family $\beta$ with the rate bound $B\ge0$ of that lemma; the twice continuously differentiable extension $(U,V,\bar\beta)$ of $\beta$ belonging to the common data, with derivative bound $K\ge0$; the observation-rate family $\tilde\beta$ with $\tilde{l}$ channels and rate bound $\tilde{B}\ge0$, with aggregate observation drift $\tilde{b}=(\tilde{b}^{\upsilon})_{\upsilon=1}^{\tilde{l}}$; the horizon $T>0$; the stationary mean-field triple $(S,A,P)$ with initial point $S_{0}$, whose first two components form a mean-field trajectory pair for $\beta$ with horizon $T$, with values $S_{t}$ in the probability simplex $\Delta^{l}$ and $A_{t}\in\mathcal{A}$ and with continuous components (as recorded in 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 bound $\kappa^{\sharp}\ge1$ of (I$'$); the constants $C_{\mathrm{flw}}$, $C_{\mathrm{ctl}}$ and $c_{Q}$; the natural number $N_{\mathrm{cl}}$ and the fixed natural number $N\ge N_{\mathrm{cl}}$ with the $N$-th solution, carried by the driving system $(\Omega^{\mathrm{ag}},\mathcal{F}^{\mathrm{ag}},P^{\mathrm{ag}})$, for the $\mathcal{A}$-valued observation-driven control policy $h=h^{(N)}$ with horizon $T$, its empirical state measure $\Sigma$, its observation record $W$ and its realized control $\hat{\alpha}$; the tolerances $\varepsilon_{S}(N)=(C_{\mathrm{flw}}+1)N^{-1/4}$ and $\varepsilon_{\mathrm{ctl}}(N)=(TC_{\mathrm{ctl}})^{1/2}N^{-1/4}$; the aggregate lattice $\mathbb{G}_{N}\subseteq\Delta^{l}$; and the intermediate time $s$ with $0<s\le T$, together with the objects 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 adopted there: the observation record space $(\mathbf{R}_{s},\mathcal{R}_{s},\rho)$ with horizon $s$ and $\tilde{l}$ channels (its reference measure written $\rho$, 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), the record prefix $W^{(s)}$, the truncated policy $h^{(s)}$ (an $\mathcal{A}$-valued observation-driven control policy with horizon $s$ by claim 1 of Restriction of a Solution of the Controlled N-Agent Dynamics to a Shorter Horizon: the Truncated Policy, the Restricted Solution, Its Filtrations, and Its Record as the Prefix of the Record), the restricted solution, the fixed dense sequence $(\mathsf{v}_{n})_{n\in\mathbb{N}}$ in $L^{2}([0,s];\mathbb{R}^{m})$, the fixed reconstruction data for $h^{(s)}$, the record-frozen control paths $a^{(s),r}$ ($r\in\mathbf{R}_{s}$) of $h^{(s)}$, and the observation-centred fluctuation $X'_{t}$ ($t\in[0,s]$). Assume in addition:

(D) $\mathsf{x}_{0}\in\mathbb{G}_{N}$ is a point with $P^{\mathrm{ag}}(\Sigma_{0}=\mathsf{x}_{0})=1$;

(OC) $\underline{b}>0$ is a real number with $\tilde{b}^{\upsilon}(\Sigma)\ge\underline{b}$ for all $\Sigma\in\Delta^{l}$ and all $\upsilon\in\{1,\dots,\tilde{l}\}$;

(X$'$) $(\tilde{U},\bar{\tilde\beta})$ is a twice continuously differentiable extension of $\tilde\beta$ with derivative bound $\tilde{K}\ge0$;

and let $\lambda$ assign to each $u\in[0,T]$ a vector $\lambda(u)\in\mathbb{R}^{l}$ with continuous components on $[0,T]$ (the profile) and let $\mathbf{c}\in\mathbb{R}^{l}$ (the estimand direction).

Conventions. Write $|\cdot|$ for the Euclidean norm on $\mathbb{R}^{l}$ and for the absolute value, $\lVert\cdot\rVert$ for the Euclidean norm on $\mathbb{R}^{d}$ (once $d$ is defined), $x\cdot y$ for the dot product, $\exp$ and $\log$ for the exponential function and the natural logarithm, $\sqrt{t}=t^{1/2}$ for the nonnegative square root, $t^{1/4}=(t^{1/2})^{1/2}$, real powers $t^{a}$ of a real $t>0$ as in Real Power of a Positive Real Number (consistent with these by claim 1 of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities), $\lfloor\cdot\rfloor$ for the integer part, $\lceil x\rceil$ for the least natural number $\ge x$ ($x\ge0$ real), as in Moment Toolkit for the Synthetic Copy: Square-Integrable Majorants of the Window Discrepancies of the Copy Clocks and the Fourth Moment of a Weighted Centred Cell-Count Sum, $\int_{[0,s]}\cdot\,du$ for the Lebesgue integral over the compact interval $[0,s]$ with respect to the restricted Lebesgue measure $\lambda_{[0,s]}$, $\mathbb{N}$ for the natural numbers regarded as real numbers, $\lVert\cdot\rVert_{2}$ for the mean-square norm on the probability space indicated, and $\varpi_{k}(x)=\min(x^{2}/(4k),x/2)$ ($k>0$, $x\ge0$) for the Chernoff exponent in the notation of The Scale Set for the Van Trees Assembly: Eventual Validity of the Copy-Side Constraints, Lower Bounds for the Chernoff Exponents, and Vanishing of the Error Majorants. Notational cautions: $\lambda$ is the profile, $\lambda_{[0,s]}$ and $\lambda_{d}$ are Lebesgue measures and $\lambda^{\sharp,\omega}$ denotes the intensities of the copy setting; the Chernoff exponent always appears as $\varpi_{k}(x)$ with a parenthesised argument, while $\varpi$ (with values $\varpi_{u}$) is the profile restricted to $[0,s]$; $P$ is the probability measure of the copy side, $P^{\mathrm{ag}}$ that of the $N$-agent side, and the co-state of the triple always carries a time subscript; $\Theta$ is the parameter of the copy and $\Theta^{\mathrm{fl}}$ the aggregate fluctuation covariance (written $\Theta$ in Mean-Field-Side Data for the Van Trees Assembly: Regularity of the Drift Jacobian and Fluctuation Covariance Along a Trajectory Pair, the Fundamental-Solution Bound, the Profile Response and Its Restriction, and the Observation Information Matrix of the Fluctuation LQG Data); $A$ with a time subscript is the mean-field control, $\alpha$ the vector of cell coefficients, $\hat\alpha$ the realized control and $\alpha_{N}$ a scalar of the scale set; $\mathsf{M}$ is the profile-response bound, $M_{N}$ the window length and $\mathsf{M}_{4}(R)$ the constant of Moment Toolkit for the Synthetic Copy: Square-Integrable Majorants of the Window Discrepancies of the Copy Clocks and the Fourth Moment of a Weighted Centred Cell-Count Sum; $D_{N}$ is the discrepancy tolerance, $\mathsf{D}$ the record coordinate and $\tilde{D}$ the observation information matrix; $\theta_{N}$ is the integer of (P) (written $\theta_{\mathrm{P}}$ in 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) and $\theta$ the second coordinate of a point of $\Omega^{\sharp}$; $\delta_{N}$ is the real number of (P), the standard basis vectors of $\mathbb{R}^{l}$ being written $\delta_{1},\dots,\delta_{l}$ only inside $v_{c}=\delta_{\gamma}-\delta_{\sigma}$; $x_{N}$ is a scale, $\mathsf{x}_{0}$ the lattice point of (D); $J_{N}$ is the number of cells per clock and $\mathsf{J}_{N}$, $\mathsf{J}^{\mathrm{mf}}$ are information majorants; $R_{N}$ is the clock horizon, $R^{\mathcal{A}}$ the control bound and $\mathbf{R}_{s}$ the record space; $\mathsf{c}_{0}$, $\mathsf{c}_{\star}$, $\mathsf{c}_{N}$ and $\mathsf{c}_{4}$ are constants and $\mathbf{c}$ the estimand direction; $\mathsf{g}_{N}$, $\mathsf{g}$ are probabilities and $g^{c}$, $g_{\upsilon}$, $g$, $g^{\mathrm{tr}}$ are the state gradients, observation gradients and densities of the adopted settings; $\mathsf{w}_{N}$, $w_{1,N}$, $w_{2,N}$ and $\mathsf{w}^{\mathrm{clk}}$ are scalars and $w$ the vector of injection weights; $A^{\mathrm{ins}}$ denotes the constant of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect written $A_{0}$ throughout the adopted copy settings (in hypothesis (W) 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 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, in 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 and 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), exactly 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, so that $A_{0}$ would denote the value of the mean-field control at time $0$ (not used); $\mathcal{E}$ with two arguments is the drift Jacobian and $\mathcal{E}_{t}$, $\mathcal{E}^{\star}_{u}$ the matrices of Mean-Field-Side Data for the Van Trees Assembly: Regularity of the Drift Jacobian and Fluctuation Covariance Along a Trajectory Pair, the Fundamental-Solution Bound, the Profile Response and Its Restriction, and the Observation Information Matrix of the Fluctuation LQG Data, the control energy process written $\mathcal{E}_{t}$ in Closeness of the Realized Control and the Realized Mean-Field Flow on a High-Probability Event under the Cost Bound not being used; $\mathcal{P}$ is the profile energy of the copy instance, the sum of leave probabilities written $\mathcal{P}$ in that lemma not being used; $\mathsf{P}$ is a real number (the first integral above), unrelated to the copy clocks $\mathsf{P}^{\sharp}$, $\mathsf{P}^{(y)}$ and to the count mass function $\mathsf{p}$; $H$ is the scalar $\sqrt{2}\,\bar\Phi^{2}\Lambda_{\mathcal{E}}s$ of the scale set, $H^{c}$ are the maps of (CP) and $\mathsf{H}=\mathbb{R}^{d}\times\mathsf{R}_{+}$ the set 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; $L_{N}$ is the real number of (W), $\mathsf{L}$ the cell index set and $G$ the good event, the population cost data of Asymptotic Lower Bound for the Recentred N-Agent Cost without Uniform Control-Moment Hypotheses (written $(L,G)$ there) not being used; $w_{1}$ and $w_{2}$ in claim 5 are the two scalars so written 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, never coordinates of the vector $w$, whose entries are always written $w_{q}$ ($q\in\mathsf{L}$); $V$ is the control-side open set of the extension $(U,V,\bar\beta)$, while the channel set 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, written $V$ there, is $\{1,\dots,\tilde{l}\}$ and is never written $V$ here; $Q$ is the noise majorant 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 $1-O(N^{-1/2})$, unrelated to $\mathsf{Q}$, $\mathsf{Q}^{\mathrm{cl}}$ and $\mathsf{Q}_{N}$; $K_{1}$, $K_{2}$, $\Lambda_{b}$, $C_{S}$ and $\mathcal{Z}^{\sharp}$ are constants of the $N$-agent side (The Transition-Rate Family and Aggregate Drift of Affine-Controlled Data, Closeness of the Realized Control and the Realized Mean-Field Flow on a High-Probability Event under the Cost Bound), used only in the proof of claim 1, while $K$, $\tilde{K}$ are the derivative bounds and $\mathsf{K}$ the cell-count vector; and the remaining letters of the adopted settings keep the meanings recorded 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 and 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.

The mean-field-side data. Let $\mathcal{L}$ be the set of transition labels $c=(\sigma,\gamma)$, $v_{c}=\delta_{\gamma}-\delta_{\sigma}$, and let $\Theta^{\mathrm{fl}}$ be the aggregate fluctuation covariance of $\beta$. The data $l$, $m$, $\tilde{l}$, $\mathcal{A}$, $\beta$ with $B$, $(U,V,\bar\beta)$ with $K$, $\tilde\beta$ with $\tilde{B}$, $(\tilde{U},\bar{\tilde\beta})$ with $\tilde{K}$, $\underline{b}$, $T$, the pair $(S,A)$, $s$ and $\lambda$ are of the kind required by Mean-Field-Side Data for the Van Trees Assembly: Regularity of the Drift Jacobian and Fluctuation Covariance Along a Trajectory Pair, the Fundamental-Solution Bound, the Profile Response and Its Restriction, and the Observation Information Matrix of the Fluctuation LQG Data (claim 1 below records that its hypotheses hold); adopt from that lemma, for these data, the label rates $\psi_{c}$, state gradients $g^{c}$ and drift Jacobian $\mathcal{E}$, the observation gradients $g_{\upsilon}$ and observation information matrices $\tilde{D}(x)$ ($x\in\Delta^{l}$), the matrices $\mathcal{E}_{t}$, $\Theta^{\star}_{t}$, $\tilde{\mathcal{E}}_{t}$, $\tilde\Theta^{\star}_{t}$, $\tilde{D}_{t}$ ($t\in[0,T]$), the two-parameter fundamental solution $\Phi^{\mathcal{E}}(t,u)$ ($t,u\in[0,s]$) of $u\mapsto\mathcal{E}(S_{u},A_{u})$ and a real number $\bar\Phi\ge0$ as in its claim 3, fixed from now on, a real number $\Lambda\ge0$ with $|\lambda(u)|\le\Lambda$ ($u\in[0,T]$), the profile response $\psi_{\lambda}:[0,T]\to\mathbb{R}^{l}$ and a real number $\mathsf{M}\ge0$ with $|\psi_{\lambda}(u)|\le\mathsf{M}$ ($u\in[0,T]$) as in its claim 4, fixed from now on, the restrictions $\varpi=\lambda|_{[0,s]}$ and $\bar\psi=\psi_{\lambda}|_{[0,s]}$ with values $\varpi_{u}$, $\bar\psi_{u}$, and the information functional $\mathcal{A}_{s}(\lambda)$ of its claim 6. Put

real numbers, nonnegative and with $\mathsf{P}+\mathsf{Q}=\mathcal{A}_{s}(\lambda)$ by claim 6 of that lemma (available by claim 1 below).

The scale set. Put $\mathsf{c}_{0}=|\mathbf{c}|$. The scalar data $l$, $m$, $\tilde{l}$, $B$, $\tilde{B}$, $K$, $\tilde{K}$, $\underline{b}$, $T$, $s$, $\Lambda$, $\mathsf{M}$, $\bar\Phi$, $\mathsf{c}_{0}$, $c_{Q}$, $\kappa^{\sharp}$, $C_{\mathrm{flw}}$, $C_{\mathrm{ctl}}$, $\mathsf{P}$ and $\mathsf{Q}$ are of the kind required by The Scale Set for the Van Trees Assembly: Eventual Validity of the Copy-Side Constraints, Lower Bounds for the Chernoff Exponents, and Vanishing of the Error Majorants (claim 1 below records that its hypotheses hold, in particular that $c_{Q}\ge0$, $C_{\mathrm{flw}}\ge0$ and $C_{\mathrm{ctl}}\ge0$); adopt from that lemma, for these data, the constants $\Lambda_{1}$, $\Lambda_{2}$, $\Lambda_{3}$, $\Lambda_{\mathcal{E}}$, $\Gamma$, $C_{\mathrm{Lip}}$, $\mathsf{c}_{\star}=c_{Q}\kappa^{\sharp}$, $C_{A}$, $C_{M}$, $c_{\mathrm{w}}$, $H$, the scale set $\mathsf{m}_{N}$, $D_{N}$, $A_{N}$, $L_{N}$, $M_{N}$, $R_{N}$, $J_{N}$, $\mu_{N}$, $d_{N}$, $\eta_{N}$, $\delta_{N}$, $\zeta_{N}$, $\theta_{N}$, $x_{N}$, $\varepsilon_{S,N}$, $\varepsilon_{\mathrm{ctl},N}$, $w_{1,N}$, $w_{2,N}$, all its derived quantities (in particular $\varepsilon_{0,N}$, $\bar{E}_{N}$, $\mathsf{c}_{N}$, $\mathsf{E}^{\star}_{N}$, $\mathsf{e}^{\star}_{N}$, $\mathsf{E}^{\mathrm{ch}}_{N}$, $\bar\Pi_{N}$, $\mathsf{g}_{N}$, $\kappa_{0,N}$, $\bar{\mathsf{j}}_{N}$, $\mathsf{j}^{\star}_{N}$, $\mathsf{B}_{N}$, $\mathsf{w}_{N}$, $\kappa^{\mathrm{mv}}_{N}$, $\alpha_{N}$, $\mathsf{k}_{N}$, $\mathsf{e}_{F,N}$, $\epsilon_{\psi,N}$, $\kappa_{N}$, $\mathsf{Q}_{N}$, $\mathsf{J}_{N}$, $\mathcal{I}_{N}$, $\Xi_{N}$) and error majorants $\mathsf{e}_{2,N}$, $\mathsf{e}_{3,N}$, $\mathsf{e}_{4,N}$, $\mathsf{e}_{5,N}$, $\mathsf{e}_{\mathrm{inj},N}$, $\mathsf{a}_{N}$, and the natural number $N_{\mathrm{c}}$ fixed in its claim 1 (which depends only on the scalar data and is defined once claim 1 below is established). Assume $N\ge N_{\mathrm{c}}$. (In this paragraph and the next, the scalar $\bar{E}_{N}$ of the scale set and the constant of the same name of the copy setting are shown equal in claim 4; until then the phrase "of the scale set" or "of the copy instance" distinguishes them.)

The copy side. For $c\in\mathcal{L}$ and $0\le j\le J_{N}$ put $\mathsf{b}^{c}_{j}=j\mu_{N}$ and, for $1\le j\le J_{N}$, $I_{c,j}=(\mathsf{b}^{c}_{j-1},\mathsf{b}^{c}_{j}]$. Let $(\Omega,\mathcal{F},P)$ be a probability space carrying an independent family of random variables $K^{c}$ ($c\in\mathcal{L}$), $V^{c}_{i}$ ($c\in\mathcal{L}$, $i\in\mathbb{N}$) and $U^{c,j}_{i}$ ($c\in\mathcal{L}$, $1\le j\le J_{N}$, $i\in\mathbb{N}$), where $K^{c}$ has the Poisson distribution with parameter $R_{N}$, $V^{c}_{i}$ has the uniform law on $(0,R_{N}]$ and $U^{c,j}_{i}$ has the uniform law on $I_{c,j}$, in the sense 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; such a space exists by Existence of Independent Sequences with Prescribed Distributions, 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. The copy instance is the instance of the setting 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 (that is, of the copy side 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, hence 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, 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, The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record and 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 horizon $s$, together with the additional objects and the hypothesis (P) 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) specified by the following choices: the numbers $N$, $l$, $m$, $\tilde{l}$, the control set $\mathcal{A}$, the families $\beta$ and $\tilde\beta$ with the bounds $B$ and $\tilde{B}$, the lattice $\mathbb{G}_{N}$ with the point $\mathsf{x}_{0}$ in the role of $x_{0}$, the record space $(\mathbf{R}_{s},\mathcal{R}_{s},\rho)$, the policy $h^{(s)}$ with the record-frozen control paths $a^{(s),r}$ in the role of $a^{r}$; the clock horizon $R=R_{N}$, the natural numbers $J_{c}=J_{N}$ ($c\in\mathcal{L}$), the cell boundaries $\mathsf{b}^{c}_{j}$ and cells $I_{c,j}$ above, the move size $\mathsf{m}=\mathsf{m}_{N}$, the smoothing parameter $\eta=\eta_{N}$ and the driving variables above on $(\Omega,\mathcal{F},P)$; for (OC) the number $\underline{b}$; for (X) the extension $(\tilde{U},\bar{\tilde\beta})$ with $\tilde{K}$ and the extension $(U,V,\bar\beta)$ with $K$ (in the role of $(U,W_{\beta},\bar\beta)$); for (W) the real numbers $L=L_{N}$ and $D=D_{N}$; for (AF) the affine family $(\beta_{0},\beta_{1})$, the base point $z_{0}=S_{0}$ and the sequence $(\mathsf{v}_{n})$; the comparison pair $S^{\mathrm{cp}}_{t}=S_{t}$, $\mathsf{A}_{t}=A_{t}$ ($t\in[0,s]$) with the mean-field label rates $\phi_{c}(t)=\psi_{c}(S_{t},A_{t})$; the profile $\varpi$ with bound $\Lambda$ and the profile response $\bar\psi$ with bound $\mathsf{M}$; for (CP) the fundamental solution $\Phi^{\mathcal{E}}$ with the bound $\bar\Phi$ and the estimand direction $\mathbf{c}$; the real numbers $\varepsilon_{S}=\varepsilon_{S}(N)$ and $\varepsilon_{\mathrm{ctl}}=C_{\mathrm{Lip}}\,\varepsilon_{\mathrm{ctl}}(N)$ 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 for (CL); the event $G=G^{\mathsf{m}}\cap\Omega'$ for (G) and (G$'$), where $G^{\mathsf{m}}$ is the good event of 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 and $\Omega'$ the event of claim 3 of Almost Sure Tracking on the Synthetic Copy: Jump Times of the Deterministic-Count Clocks, Almost Sure Conflict-Freeness of Every Record, Almost Sure Null Mass of the Untracked Records, and Trimming an Event to the Tracked Set; the real number $\zeta=\zeta_{N}$; and for (P) the real number $\delta=\delta_{N}$, the integer $\theta_{\mathrm{P}}=\theta_{N}$ and the real numbers $x^{\mathrm{P}}_{q}=x_{N}$ ($q\in\mathsf{L}$). That these choices satisfy the requirements and hypotheses of the settings named is the content of claims 1--4; an object of an adopted setting that is defined only under one of those hypotheses is used only after the claim establishing it. All objects of the copy instance are written 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 and 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: in particular the index set $\mathsf{L}$ of the cells with $d$ elements, the cell lengths $\mu_{q}$, $\mu_{\max}$, $\mu_{\min}$, the cell-count vector $\mathsf{K}$, the clock-good event $G_{L,D}$, the tracked records $\mathsf{T}_{\omega}$, the synthetic copy $(\Omega^{\sharp},\mathcal{F}^{\sharp},\mu^{\sharp})$ with parameter $\Theta$ and record $\mathsf{D}$, $\mathsf{g}=P(\Omega\setminus G)$, the injection weights $w$ with $\lVert w\rVert_{1}$, the profile energy $\mathcal{P}$, the cell coefficients $\alpha\in\mathbb{R}^{d}$, the maps $H^{c}$ with $\lVert H^{c}\rVert_{1}$, the constant map $\varsigma$, $\mathsf{k}_{4}$, $\mathsf{w}^{\mathrm{clk}}$, $A^{\mathrm{ins}}$, $\Gamma$, $\varepsilon_{0}$, $\bar{E}_{N}$, $\mathsf{c}_{N}$, $\mathsf{E}^{\star}_{N}$, $\mathsf{e}^{\star}_{N}$, $\kappa^{\mathrm{mf}}_{0}$, $\mathsf{j}_{q}$, $\bar{\mathsf{j}}$, $\mathsf{j}^{\star}$, $\mathsf{Q}^{\mathrm{cl}}$, $\mathsf{E}^{\mathrm{ch}}_{\theta_{\mathrm{P}}}$, $\bar\Pi$, $\mathsf{B}$, $\mathsf{e}_{F}$, $\epsilon_{\psi}$, $\kappa$, the constants $\mathsf{e}_{1},\dots,\mathsf{e}_{4}$, $\hat{\mathsf{e}}_{5}$ 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 (for data $y_{1}$, $y_{2}$, $(\Xi^{c})_{c}$ specified in claim 5), and, from 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, the record support $\mathsf{R}_{+}$ with $(\mathcal{R}_{+},\rho_{+})$, the trimmed copy $(\Omega^{\mathrm{tr}},\mathcal{F}^{\mathrm{tr}},\mu^{\mathrm{tr}})$ with $\Theta^{\mathrm{tr}}$, $\mathsf{D}^{\mathrm{tr}}$, $X^{\mathrm{tr}}$, the density $g^{\mathrm{tr}}$, the moves $a_{q}=\mathsf{m}e_{q}/\sqrt{N}$, the directions $\mathsf{u}$ and $\mathsf{z}=\mathsf{m}w/N$, the mixture-weight informations $\mathcal{I}_{\mathsf{u}}$, $\mathcal{I}_{\mathsf{z}}$, $\kappa_{\mathrm{mv}}$, $\mathsf{J}^{\mathrm{mf}}$ and $\mathsf{e}_{\mathrm{inj}}$.

Then the following hold.

1. (The two auxiliary lemmas apply; admissibility of the scales.) The hypotheses of Mean-Field-Side Data for the Van Trees Assembly: Regularity of the Drift Jacobian and Fluctuation Covariance Along a Trajectory Pair, the Fundamental-Solution Bound, the Profile Response and Its Restriction, and the Observation Information Matrix of the Fluctuation LQG Data hold for the data named above; the real numbers $c_{Q}$, $C_{\mathrm{flw}}$ and $C_{\mathrm{ctl}}$ are nonnegative, and the hypotheses of The Scale Set for the Van Trees Assembly: Eventual Validity of the Copy-Side Constraints, Lower Bounds for the Chernoff Exponents, and Vanishing of the Error Majorants hold for the scalar data named above; its constants $\Lambda_{1}$, $\Lambda_{2}$, $\Lambda_{3}$, $\Lambda_{\mathcal{E}}$, $\Gamma$ and $C_{\mathrm{Lip}}$ coincide with the constants of the same names of the copy instance. Moreover: $R_{N}\in\mathbb{N}$ and $R_{N}>NBs$; $J_{N}\in\mathbb{N}$, $\mu_{N}>0$, $0=\mathsf{b}^{c}_{0}<\mathsf{b}^{c}_{1}<\dots<\mathsf{b}^{c}_{J_{N}}=R_{N}$, every cell has length $\mu_{q}=\mu_{N}$, so that $\mu_{\max}=\mu_{\min}=\mu_{N}$, and $d=d_{N}$; $\mathsf{m}_{N}\in\mathbb{N}$ and $\eta_{N}\in(0,1]$. Consequently the settings 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 and The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record are instantiated by the copy instance, and so are the settings 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 Pathwise Reduction of the Symmetrised Move Information on the Synthetic Copy: Removed-Clock Likelihoods, the Shifted Family, and the Averaging Bound once $\phi_{c}$, $\varpi$ and the injection weights $w$ are available (claim 2). Hypothesis (W) holds with $L=L_{N}$, $D=D_{N}$, its constant (written $A_{0}$ there and $A^{\mathrm{ins}}$ here) being equal to $A_{N}$ (an assertion involving only $\mathsf{m}_{N}$, $D_{N}$, $L_{N}$, $\Lambda_{1}$ and $s$).

2. (The regularity hypotheses and the profile data.) Hypotheses (OC), (X), (AF) and (CP) hold for the choices made; in particular the tolerances $\varepsilon_{S}=\varepsilon_{S}(N)$ and $\varepsilon_{\mathrm{ctl}}=C_{\mathrm{Lip}}\varepsilon_{\mathrm{ctl}}(N)$ are nonnegative real numbers. The comparison pair has continuous, hence measurable, components on $[0,s]$; each $\phi_{c}$ is measurable with values in $[0,B]$; $\varpi$ and $\bar\psi$ have measurable components with $|\varpi_{u}|\le\Lambda$ and $|\bar\psi_{u}|\le\mathsf{M}$ for $u\in[0,s]$, and $\bar\psi$ satisfies the profile response equation 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 comparison pair and the profile $\varpi$; the observation information matrix $\tilde{D}(x)$ of the copy instance is the matrix $\tilde{D}(x)$ of Mean-Field-Side Data for the Van Trees Assembly: Regularity of the Drift Jacobian and Fluctuation Covariance Along a Trajectory Pair, the Fundamental-Solution Bound, the Profile Response and Its Restriction, and the Observation Information Matrix of the Fluctuation LQG Data. Consequently the profile energy of the copy instance is $\mathcal{P}=\mathsf{P}$, the integral $\int_{[0,s]}\bar\psi_{u}\cdot(\tilde{D}(S_{u})\bar\psi_{u})\,du$ appearing in $\mathsf{Q}^{\mathrm{cl}}$ equals $\mathsf{Q}$, and

3. (The good event.) $G^{\mathsf{m}}\in\mathcal{F}$ and $G=G^{\mathsf{m}}\cap\Omega'\in\mathcal{F}$; hypotheses (G) and (G$'$) hold for $G$; and

4. (Hypotheses (P) and (L); availability of the chain.) 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 holds with $\delta=\delta_{N}$, $\theta_{\mathrm{P}}=\theta_{N}$ and $x^{\mathrm{P}}_{q}=x_{N}$; the constants $\varepsilon_{0}$, $\bar{E}_{N}$, $\mathsf{c}_{N}$, $\mathsf{E}^{\star}_{N}$, $\mathsf{e}^{\star}_{N}$, $\kappa^{\mathrm{mf}}_{0}$, $\mathsf{E}^{\mathrm{ch}}_{\theta_{\mathrm{P}}}$, $\bar\Pi$ and $\kappa_{\mathrm{mv}}$ of the copy instance equal, respectively, $\varepsilon_{0,N}$, $\bar{E}_{N}$, $\mathsf{c}_{N}$, $\mathsf{E}^{\star}_{N}$, $\mathsf{e}^{\star}_{N}$, $\kappa_{0,N}$, $\mathsf{E}^{\mathrm{ch}}_{N}$, $\bar\Pi_{N}$ and $\kappa^{\mathrm{mv}}_{N}$ of the scale set. Hypothesis (L) 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 holds. Consequently the whole setting 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 is instantiated by the $N$-agent side, the intermediate time and the copy instance, so that its claims 1--4, the claims 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 (by claim 3 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) the claims 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 are available for it. The constant $\mathsf{c}_{4}=c_{Q}\kappa^{\sharp}$ of claim 4 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 equals $\mathsf{c}_{\star}$, and the tolerances of the copy instance satisfy $\varepsilon_{S}=\varepsilon_{S}(N)=\varepsilon_{S,N}$ and $\varepsilon_{\mathrm{ctl}}=C_{\mathrm{Lip}}\varepsilon_{\mathrm{ctl}}(N)=\varepsilon_{\mathrm{ctl},N}$.

5. (Domination of the constants by the scale-set majorants.) Put $w_{1}=\mathsf{w}^{\mathrm{clk}}$ and $w_{2}=\mu_{\max}$; then $w_{1}=w_{1,N}\ge0$ (claim 2) and $w_{2}=w_{2,N}>0$, so that $y_{i}=(\lceil w_{i}\rceil+2)^{1/2}N^{1/32}$ ($i\in\{1,2\}$) are defined and $y_{1},y_{2}>0$. Let $(\Xi^{c})_{c\in\mathcal{L}}$ be a family as furnished by claim 2 of Moment Toolkit for the Synthetic Copy: Square-Integrable Majorants of the Window Discrepancies of the Copy Clocks and the Fourth Moment of a Weighted Centred Cell-Count Sum for the data $w_{1}$, $w_{2}$, $y_{1}$, $y_{2}$ (such a family exists by that claim), so that the constants $\mathsf{e}_{1},\dots,\mathsf{e}_{4}$, $\hat{\mathsf{e}}_{5}$ 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 are formed with $y_{1}$, $y_{2}$ and this family. Then

and, for the information,

6. (The trimmed-copy data in scale-set form.) With the data of claim 5: (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 this pair equals the image measure of $P^{\mathrm{ag}}$ under $(\mathbf{c}\cdot X'_{s},W^{(s)})$; (b) the probability space $(\Omega^{\mathrm{tr}},\mathcal{F}^{\mathrm{tr}},\mu^{\mathrm{tr}})$, the natural number $d$, the $\sigma$-finite measure space $(\mathsf{R}_{+},\mathcal{R}_{+},\rho_{+})$, the square-integrable random variables $\Theta^{\mathrm{tr}}_{q}$ ($q\in\mathsf{L}$), the map $\mathsf{D}^{\mathrm{tr}}$ (measurable with respect to $\mathcal{F}^{\mathrm{tr}}$ and $\mathcal{R}_{+}$) and the density $g^{\mathrm{tr}}$ satisfy hypotheses (i), (ii), (iii) of The Multivariate van Trees Inequality and (iv$'$) of Score Identities and the Mixture-Weight Directional van Trees Inequality, and $\mathcal{I}_{\mathsf{z}}$ is the mixture-weight information of that lemma in the direction $\mathsf{z}$ for the shifts $a_{q}$ ($q\in\mathsf{L}$, $n=d$); (c) the restriction $\varsigma|_{\mathsf{R}_{+}}$ is measurable with respect to $\mathcal{R}_{+}$ and the Borel $\sigma$-algebra, $\varsigma(\mathsf{D}^{\mathrm{tr}})=\varsigma|_{\mathsf{R}_{+}}\circ\mathsf{D}^{\mathrm{tr}}$ is square-integrable on the trimmed copy, and

(d) $\alpha\cdot\mathsf{z}\ge\mathbf{c}\cdot\psi_{\lambda}(s)-\mathsf{e}_{\mathrm{inj},N}$, $\max_{q\in\mathsf{L}}|\alpha\cdot a_{q}|\le\mathsf{a}_{N}$, and $\mathcal{I}_{\mathsf{z}}\le\mathcal{I}_{N}$.