Injection Certificates on the Trimmed Synthetic Copy for the Family of N-Agent Solutions: the Law-Transported Van Trees Certificate Hypothesis Holds under Deterministic Initial States, Uniform Observation Positivity and a $C^2$ Observation Extension

Setting. Adopt 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 hence of Closeness of the Realized Control and the Realized Mean-Field Flow on a High-Probability Event under the Cost Bound and Asymptotic Lower Bound for the Recentred N-Agent Cost without Uniform Control-Moment Hypotheses), with the renamings of the paragraph The $N$-agent 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 applied to every member of the family of solutions: 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 $\mathcal{A}\subseteq\mathbb{R}^{m}$, its transition-rate family $\beta$ with the rate bound $B\ge0$ of that lemma, and the twice continuously differentiable extension $(U,V,\bar\beta)$ of $\beta$ of the common data, with derivative bound $K\ge0$; the observation-rate family $\tilde\beta$ with $\tilde{l}$ channels and rate bound $\tilde{B}\ge0$ and its 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 $S_{0}$ in the probability simplex $\Delta^{l}$, whose first two components form a mean-field trajectory pair for $\beta$ with horizon $T$, with values $S_{t}\in\Delta^{l}$ and $A_{t}\in\mathcal{A}$ and with continuous components; hypotheses (I$'$) with the bound $\kappa^{\sharp}$ and (CB); the constants $C_{\mathrm{flw}}$, $C_{\mathrm{ctl}}$, $C_{\mathrm{esc}}$, $c_{Q}$ and the natural number $N_{1}$; and, for every natural number $N\ge1$, the $N$-th solution of the controlled $N$-agent dynamics on $[0,T]$, carried by the $N$-agent driving system $(\Omega^{\mathrm{ag}},\mathcal{F}^{\mathrm{ag}},P^{\mathrm{ag}})$ with expectation $\mathbb{E}^{\mathrm{ag}}$, for the $\mathcal{A}$-valued observation-driven control policy $h^{(N)}$ with horizon $T$, with regular event $\Omega^{\mathrm{ag}}_{0}$, empirical state measure $\Sigma$, observation record $W$, observation filtration $(\mathcal{G}_{t})_{t\in[0,T]}$ and realized control $\hat\alpha$ (the dependence of these objects on $N$ is suppressed, as in the adopted setting), and the aggregate lattice $\mathbb{G}_{N}\subseteq\Delta^{l}$. Fix, 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, a natural number $N_{\mathrm{cl}}$ with $N_{\mathrm{cl}}\ge N_{1}$ and $N_{\mathrm{cl}}\ge C_{\mathrm{esc}}^{2}$ (such a number exists). For $0<s\le T$ let $(\mathbf{R}_{s},\mathcal{R}_{s},\rho)$ be the observation record space with horizon $s$ and $\tilde{l}$ channels with its reference measure (written $\rho$ for every $s$, 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; it is the measure written $\varrho_{s}$ in 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, whose weak metric $\rho$ is not used), let $\pi_{s}$ be the prefix map $\mathbf{R}_{T}\to\mathbf{R}_{s}$ when $s<T$ and the identity map when $s=T$, let $W^{(s)}=\pi_{s}\circ W$ be the record prefix, let $h^{(s)}$ be the truncated policy ($h^{(T)}=h^{(N)}$), and let $X'_{s}=\sqrt{N}(\Sigma_{s}-\Phi_{s})$ be the observation-centred fluctuation of the $N$-th solution at time $s$ formed with the base point $S_{0}$, as in 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 and in Asymptotic Lower Bound for the Recentred N-Agent Cost by the Fluctuation LQG Value, under Law-Transported Injection Certificates, where $\Phi_{t}=\mathsf{S}_{t}(S_{0},\hat\alpha)$ is the realized mean-field flow formed with the two-argument mean-field flow $\mathsf{S}$. For every $s\in(0,T]$ fix a sequence in $L^{2}([0,s];\mathbb{R}^{m})$ whose set of terms is dense (one exists, as asserted in the setting 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; for $s=T$ take the sequence already fixed in the adopted setting), and for every $N\ge1$ and every $s\in(0,T]$ fix reconstruction data for the driving system of the $N$-th solution and the policy $h^{(s)}$ with horizon $s$ (they exist by that lemma). Assume in addition:

(D0) (Deterministic initial states.) $(\mathsf{x}^{N}_{0})_{N\ge1}$ is a family of points $\mathsf{x}^{N}_{0}\in\mathbb{G}_{N}$ with $P^{\mathrm{ag}}(\Sigma_{0}=\mathsf{x}^{N}_{0})=1$ for the $N$-th solution (for each $N$ there is at most one such point, two distinct points giving disjoint events of probability one), such that the real sequence $\bigl(\sqrt{N}\,|\mathsf{x}^{N}_{0}-S_{0}|\bigr)_{N\in\mathbb{N}}$ has limit $0$;

(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 $\tilde{K}\ge0$.

The mean-field-side data. For a real number $s\in(0,T]$ and an assignment $\lambda$ of a vector $\lambda(u)\in\mathbb{R}^{l}$ to each $u\in[0,T]$ with continuous components (a profile), 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 $\tilde{K}$, $\underline{b}$, $T$, $(S,A)$, $s$ 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); write $\psi_{\lambda}:[0,T]\to\mathbb{R}^{l}$ for its profile response (the unique bounded measurable map with $\psi_{\lambda}(u)=\int_{[0,u]}(\mathcal{E}_{r}\psi_{\lambda}(r)+\Theta^{\star}_{r}\lambda(r))\,dr$ for $u\in[0,T]$, with the state matrix $\mathcal{E}_{r}$ and state noise covariance $\Theta^{\star}_{r}$ of the fluctuation LQG data along $(S,A)$, as in claim 4 there) and $\mathcal{A}_{s}(\lambda)\ge0$ for the information functional of its claim 6, both independent of $N$. (Claim 1 records that under (D0) hypothesis (I) of Asymptotic Lower Bound for the Recentred N-Agent Cost without Uniform Control-Moment Hypotheses holds with the zero matrix as $\Pi_{0}$, and that these are then the maps written $\psi_{\lambda}$ and $\mathcal{A}_{s}(\lambda)$ in Asymptotic Lower Bound for the Recentred N-Agent Cost by the Fluctuation LQG Value, under Law-Transported Injection Certificates.)

Conventions. Write $|\cdot|$ for the Euclidean norm and the absolute value, $x\cdot y$ for the dot product, $\lVert\cdot\rVert_{2}$ for the mean-square norm on the probability space indicated, $\mathcal{B}(\mathbb{R})$ for the Borel $\sigma$-algebra of the real line, $\otimes$ for the product $\sigma$-algebra, and measurability and image measures as in those items. For a map $\mathsf{D}'$ from a set $\Omega'$ into $\mathbf{R}_{s}$ write $\sigma(\mathsf{D}')=\{\mathsf{D}'^{-1}(E):E\in\mathcal{R}_{s}\}$, as in Factorisation of Random Variables Through a Measurable Map, the Variational Form of the Mean-Square Filtering Error, and Its Invariance Under the Joint Law. Notational cautions: $\lambda$ is the profile, unrelated to Lebesgue measure; $\vartheta_{1},\dots,\vartheta_{d}$ are the certificate random variables, the letter $\Theta$ being reserved for the covariance matrices; the certificate vectors $\alpha$, $z$, $a_{1},\dots,a_{d}$ are unrelated to the realized control $\hat\alpha$ and to the record-frozen control paths; the certificate map $G$ is unrelated to the terminal cost of the common data; $P$ with a time subscript is the co-state of the triple, $P^{\mathrm{ag}}$ and $P'$ are probability measures, and in claim 3 and in the proof $P$ without subscript is the probability measure of the copy side of 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; in claim 3, $\Lambda$ is the profile bound 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 affine Lipschitz constant being written $\Lambda^{\mathrm{aff}}$, 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 certificate map $G$ of claim 2 is unrelated to the good event written $G$ in 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, and the certificate space $\Omega'$ of claim 2 is unrelated to the event written $\Omega'$ in 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 (in claim 3 the certificate space is written $\Omega^{\mathrm{tr}}$); the hypothesis (OC) above (uniform positivity of $\tilde{b}$ on the whole simplex) is that 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 and 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, and is distinct from, and stronger than, the hypothesis of the same label in Asymptotic Lower Bound for the Recentred N-Agent Cost by the Fluctuation LQG Value, under Law-Transported Injection Certificates; and the intermediate time is restricted to $0<s\le T$ (the record space, the profile response equation on $[0,s]$ and the copy construction all require $s>0$), so that the case $s=0$ of the hypothesis (VT) of that theorem is not covered by claim 2 and must be handled separately where it is needed.

Then the following hold.

1. (The mean-field-side lemma applies; the initial covariance vanishes.) For every $s\in(0,T]$ and every profile $\lambda$, 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; in particular $\psi_{\lambda}$ and $\mathcal{A}_{s}(\lambda)$ are defined, and $\psi_{\lambda}$ does not depend on $s$. Moreover, for every $N\ge1$ and all $\gamma,\delta\in\{1,\dots,l\}$, the state fluctuation $\mathfrak{s}_{0}=\sqrt{N}(\Sigma_{0}-S_{0})$ of the $N$-th solution satisfies $\mathbb{E}^{\mathrm{ag}}[\mathfrak{s}^{\gamma}_{0}\mathfrak{s}^{\delta}_{0}]=N(\mathsf{x}^{N,\gamma}_{0}-S^{\gamma}_{0})(\mathsf{x}^{N,\delta}_{0}-S^{\delta}_{0})$, and the sequence $(\mathbb{E}^{\mathrm{ag}}[\mathfrak{s}^{\gamma}_{0}\mathfrak{s}^{\delta}_{0}])_{N\ge1}$ has limit $0$; consequently hypothesis (I) of Asymptotic Lower Bound for the Recentred N-Agent Cost without Uniform Control-Moment Hypotheses holds for the family of solutions with the zero matrix as $\Pi_{0}$. Finally, hypothesis (OC) of Asymptotic Lower Bound for the Recentred N-Agent Cost by the Fluctuation LQG Value, under Law-Transported Injection Certificates holds with $\tilde\beta_{\min}=\underline{b}$, and the maps $\psi_{\lambda}$ and $\mathcal{A}_{s}(\lambda)$ of that theorem, formed with the zero matrix as $\Pi_{0}$ and with the observation extension $(\tilde{U},\bar{\tilde\beta})$, coincide with the $\psi_{\lambda}$ and $\mathcal{A}_{s}(\lambda)$ above.

2. (Law-transported van Trees certificates: hypothesis (VT$'$).) For every $N\ge1$ and every $s\in(0,T]$, the observation filtration satisfies:

(C0) (The conditioning $\sigma$-algebra is generated by the record prefix.) $\mathcal{G}_{s}$ is $W^{(s)}$-generated up to null sets in $(\Omega^{\mathrm{ag}},\mathcal{F}^{\mathrm{ag}},P^{\mathrm{ag}})$, in the sense of Factorisation of Random Variables Through a Measurable Map, the Variational Form of the Mean-Square Filtering Error, and Its Invariance Under the Joint Law applied with the measurable space $(\mathbf{R}_{s},\mathcal{R}_{s})$ and the map $W^{(s)}$.

Moreover, for every real number $s$ with $0<s\le T$, every $\mathbf{c}\in\mathbb{R}^{l}$, every profile $\lambda$ and every real number $\epsilon>0$ there is a natural number $N_{2}$ such that for every natural number $N\ge N_{2}$ there exist: a probability space $(\Omega',\mathcal{F}',P')$; a set $\mathsf{Y}\in\mathcal{R}_{s}$ with its trace $\sigma$-algebra $\mathcal{Y}=\{E\in\mathcal{R}_{s}:E\subseteq\mathsf{Y}\}$ (a $\sigma$-algebra on $\mathsf{Y}$ by claim 1 of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions) and a $\sigma$-finite measure $\varrho_{0}$ on $(\mathsf{Y},\mathcal{Y})$; a natural number $d\ge1$; square-integrable random variables $\vartheta_{1},\dots,\vartheta_{d}$ on $(\Omega',\mathcal{F}',P')$, collected in $\vartheta=(\vartheta_{1},\dots,\vartheta_{d})$; a map $\mathsf{D}':\Omega'\to\mathsf{Y}$ measurable with respect to $\mathcal{F}'$ and $\mathcal{Y}$; a square-integrable random variable $X^{\dagger}$ on $(\Omega',\mathcal{F}',P')$; a function $p:\mathbb{R}^{d}\times\mathsf{Y}\to[0,\infty)$; a map $G:\mathsf{Y}\to\mathbb{R}$; and vectors $\alpha$, $z$, $a_{1},\dots,a_{d}$ in Euclidean space $\mathbb{R}^{d}$, such that:

(a) (Law identity.) The pair $(X^{\dagger},\mathsf{D}')$, with $\mathsf{D}'$ regarded as $\mathbf{R}_{s}$-valued, is measurable with respect to $\mathcal{F}'$ and $\mathcal{B}(\mathbb{R})\otimes\mathcal{R}_{s}$, the pair $(\mathbf{c}\cdot X'_{s},W^{(s)})$ is measurable with respect to $\mathcal{F}^{\mathrm{ag}}$ and $\mathcal{B}(\mathbb{R})\otimes\mathcal{R}_{s}$, $\mathbf{c}\cdot X'_{s}$ is square-integrable on $(\Omega^{\mathrm{ag}},\mathcal{F}^{\mathrm{ag}},P^{\mathrm{ag}})$, and the image measure of $P'$ under $(X^{\dagger},\mathsf{D}')$ equals the image measure of $P^{\mathrm{ag}}$ under $(\mathbf{c}\cdot X'_{s},W^{(s)})$ on $\mathcal{B}(\mathbb{R})\otimes\mathcal{R}_{s}$.

(C1) (Factorisation through the record.) $\sigma(\mathsf{D}')=\{\mathsf{D}'^{-1}(E):E\in\mathcal{Y}\}$, and every random variable on $(\Omega',\mathcal{F}',P')$ that is measurable with respect to $\sigma(\mathsf{D}')$ and square-integrable equals $g\circ\mathsf{D}'$ for some $g:\mathsf{Y}\to\mathbb{R}$ measurable with respect to $\mathcal{Y}$ and $\mathcal{B}(\mathbb{R})$, for which $g\circ\mathsf{D}'$ is then square-integrable.

(C2$'$) (Van Trees regularity and mixture-weight information.) The probability space $(\Omega',\mathcal{F}',P')$, the natural number $d$ (in the role of $l$), the measurable space $(\mathsf{Y},\mathcal{Y})$ carrying $\varrho_{0}$, the random variables $\vartheta_{1},\dots,\vartheta_{d}$, the map $\mathsf{D}'$ and the density $p$ 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; write $\mathcal{I}_{z}\in[0,\infty)$ for the mixture-weight information of that lemma in the direction $z$ for the shifts $a_{1},\dots,a_{d}$ (with $n=d$).

(C3) (Approximation of the estimand.) $G$ is measurable with respect to $\mathcal{Y}$ and $\mathcal{B}(\mathbb{R})$, $G(\mathsf{D}')=G\circ\mathsf{D}'$ is square-integrable, and, with $\alpha\cdot\vartheta=\sum_{j=1}^{d}\alpha_{j}\vartheta_{j}$,

(C4$'$) (Pairing, information and shift bounds.)

3. (The certificate data of the construction.) Let $s\in(0,T]$, $\mathbf{c}\in\mathbb{R}^{l}$, a profile $\lambda$ and a real number $\epsilon>0$ be given. Fix real numbers $\Lambda\ge0$, $\mathsf{M}\ge0$ and $\bar\Phi\ge0$ as furnished by claims 3 and 4 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 for the data named above, and, for $N\ge N_{\mathrm{cl}}$, let $(\Omega,\mathcal{F},P)$ be a probability space carrying the driving variables of the copy side of 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 for the $N$-th solution, the point $\mathsf{x}_{0}=\mathsf{x}^{N}_{0}$, the intermediate time $s$ with the dense sequence and reconstruction data fixed above, the profile $\lambda$, the estimand direction $\mathbf{c}$ and these $\Lambda$, $\mathsf{M}$, $\bar\Phi$ (the parameters $R_{N}$, $J_{N}$, $\mu_{N}$, $\mathsf{m}_{N}$, $\eta_{N}$ of the scale set 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 that specify these variables are defined by their formulas for every $N\ge1$, and such a space exists by Existence of Independent Sequences with Prescribed Distributions). Then in claim 2, for these $s$, $\mathbf{c}$, $\lambda$, $\epsilon$, the number $N_{2}$ can be chosen with $N_{2}\ge N_{\mathrm{cl}}$ such that for every $N\ge N_{2}$ the hypotheses of 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 hold for these choices and, in its notation and that 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 data of claim 2 can be taken to be: $(\Omega',\mathcal{F}',P')$ the trimmed copy $(\Omega^{\mathrm{tr}},\mathcal{F}^{\mathrm{tr}},\mu^{\mathrm{tr}})$; $\mathsf{Y}=\mathsf{R}_{+}$ the record support with $(\mathcal{Y},\varrho_{0})=(\mathcal{R}_{+},\rho_{+})$; $d$ the number of cells; $\vartheta_{q}=\Theta^{\mathrm{tr}}_{q}$ ($q\in\mathsf{L}$, identified with $\{1,\dots,d\}$ by the fixed bijection); $\mathsf{D}'=\mathsf{D}^{\mathrm{tr}}$; $X^{\dagger}=\mathbf{c}\cdot X^{\mathrm{tr}}$; $p=g^{\mathrm{tr}}$; $G=\varsigma|_{\mathsf{R}_{+}}$; $\alpha$ the vector of cell coefficients; $z=\mathsf{z}=\mathsf{m}w/N$ the injection direction; and $a_{q}=\mathsf{m}e_{q}/\sqrt{N}$ the moves.