Proof 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

Tools and conventions. We use freely: claims 1, 2, 4, 5 and 6 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 (algebra and monotonicity of real powers, integer rounding, square roots, and the monotonicity of $\exp$); the identity $\log(\exp x)=x$ for real $x$, by The Natural Logarithm; claims 1 and 2 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers (termwise comparison of finite sums, and the triangle inequality for finite sums); claims 3 and 5 of Properties of Finite Sums (homogeneity of finite sums, and nonnegativity of a sum of nonnegative terms); and the fact that a finite sum of $n$ terms all equal to a real number $t$ equals $nt$ (homogeneity with the sum of $n$ ones, which is $n$ by the recursion of claim 1 of Properties of Finite Sums and The Canonical Map from the Natural Numbers to a Field). Sums over the finite index sets $\mathcal{L}$ and $\mathsf{L}$ are sums over a finite index set, computed along any enumeration by A Sum over a Finite Index Set Does Not Depend on the Enumeration. Throughout, "1b" refers to 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, "2a" to 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, "P8.4b" to 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 "P8.4c" to 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. Once the hypotheses of 1b are verified (claim 1), the assumption $N\ge N_{\mathrm{c}}$ makes items (i)--(viii) of claim 1 of 1b available for the present $N$; we refer to them as (i)--(viii).

Claim 1. The hypotheses of 2a. The natural numbers $l\ge2$, $m\ge1$, $\tilde{l}\ge1$ and the nonempty subset $\mathcal{A}$ of $\mathbb{R}^{m}$ are given; $\beta$ is a transition-rate family on $l$ states with control set $\mathcal{A}$ and rate bound $B\ge0$ by claim 2 of The Transition-Rate Family and Aggregate Drift of Affine-Controlled Data; $(U,V,\bar\beta)$ is a twice continuously differentiable extension of $\beta$ with derivative bound $K\ge0$ (part of the common data of Asymptotic Lower Bound for the Recentred N-Agent Cost without Uniform Control-Moment Hypotheses); $\tilde\beta$ is an observation-rate family on $l$ states with $\tilde{l}$ channels and rate bound $\tilde{B}\ge0$, and $(\tilde{U},\bar{\tilde\beta})$ is a twice continuously differentiable extension of it with derivative bound $\tilde{K}\ge0$ by (X$'$); (OC) is assumed with the same wording; $T>0$; $(S,A)$ is a mean-field trajectory pair for $\beta$ with horizon $T$ with $S_{t}\in\Delta^{l}$ and $A_{t}\in\mathcal{A}$, as recorded in the Data 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 $1-O(N^{-1/2})$; $0<s\le T$; and $\lambda$ has continuous components on $[0,T]$. These are exactly the hypotheses of 2a, whose claims 1--6 are therefore available; in particular the objects adopted from it in the statement are defined, its $\Theta$ is $\Theta^{\mathrm{fl}}$, the numbers $\Lambda$, $\mathsf{M}$, $\bar\Phi$ exist and are fixed, and $\mathsf{P}\ge0$, $\mathsf{Q}\ge0$, $\mathsf{P}+\mathsf{Q}=\mathcal{A}_{s}(\lambda)$ by its claim 6.

The hypotheses of 1b. The scalar data must be: natural numbers $l\ge2$, $m\ge1$, $\tilde{l}\ge1$ (given); real numbers $B\ge0$, $\tilde{B}\ge0$, $K\ge0$, $\tilde{K}\ge0$ (as above), $\underline{b}>0$ ((OC)), $T>0$, $s\in(0,T]$, $\Lambda\ge0$, $\mathsf{M}\ge0$, $\bar\Phi\ge0$ (2a, claims 3 and 4), $\mathsf{c}_{0}=|\mathbf{c}|\ge0$, $\kappa^{\sharp}\ge1$ (hypothesis (I$'$) of Asymptotic Lower Bound for the Recentred N-Agent Cost without Uniform Control-Moment Hypotheses), $\mathsf{P}\ge0$ and $\mathsf{Q}\ge0$ (2a, claim 6), and the three constants $c_{Q}$, $C_{\mathrm{ctl}}$, $C_{\mathrm{flw}}$, whose nonnegativity we now verify. By claim 1 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 $1-O(N^{-1/2})$ for the present $N$, the noise majorant $Q$ of the $N$-th solution satisfies $\mathbb{E}^{\mathrm{ag}}[Q^{4}]\le c_{Q}\kappa^{\sharp}N^{-2}$ (the second inequality of that claim); as $\mathbb{E}^{\mathrm{ag}}[Q^{4}]\ge0$ (the integral of a nonnegative function) and $\kappa^{\sharp}N^{-2}>0$, $c_{Q}\ge0$. By claims 3 and 4 of Closeness of the Realized Control and the Realized Mean-Field Flow on a High-Probability Event under the Cost Bound, $C_{\mathrm{ctl}}=2(\mathcal{Z}^{\sharp}+1)$ with $\mathcal{Z}^{\sharp}\ge0$, so $C_{\mathrm{ctl}}\ge2>0$, and $C_{\mathrm{flw}}=C_{S}C_{\mathrm{ctl}}^{1/2}$, where, by the Data of that lemma, $C_{S}=e^{\Lambda_{b}T}\sqrt{l}\,K_{2}\sqrt{T}$ with $K_{2}=2\sqrt{l}\,(l-1)K_{1}$ and $K_{1}\ge0$ (claim 1 of The Transition-Rate Family and Aggregate Drift of Affine-Controlled Data); every factor being nonnegative ($e^{\Lambda_{b}T}>0$), $C_{S}\ge0$ and hence $C_{\mathrm{flw}}\ge0$. Hence the claims of 1b are available for these data. Its constants are formed by the formulas $\Lambda_{1}=\sqrt{l+m}\,(B+K)$, $\Lambda_{2}=\tfrac32(l+m)K$, $\Lambda_{3}=3K\sqrt{l(l+m)}$, $\Lambda_{\mathcal{E}}=\sqrt{2}\,l(l-1)\sqrt{l}\,(B+K)$, $\Gamma=\sqrt{l}\,(\tilde{B}+\tilde{K})$ and $C_{\mathrm{Lip}}=l(l-1)(\Lambda_{1}+\Lambda_{3})$, which are the defining formulas of the constants of the same names in hypothesis (X) 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 setting 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 and in the conventions of P8.4b, all formed from the same numbers $l$, $m$, $B$, $K$, $\tilde{B}$, $\tilde{K}$; so they coincide.

The scales. By (ii), $R_{N}\in\mathbb{N}$ and $NBs<R_{N}$. By (iii), $J_{N}\in\mathbb{N}$ and $\mu_{N}=R_{N}/J_{N}\ge2>0$. Hence $\mathsf{b}^{c}_{0}=0$, $\mathsf{b}^{c}_{J_{N}}=J_{N}\mu_{N}=R_{N}$, and $\mathsf{b}^{c}_{j-1}<\mathsf{b}^{c}_{j}$ with $\mathsf{b}^{c}_{j}-\mathsf{b}^{c}_{j-1}=\mu_{N}$ for $1\le j\le J_{N}$; so the cells $I_{c,j}=(\mathsf{b}^{c}_{j-1},\mathsf{b}^{c}_{j}]$ all have length $\mu_{q}=\mu_{N}$, whence $\mu_{\max}=\mu_{\min}=\mu_{N}$, and $d=\sum_{c\in\mathcal{L}}J_{c}$ is a sum of $l(l-1)$ terms equal to $J_{N}$, that is, $d=l(l-1)J_{N}=d_{N}$. By (i) and (v), $\mathsf{m}_{N}\in\mathbb{N}$ and $0<\eta_{N}\le1$. The setting of The Record-Driven Causal Intensity of the Open-Loop Aggregate Solution: Joint Measurability of the Record-Frozen Control, the Regularised Recursion Path, Non-Anticipation, and Measurability of the Likelihood with horizon $s$ requires: the natural numbers $N\ge1$, $l\ge2$, $m\ge1$, $\tilde{l}\ge1$; the nonempty set $\mathcal{A}\subseteq\mathbb{R}^{m}$; the real numbers $B\ge0$, $\tilde{B}\ge0$ and $s>0$; the transition-rate family $\beta$ with rate bound $B$; the lattice $\mathbb{G}_{N}$ and a point of it, here $\mathsf{x}_{0}$ ((D)); the observation-rate family $\tilde\beta$ with $\tilde{l}$ channels and rate bound $\tilde{B}$; the observation record space with horizon $s$ and $\tilde{l}$ channels, here $(\mathbf{R}_{s},\mathcal{R}_{s},\rho)$; and an observation-driven control policy with horizon $s$, control dimension $m$ and $\tilde{l}$ channels all of whose members take values in $\mathcal{A}$, here $h^{(s)}$ (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), with its record-frozen control paths $a^{(s),r}$. Its clock family is supplied, 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 which adopts it, by the deterministic-count and copy clocks constructed there, all of whose paths are counting paths (as recorded in the setting 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 setting of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record requires in addition a real number $R>0$ with $R\ge NBs$, natural numbers $J_{c}\ge1$, strictly increasing boundaries $0=\mathsf{b}^{c}_{0}<\dots<\mathsf{b}^{c}_{J_{c}}=R$, a real $\eta\in(0,1]$ and a probability space carrying the independent driving variables with the laws stated: all of these hold for $R=R_{N}$, $J_{c}=J_{N}$, the boundaries $\mathsf{b}^{c}_{j}$, $\eta=\eta_{N}$ and the space $(\Omega,\mathcal{F},P)$ of the statement. The setting of Pathwise Reduction of the Symmetrised Move Information on the Synthetic Copy: Removed-Clock Likelihoods, the Shifted Family, and the Averaging Bound requires moreover a natural number $\mathsf{m}\ge1$, here $\mathsf{m}_{N}$, and a vector of weights in $\mathbb{R}^{d}$, here the injection weights $w$ (as in P8.4c); it is therefore instantiated as soon as $w$ is defined, which happens in claim 2 once $\phi_{c}$ and $\varpi$ are shown admissible. The setting 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 with horizon $s$ requires the natural numbers $l$, $m$, $N$, $\mathsf{m}=\mathsf{m}_{N}$, the real numbers $s>0$, $B\ge0$ and $R=R_{N}>0$ with $R\ge NBs$, the cells above, measurable mean-field label rates $\phi_{c}:[0,s]\to[0,B]$ and a profile with measurable components bounded by a real $\Lambda\ge0$; the first group holds, and the admissibility of $\phi_{c}$ and $\varpi$ is shown in claim 2.

Hypothesis (W). By (iv), $0\le\Lambda_{1}sA_{N}<L_{N}$, so $L_{N}\ge0$; by (v), $D_{N}\ge4\ge0$. The constant of (W) written $A_{0}$ there and $A^{\mathrm{ins}}$ here (the constant of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect for the move size $\mathsf{m}$ and the discrepancy tolerance $D+\mathsf{m}$, with horizon $s$) is $\sqrt{2}(\mathsf{m}+l(l-1)(D+\mathsf{m}))\exp(\sqrt{2}\,l(l-1)\Lambda_{1}s)$, which for $\mathsf{m}=\mathsf{m}_{N}$ and $D=D_{N}$ is the defining formula of $A_{N}$; so $A^{\mathrm{ins}}=A_{N}$ and $\Lambda_{1}sA^{\mathrm{ins}}<L=L_{N}$, which is (W).

Claim 2. (OC) and (X). (OC) 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 is the assumed (OC), the channel set of that hypothesis being $\{1,\dots,\tilde{l}\}$. (X) requires a twice continuously differentiable extension $(\tilde{U},\bar{\tilde\beta})$ of $\tilde\beta$ with derivative bound $\tilde{K}$ ((X$'$)) and a twice continuously differentiable extension $(U,W_{\beta},\bar\beta)$ of $\beta$ with derivative bound $K$, supplied by $(U,V,\bar\beta)$ with $W_{\beta}=V$.

(AF). $(\beta_{0},\beta_{1})$ is an affine-controlled transition-rate family on $l$ states with control set $\mathcal{A}$, nonempty, convex and compact, and Lipschitz constant $\Lambda^{\mathrm{aff}}$ (common data, with the renaming of P8.4b); $\beta$ is the transition-rate family of claim 2 of The Transition-Rate Family and Aggregate Drift of Affine-Controlled Data formed from it and $B$ is the number $B$ of that lemma (the rate bound named in the common data is that of this lemma, as recorded in the $N$-agent side of P8.4b); the control bound is written $R^{\mathcal{A}}$; the policy $h^{(s)}$ is $\mathcal{A}$-valued (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 base point $z_{0}=S_{0}$ lies in $\Delta^{l}$ (the initial point of the triple); and $(\mathsf{v}_{n})_{n\in\mathbb{N}}$ is a sequence in $L^{2}([0,s];\mathbb{R}^{m})$ with dense set of terms, fixed 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. This is (AF).

The comparison pair and the label rates. By claim 2 of 2a, the components of $t\mapsto S_{t}$ and $t\mapsto A_{t}$ restricted to $[0,s]$ are continuous on $[0,s]$, hence measurable with respect to $\mathcal{B}_{[0,s]}$ (claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval), with values in $\Delta^{l}$ and $\mathcal{A}$; so $(S^{\mathrm{cp}},\mathsf{A})=(S,A)|_{[0,s]}$ is a comparison pair in the sense 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 label rates $\psi_{c}$ of the copy instance are those of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect formed from $(U,V,\bar\beta)$, which are the $\psi_{c}$ of 2a (its paragraph The copy-side objects identifies them, together with $g^{c}$ and $\mathcal{E}$, with the objects of that lemma). For $c=(\sigma,\gamma)$ and $t\in[0,s]$, $\phi_{c}(t)=\psi_{c}(S_{t},A_{t})=S^{\sigma}_{t}\bar\beta(\sigma,\gamma,S_{t},A_{t})=S^{\sigma}_{t}\beta(\sigma,\gamma,S_{t},A_{t})$, since $\bar\beta$ agrees with $\beta$ on $\Delta^{l}\times\mathcal{A}$ (clause 1 of Twice Continuously Differentiable Extension of a Transition-Rate Family); as $0\le S^{\sigma}_{t}\le1$ ($S_{t}\in\Delta^{l}$) and $0\le\beta\le B$ (clause 1 of Transition-Rate Family), $\phi_{c}(t)\in[0,B]$. Moreover $\psi_{c}$ is of class $C^{2}$, hence continuous at every point of $U\times V$ (claim 1 of 2a and clause 1 of C^k Maps on a Euclidean Open Set), and $t\mapsto(S_{t},A_{t})$ is continuous at every point of $[0,s]$ as a map into $\mathbb{R}^{l+m}$ with values in $U\times V$ (claim 2 of 2a, restricted to $[0,s]$ by claim 1 of Restriction Stability of Continuity and of the Derivative); by Composition of Continuous Euclidean Maps and claim 1 of Euclidean Continuity Agrees with Metric Continuity for Real-Valued Functions, $\phi_{c}$ is continuous on $[0,s]$, hence measurable (claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval). This is the admissibility of $\phi_{c}$ required by 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 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, in agreement with 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 and its response. By claim 4 of 2a, $\varpi$ and $\bar\psi$ have continuous, hence bounded measurable, components on $[0,s]$, $|\varpi_{u}|\le\Lambda$ and $|\bar\psi_{u}|\le\mathsf{M}$ for $u\in[0,s]$, and

where $\mathcal{E}$ is the drift Jacobian of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect and $\Theta^{\mathrm{fl}}$ the aggregate fluctuation covariance of $\beta$. This is 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 $(S,A)|_{[0,s]}$ and the profile $\varpi$ (its $\Theta$ being the aggregate fluctuation covariance of $\beta$ and its $\mathcal{E}$ the drift Jacobian, as recorded in the setting 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); so $\varpi$ with bound $\Lambda$ is an admissible profile and $\bar\psi$ with bound $\mathsf{M}$ an admissible profile response. The observation information matrix $\tilde{D}(x)$ 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 is defined by the same formula as the matrix $\tilde{D}(x)$ of 2a (as its paragraph The copy-side objects records, both being formed from $\bar{\tilde{b}}$ and $\tilde{b}$ alone), so the two agree.

(CP). By claim 3 of 2a, every entry of $u\mapsto\mathcal{E}^{\star}_{u}=\mathcal{E}(S_{u},A_{u})$ is continuous on $[0,s]$; $\Phi^{\mathcal{E}}(t,u)$ ($t,u\in[0,s]$) is by definition the two-parameter fundamental solution of Variation of Constants with Bounded Measurable Forcing and the Two-Parameter Fundamental Solution for this coefficient matrix with horizon $s$, and $|\Phi^{\mathcal{E}}(t,u)y|\le\bar\Phi^{2}|y|$ for all $t,u\in[0,s]$ and $y\in\mathbb{R}^{l}$ by that claim. The tolerances $\varepsilon_{S}=\varepsilon_{S}(N)=(C_{\mathrm{flw}}+1)N^{-1/4}$ and $\varepsilon_{\mathrm{ctl}}=C_{\mathrm{Lip}}\varepsilon_{\mathrm{ctl}}(N)=C_{\mathrm{Lip}}(TC_{\mathrm{ctl}})^{1/2}N^{-1/4}$ are nonnegative real numbers, since $C_{\mathrm{flw}}\ge0$, $C_{\mathrm{ctl}}\ge0$ (claim 1), $C_{\mathrm{Lip}}=l(l-1)(\Lambda_{1}+\Lambda_{3})\ge0$ (as $\Lambda_{1}=\sqrt{l+m}\,(B+K)\ge0$ and $\Lambda_{3}=3K\sqrt{l(l+m)}\ge0$, $B$ and $K$ being nonnegative), $T>0$ and $N^{-1/4}>0$; this is the first clause of (CL) ($\varepsilon_{S}\ge0$ and $\varepsilon_{\mathrm{ctl}}\ge0$ are real numbers), which is all that the final display of (CP) consumes, as recorded in the scope statement 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. With the estimand direction $\mathbf{c}$ and these tolerances, this is (CP) in the horizon-$s$ form adopted by P8.4b, with $\mathsf{w}^{\mathrm{clk}}=N(\Lambda_{1}s\varepsilon_{S}+\varepsilon_{\mathrm{ctl}})$.

Energy, observation integral and weights. 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 (whose setting requires only that 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 with (OC), (X), (W) and the comparison data, all in place by claim 1 and the present claim), the profile energy is $\mathcal{P}=\int_{[0,s]}\varpi_{u}\cdot(\Theta^{\mathrm{fl}}(S_{u},A_{u})\varpi_{u})\,du=\mathsf{P}$. The integral appearing in $\mathsf{Q}^{\mathrm{cl}}$ is $\int_{[0,s]}\bar\psi_{u}\cdot(\tilde{D}(S_{u})\bar\psi_{u})\,du$, which is $\mathsf{Q}$ by definition, the two matrices $\tilde{D}(S_{u})$ agreeing. Finally, the setting 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 being instantiated (claim 1 and the admissibility just shown), its claim 2 gives $\lVert w\rVert_{1}=\sum_{q\in\mathsf{L}}|w_{q}|\le\sqrt{2}\,\Lambda\,l(l-1)\,R_{N}/\mathsf{m}_{N}=\mathsf{w}_{N}$.

Claim 3. The setting of Probability of the Good Event on the Synthetic Copy: Poisson Tail for the Cell Counts, the Window Discrepancy Bound for the Clock-Good Event, and the Bound on the Complement of the Good Event is that 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 with (OC), (X), (W), instantiated by claims 1 and 2; its $G^{\mathsf{m}}$ 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, the set of $\omega\in G_{L,D}$ with $\mathsf{K}_{q}(\omega)\ge\mathsf{m}$ for all $q$. The hypotheses of its claim 2 hold: $R=R_{N}$ is a natural number (ii); $M=M_{N}$ is a natural number with $L_{N}<M_{N}$, so $L\le M$ (iv); and $D=D_{N}\ge4>2$ (v). Moreover $\mathsf{m}=\mathsf{m}_{N}<\mu_{N}/2<\mu_{N}=\mu_{q}$ for every $q$ (vi). Hence its claim 3 gives $G^{\mathsf{m}}\in\mathcal{F}$ and

the sum having $d_{N}$ equal terms, the arguments $\mu_{N}-\mathsf{m}_{N}$ and $D_{N}-2$ being positive (so that the exponent of Series Formula, Exponential Moments, and Chernoff Tail Bounds for the Poisson Distribution used there agrees with the $\varpi_{k}(x)$ of 1b, and the formula for $\mathsf{g}_{N}$ is read as displayed, its two arguments being nonnegative). The setting 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 is that of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record with the move size $\mathsf{m}$, and its tracked records are the $\mathsf{T}_{\omega}$ 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; by its claim 3 the event $\Omega'$ belongs to $\mathcal{U}\subseteq\mathcal{F}$, and by its claim 4 applied to $G_{0}=G^{\mathsf{m}}$, the event $G=G^{\mathsf{m}}\cap\Omega'$ belongs to $\mathcal{F}$, satisfies $\rho(\mathbf{R}_{s}\setminus\mathsf{T}_{\omega})=0$ for every $\omega\in G$, and $P(\Omega\setminus G)=P(\Omega\setminus G^{\mathsf{m}})$. Since $G\subseteq G^{\mathsf{m}}\subseteq G_{L,D}$, hypothesis (G) 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 ($G\in\mathcal{F}$, $G\subseteq G_{L,D}$, $\rho(\mathbf{R}_{s}\setminus\mathsf{T}_{\omega})=0$ on $G$) and hypothesis (G$'$) 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 ($G\subseteq G^{\mathsf{m}}$) hold, and $\mathsf{g}=P(\Omega\setminus G)=P(\Omega\setminus G^{\mathsf{m}})\le\mathsf{g}_{N}$.

Claim 4. (P). 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 is that 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 with (OC), (X), (W) (claims 1, 2) together with (G) (claim 3). Its constants are $\varepsilon_{0}=\Gamma A^{\mathrm{ins}}/(N\underline{b})=\Gamma A_{N}/(N\underline{b})=\varepsilon_{0,N}$ (claim 1), $\bar{E}_{N}=\tilde{l}\,s\,\Gamma^{2}(A^{\mathrm{ins}})^{2}/(N\underline{b})$ (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 with horizon $s$), which is the $\bar{E}_{N}$ of the scale set, and $\mathsf{c}_{N}=\exp(\bar{E}_{N})-1$, likewise. Hypothesis (P) requires: a real $\delta$ with $0<\delta\le1$, satisfied by $\delta_{N}$ (v); $\varepsilon_{0}\le\tfrac12$, which is $\varepsilon_{0,N}\le\tfrac12$ (vii); an integer $\theta\ge1$ with $2\theta\varepsilon_{0}\le1$, satisfied by $\theta_{N}\in\mathbb{N}$ (v) with $2\theta_{N}\varepsilon_{0,N}\le1$ (vii); and reals $x_{q}>0$ with $\mathsf{m}(x_{q}+\mathsf{m})\le\delta\mu_{q}/2$, satisfied by $x_{q}=x_{N}=\mu_{N}^{1/2}N^{1/32}>0$ (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) with $\mathsf{m}_{N}(x_{N}+\mathsf{m}_{N})\le\delta_{N}\mu_{N}/2$ (viii), $\mu_{q}=\mu_{N}$. Its constants then read $\mathsf{E}^{\mathrm{ch}}_{\theta_{\mathrm{P}}}=8\tilde{l}\,s\,N\tilde{B}\theta_{N}^{2}\varepsilon_{0,N}^{2}=\mathsf{E}^{\mathrm{ch}}_{N}$ and $\bar\Pi=\sum_{q\in\mathsf{L}}(\exp(-\varpi_{\mu_{N}}(x_{N}))+\exp(-\theta_{N}\delta_{N}/2+\mathsf{E}^{\mathrm{ch}}_{N}))=d_{N}(\dots)=\bar\Pi_{N}$, a sum of $d_{N}$ equal terms. The constants 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 read $\mathsf{E}^{\star}_{N}=\tilde{l}\,s\,N\tilde{B}\varepsilon_{0,N}^{2}$, $\mathsf{e}^{\star}_{N}=\tfrac12(\mathsf{E}^{\star}_{N})^{2}\exp(\mathsf{E}^{\star}_{N})+\mathsf{E}^{\star}_{N}(\exp(9\mathsf{E}^{\star}_{N})-1)^{1/2}$ and $\kappa^{\mathrm{mf}}_{0}=1+\frac{\mathsf{m}_{N}^{2}}{2\mu_{N}}\exp(\mathsf{m}_{N}^{2}/\mu_{N})$ (as $\mu_{\min}=\mu_{N}$), which are the defining formulas of $\mathsf{E}^{\star}_{N}$, $\mathsf{e}^{\star}_{N}$ and $\kappa_{0,N}$ in 1b; and $\kappa_{\mathrm{mv}}=\mathsf{m}_{N}^{2}/(N\eta_{N})=\kappa^{\mathrm{mv}}_{N}$ by the definitions in P8.4c and 1b.

(L). The numbers $N$, $l$, $m$, $\tilde{l}$, the control set $\mathcal{A}$, the affine family $(\beta_{0},\beta_{1})$ with $\beta$ and $B$, and $\tilde\beta$ with $\tilde{B}$ of the copy instance are those of the $N$-agent side by construction; the policy of the copy instance is $h^{(s)}$, so its record-frozen control paths are the $a^{(s),r}$ (The Record-Frozen Control Path and Record-Frozen Policy), and the sequence of (AF) is $(\mathsf{v}_{n})$, the one fixed for the horizon $s$; the comparison pair is $(S,A)|_{[0,s]}$; the base point is $z_{0}=S_{0}$; $\mathsf{x}_{0}\in\mathbb{G}_{N}$ satisfies $P^{\mathrm{ag}}(\Sigma_{0}=\mathsf{x}_{0})=1$ ((D)); and $R=R_{N}$ is a natural number with $R_{N}>NBs$ (claim 1). This is (L).

Availability of the chain. The setting of P8.4b consists of its $N$-agent side and intermediate time (adopted in the statement), of its Conventions paragraph (which introduces, without hypotheses, $C_{\mathrm{Lip}}$, the objects $\mathsf{s}(p)$, $\mathsf{d}(r)$, $\mathsf{E}(\varepsilon,\varepsilon')$ of The Realized Control as the Record-Frozen Control at the Observation Record, and Measurability of the Path-and-Record Closeness Set, the path space $\mathsf{Path}$ and the event $\Omega'$ 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, all defined once the copy side is), of the setting 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 with horizon $s$ and hypotheses (OC), (X), (W), (G), (G$'$), (AF), (CP), with (CL), (FM), (DM) not assumed and the two real numbers of the first clause of (CL) equal to $\varepsilon_{S}(N)$ and $C_{\mathrm{Lip}}\varepsilon_{\mathrm{ctl}}(N)$ (claims 1--3 and the choices made), and of (L); hence P8.4b's claims 1--5 are available for the copy instance, in particular its claim 3 supplies (CL) with the close records $\mathsf{R}^{\mathrm{cl}}_{\omega}$ named in the statement, and its claim 5, for each admissible choice of $y_{1}$, $y_{2}$, $(\Xi^{c})_{c}$, the constants $\mathsf{e}_{1},\dots,\hat{\mathsf{e}}_{5}$. The setting of P8.4c consists of that of P8.4b, of objects defined within the adopted settings (the parameter lattice, the count mass function, the record kernel, the smoothed density, the moves, $\mathsf{J}^{\mathrm{sym}}$, the profile data 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 constants 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 and 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) and of hypothesis (P), verified above; hence P8.4c's claims 1--4 are available. Its claim 3 asserts that 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 these data, so that theorem's claims 1--4 are available as well.

Consequences. Claim 4 of P8.4b asserts that (FM) holds with $\mathsf{c}_{4}=c_{Q}\kappa^{\sharp}$, which is $\mathsf{c}_{\star}$ by definition. Moreover $\varepsilon_{S}(N)=(C_{\mathrm{flw}}+1)N^{-1/4}=\varepsilon_{S,N}$ and $C_{\mathrm{Lip}}\varepsilon_{\mathrm{ctl}}(N)=C_{\mathrm{Lip}}(TC_{\mathrm{ctl}})^{1/2}N^{-1/4}=\varepsilon_{\mathrm{ctl},N}$ by the defining formulas (the two symbols $N^{-1/4}$ agreeing: $N^{-1/4}=1/N^{1/4}$ with $N^{1/4}=\sqrt{\sqrt{N}}$ in both P8.4b and 1b, 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).

Claim 5. The data $w_{i}$, $y_{i}$, $\Xi^{c}$. $w_{1}=\mathsf{w}^{\mathrm{clk}}=N(\Lambda_{1}s\varepsilon_{S}(N)+C_{\mathrm{Lip}}\varepsilon_{\mathrm{ctl}}(N))=N(\Lambda_{1}s\varepsilon_{S,N}+\varepsilon_{\mathrm{ctl},N})=w_{1,N}$ by claim 4, and $w_{1}\ge0$ because both tolerances are nonnegative (claim 2) and $\Lambda_{1}s\ge0$. Also $w_{2}=\mu_{\max}=\mu_{N}=w_{2,N}>0$. For $i\in\{1,2\}$, $\lceil w_{i}\rceil\ge1$ (the least natural number $\ge w_{i}$), so $y_{i}=(\lceil w_{i}\rceil+2)^{1/2}N^{1/32}>0$. The setting of 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 is that of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record (claim 1) with $R=R_{N}$ a natural number and the reals $w_{1},w_{2}\ge0$, $y_{1},y_{2}>0$; that claim furnishes a family $(\Xi^{c})_{c\in\mathcal{L}}$ of the kind required by claim 5 of P8.4b for the data $w_{1}=\mathsf{w}^{\mathrm{clk}}$, $w_{2}=\mu_{\max}$, $x_{1}=y_{1}$, $x_{2}=y_{2}$, and we fix one. Claim 5 of P8.4b then gives the constants $\mathsf{e}_{1},\dots,\mathsf{e}_{4}$, $\hat{\mathsf{e}}_{5}$ and the bound on $\lVert\Xi^{c}\rVert_{2}$ recorded there.

$\mathsf{e}_{1}$ and $\mathsf{e}_{2}$. By claim 5 of P8.4b, $\mathsf{e}_{1}=|\mathbf{c}|\bar\Phi^{2}\sqrt{N}|\mathsf{x}_{0}-S_{0}|$ (its $x_{0}$ being $\mathsf{x}_{0}$ and $z_{0}=S_{0}$), and

With $\mathsf{c}_{4}=\mathsf{c}_{\star}$, $|\mathbf{c}|=\mathsf{c}_{0}$, $C_{\mathrm{Lip}}\varepsilon_{\mathrm{ctl}}(N)=\varepsilon_{\mathrm{ctl},N}$, $\varepsilon_{S}(N)=\varepsilon_{S,N}$ (claim 4) and $1/\sqrt{N}=N^{-1/2}$, this is the defining formula of $\mathsf{e}_{2,N}$; so $\mathsf{e}_{2}=\mathsf{e}_{2,N}$, in particular $\mathsf{e}_{2}\le\mathsf{e}_{2,N}$.

$\Xi^{c}$, $H^{c}$ and $\mathsf{e}_{3}$. By claim 4 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, $n_{i}=\lfloor w_{i}\rfloor+1$ is a natural number with $w_{i}<n_{i}\le w_{i}+1$; the least natural number $\ge w_{i}$ is therefore at most $n_{i}$, so $\lceil w_{i}\rceil\le w_{i}+1=w_{i,N}+1$. Hence $y_{i}\le(w_{i,N}+3)^{1/2}N^{1/32}$ (monotonicity of the square root and $N^{1/32}>0$) and $\lceil w_{i}\rceil+3\le w_{i,N}+4$. By claim 2 of 1b applied with the real number $n=\lceil w_{i}\rceil\ge0$, $\varpi_{\lceil w_{i}\rceil+2}(y_{i})\ge N^{1/32}/4$, so $\exp(-\varpi_{\lceil w_{i}\rceil+2}(y_{i}))\le\exp(-N^{1/32}/4)$ ($\exp$ nondecreasing). Consequently, using that $t\mapsto t^{1/4}=\sqrt{\sqrt{t}}$ is nondecreasing on $[0,\infty)$ and multiplicative (claim 5 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, twice), and that $(\exp(-N^{1/32}/4))^{1/4}=\exp(-N^{1/32}/16)$ (for $t=\exp(x)>0$, claim 1 of that lemma gives $\sqrt{\sqrt{t}}=t^{1/4}=\exp(\tfrac14\log t)=\exp(x/4)$, as $\log(\exp x)=x$),

Inserting this and $y_{i}\le(w_{i,N}+3)^{1/2}N^{1/32}$ into the bound of claim 5 of P8.4b, with $\mathsf{M}_{4}(R_{N})=(8(4096+17R_{N}^{4}))^{1/4}\ge0$, gives $\lVert\Xi^{c}\rVert_{2}\le\Xi_{N}$ for every $c\in\mathcal{L}$. Next, by claim 4 of P8.4c, $|H^{c}_{u}|\le\sqrt{2}\,\bar\Phi^{2}\Lambda_{\mathcal{E}}$ for every $u\in[0,s]$; the map $u\mapsto|H^{c}_{u}|$ is nonnegative and measurable on $[0,s]$ (as established in Pathwise Estimand Linearisation of an Open-Loop Aggregate Solution Along a Comparison Pair: Cell-Count Coefficients from the Fundamental Solution and the Three Error Terms, from which (CP) adopts $H^{c}$ and $\lVert H^{c}\rVert_{1}=\int_{[0,s]}|H^{c}_{u}|\,du$, the instantiation of that lemma being recorded in 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), so by monotonicity of the integral, The Integral of an Indicator Function is the Measure of the Set and claim 1 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval, $\lVert H^{c}\rVert_{1}=\int_{[0,s]}|H^{c}_{u}|\,du\le\sqrt{2}\,\bar\Phi^{2}\Lambda_{\mathcal{E}}\,\lambda_{[0,s]}([0,s])=\sqrt{2}\,\bar\Phi^{2}\Lambda_{\mathcal{E}}s=H$. Therefore each of the $l(l-1)$ terms of $\mathsf{e}_{3}=\frac{|\mathbf{c}|}{\sqrt{N}}\sum_{c\in\mathcal{L}}(\sqrt{2}+\lVert H^{c}\rVert_{1})\lVert\Xi^{c}\rVert_{2}$ is at most $(\sqrt{2}+H)\Xi_{N}$ (both factors nonnegative), and termwise comparison gives $\mathsf{e}_{3}\le\frac{\mathsf{c}_{0}}{\sqrt{N}}\,l(l-1)(\sqrt{2}+H)\Xi_{N}=\mathsf{e}_{3,N}$.

$\alpha$, $\mathsf{e}_{4}$, $\mathsf{k}_{4}$ and $\hat{\mathsf{e}}_{5}$. By claim 4 of P8.4c, $\lVert\alpha\rVert\le\sqrt{2d}\,|\mathbf{c}|\bar\Phi^{2}=\sqrt{2d_{N}}\,\mathsf{c}_{0}\bar\Phi^{2}=\alpha_{N}$ and $\max_{q}|\alpha\cdot a_{q}|\le\sqrt{2}\,|\mathbf{c}|\bar\Phi^{2}\mathsf{m}_{N}/\sqrt{N}=\mathsf{a}_{N}$. Hence $\mathsf{e}_{4}=\sqrt{\eta_{N}}\,\lVert\alpha\rVert\le\eta_{N}^{1/2}\alpha_{N}=\mathsf{e}_{4,N}$ (the quantity written $|\alpha|$ in $\mathsf{e}_{4}$ being $\lVert\alpha\rVert$, as recorded in the conventions of P8.4c). By claim 5 of P8.4b, $\mathsf{k}_{4}^{4}\le3(\sum_{q}\alpha_{q}^{2}\mu_{q})^{2}+\sum_{q}\alpha_{q}^{4}\mu_{q}$; with $\mu_{q}=\mu_{N}$ and homogeneity, $\sum_{q}\alpha_{q}^{2}\mu_{q}=\mu_{N}\lVert\alpha\rVert^{2}$ and $\sum_{q}\alpha_{q}^{4}\mu_{q}=\mu_{N}\sum_{q}\alpha_{q}^{4}\le\mu_{N}\lVert\alpha\rVert^{4}$, the last because $\sum_{q}\alpha_{q}^{4}\le(\sum_{q}\alpha_{q}^{2})^{2}$ (expanding the square of the sum by homogeneity and additivity, the cross terms $\alpha_{q}^{2}\alpha_{q'}^{2}$, $q\ne q'$, being nonnegative; here $\lVert\alpha\rVert^{2}=\sum_{q}\alpha_{q}^{2}$ by claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$). Thus $\mathsf{k}_{4}^{4}\le(3\mu_{N}^{2}+\mu_{N})\lVert\alpha\rVert^{4}\le(3\mu_{N}^{2}+\mu_{N})\alpha_{N}^{4}$, and taking fourth roots (monotone and multiplicative, as above) $\mathsf{k}_{4}\le(3\mu_{N}^{2}+\mu_{N})^{1/4}\alpha_{N}=\mathsf{k}_{N}$. Finally, by claim 5 of P8.4b,

with $\mathsf{g}\le\mathsf{g}_{N}$ (claim 3), $c_{Q}\kappa^{\sharp}=\mathsf{c}_{\star}=\mathsf{c}_{4}$, $\mathsf{k}_{4}\le\mathsf{k}_{N}$ and the monotonicity of fourth roots, both factors are at most the corresponding factors of $\mathsf{e}_{5,N}$, all quantities being nonnegative; so $\hat{\mathsf{e}}_{5}\le\mathsf{e}_{5,N}$.

$\mathsf{e}_{\mathrm{inj}}$. By definition (P8.4c), $\mathsf{e}_{\mathrm{inj}}=2\,l(l-1)\,|\mathbf{c}|\,\Lambda\,\bar\Phi^{2}(\Lambda_{\mathcal{E}}s+1)\mu_{\max}/N$, which with $\mu_{\max}=\mu_{N}$ and $|\mathbf{c}|=\mathsf{c}_{0}$ is the defining formula of $\mathsf{e}_{\mathrm{inj},N}$.

The information constants. By claim 1 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 (available by claim 4), $\bar{\mathsf{j}}\le\kappa^{\mathrm{mf}}_{0}\mathsf{m}^{2}/\mu_{\min}=\kappa_{0,N}\mathsf{m}_{N}^{2}/\mu_{N}=\bar{\mathsf{j}}_{N}$; also $\bar{\mathsf{j}}=\max_{q}\mathsf{j}_{q}\ge0$, each $\mathsf{j}_{q}$ being a sum of nonnegative terms (claim 5 of Properties of Finite Sums; the binomial coefficients, factorials and powers $\mu_{q}^{-i}$ are positive). The map $x\mapsto((1+x)^{1/2}+1)x^{1/2}$ is nondecreasing on $[0,\infty)$ (a product of two nonnegative nondecreasing functions, by monotonicity of the square root), so $\mathsf{j}^{\star}\le\mathsf{j}^{\star}_{N}$; and $\mathsf{j}_{q}\le\bar{\mathsf{j}}\le\bar{\mathsf{j}}_{N}$ for every $q$. In the bad part $\mathsf{B}=\bar\Pi+\sum_{q}(1+\mathsf{j}_{q})^{1/2}((d\bar\Pi)^{1/2}+(\mathsf{c}_{N}\bar\Pi)^{1/2})+\mathsf{g}+\mathsf{g}^{1/2}\sum_{q}(1+\mathsf{j}_{q})^{1/2}$ 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, all terms are nonnegative ($\bar\Pi\ge0$ and $\mathsf{c}_{N}=\exp(\bar{E}_{N})-1\ge0$ as $\bar{E}_{N}\ge0$ and $\exp$ is nondecreasing with $\exp(0)=1$); replacing $\bar\Pi$ by $\bar\Pi_{N}$ (equal), each $(1+\mathsf{j}_{q})^{1/2}$ by $(1+\bar{\mathsf{j}}_{N})^{1/2}$ (larger, by monotonicity of the square root; the two sums over $\mathsf{L}$ then have $d_{N}$ equal terms), $d$ by $d_{N}$ and $\mathsf{g}$ by $\mathsf{g}_{N}$ (larger) gives, by termwise comparison, $\mathsf{B}\le\mathsf{B}_{N}$. Next, the constants of claims 2--4 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 copy instance read, with $\mu_{\max}=\mu_{\min}=\mu_{N}$, $A^{\mathrm{ins}}=A_{N}$ (the constant written $A_{0}$ in that lemma), $D=D_{N}$, $\varepsilon_{S}=\varepsilon_{S}(N)=\varepsilon_{S,N}$ and $\varepsilon_{\mathrm{ctl}}=\varepsilon_{\mathrm{ctl},N}$ (claim 4):