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 $N$-agent side. 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 following renamings: the $N$-agent driving system carrying the $N$-th solution is written $(\Omega^{\mathrm{ag}},\mathcal{F}^{\mathrm{ag}},P^{\mathrm{ag}})$ with expectation $\mathbb{E}^{\mathrm{ag}}$ (extended to $[0,\infty]$-valued measurable maps as their integrals), its regular event $\Omega^{\mathrm{ag}}_{0}$, the Lipschitz constant of the affine-controlled transition-rate family $(\beta_{0},\beta_{1})$ is written $\Lambda^{\mathrm{aff}}$, the control bound $\sup_{a\in\mathcal{A}}|a|$ is written $R^{\mathcal{A}}$, the initial point of the stationary mean-field triple $(S,A,P)$ (written $x_{0}$ there) is written $S_{0}$, and, with the numbers $N_{1}$ and $C_{\mathrm{esc}}$ of that setting, a natural number $N_{\mathrm{cl}}$ with $N_{\mathrm{cl}}\ge N_{1}$ and $N_{\mathrm{cl}}\ge C_{\mathrm{esc}}^{2}$ is fixed (such a number exists), in the role of the number written $N_{2}$ in claim 2 there. Thus we have: natural numbers $l\ge2$, $m\ge1$, $\tilde{l}\ge1$; the 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 $B$ of that lemma and its state-Lipschitz constant $\Lambda_{b}$; the observation-rate family $\tilde{\beta}$ with $\tilde{l}$ channels and rate bound $\tilde{B}$; the horizon $T>0$; the triple $(S,A,P)$, whose first two components have continuous components on $[0,T]$ and whose third component, the co-state, always carries a time subscript; the hypotheses (I$'$) with the bound $\kappa^{\sharp}$ and (CB); the constants $C_{\mathrm{flw}}$, $C_{\mathrm{ctl}}$ and $c_{Q}$; and, for the fixed natural number $N\ge N_{\mathrm{cl}}$, the $N$-th solution for the $\mathcal{A}$-valued observation-driven control policy $h=h^{(N)}$ with horizon $T$, its empirical state measure $\Sigma$, its observation record $W$, the realized control $\hat{\alpha}$ (formed as in claim 2 of The Realized Control of the Controlled N-Agent Dynamics as a Random Element of the Control Set from the fixed family furnished by its claim 1, the dense sequence in $L^{2}([0,T];\mathbb{R}^{m})$ of that setting being fixed as there), the realized mean-field flow $\Phi_{t}(\omega)=\mathsf{S}_{t}(S_{0},\hat{\alpha}(\omega))$ formed with the two-argument mean-field flow $\mathsf{S}$ (written $S(\cdot,\cdot)$ there) of claim 2 of Stability of the Mean-Field Flow under Perturbation of the Initial State and Weak Convergence of Controls, the noise majorant $Q$, the tolerances $\varepsilon_{S}(N)=(C_{\mathrm{flw}}+1)N^{-1/4}$ and $\varepsilon_{\mathrm{ctl}}(N)=(TC_{\mathrm{ctl}})^{1/2}N^{-1/4}$, and the closeness event $\Omega^{\mathrm{cl}}_{N}$ of claim 2 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 $1-O(N^{-1/2})$ formed with $N_{\mathrm{cl}}$ in the role of its $N_{2}$. The values of the mean-field control are written $A_{t}$ ($t\in[0,T]$).

The intermediate time. Fix a real number $s$ with $0<s\le T$, and adopt the setting and notation 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 the $N$-th solution, with the same realized control $\hat{\alpha}$ and with the base point $z_{0}=S_{0}$, so that its realized mean-field flow and its observation-centred fluctuation are $\Phi$ and $X'_{t}=\sqrt{N}(\Sigma_{t}-\Phi_{t})$: the record spaces $(\mathbf{R}_{s},\mathcal{R}_{s})$ and $(\mathbf{R},\mathcal{R})$, the prefix $W^{(s)}$, the truncated policy $h^{(s)}$, the restricted solution, the fixed dense sequence in $L^{2}([0,s];\mathbb{R}^{m})$, the fixed reconstruction data for $h^{(s)}$ with horizon $s$, the realized control $\hat{\alpha}^{(s)}$ of the restricted solution, the record-frozen control paths $a^{(s),r}$ ($r\in\mathbf{R}_{s}$) of $h^{(s)}$, the record-frozen flow $\Phi^{(s),r}_{t}$ ($t\in[0,s]$), the maps $\Psi_{t}$ and the recentred copy endpoint $X''_{s}$ of its claims 4 and 5.

The copy side. Adopt the setting, hypotheses (OC), (X), (W), (G), (G$'$), (AF), (CP) and notation 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 hence 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, 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, 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, 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 its horizon (written $T$ there) equal to the intermediate time $s$, and with its hypotheses (CL), (FM) and (DM) not assumed: the two real numbers $\varepsilon_{S}\ge0$ and $\varepsilon_{\mathrm{ctl}}\ge0$ of the first clause of (CL), which the final display of (CP) consumes, are those fixed in claim 3, whose formulas involve only data introduced above, so that the final display of (CP) is well formed and (CP) holds in full, with the clock tolerance $\mathsf{w}^{\mathrm{clk}}=N(\Lambda_{1}s\,\varepsilon_{S}+\varepsilon_{\mathrm{ctl}})$ formed by its formula; the remaining objects introduced through (CL), (FM) and (DM) there, namely the close records $\mathsf{R}^{\mathrm{cl}}_{\omega}$, the non-close masses $\pi^{\mathrm{nc}}_{\omega}$ and $\bar\pi^{\mathrm{nc}}$, the fourth-moment bound $\mathsf{c}_{4}$ and the majorants $\Xi^{c}$, are introduced in the claims below; the constants $\mathsf{e}_{1},\dots,\mathsf{e}_{5}$ of that lemma are formed from them; and its claims 1 and 2 are available throughout by its scope statement. Thus we have in particular: the probability space $(\Omega,\mathcal{F},P)$ with expectation $\mathbb{E}$ carrying the driving variables; the real numbers $K\ge0$ (the derivative bound of (X), with the extension $(U,W_{\beta},\bar\beta)$), $\tilde{K}\ge0$, $\underline{b}>0$, the real number $L\ge0$ and the discrepancy tolerance $D\ge0$ of (W) with the clock-good event $G_{L,D}$, the events $\Omega^{U}_{0}$ and $G^{\mathsf{m}}$; the set $\mathcal{L}$ of transition labels $c=(\sigma,\gamma)$, which has $l(l-1)$ elements, the label rates $\psi_{c}$, state gradients $g^{c}$ and drift Jacobian $\mathcal{E}$ of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect (in the instance of its setting fixed by the adopted setting), and the constants $\Lambda_{1}=\sqrt{l+m}\,(B+K)$, $\Lambda_{2}=\tfrac32(l+m)K$ and $\Lambda_{3}=3K\sqrt{l(l+m)}$; the probability simplex $\Delta^{l}$ and the aggregate lattice $\mathbb{G}_{N}$ with the point of $\mathbb{G}_{N}$ written $x_{0}$ there and $\mathsf{x}_{0}$ here; the record space $(\mathbf{R}_{s},\mathcal{R}_{s},\rho)$ with horizon $s$ (its reference 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, is written $\rho$ here, as in Law Identity between the N-Agent Aggregate Path with Its Observation Record and the Synthetic Copy's Regularised Path with Its Record and on the copy side); the clock horizon $R$, the index set $\mathsf{L}$ of the cells with $d$ elements, the cells with lengths $\mu_{q}$ ($q\in\mathsf{L}$), $\mu_{\max}$ and the vector $\mu$ of cell lengths, the cell-count vector $\mathsf{K}$, the move size $\mathsf{m}$, the copy clocks $\mathsf{P}^{\sharp}$, the regularised paths $\bar\Sigma^{\sharp,r}(\omega)$, the tracked records $\mathsf{T}_{\omega}$, the likelihoods $\ell^{\sharp,\omega}$, the smoothing parameter $\eta$, the synthetic copy $(\Omega^{\sharp},\mathcal{F}^{\sharp},\mu^{\sharp})$ with $\Omega^{\sharp}=\Omega\times\mathbb{R}^{d}\times\mathbf{R}_{s}$, its parameter $\Theta$ and record $\mathsf{D}$; the event $G$ of (G) with $\mathsf{g}=P(\Omega\setminus G)$; the comparison pair $(S^{\mathrm{cp}},\mathsf{A})$ (written $(S,\mathsf{A})$ there) with $\phi_{c}(t)=\psi_{c}(S^{\mathrm{cp}}_{t},\mathsf{A}_{t})$, the control discrepancy $\mathsf{D}^{r}_{\mathrm{ctl}}$ of a record, the fundamental solution $\Phi^{\mathcal{E}}$ with bound $\bar\Phi$, the estimand direction $\mathbf{c}\in\mathbb{R}^{l}$, the maps $H^{c}$ with $\lVert H^{c}\rVert_{1}$, the cell coefficients $\alpha=(\alpha_{q})_{q\in\mathsf{L}}\in\mathbb{R}^{d}$, the window discrepancies $\mathrm{Disc}_{w}$, the base point $z_{0}$ and the record-frozen flow $\Phi^{r}_{t}$ of (AF), the deviation $\bar{\mathsf{e}}(\omega,r)$, the recentred endpoint $X''(\omega,r)$, the quantity $\mathsf{k}_{4}$ and the constant map $\varsigma$.

(L) (Linking of the two sides.) Assume: the numbers $N$, $l$, $m$, $\tilde{l}$, the control set $\mathcal{A}$, the affine family $(\beta_{0},\beta_{1})$ with its transition-rate family $\beta$ and rate bound $B$, and the observation-rate family $\tilde{\beta}$ with rate bound $\tilde{B}$ of the copy side are those of the $N$-agent side; the policy of the copy side is $h^{(s)}$ (so that its record-frozen control paths are the $a^{(s),r}$), and the dense sequence of (AF) is the one fixed for the horizon $s$; the comparison pair is the restriction of the mean-field pair to $[0,s]$, that is, $S^{\mathrm{cp}}_{t}=S_{t}$ and $\mathsf{A}_{t}=A_{t}$ for every $t\in[0,s]$; the base point of (AF) is $z_{0}=S_{0}$; the point $\mathsf{x}_{0}\in\mathbb{G}_{N}$ satisfies $P^{\mathrm{ag}}(\Sigma_{0}=\mathsf{x}_{0})=1$; and the clock horizon $R$ is a natural number with $R>NBs$. (Hypotheses (AF) and (CP) are assumed as stated; under (L) several of their clauses are consequences of the $N$-agent data, but no use is made of this.)

Conventions. Adopt from The Realized Control as the Record-Frozen Control at the Observation Record, and Measurability of the Path-and-Record Closeness Set, in the instance with horizon $s$ described in claim 1, the path deviation $\mathsf{s}(p)$, the control discrepancy $\mathsf{d}(r)$ and the closeness sets $\mathsf{E}(\varepsilon,\varepsilon')$; and from 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 (whose setting 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, adopted above, with the move size $\mathsf{m}$) the event $\Omega'$ of its claim 3. Write $\mathsf{Path}=\mathsf{Path}(\mathbb{G}_{N},s)$ for the space of piecewise constant paths in $\mathbb{G}_{N}$ with horizon $s$ with its $\sigma$-algebra $\mathcal{C}$ generated by the sets $\{p:p(t)=y\}$ ($t\in[0,s]$, $y\in\mathbb{G}_{N}$); $\bar\Sigma^{\sharp,r}(\omega)$ denotes also the element $t\mapsto\bar\Sigma^{\sharp,r}_{t}(\omega)$ of $\mathsf{Path}$. Write $|\cdot|$ for the Euclidean norm, $\int_{[0,s]}\cdot\,dt$ for the Lebesgue integral over the compact interval $[0,s]$, $\lVert\cdot\rVert_{2}$ for the mean-square norm (on the copy for random variables on $(\Omega^{\sharp},\mathcal{F}^{\sharp},\mu^{\sharp})$, on $(\Omega,\mathcal{F},P)$ for the $\Xi^{c}$), $t^{1/2}=\sqrt{t}$ for the nonnegative square root and $t^{1/4}=(t^{1/2})^{1/2}$, $N^{-1/2}=1/\sqrt{N}$, $N^{-1}=1/N$, $\mathbf{1}_{A}$ for the indicator of a set $A$, and $\otimes$ for the product $\sigma$-algebra. Put

Notational cautions: $P$ is the probability measure of the copy side, $P^{\mathrm{ag}}$ that of the $N$-agent side, and the co-state of the triple is written $P_{t}$; $R$ is the clock horizon and $R^{\mathcal{A}}$ the control bound; $K$ is the derivative bound of (X); $\Lambda$ is the profile bound of the copy setting and $\Lambda^{\mathrm{aff}}$ the affine Lipschitz constant; $\kappa^{\sharp}$ is the bound of (I$'$), unrelated to the constant $\kappa$ of Closeness to a Mean-Field Pair on the Synthetic Copy: Clock Discrepancy, Gronwall Comparison of the Weighted Response with the Profile Response, and the Observation-Information Bound with Mean-Field Data; $D$ is the discrepancy tolerance of (W) and $\mathsf{D}$ the record coordinate; the mean-field control of the triple is written $A$, with values $A_{t}$, while the constant written $A_{0}$ in the copy setting (the constant of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect) is written $A^{\mathrm{ins}}$ here and is not used, so that $A_{0}$ denotes the value of the mean-field control at time $0$; $\mathsf{A}$ is the copy-side comparison control, equal to $A$ on $[0,s]$ by (L); $d$ is the number of cells, unrelated to the control discrepancy $\mathsf{d}$; $\varepsilon_{S}(N)$ and $\varepsilon_{\mathrm{ctl}}(N)$, always written with the argument $N$, are the tolerances 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 $1-O(N^{-1/2})$, while $\varepsilon_{S}$ and $\varepsilon_{\mathrm{ctl}}$ are the tolerances of (CL) fixed in claim 3; $\mathsf{s}$, $\mathsf{d}$ and $\mathsf{E}(\varepsilon,\varepsilon')$ always denote the horizon-$s$ objects of claim 1(b) (the constant $\hat{\mathsf{e}}_{5}$ of claim 5 is unrelated), the horizon-$T$ instance of The Realized Control as the Record-Frozen Control at the Observation Record, and Measurability of the Path-and-Record Closeness Set carried by 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 $1-O(N^{-1/2})$ not being used; the weak metric on the control set (written $\rho$ in The Realized Control as the Record-Frozen Control at the Observation Record, and Measurability of the Path-and-Record Closeness Set) is not used, $\rho$ denoting the reference measure of $(\mathbf{R}_{s},\mathcal{R}_{s})$; $\mathcal{C}$ denotes the $\sigma$-algebra of $\mathsf{Path}$, the record part $\mathcal{C}$ and the pathwise record parts $\mathcal{C}^{\omega}$ of the adopted copy setting not being used; the letter $\varpi$ occurs only, in claim 5, as the Chernoff exponent $\varpi_{k}$ 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, the profile $\varpi$ of the adopted copy setting not being used; $\mathsf{M}_{4}(R)$ is the constant of that lemma, unrelated to the profile-response bound $\mathsf{M}$ of the copy setting; $Q$ is the noise majorant, unrelated to the weighted squares $Q^{\omega}$ 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; the constant written $\kappa_{0}$ in each of the two adopted settings is not used; $\Pi$ and $\Pi^{\sharp}$ are the path-valued maps of Law Identity between the N-Agent Aggregate Path with Its Observation Record and the Synthetic Copy's Regularised Path with Its Record, and the real numbers $\bar\Pi$ and $\mathsf{B}$ of Assembly of the Symmetrised Move Information on the Synthetic Copy: Prior Part, Record Part, and the Bad Part with Exponentially Small Under-Likelihood Mass and the leave events $D_{0},\dots,D_{K-1}$ of Closeness of the Realized Control and the Realized Mean-Field Flow on a High-Probability Event under the Cost Bound are not used, so that $\bar\Pi$, $\mathsf{B}$ and $D_{0}$ are free for the objects introduced in the proof; and $\mathsf{S}$ is the mean-field flow, the parameter lattice of the copy setting not being used.

Then the following hold.

1. (Instantiation.) (a) The setting of Law Identity between the N-Agent Aggregate Path with Its Observation Record and the Synthetic Copy's Regularised Path with Its Record is instantiated with horizon $s$ by the copy side, the driving system $(\Omega^{\mathrm{ag}},\mathcal{F}^{\mathrm{ag}},P^{\mathrm{ag}})$, the restricted solution and the fixed reconstruction data, with the probability space of the copy side in the role of its $(\Omega^{\flat},\mathcal{F}^{\flat},P^{\flat})$ and $\mathsf{x}_{0}$ in the role of its $x_{0}$; the additional hypotheses of claim 5 of The Observation-Centred Fluctuation at an Intermediate Time: Restriction of the Mean-Field Flow and of the Realized Control to a Shorter Horizon, Representation through the Aggregate State and the Record Prefix, and Transport of Its Joint Law with the Record to the Synthetic Copy hold, so that its claims 1--5 are available; the record-frozen flow $\Phi^{r}$ of (AF) is the flow $\Phi^{(s),r}$ of that lemma for every $r\in\mathbf{R}_{s}$; and $X''(\omega,r)=X''_{s}(\omega,\theta,r)$ for all $(\omega,\theta,r)\in\Omega^{\sharp}$. (b) The setting of The Realized Control as the Record-Frozen Control at the Observation Record, and Measurability of the Path-and-Record Closeness Set is instantiated with horizon $s$ by the $N$-agent data, the policy $h^{(s)}$ and the restricted solution, with the comparison data $S^{*}_{t}=S_{t}$ ($t\in[0,s]$), $K^{*}=1$ and the control $t\mapsto A_{t}$ ($t\in[0,s]$), and with the path data $E=\mathbb{G}_{N}$; hence its objects $\mathsf{s}(p)=\sup_{t\in[0,s]}|p(t)-S_{t}|$ ($p\in\mathsf{Path}$), $\mathsf{d}(r)=\int_{[0,s]}|a^{(s),r}(t)-A_{t}|\,dt$ ($r\in\mathbf{R}_{s}$) and $\mathsf{E}(\varepsilon,\varepsilon')=\{(p,r):\mathsf{s}(p)\le\varepsilon,\ \mathsf{d}(r)\le\varepsilon'\}\in\mathcal{C}\otimes\mathcal{R}_{s}$ are defined, and for every $\omega\in\Omega^{\mathrm{ag}}_{0}$,

2. (Control-Lipschitz bound on the record discrepancy.) For every $r\in\mathbf{R}_{s}$ and every $t\in[0,s]$,

and consequently $\mathsf{D}^{r}_{\mathrm{ctl}}\le C_{\mathrm{Lip}}\,\mathsf{d}(r)$.

3. (The close records; hypothesis (CL) holds.) Put $\varepsilon_{S}=\varepsilon_{S}(N)$ and $\varepsilon_{\mathrm{ctl}}=C_{\mathrm{Lip}}\,\varepsilon_{\mathrm{ctl}}(N)$, and for $\omega\in\Omega$ put

Then hypothesis (CL) 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, as adopted by Mean-Square Assembly of the Estimand Linearisation on the Synthetic Copy: Approximation of the Recentred Copy Endpoint by an Affine Function of the Parameter, holds for the tolerances $\varepsilon_{S}$, $\varepsilon_{\mathrm{ctl}}$ and the family $(\mathsf{R}^{\mathrm{cl}}_{\omega})_{\omega\in\Omega}$: each $\mathsf{R}^{\mathrm{cl}}_{\omega}$ belongs to $\mathcal{R}_{s}$ and is contained in $\mathsf{T}_{\omega}$; for every $\omega\in G$ and every $r\in\mathsf{R}^{\mathrm{cl}}_{\omega}$ one has $|\bar\Sigma^{\sharp,r}_{t}(\omega)-S_{t}|\le\varepsilon_{S}$ for every $t\in[0,s]$ and $\mathsf{D}^{r}_{\mathrm{ctl}}\le\varepsilon_{\mathrm{ctl}}$; and the non-close mass $\pi^{\mathrm{nc}}_{\omega}=\int_{\mathbf{R}_{s}}\ell^{\sharp,\omega}\,\mathbf{1}_{\mathbf{R}_{s}\setminus\mathsf{R}^{\mathrm{cl}}_{\omega}}\,d\rho$ takes values in $[0,1]$ and is an $\mathcal{F}$-measurable function of $\omega$. Moreover

4. (Hypothesis (FM) holds.) $\int_{\Omega^{\sharp}}\bar{\mathsf{e}}^{4}\,d\mu^{\sharp}\le c_{Q}\,\kappa^{\sharp}$; hence hypothesis (FM) 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 holds with $\mathsf{c}_{4}=c_{Q}\kappa^{\sharp}$.

5. (Hypothesis (DM), the fourth moment of the cell-count sum, and the explicit mean-square bound.) Let $x_{1}>0$ and $x_{2}>0$ be real numbers, and 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 $w_{1}=\mathsf{w}^{\mathrm{clk}}=N(\Lambda_{1}s\,\varepsilon_{S}(N)+C_{\mathrm{Lip}}\varepsilon_{\mathrm{ctl}}(N))$, $w_{2}=\mu_{\max}$, $x_{1}$ and $x_{2}$. Then hypothesis (DM) 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 holds for this family, with

in the notation of that lemma; the quantity $\mathsf{k}_{4}$ satisfies $\mathsf{k}_{4}^{4}\le3(\sum_{q\in\mathsf{L}}\alpha_{q}^{2}\mu_{q})^{2}+\sum_{q\in\mathsf{L}}\alpha_{q}^{4}\mu_{q}$; all hypotheses 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 hold for the data of claims 3, 4 and of the present claim; and consequently, by its claim 4 and the bound on $\bar\pi^{\mathrm{nc}}$ of claim 3,

where $\mathsf{e}_{1},\dots,\mathsf{e}_{5}$ are the constants of that lemma for the present data, which read, with $\mathsf{c}_{4}=c_{Q}\kappa^{\sharp}$,

and where

the left-hand side is, by claim 1(a), the mean-square norm on the copy of $\alpha\cdot\Theta+\varsigma(\mathsf{D})-\mathbf{c}\cdot X''_{s}$.