Proof of 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

Throughout, "2a" refers 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, "1b" 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, "2b" to 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, "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.

Claim 1. Fix $s\in(0,T]$ and a profile $\lambda$. The natural numbers $l\ge2$, $m\ge1$, $\tilde{l}\ge1$ and the nonempty set $\mathcal{A}\subseteq\mathbb{R}^{m}$ are given; $\beta$ is a transition-rate family on $l$ states with control set $\mathcal{A}$ and rate bound $B\ge0$ (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 $K\ge0$ (common data); $\tilde\beta$ is an observation-rate family on $l$ states with $\tilde{l}$ channels and rate bound $\tilde{B}\ge0$; $(\tilde{U},\bar{\tilde\beta})$ is a twice continuously differentiable extension of $\tilde\beta$ with derivative bound $\tilde{K}\ge0$ ((X$'$)); (OC) is assumed with the wording of 2a; $T>0$; $(S,A)$ 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 the hypotheses of 2a, none of which involves $N$; so its claims are available, $\psi_{\lambda}$ is the map of its claim 4 (defined on $[0,T]$ from $\lambda$, $\mathcal{E}_{r}$ and $\Theta^{\star}_{r}$ alone, hence independent of $s$), and $\mathcal{A}_{s}(\lambda)\ge0$ is the real number of its claim 6.

The initial covariance. Fix $N\ge1$ and $\gamma,\delta$. By (D0) the event $\Omega_{*}=\{\Sigma_{0}=\mathsf{x}^{N}_{0}\}$ has $P^{\mathrm{ag}}(\Omega_{*})=1$ (an event of $\mathcal{F}^{\mathrm{ag}}$, as (D0) presupposes; $\Sigma_{0}$ is a random variable with values in $\mathbb{G}_{N}\subseteq\mathbb{R}^{l}$ and $\{\mathsf{x}^{N}_{0}\}$ is a Borel set), and on $\Omega_{*}$ one has $\mathfrak{s}_{0}=\sqrt{N}(\mathsf{x}^{N}_{0}-S_{0})$, so $\mathfrak{s}^{\gamma}_{0}\mathfrak{s}^{\delta}_{0}=N(\mathsf{x}^{N,\gamma}_{0}-S^{\gamma}_{0})(\mathsf{x}^{N,\delta}_{0}-S^{\delta}_{0})=:\mathsf{v}_{N}$ there. Both $\mathfrak{s}^{\gamma}_{0}\mathfrak{s}^{\delta}_{0}$ and the constant random variable $\mathsf{v}_{N}$ are bounded in absolute value by $4N$ (any two points of $\Delta^{l}$ are at Euclidean distance at most $2$, as recorded in Asymptotic Lower Bound for the Recentred N-Agent Cost by the Fluctuation LQG Value, under Law-Transported Injection Certificates, and $|\mathfrak{s}^{\gamma}_{0}|\le|\mathfrak{s}_{0}|$); by claim 2 of Almost Sure Inequalities Between Bounded Random Variables Pass to Expectations, $\mathbb{E}^{\mathrm{ag}}[\mathfrak{s}^{\gamma}_{0}\mathfrak{s}^{\delta}_{0}]=\mathbb{E}^{\mathrm{ag}}[\mathsf{v}_{N}]=\mathsf{v}_{N}$ (the expectation of a constant on a probability space being that constant, by The Integral of an Indicator Function is the Measure of the Set and the homogeneity of the integral). Now $|\mathsf{v}_{N}|\le N|\mathsf{x}^{N}_{0}-S_{0}|^{2}=(\sqrt{N}|\mathsf{x}^{N}_{0}-S_{0}|)^{2}$ (components are bounded by the norm, claim 1 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square), and the right-hand side converges to $0$ by (D0) and Arithmetic of Limits of Real Sequences (product of two null sequences); by claim 3 of Order Properties of Limits of Real Sequences, $\mathsf{v}_{N}\to0$. Hypothesis (I) of Asymptotic Lower Bound for the Recentred N-Agent Cost without Uniform Control-Moment Hypotheses asks for a symmetric real matrix $\Pi_{0}$ with $l$ rows and columns such that, for all $\gamma,\delta$, the sequence $(\mathbb{E}^{\mathrm{ag}}[\mathfrak{s}^{\gamma}_{0}\mathfrak{s}^{\delta}_{0}])_{N\ge1}$ converges to $\Pi^{\gamma\delta}_{0}$; the zero matrix (symmetric) satisfies this.

The maps of the LQG lower bound theorem. The observation data of Asymptotic Lower Bound for the Recentred N-Agent Cost by the Fluctuation LQG Value, under Law-Transported Injection Certificates are supplied by $(\tilde{U},\bar{\tilde\beta})$; its matrices $\tilde{\mathcal{E}}_{t}$, $\tilde\Theta^{\star}_{t}$ are the observation matrix and observation noise covariance of the fluctuation LQG data relative to these extensions, which are the matrices of the same names in 2a (as its paragraph The fluctuation LQG matrices records), and its $E_{t}$ and $\Theta^{\star}_{t}$ coincide with the state matrix $\mathcal{E}_{t}$ and state noise covariance $\Theta^{\star}_{t}$ of those data (as stated there), hence with those of 2a. By claim 2 of 2a every entry of $t\mapsto\tilde{\mathcal{E}}_{t}$ and every $t\mapsto\tilde{b}^{\upsilon}(S_{t})$ is continuous on $[0,T]$, and $\tilde{b}^{\upsilon}(S_{t})\ge\underline{b}>0$ by (OC); so hypothesis (OC) of that theorem holds with $\tilde\beta_{\min}=\underline{b}$, and its matrix $\tilde{D}_{t}=\tilde{\mathcal{E}}_{t}^{\top}(\tilde\Theta^{\star}_{t})^{-1}\tilde{\mathcal{E}}_{t}$ is the $\tilde{D}_{t}$ of 2a (same defining formula; the inverse is that of claim 5 of 2a). With the zero matrix as $\Pi_{0}$, the map $\psi_{\lambda}$ of that theorem is the unique assignment with continuous components satisfying $\psi_{\lambda}(u)=\int_{[0,u]}(\mathcal{E}_{r}\psi_{\lambda}(r)+\Theta^{\star}_{r}\lambda(r))\,dr$ on $[0,T]$; having continuous components on the compact interval $[0,T]$ it is bounded (Continuous Real-Valued Functions on a Compact Interval are Bounded) and measurable (claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval), so by the uniqueness assertion of claim 4 of 2a it equals the $\psi_{\lambda}$ of 2a. Its $\mathcal{A}_{s}(\lambda)$, with the term $\lambda(0)\cdot(\Pi_{0}\lambda(0))=0$, is then $\int_{[0,s]}(\lambda(r)\cdot(\Theta^{\star}_{r}\lambda(r))+\psi_{\lambda}(r)\cdot(\tilde{D}_{r}\psi_{\lambda}(r)))\,dr$, the $\mathcal{A}_{s}(\lambda)$ of claim 6 of 2a.

Claim 2. (C0). Fix $N\ge1$ and $s\in(0,T]$. The setting of The Observation Filtration of a Solution of the Controlled N-Agent Dynamics is Generated, up to Null Sets, by the Observation Record up to that Time is instantiated by the $N$-th solution: the control set is convex and compact, the policy $h^{(N)}$ is $\mathcal{A}$-valued with horizon $T$, the event times and channels $(\tau_{j},\upsilon_{j})_{j\ge1}$ are the fixed family furnished by claim 1 of The Realized Control of the Controlled N-Agent Dynamics as a Random Element of the Control Set from which the realized control is formed (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 $1-O(N^{-1/2})$), and reconstruction data for the driving system and $h^{(s)}$ with horizon $s$ are fixed in the statement. Its claim 4 states that $\mathcal{G}_{s}$ is $W^{(s)}$-generated up to null sets 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, with $\sigma(W^{(s)})$ formed from the sets of $\mathcal{R}_{s}$; this is (C0).

The certificates. Fix $s\in(0,T]$, $\mathbf{c}\in\mathbb{R}^{l}$, a profile $\lambda$ and $\epsilon>0$.

The $N$-independent data. By claim 1, 2a applies; adopt from it, as in the paragraph The mean-field-side data of 2b, the real numbers $\Lambda\ge0$, $\mathsf{M}\ge0$ and $\bar\Phi\ge0$ (its claims 3 and 4; fix one choice of each), the restrictions $\varpi$, $\bar\psi$ of $\lambda$, $\psi_{\lambda}$ to $[0,s]$, and the real numbers $\mathsf{P}=\int_{[0,s]}\varpi_{u}\cdot(\Theta^{\mathrm{fl}}(S_{u},A_{u})\varpi_{u})\,du$ and $\mathsf{Q}=\int_{[0,s]}\bar\psi_{u}\cdot(\tilde{D}(S_{u})\bar\psi_{u})\,du$ defined in the paragraph The mean-field-side data of 2b ($\Theta^{\mathrm{fl}}$ the aggregate fluctuation covariance of $\beta$), which are nonnegative with $\mathsf{P}+\mathsf{Q}=\mathcal{A}_{s}(\lambda)$ by claim 6 of 2a. Put $\mathsf{c}_{0}=|\mathbf{c}|$. The scalar data $l$, $m$, $\tilde{l}$, $B$, $\tilde{B}$, $K$, $\tilde{K}$, $\underline{b}$, $T$, $s$, $\Lambda$, $\mathsf{M}$, $\bar\Phi$, $\mathsf{c}_{0}$, $c_{Q}$, $\kappa^{\sharp}$, $C_{\mathrm{flw}}$, $C_{\mathrm{ctl}}$, $\mathsf{P}$, $\mathsf{Q}$ satisfy the hypotheses of 1b: $l\ge2$, $m\ge1$, $\tilde{l}\ge1$ are natural numbers; $B$, $\tilde{B}$, $K$, $\tilde{K}\ge0$; $\underline{b}>0$; $T>0$ and $s\in(0,T]$; $\Lambda$, $\mathsf{M}$, $\bar\Phi\ge0$ and $\mathsf{c}_{0}\ge0$; $\kappa^{\sharp}\ge1$ by (I$'$); $\mathsf{P}$, $\mathsf{Q}\ge0$; $c_{Q}\ge0$, because 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 any one $N\ge1$) gives $0\le\mathbb{E}^{\mathrm{ag}}[Q^{4}]\le c_{Q}\kappa_{0}N^{-2}$ with $\kappa_{0}=1+\mathbb{E}^{\mathrm{ag}}[|\mathfrak{s}_{0}|^{4}]\ge1$; $C_{\mathrm{ctl}}=2(\mathcal{Z}^{\sharp}+1)\ge2$ with $\mathcal{Z}^{\sharp}\ge0$ 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; and $C_{\mathrm{flw}}=C_{S}C_{\mathrm{ctl}}^{1/2}\ge0$ by claim 4 there, since by its Data $C_{S}=e^{\Lambda_{b}T}\sqrt{l}\,K_{2}\sqrt{T}$ 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). None of these data depends on $N$. (We do not invoke claim 1 of 2b here, since the setting of 2b presupposes the number $N_{\mathrm{c}}$ about to be obtained from 1b.) Let $N_{\mathrm{c}}$ be the natural number fixed in claim 1 of 1b for these data, and let $N_{\mathrm{sc}}\ge N_{\mathrm{c}}$ be a natural number furnished by claim 4 of 1b for the tolerance $\epsilon/2$, so that for every $N\ge N_{\mathrm{sc}}$

$$\mathsf{e}_{2,N}+\mathsf{e}_{3,N}+\mathsf{e}_{4,N}+\mathsf{e}_{5,N}\le\tfrac{\epsilon}{2},\qquad \mathsf{e}_{\mathrm{inj},N}\le\tfrac{\epsilon}{2},\qquad \mathsf{a}_{N}\le\tfrac{\epsilon}{2},\qquad \mathcal{I}_{N}\le\mathsf{P}+\mathsf{Q}+\tfrac{\epsilon}{2}.$$

By (D0) the sequence $\varepsilon^{\mathrm{ini}}_{N}=\sqrt{N}|\mathsf{x}^{N}_{0}-S_{0}|$ converges to $0$, hence so does $|\mathbf{c}|\bar\Phi^{2}\varepsilon^{\mathrm{ini}}_{N}$ (limit of a scalar multiple), and by claim 3(d) 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 $L=0<\epsilon/2$) there is a natural number $N_{\mathrm{D}}$ with $|\mathbf{c}|\bar\Phi^{2}\sqrt{N}|\mathsf{x}^{N}_{0}-S_{0}|<\epsilon/2$ for every $N\ge N_{\mathrm{D}}$. Put $N_{2}=\max(N_{\mathrm{cl}},N_{\mathrm{sc}},N_{\mathrm{D}})$, a natural number with $N_{2}\ge N_{\mathrm{cl}}$.

The instance of 2b. Let $N\ge N_{2}$. The $N$-agent side of P8.4b is instantiated by the $N$-th solution: its setting is that 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 $1-O(N^{-1/2})$ with the renamings listed, the fixed number $N_{\mathrm{cl}}$ and the natural number $N\ge N_{\mathrm{cl}}$. For its intermediate-time paragraph, take the dense sequence in $L^{2}([0,s];\mathbb{R}^{m})$ and the reconstruction data for the driving system and $h^{(s)}$ fixed in the statement; with the base point $z_{0}=S_{0}$, 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 is then instantiated as in P8.4b, its observation-centred fluctuation at time $s$ being $X'_{s}$ and its record prefix $W^{(s)}$. Hypothesis (D) of 2b holds with $\mathsf{x}_{0}=\mathsf{x}^{N}_{0}$ ((D0)), and (OC), (X$'$) of 2b are the present hypotheses; the profile is $\lambda$ and the estimand direction $\mathbf{c}$. The mean-field-side data and the scale-set data of 2b are then those fixed above (2b adopts $\Lambda$, $\mathsf{M}$, $\bar\Phi$ from 2a as fixed choices, and we take the choices made above), so the scale set, its derived quantities and its number $N_{\mathrm{c}}$ in 2b are those of the present paragraph, and $N\ge N_{\mathrm{sc}}\ge N_{\mathrm{c}}$. Let $(\Omega,\mathcal{F},P)$ be a probability space carrying the independent driving variables of the copy side of 2b for the present $N$ (it exists by Existence of Independent Sequences with Prescribed Distributions, as recorded there). All hypotheses of 2b hold, so its claims 1--6 are available for this copy instance; we use its notation.

The certificate data. Take $(\Omega',\mathcal{F}',P')=(\Omega^{\mathrm{tr}},\mathcal{F}^{\mathrm{tr}},\mu^{\mathrm{tr}})$, $\mathsf{Y}=\mathsf{R}_{+}$, $\mathcal{Y}=\mathcal{R}_{+}$, $\varrho_{0}=\rho_{+}$, the natural number $d$ (with $\mathsf{L}$ identified with $\{1,\dots,d\}$ by the fixed bijection), $\vartheta_{q}=\Theta^{\mathrm{tr}}_{q}$, $\mathsf{D}'=\mathsf{D}^{\mathrm{tr}}$, $X^{\dagger}=\mathbf{c}\cdot X^{\mathrm{tr}}$, $p=g^{\mathrm{tr}}$, $G=\varsigma|_{\mathsf{R}_{+}}$, $\alpha$ the vector of cell coefficients, $z=\mathsf{z}$ and $a_{q}=\mathsf{m}e_{q}/\sqrt{N}$. By claim 1 of P8.4c (available by claim 4 of 2b), $\mathsf{R}_{+}\in\mathcal{R}_{s}$, $\mathcal{R}_{+}=\{E\in\mathcal{R}_{s}:E\subseteq\mathsf{R}_{+}\}$ is its trace $\sigma$-algebra, the trimmed copy is a probability space, and $\mathsf{D}^{\mathrm{tr}}$ takes its values in $\mathsf{R}_{+}$; by claim 6(b) of 2b, $\rho_{+}$ is $\sigma$-finite on $(\mathsf{R}_{+},\mathcal{R}_{+})$, each $\vartheta_{q}$ is a square-integrable random variable on $(\Omega',\mathcal{F}',P')$, and $\mathsf{D}'$ is measurable with respect to $\mathcal{F}'$ and $\mathcal{Y}$; by claim 6(a) of 2b, $X^{\dagger}$ is square-integrable; $d\ge1$ (the index set $\mathsf{L}$ is nonempty, $l(l-1)\ge2$ labels each carrying at least one cell); and $p$ takes values in $[0,\infty)$ by claim 2 of P8.4c. We verify (a), (C1), (C2$'$), (C3), (C4$'$).

(a). This is claim 6(a) of 2b, together with the measurability of $\omega\mapsto(\mathbf{c}\cdot X'_{s}(\omega),W^{(s)}(\omega))$ with respect to $\mathcal{F}^{\mathrm{ag}}$ and $\mathcal{B}(\mathbb{R})\otimes\mathcal{R}_{s}$ and the square-integrability of $\mathbf{c}\cdot X'_{s}$ on the $N$-agent side, which are part of claim 4 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 (with $t=s$), available from the instantiation of its setting made in the preceding paragraph.

(C1). Since $\mathsf{D}'$ takes its values in $\mathsf{Y}=\mathsf{R}_{+}$, for every $E\in\mathcal{R}_{s}$ one has $\mathsf{D}'^{-1}(E)=\mathsf{D}'^{-1}(E\cap\mathsf{R}_{+})$ with $E\cap\mathsf{R}_{+}\in\mathcal{Y}$, and conversely every $E\in\mathcal{Y}$ belongs to $\mathcal{R}_{s}$; hence $\{\mathsf{D}'^{-1}(E):E\in\mathcal{R}_{s}\}=\{\mathsf{D}'^{-1}(E):E\in\mathcal{Y}\}$, which is $\sigma(\mathsf{D}')$ in both readings. Now let $Z$ be a random variable on $(\Omega',\mathcal{F}',P')$ measurable with respect to $\sigma(\mathsf{D}')$ and square-integrable. By claim 2 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 to the probability space $(\Omega',\mathcal{F}',P')$, the measurable space $(\mathsf{Y},\mathcal{Y})$ and the map $\mathsf{D}'$ (measurable with respect to $\mathcal{F}'$ and $\mathcal{Y}$), there is a $\mathcal{Y}$-measurable $g:\mathsf{Y}\to\mathbb{R}$ with $Z=g\circ\mathsf{D}'$ everywhere on $\Omega'$; then $g\circ\mathsf{D}'=Z$ is square-integrable. This is (C1).

(C2$'$). This is claim 6(b) of 2b: the data named satisfy (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 $z=\mathsf{z}$ for the shifts $a_{q}$ with $n=d$; so $\mathcal{I}_{z}=\mathcal{I}_{\mathsf{z}}$, which is finite by claim 2 of P8.4c ($\mathcal{I}_{\mathsf{u}}<\infty$ and $\mathcal{I}_{\mathsf{z}}=\mathcal{I}_{\mathsf{u}/\sqrt{N}}=\mathcal{I}_{\mathsf{u}}/N$ by the scaling identity there).

(C3). By claim 6(c) of 2b, $G=\varsigma|_{\mathsf{R}_{+}}$ is measurable with respect to $\mathcal{Y}$ and $\mathcal{B}(\mathbb{R})$, $G(\mathsf{D}')=\varsigma(\mathsf{D}^{\mathrm{tr}})$ is square-integrable, and, with $\alpha\cdot\vartheta=\alpha\cdot\Theta^{\mathrm{tr}}$,

$$\bigl\lVert\alpha\cdot\vartheta+G(\mathsf{D}')-X^{\dagger}\bigr\rVert_{2}\le|\mathbf{c}|\,\bar\Phi^{2}\sqrt{N}\,|\mathsf{x}^{N}_{0}-S_{0}|+\mathsf{e}_{2,N}+\mathsf{e}_{3,N}+\mathsf{e}_{4,N}+\mathsf{e}_{5,N}<\tfrac{\epsilon}{2}+\tfrac{\epsilon}{2}=\epsilon,$$

by the choice of $N_{\mathrm{D}}$ and $N_{\mathrm{sc}}$ ($N\ge N_{2}$).

(C4$'$). By claim 6(d) of 2b and the choice of $N_{\mathrm{sc}}$: $\alpha\cdot z\ge\mathbf{c}\cdot\psi_{\lambda}(s)-\mathsf{e}_{\mathrm{inj},N}\ge\mathbf{c}\cdot\psi_{\lambda}(s)-\epsilon$; $\max_{q}|\alpha\cdot a_{q}|\le\mathsf{a}_{N}\le\epsilon$; and $\mathcal{I}_{z}=\mathcal{I}_{\mathsf{z}}\le\mathcal{I}_{N}\le\mathsf{P}+\mathsf{Q}+\epsilon/2\le\mathcal{A}_{s}(\lambda)+\epsilon$, using $\mathsf{P}+\mathsf{Q}=\mathcal{A}_{s}(\lambda)$. This completes the proof of claim 2.

Claim 3. Let $s$, $\mathbf{c}$, $\lambda$, $\epsilon$, $\Lambda$, $\mathsf{M}$, $\bar\Phi$ and the spaces $(\Omega,\mathcal{F},P)$ be as in the statement of claim 3. Run the proof of claim 2 with these $\Lambda$, $\mathsf{M}$, $\bar\Phi$ in place of the choices fixed there (nothing in that argument depends on which admissible triple is chosen, only on its being fixed before the scale set is formed) and, for each $N$, with the given space $(\Omega,\mathcal{F},P)$ as the probability space carrying the driving variables. It produces $N_{\mathrm{c}}$, $N_{\mathrm{sc}}$, $N_{\mathrm{D}}$ and $N_{2}=\max(N_{\mathrm{cl}},N_{\mathrm{sc}},N_{\mathrm{D}})\ge N_{\mathrm{cl}}$, verifies the hypotheses of 2b for every $N\ge N_{2}$ with exactly the choices listed in claim 3, and takes as certificate data exactly the objects listed there. $\qquad\blacksquare$