Proof of Ledger Decomposition of the Recentred N-Agent Cost over the Block Cascade and Its Near-Field Filtering Lower Bound

Throughout, "the cascade lemma" is the block cascade lemma, "the energy lemma" is the tracked energy bound lemma, "the envelope lemma" is the anchored pre-stopping envelope lemma, "the pre-stopping lemma" is the pre-stopping envelope lemma, "the extended lemma" is the extended good-set stopping-time lemma, "the coercivity lemma" is the localized joint coercivity lemma, "the exit lemma" is the post-exit comparison lemma, "the block lemma" is the stopped completion-of-squares lemma, "the filtering lemma" is the cascade filtering lemma, "the first-order lemma" is the first-order expansion lemma for the recentred $N$-agent cost, "the squares theorem" is the completion-of-squares theorem, "the expansion theorem" is the second-order expansion theorem, "the covariance lemma" is the covariance deviation lemma, and "the integral toolkit" is the toolkit for Lebesgue integrals over compact intervals. All notation is that of the statement.

Preliminary P0 (the partition). By claim 2 of the cascade lemma the sets $D_{0},\dots,D_{K-1},G_{K}$ are pairwise disjoint with union $\Omega_{0}$, and $G_{0}\supseteq G_{1}\supseteq\dots\supseteq G_{K}$ with $G_{0}=\Omega_{0}$. Since $G_{k}=G_{k+1}\cup D_{k}$ disjointly for every $k$ (the two sets being $G_{k}\cap\{\sigma^{(k)}\ge t_{k+1}\}$ and $G_{k}\cap\{\sigma^{(k)}<t_{k+1}\}$ by their definitions in the cascade lemma), induction gives $G_{k}=D_{k}\cup\dots\cup D_{K-1}\cup G_{K}$ disjointly, and hence $\Omega_{0}\setminus G_{k}=D_{0}\cup\dots\cup D_{k-1}$ disjointly. The regular event of the solution definition has probability $1$, so $P(\Omega\setminus\Omega_{0})=0$.

Preliminary P1 (tracked envelope). Let $k\in\{0,\dots,K-1\}$, $s\in[t_{k},T]$ and $\omega\in\mathcal{T}_{k}(s)$. By claim 3 of the cascade lemma $G_{k}\subseteq\{Y_{t_{k}}\le L_{k}/(2C_{a})\}\subseteq\{Y_{t_{k}}<L_{k}\}$, and $G_{k}\subseteq G_{0}=\Omega_{0}$. The clock $\sigma^{(k)}$ is the anchored good-set clock with anchor $t_{k}$ and level $L_{k}$, so claim 1 of the envelope lemma, applied with $t_{0}=t_{k}$ and $\varepsilon_{1}$ there equal to $L_{k}$, gives $\mathbf{1}_{\{s<\sigma^{(k)}\}}(\omega)|\mathfrak{s}_{s}(\omega)|\le\sqrt{N}(L_{k}+Q(\omega))$, whence

$$|\mathfrak{s}_{s}(\omega)|\ \le\ \sqrt{N}\bigl(L_{k}+Q(\omega)\bigr)\ \le\ \sqrt{N}\bigl(\varepsilon_{1}+Q(\omega)\bigr),$$

using $L_{k}\le\varepsilon_{1}$ from claim 1 of the cascade lemma. In particular, if moreover $\omega\notin\mathcal{N}$, that is $Q(\omega)\le q_{0}$, then $|\mathfrak{s}_{s}(\omega)|\le\sqrt{N}(\varepsilon_{1}+q_{0})$. The same argument at $t=T$ on $G_{K}$ gives $|\mathfrak{s}_{T}|\le\sqrt{N}(\varepsilon_{1}+Q)$ there: on $G_{K}$ one has $\sigma^{(K-1)}\ge t_{K}=T$, so $\min(T,\sigma^{(K-1)})=T$ and the first inequality of claim 1 of the envelope lemma applies to the sampled point $T$ with $L_{K-1}=\varepsilon_{1}$.

Preliminary P2 (crude bounds). At every point of $[0,T]\times\Omega$ one has $\Sigma_{t}\in\Delta^{l}$ and $\alpha_{t}\in\mathcal{A}$ by the solution definition, so $|\mathfrak{s}_{t}|=\sqrt{N}|\Sigma_{t}-S_{t}|\le2\sqrt{N}$ (two points of the probability simplex being at Euclidean distance at most $2$) and $|\mathfrak{a}_{t}|=\sqrt{N}|\alpha_{t}-A_{t}|\le2\sqrt{N}R$; hence $|\mathfrak{z}_{t}|^{2}=|\mathfrak{s}_{t}|^{2}+|\mathfrak{a}_{t}|^{2}\le4N(1+R^{2})$. For $x,y\in\mathbb{R}^{p}$ and a real $p\times p$ matrix $M$ with $|M^{pq}|\le c$ one has $|x\cdot My|\le c(\sum_{p}|x^{p}|)(\sum_{q}|y^{q}|)\le c\,p\,|x||y|$, the inner estimate being the Cauchy--Schwarz inequality against the vector of ones; applied to $H$ this gives $|\mathfrak{h}_{t}|\le\tfrac{1}{2}C_{H}(l+m)|\mathfrak{z}_{t}|^{2}=C_{\mathfrak{h}}|\mathfrak{z}_{t}|^{2}\le4C_{\mathfrak{h}}(1+R^{2})N$, and applied to $Z_{t}$ it gives $|x\cdot Z_{t}y|\le lC_{Z}|x||y|$. By conclusion (b) of the squares theorem, $|e_{s}|\le c_{e}N^{-1/2}|\mathfrak{z}_{s}|^{2}$, so

$$2\,|\mathfrak{s}_{s}\cdot Z_{s}e_{s}|\ \le\ 2l\,C_{Z}\,c_{e}\,N^{-1/2}|\mathfrak{s}_{s}|\,|\mathfrak{z}_{s}|^{2}\ \le\ 16\,l\,C_{Z}\,c_{e}\,(1+R^{2})\,N .$$

Finally $|N\mathcal{D}_{t}|\le NC_{\mathcal{D}}$ and $|N\mathcal{D}_{G}|\le NC_{\mathcal{D}G}$ by part (a) of the first-order lemma, and $|\mathfrak{s}_{T}\cdot Z_{T}\mathfrak{s}_{T}|\le4lC_{Z}N$.

Preliminary P3 (measurability and interchange). For each $k$ the function $(s,\omega)\mapsto\mathbf{1}_{\mathcal{T}_{k}(s)}(\omega)=\mathbf{1}_{G_{k}}(\omega)\mathbf{1}_{\{s<\sigma^{(k)}\}}(\omega)$ is measurable for the product $\sigma$-algebra of the trace Borel $\sigma$-algebra on $[0,T]$ and $\mathcal{F}$: $G_{k}$ is an event and the pre-stopping-time indicator is jointly measurable by the stopped-time integral lemma. The processes $\mathfrak{s}$ and $\mathfrak{a}$ are jointly measurable and $\mathbf{1}_{\Omega_{0}}\mathcal{D}_{s}$ is jointly measurable and bounded by part (a) of the first-order lemma; $|\mathfrak{a}_{s}|$ is jointly measurable, so the sets $\{|\mathfrak{a}_{s}|\le\sqrt{N}\varrho\}$ define a jointly measurable indicator as well. Every integrand appearing below is therefore jointly measurable and bounded in absolute value by a constant multiple of $N$, so all the interchanges of $\int_{[a,b]}\cdot\,ds$ with $\mathbb{E}$ used below are legitimate by the Tonelli--Fubini theorem, applied separately to the positive and negative parts of the integrand, each of which is nonnegative, jointly measurable and bounded.

Claim 1(a). Fix $k$ and $\omega\in G_{k}\subseteq\Omega_{0}$. Since $\sigma^{(k)}(\omega)\ge t_{k}$ by claim 2 of the cascade lemma, $\{s\in[t_{k},t_{k+1}]:s<\sigma^{(k)}(\omega)\}=[t_{k},\min(t_{k+1},\sigma^{(k)}(\omega)))$. By claim 5 of the pre-stopping lemma, $\mathcal{E}_{t}(\omega)=N^{-1}\int_{[0,t]}|\mathfrak{a}_{s}(\omega)|^{2}ds$ for every $t$, so by additivity of the integral over subintervals and the fact that a single point is Lebesgue-null (claims 1 and 2 of the integral toolkit),

$$\int_{[t_{k},t_{k+1}]}\mathbf{1}_{\{s<\sigma^{(k)}\}}(\omega)|\mathfrak{a}_{s}(\omega)|^{2}ds=\int_{[t_{k},\min(t_{k+1},\sigma^{(k)}(\omega))]}|\mathfrak{a}_{s}(\omega)|^{2}ds=N\bigl(\mathcal{E}_{\min(t_{k+1},\sigma^{(k)})}(\omega)-\mathcal{E}_{t_{k}}(\omega)\bigr)=N\,\Delta_{k}\mathcal{E}(\omega).$$

Multiplying by $\mathbf{1}_{G_{k}}$, taking expectations, interchanging by P3 and summing over $k$ gives (a); each term is finite because $0\le\Delta_{k}\mathcal{E}\le4R^{2}h_{k}$ by claim 6 of the anchored good-set clocks lemma.

Claim 1(b). Fix $k$ and $s\in[t_{k},t_{k+1}]$, and let $\omega\in\mathcal{T}_{k}(s)$. By claim 6 of the pre-stopping lemma $|\mathfrak{s}_{s}(\omega)|\le\sqrt{N}(Y_{s}(\omega)+Q(\omega))$, and by claim 4 of the extended lemma $Y_{s}(\omega)\le C_{S}(\mathcal{E}_{s}(\omega))^{1/2}$; since $(a+b)^{2}\le2a^{2}+2b^{2}$ for reals,

$$|\mathfrak{s}_{s}(\omega)|^{2}\ \le\ 2N\bigl(C_{S}^{2}\,\mathcal{E}_{s}(\omega)+Q(\omega)^{2}\bigr).$$

Next, on $\mathcal{T}_{k}(s)$ one has $\mathcal{E}_{s}\le\sum_{j\le k}\mathbf{1}_{G_{j}}\Delta_{j}\mathcal{E}$. Indeed $\mathcal{E}$ has nondecreasing paths, $\mathcal{E}_{0}=0$, and $s\le\min(t_{k+1},\sigma^{(k)})$ there, so $\mathcal{E}_{s}-\mathcal{E}_{t_{k}}\le\Delta_{k}\mathcal{E}$; while for $j<k$ one has $G_{k}\subseteq G_{j+1}$, hence $\sigma^{(j)}\ge t_{j+1}$ and $\Delta_{j}\mathcal{E}=\mathcal{E}_{t_{j+1}}-\mathcal{E}_{t_{j}}$, so that $\mathcal{E}_{t_{k}}=\sum_{j<k}(\mathcal{E}_{t_{j+1}}-\mathcal{E}_{t_{j}})=\sum_{j<k}\Delta_{j}\mathcal{E}$; and $G_{k}\subseteq G_{j}$ for $j\le k$. Therefore

$$N\,\mathbb{E}\bigl[\mathbf{1}_{\mathcal{T}_{k}(s)}\mathcal{E}_{s}\bigr]\ \le\ \sum_{j\le k}N\,\mathbb{E}\bigl[\mathbf{1}_{G_{j}}\Delta_{j}\mathcal{E}\bigr]\ \le\ \mathcal{Z},$$

all the summands being nonnegative. Also $\mathbb{E}[Q^{2}]\le(\mathbb{E}[Q^{4}])^{1/2}\le c_{Q}^{1/2}\kappa_{0}^{1/2}N^{-1}$ by the Cauchy--Schwarz inequality for square-integrable random variables applied to $Q^{2}$ and the constant $1$, together with claim 2 of the pre-stopping lemma. Hence $\mathbb{E}[\mathbf{1}_{\mathcal{T}_{k}(s)}|\mathfrak{s}_{s}|^{2}]\le2C_{S}^{2}\mathcal{Z}+2c_{Q}^{1/2}\kappa_{0}^{1/2}$ for every such $s$; integrating over $[t_{k},t_{k+1}]$ and summing over $k$, the total length being $t_{K}-t_{0}=T$, gives (b).

Claim 1(c). On $\mathcal{T}^{\mathrm{fr}}_{k}(s)$ one has $|\mathfrak{a}_{s}|>\sqrt{N}\varrho>0$, so $\mathbf{1}_{\mathcal{T}^{\mathrm{fr}}_{k}(s)}\le\mathbf{1}_{\mathcal{T}_{k}(s)}|\mathfrak{a}_{s}|^{2}(N\varrho^{2})^{-1}$ and $\mathbf{1}_{\mathcal{T}^{\mathrm{fr}}_{k}(s)}|\mathfrak{a}_{s}|\le\mathbf{1}_{\mathcal{T}_{k}(s)}|\mathfrak{a}_{s}|^{2}(\sqrt{N}\varrho)^{-1}$ everywhere. Taking expectations, integrating, summing and using (a) gives both bounds.

Claim 1(d). By claim 4 of the cascade lemma, $NL_{k}^{2}P(D_{k})\le\Lambda_{\star}N\mathbb{E}[\mathbf{1}_{G_{k}}\Delta_{k}\mathcal{E}]$ for each $k$; dividing by $NL_{k}^{2}$ and summing, and bounding each $L_{k}^{-2}$ by $\Upsilon_{\mathrm{lev}}$ (all terms of that sum being positive),

$$\mathcal{P}=\sum_{k}P(D_{k})\ \le\ \Lambda_{\star}N^{-1}\sum_{k}L_{k}^{-2}\,N\mathbb{E}[\mathbf{1}_{G_{k}}\Delta_{k}\mathcal{E}]\ \le\ \Lambda_{\star}\,\Upsilon_{\mathrm{lev}}\,N^{-1}\,\mathcal{Z}.$$

For the second assertion, hypothesis (EB) of claim 6 of the cascade lemma holds with $C_{\dagger}=\mathcal{Z}$, this being by definition the quantity there bounded, and $\mathcal{Z}$ being a finite nonnegative real by claim 1 of the energy lemma; claim 6 of the cascade lemma then gives $\sum_{k}\sqrt{N}P(D_{k})^{3/4}\le(\Lambda_{\star}\mathcal{Z})^{3/4}N^{-1/4}\Upsilon_{\mathrm{esc}}$. For the third, fix $k$ and $s\in[t_{k},t_{k+1})$. By P0, $\Omega_{0}\setminus G_{k}=D_{0}\cup\dots\cup D_{k-1}$, and $G_{k}\setminus\{s<\sigma^{(k)}\}=G_{k}\cap\{\sigma^{(k)}\le s\}\subseteq G_{k}\cap\{\sigma^{(k)}<t_{k+1}\}=D_{k}$; hence $\Omega\setminus\mathcal{T}_{k}(s)\subseteq(\Omega\setminus\Omega_{0})\cup D_{0}\cup\dots\cup D_{k}$ and $\mathbb{E}[1-\mathbf{1}_{\mathcal{T}_{k}(s)}]\le\mathcal{P}$, since $P(\Omega\setminus\Omega_{0})=0$. Integrating over $[t_{k},t_{k+1}]$ (the endpoint $t_{k+1}$ being Lebesgue-null) and summing gives at most $T\mathcal{P}$.

Claim 1(e). Fix $k$. By claim 3 of the cascade lemma $D_{k}\subseteq G_{k}\subseteq\{Y_{t_{k}}<L_{k}\}$, and $D_{k}\subseteq\Omega_{0}$, so claim 3 of the envelope lemma applies with anchor $t_{k}$, level $L_{k}$, $t=t_{k+1}$ and $D=D_{k}$ and gives, in its middle form,

$$\mathbb{E}\bigl[\mathbf{1}_{D_{k}}\mathbf{1}_{\Omega_{0}}|\mathfrak{s}_{\min(t_{k+1},\sigma^{(k)})}|^{2}\bigr]\ \le\ N\,\mathbb{E}\bigl[\mathbf{1}_{D_{k}}(L_{k}+Q)^{2}\bigr]\ \le\ 2NL_{k}^{2}P(D_{k})+2N\,\mathbb{E}\bigl[\mathbf{1}_{D_{k}}Q^{2}\bigr].$$

On $D_{k}$ one has $\sigma^{(k)}<t_{k+1}$, so $\min(t_{k+1},\sigma^{(k)})=\sigma^{(k)}$ and the left side is $\mathbb{E}[\mathbf{1}_{D_{k}}|\mathfrak{s}_{\sigma^{(k)}}|^{2}]$. Summing over $k$: the first term sums to at most $2\Lambda_{\star}\mathcal{Z}$ by claim 4 of the cascade lemma, and, the $D_{k}$ being pairwise disjoint, the second sums to $2N\mathbb{E}[\mathbf{1}_{D}Q^{2}]$ with $D=D_{0}\cup\dots\cup D_{K-1}$, which by the Cauchy--Schwarz inequality for square-integrable random variables, applied to $Q^{2}$ and $\mathbf{1}_{D}$, is at most $2N(\mathbb{E}[Q^{4}])^{1/2}P(D)^{1/2}\le2c_{Q}^{1/2}\kappa_{0}^{1/2}\mathcal{P}^{1/2}$.

Claim 2. Fix $k$ and $s\in[t_{k},T]$. By claim 1 of the filtering lemma, $\mathcal{T}_{k}(s)\in\mathcal{G}_{s}$. By the observation-adaptedness lemma, for each $j$ there is a $\mathcal{G}_{s}$-measurable random variable $\tilde{\mathfrak{a}}^{j}_{s}$ with $\mathfrak{a}^{j}_{s}=\tilde{\mathfrak{a}}^{j}_{s}$ almost surely. Put $\mathcal{H}'=\{\sum_{j}(\tilde{\mathfrak{a}}^{j}_{s})^{2}\le N\varrho^{2}\}\in\mathcal{G}_{s}$ and $\mathcal{H}=\{|\mathfrak{a}_{s}|\le\sqrt{N}\varrho\}\in\mathcal{F}$. The two differences $\mathcal{H}\setminus\mathcal{H}'$ and $\mathcal{H}'\setminus\mathcal{H}$ are events contained in the union over $j$ of the null events $\{\mathfrak{a}^{j}_{s}\ne\tilde{\mathfrak{a}}^{j}_{s}\}$, hence are $\mathcal{F}$-events of probability zero; by the description of the observation filtration in the solution definition, which adjoins to the observation $\sigma$-algebras every event of $\mathcal{F}$ of probability zero (the same feature that yields claim 0 of the filtering lemma), every such event belongs to $\mathcal{G}_{s}$. Therefore $\mathcal{H}=(\mathcal{H}'\cup(\mathcal{H}\setminus\mathcal{H}'))\setminus(\mathcal{H}'\setminus\mathcal{H})\in\mathcal{G}_{s}$ and $\mathcal{T}^{\mathrm{nr}}_{k}(s)=\mathcal{T}_{k}(s)\cap\mathcal{H}\in\mathcal{G}_{s}$. Claim 3 of the filtering lemma, applied with this event, gives the displayed inequalities.

Claim 3. Fix $\omega\in\Omega_{0}$ and set $\varsigma(\omega)=\sigma^{(k)}(\omega)$ if $\omega\in D_{k}$ and $\varsigma(\omega)=T$ if $\omega\in G_{K}$; by P0 this defines $\varsigma$ on all of $\Omega_{0}$. Suppose first $\omega\in D_{k}$. For $j<k$ one has $\omega\in G_{j+1}$, so $\sigma^{(j)}(\omega)\ge t_{j+1}$ and $\{s\in[t_{j},t_{j+1}]:s<\sigma^{(j)}(\omega)\}=[t_{j},t_{j+1})$; for $j=k$ this set is $[t_{k},\sigma^{(k)}(\omega))$; and for $j>k$ one has $\omega\notin G_{j}$, so $\mathbf{1}_{\mathcal{T}_{j}(s)}(\omega)=0$. Hence the sets over which the $j$-th integral is taken are pairwise disjoint up to finitely many points and have union $[0,\sigma^{(k)}(\omega))$ up to finitely many points, and, by additivity of the integral and the Lebesgue-nullity of finite sets,

$$\sum_{j=0}^{K-1}\int_{[t_{j},t_{j+1}]}\mathbf{1}_{\mathcal{T}_{j}(s)}(\omega)\,N\mathcal{D}_{s}(\omega)\,ds=N\int_{[0,\varsigma(\omega)]}\mathcal{D}_{s}(\omega)\,ds .$$

If instead $\omega\in G_{K}$ then $\omega\in G_{j}$ and $\sigma^{(j)}(\omega)\ge t_{j+1}$ for every $j$, the union is $[0,T)$, and the same identity holds with $\varsigma(\omega)=T$. By P3 the left-hand side integrates term by term to $\sum_{j}\int_{[t_{j},t_{j+1}]}\mathbb{E}[\mathbf{1}_{\mathcal{T}_{j}(s)}N\mathcal{D}_{s}]ds$. Using $\int_{[0,T]}=\int_{[0,\varsigma]}+\int_{[\varsigma,T]}$ pathwise on $\Omega_{0}$ (again by additivity, the overlap being a single point), together with the convention of the exit lemma that $\int_{[\varsigma,T]}$ is $0$ when $\varsigma=T$, and taking expectations against $\mathbf{1}_{\Omega_{0}}$,

$$\int_{[0,T]}\mathbb{E}\bigl[\mathbf{1}_{\Omega_{0}}N\mathcal{D}_{t}\bigr]dt=\sum_{j=0}^{K-1}\int_{[t_{j},t_{j+1}]}\mathbb{E}\bigl[\mathbf{1}_{\mathcal{T}_{j}(s)}N\mathcal{D}_{s}\bigr]ds+\sum_{k=0}^{K-1}\mathbb{E}\Bigl[\mathbf{1}_{D_{k}}N\int_{[\sigma^{(k)},T]}\mathcal{D}_{t}\,dt\Bigr].$$

Adding $\mathbb{E}[\mathbf{1}_{\Omega_{0}}N\mathcal{D}_{G}]=\sum_{k}\mathbb{E}[\mathbf{1}_{D_{k}}N\mathcal{D}_{G}]+\mathbb{E}[\mathbf{1}_{G_{K}}N\mathcal{D}_{G}]$ (P0) and invoking part (c) of the first-order lemma, which states $\mathcal{J}_{N}=\int_{[0,T]}\mathbb{E}[\mathbf{1}_{\Omega_{0}}N\mathcal{D}_{t}]dt+\mathbb{E}[\mathbf{1}_{\Omega_{0}}N\mathcal{D}_{G}]$, gives claim 3. Finiteness of every term follows from P2.

Claim 4. Fix $k$ and set $D^{-}_{k}=D_{k}\cap\{|\mathfrak{s}_{\sigma^{(k)}}|\le\sqrt{N}\varepsilon_{tg}\}$ and $D^{+}_{k}=D_{k}\setminus D^{-}_{k}$. The process $\mathbf{1}_{\Omega_{0}}\mathfrak{s}$ is progressively measurable for $(\mathcal{F}^{\mathrm{sys}}_{t})_{t\in[0,T]}$ — $\Sigma$ is adapted with right-continuous paths and $S$ is deterministic and continuous, so this follows from claim 4 of the progressive measurability toolkit — so $\mathfrak{s}_{\sigma^{(k)}}$ is measurable for the $\sigma$-algebra of events prior to $\sigma^{(k)}$ by claim 4 of the stopping-time toolkit, and, $D_{k}$ belonging to that $\sigma$-algebra by claim 2 of the cascade lemma, so does $D^{-}_{k}$. Moreover $D^{-}_{k}\subseteq\Omega_{0}$ and $|\Sigma_{\sigma^{(k)}}-S_{\sigma^{(k)}}|=N^{-1/2}|\mathfrak{s}_{\sigma^{(k)}}|\le\varepsilon_{tg}$ on it. Claim 3 of the exit lemma, applied with $\varsigma=\sigma^{(k)}$ and $D=D^{-}_{k}$, therefore gives

$$\mathbb{E}\Bigl[\mathbf{1}_{D^{-}_{k}}\Bigl(N\int_{[\sigma^{(k)},T]}\mathcal{D}_{t}\,dt+N\mathcal{D}_{G}\Bigr)\Bigr]\ \ge\ -C^{\vee}_{tg}\,\mathbb{E}\bigl[\mathbf{1}_{D_{k}}|\mathfrak{s}_{\sigma^{(k)}}|^{2}\bigr]-C_{ns}\sqrt{N}\,P(D_{k})^{3/4},$$

where we enlarged $D^{-}_{k}$ to $D_{k}$ in the two right-hand terms, using that the integrand of the first is nonnegative and that $x\mapsto x^{3/4}$ is nondecreasing. Next, $D^{+}_{k}\subseteq\mathcal{N}$: on $D_{k}$ one has $\min(t_{k+1},\sigma^{(k)})=\sigma^{(k)}$, so P1 gives $|\mathfrak{s}_{\sigma^{(k)}}|\le\sqrt{N}(\varepsilon_{1}+Q)$, and $|\mathfrak{s}_{\sigma^{(k)}}|>\sqrt{N}\varepsilon_{tg}$ then forces $Q>\varepsilon_{tg}-\varepsilon_{1}\ge q_{0}$ by the third inequality of (SM). On $D^{+}_{k}$ we use the crude bounds of P2:

$$\mathbb{E}\Bigl[\mathbf{1}_{D^{+}_{k}}\Bigl(N\int_{[\sigma^{(k)},T]}\mathcal{D}_{t}\,dt+N\mathcal{D}_{G}\Bigr)\Bigr]\ \ge\ -\bigl(TC_{\mathcal{D}}+C_{\mathcal{D}G}\bigr)N\,P(D^{+}_{k}).$$

Summing over $k$, using claims 1(d) and 1(e) for the first two right-hand terms and the disjointness of the $D^{+}_{k}\subseteq\mathcal{N}$ for the third, gives claim 4.

Claim 5. By conclusion (a) of the squares theorem and $Z_{T}=\hat{F}$ from (H2), $\mathfrak{s}_{T}\cdot Z_{T}\mathfrak{s}_{T}=\tfrac{1}{2}\sum_{\gamma,\delta}F_{\gamma\delta}\mathfrak{s}^{\gamma}_{T}\mathfrak{s}^{\delta}_{T}$ at every point. By the terminal half of claim 1 of the coercivity lemma,

$$N\mathcal{D}_{G}\ \ge\ \mathfrak{s}_{T}\cdot Z_{T}\mathfrak{s}_{T}-\tfrac{1}{2}\,l\,\omega_{G}\bigl(d(\Sigma_{T},S_{T})\bigr)|\mathfrak{s}_{T}|^{2}$$

at every point of $\Omega$. On $G_{K}\setminus\mathcal{N}$ one has $d(\Sigma_{T},S_{T})=N^{-1/2}|\mathfrak{s}_{T}|\le\varepsilon_{1}+q_{0}\le\rho^{*}_{G}$ by P1 and the standing requirement, so the modulus factor is at most $\epsilon$ by the choice of $\rho^{*}_{G}$, whence $N\mathcal{D}_{G}\ge\mathfrak{s}_{T}\cdot Z_{T}\mathfrak{s}_{T}-\epsilon|\mathfrak{s}_{T}|^{2}$ there. On $G_{K}\cap\mathcal{N}$, P2 gives $N\mathcal{D}_{G}-\mathfrak{s}_{T}\cdot Z_{T}\mathfrak{s}_{T}\ge-(C_{\mathcal{D}G}+4lC_{Z})N$. Taking expectations over $G_{K}$ and using $\mathbb{E}[\mathbf{1}_{G_{K}}|\mathfrak{s}_{T}|^{2}]\le2C_{S}^{2}\mathcal{Z}+2c_{Q}^{1/2}\kappa_{0}^{1/2}$, which is claim 3 of the energy lemma, gives claim 5.

Claim 6. Fix $k$. The block lemma applies with $t_{\flat}=t_{k}$, $t_{\sharp}=t_{k+1}$, $\sigma=\sigma^{(k)}$ and $\mathcal{G}=G_{k}$: indeed $G_{k}\subseteq\Omega_{0}$ and $G_{k}\in\mathcal{F}^{\mathrm{sys}}_{t_{k}}$ and $\sigma^{(k)}\ge t_{k}$ by claim 2 of the cascade lemma, and the integrability hypothesis $\mathcal{A}_{2}<\infty$ of its adopted setting holds because $\mathcal{A}$ is compact, as recorded in the block lemma's own preamble. Claim 3 of the block lemma then reads

$$\int_{[t_{k},t_{k+1}]}\mathbb{E}\bigl[\mathbf{1}_{\mathcal{T}_{k}(s)}\mathfrak{h}_{s}\bigr]ds=\Gamma_{k}-\Gamma'_{k}+\int_{[t_{k},t_{k+1}]}\Bigl(\mathbb{E}\bigl[\mathbf{1}_{\mathcal{T}_{k}(s)}\bigl(u_{s}\cdot R_{s}u_{s}+2\mathfrak{s}_{s}\cdot Z_{s}e_{s}\bigr)\bigr]+\sum_{\gamma,\delta=1}^{l}Z^{\gamma\delta}_{s}\mathbb{E}\bigl[\mathbf{1}_{\mathcal{T}_{k}(s)}\Theta^{\gamma\delta}(\Sigma_{s},\alpha_{s})\bigr]\Bigr)ds,$$

where $\Gamma_{k}=\mathbb{E}[\mathbf{1}_{G_{k}}\mathfrak{s}_{t_{k}}\cdot Z_{t_{k}}\mathfrak{s}_{t_{k}}]$ and $\Gamma'_{k}=\mathbb{E}[\mathbf{1}_{G_{k}}\mathfrak{s}^{\sharp}_{k}\cdot Z^{\sharp}_{k}\mathfrak{s}^{\sharp}_{k}]$.

Write $\int_{[t_{k},t_{k+1}]}\mathbb{E}[\mathbf{1}_{\mathcal{T}_{k}(s)}N\mathcal{D}_{s}]ds=\int_{[t_{k},t_{k+1}]}\mathbb{E}[\mathbf{1}_{\mathcal{T}_{k}(s)}\mathfrak{h}_{s}]ds+\mathcal{R}$ with $\mathcal{R}=\int_{[t_{k},t_{k+1}]}\mathbb{E}[\mathbf{1}_{\mathcal{T}_{k}(s)}(N\mathcal{D}_{s}-\mathfrak{h}_{s})]ds$, and split $\mathcal{T}_{k}(s)$ into $\mathcal{T}^{\mathrm{nr}}_{k}(s)\setminus\mathcal{N}$, $\mathcal{T}^{\mathrm{fr}}_{k}(s)\setminus\mathcal{N}$ and $\mathcal{T}_{k}(s)\cap\mathcal{N}$.

Near part. On $\mathcal{T}^{\mathrm{nr}}_{k}(s)\setminus\mathcal{N}$ one has $N^{-1/2}|\mathfrak{s}_{s}|\le\varepsilon_{1}+q_{0}$ by P1 and $N^{-1/2}|\mathfrak{a}_{s}|\le\varrho$, so $\rho_{s}^{2}=N^{-1}(|\mathfrak{s}_{s}|^{2}+|\mathfrak{a}_{s}|^{2})\le(\varepsilon_{1}+q_{0})^{2}+\varrho^{2}\le\rho_{\eta}^{2}$ by (SN), hence $\rho_{s}\le\rho_{\eta}$ and, by claim 1 of the coercivity lemma and the defining property of $\rho_{\eta}$, $N\mathcal{D}_{s}-\mathfrak{h}_{s}\ge-\eta|\mathfrak{z}_{s}|^{2}$ there.

Far part. On $\mathcal{T}^{\mathrm{fr}}_{k}(s)\setminus\mathcal{N}$, claim 2 of the energy lemma gives $N\mathcal{D}_{s}\ge c_{\star}|\mathfrak{a}_{s}|^{2}\ge0$ (its hypotheses are the standing ones, and $s\in[t_{k},T]$ with $s<\sigma^{(k)}$ and $\omega\in G_{k}\setminus\mathcal{N}$), so $N\mathcal{D}_{s}-\mathfrak{h}_{s}\ge-\mathfrak{h}_{s}=\Psi_{s}-u_{s}\cdot R_{s}u_{s}$. By conclusion (a) of the squares theorem, $\mathfrak{h}_{s}=\mathfrak{s}_{s}\cdot Q_{s}\mathfrak{s}_{s}+\mathfrak{s}_{s}\cdot V_{s}\mathfrak{a}_{s}+\mathfrak{a}_{s}\cdot R_{s}\mathfrak{a}_{s}$, while expanding $u_{s}=\mathfrak{a}_{s}+R_{s}^{-1}W_{s}^{T}\mathfrak{s}_{s}$ with $R_{s}$ symmetric gives $u_{s}\cdot R_{s}u_{s}=\mathfrak{a}_{s}\cdot R_{s}\mathfrak{a}_{s}+2\mathfrak{a}_{s}\cdot W_{s}^{T}\mathfrak{s}_{s}+\mathfrak{s}_{s}\cdot W_{s}R_{s}^{-1}W_{s}^{T}\mathfrak{s}_{s}$; subtracting, the two $\mathfrak{a}$-quadratic terms cancel and

$$\Psi_{s}=2\,\mathfrak{a}_{s}\cdot W_{s}^{T}\mathfrak{s}_{s}-\mathfrak{s}_{s}\cdot V_{s}\mathfrak{a}_{s}+\mathfrak{s}_{s}\cdot\bigl(W_{s}R_{s}^{-1}W_{s}^{T}-Q_{s}\bigr)\mathfrak{s}_{s},$$

every summand carrying a factor $\mathfrak{s}_{s}$. By the entry estimate of P2 and the bounds $C_{M}$, $|\Psi_{s}|\le3\sqrt{lm}\,C_{M}|\mathfrak{s}_{s}||\mathfrak{a}_{s}|+2l\,C_{M}|\mathfrak{s}_{s}|^{2}\le C_{\Psi}(|\mathfrak{s}_{s}||\mathfrak{a}_{s}|+|\mathfrak{s}_{s}|^{2})$. Since $u_{s}\cdot R_{s}u_{s}\ge0$ by (H1), enlarging $\mathcal{T}^{\mathrm{fr}}_{k}(s)\setminus\mathcal{N}$ to $\mathcal{T}^{\mathrm{fr}}_{k}(s)$ in the subtracted term only decreases the bound, so the far part contributes at least

$$-\int_{[t_{k},t_{k+1}]}\mathbb{E}\bigl[\mathbf{1}_{\mathcal{T}^{\mathrm{fr}}_{k}(s)}u_{s}\cdot R_{s}u_{s}\bigr]ds-C_{\Psi}\int_{[t_{k},t_{k+1}]}\mathbb{E}\bigl[\mathbf{1}_{\mathcal{T}^{\mathrm{fr}}_{k}(s)\setminus\mathcal{N}}\bigl(|\mathfrak{s}_{s}||\mathfrak{a}_{s}|+|\mathfrak{s}_{s}|^{2}\bigr)\bigr]ds .$$

Noise part. On $\mathcal{T}_{k}(s)\cap\mathcal{N}$, P2 gives $N\mathcal{D}_{s}-\mathfrak{h}_{s}\ge-(C_{\mathcal{D}}+4(1+R^{2})C_{\mathfrak{h}})N$, contributing at least $-(C_{\mathcal{D}}+4(1+R^{2})C_{\mathfrak{h}})Nh_{k}P(\mathcal{N})$.

The linearization term. By P2, $2|\mathfrak{s}_{s}\cdot Z_{s}e_{s}|\le2lC_{Z}c_{e}N^{-1/2}|\mathfrak{s}_{s}||\mathfrak{z}_{s}|^{2}$, which off $\mathcal{N}$ and on $\mathcal{T}_{k}(s)$ is at most $2lC_{Z}c_{e}(\varepsilon_{1}+q_{0})|\mathfrak{z}_{s}|^{2}$ by P1, and everywhere is at most $16lC_{Z}c_{e}(1+R^{2})N$. Hence

$$2\int_{[t_{k},t_{k+1}]}\bigl|\mathbb{E}[\mathbf{1}_{\mathcal{T}_{k}(s)}\mathfrak{s}_{s}\cdot Z_{s}e_{s}]\bigr|ds\le2lC_{Z}c_{e}(\varepsilon_{1}+q_{0})\int_{[t_{k},t_{k+1}]}\mathbb{E}\bigl[\mathbf{1}_{\mathcal{T}_{k}(s)}|\mathfrak{z}_{s}|^{2}\bigr]ds+16lC_{Z}c_{e}(1+R^{2})N\,h_{k}P(\mathcal{N}).$$

Combining: the block identity contributes $\int\mathbb{E}[\mathbf{1}_{\mathcal{T}_{k}(s)}u_{s}\cdot R_{s}u_{s}]ds$, from which the far part subtracts $\int\mathbb{E}[\mathbf{1}_{\mathcal{T}^{\mathrm{fr}}_{k}(s)}u_{s}\cdot R_{s}u_{s}]ds$, leaving exactly $\int\mathbb{E}[\mathbf{1}_{\mathcal{T}^{\mathrm{nr}}_{k}(s)}u_{s}\cdot R_{s}u_{s}]ds$, since $\mathcal{T}_{k}(s)$ is the disjoint union of $\mathcal{T}^{\mathrm{nr}}_{k}(s)$ and $\mathcal{T}^{\mathrm{fr}}_{k}(s)$. Collecting the four error contributions, and bounding the near part's $-\eta\int\mathbb{E}[\mathbf{1}_{\mathcal{T}^{\mathrm{nr}}_{k}(s)\setminus\mathcal{N}}|\mathfrak{z}_{s}|^{2}]$ below by $-\eta\int\mathbb{E}[\mathbf{1}_{\mathcal{T}_{k}(s)}|\mathfrak{z}_{s}|^{2}]$, yields claim 6 with $\mathcal{X}_{k}$ exactly as displayed there.

Claim 7. Sum claim 6 over $k$ and substitute into claim 3, then add claims 4 and 5.

Telescoping. For each $k$, $G_{k}=G_{k+1}\cup D_{k}$ disjointly (P0); on $G_{k+1}$ one has $\sigma^{(k)}\ge t_{k+1}$, so $\mathfrak{s}^{\sharp}_{k}=\mathfrak{s}_{t_{k+1}}$ and $Z^{\sharp}_{k}=Z_{t_{k+1}}$ there, while on $D_{k}$ one has $\sigma^{(k)}<t_{k+1}$, so $\mathfrak{s}^{\sharp}_{k}=\mathfrak{s}_{\sigma^{(k)}}$ and $Z^{\sharp}_{k}=Z_{\sigma^{(k)}}$ there. Hence $\Gamma'_{k}=\Gamma_{k+1}+\mathbb{E}[\mathbf{1}_{D_{k}}\mathfrak{s}_{\sigma^{(k)}}\cdot Z_{\sigma^{(k)}}\mathfrak{s}_{\sigma^{(k)}}]$ and

$$\sum_{k=0}^{K-1}(\Gamma_{k}-\Gamma'_{k})=\Gamma_{0}-\Gamma_{K}-\sum_{k=0}^{K-1}\mathbb{E}\bigl[\mathbf{1}_{D_{k}}\mathfrak{s}_{\sigma^{(k)}}\cdot Z_{\sigma^{(k)}}\mathfrak{s}_{\sigma^{(k)}}\bigr],$$

with $\Gamma_{0}=\mathbb{E}[\mathbf{1}_{\Omega_{0}}\mathfrak{s}_{0}\cdot Z_{0}\mathfrak{s}_{0}]$ (as $G_{0}=\Omega_{0}$ and $t_{0}=0$) and $\Gamma_{K}=\mathbb{E}[\mathbf{1}_{G_{K}}\mathfrak{s}_{T}\cdot Z_{T}\mathfrak{s}_{T}]$ (as $t_{K}=T$). By P2 and claim 1(e), $-\sum_{k}\mathbb{E}[\mathbf{1}_{D_{k}}\mathfrak{s}_{\sigma^{(k)}}\cdot Z_{\sigma^{(k)}}\mathfrak{s}_{\sigma^{(k)}}]\ge-lC_{Z}(2\Lambda_{\star}\mathcal{Z}+2c_{Q}^{1/2}\kappa_{0}^{1/2}\mathcal{P}^{1/2})$. By claim 5, $-\Gamma_{K}+\mathbb{E}[\mathbf{1}_{G_{K}}N\mathcal{D}_{G}]\ge-2\epsilon(C_{S}^{2}\mathcal{Z}+c_{Q}^{1/2}\kappa_{0}^{1/2})-(C_{\mathcal{D}G}+4lC_{Z})NP(\mathcal{N})$.

The error sum. By claim 1(a) and 1(b), $\sum_{k}\int_{[t_{k},t_{k+1}]}\mathbb{E}[\mathbf{1}_{\mathcal{T}_{k}(s)}|\mathfrak{z}_{s}|^{2}]ds=\mathcal{S}^{\mathrm{tr}}+\mathcal{Z}$. By P1, on $\mathcal{T}^{\mathrm{fr}}_{k}(s)\setminus\mathcal{N}$ one has $|\mathfrak{s}_{s}|\le\sqrt{N}(\varepsilon_{1}+q_{0})$, so by claim 1(c)

$$\sum_{k}\int_{[t_{k},t_{k+1}]}\mathbb{E}\bigl[\mathbf{1}_{\mathcal{T}^{\mathrm{fr}}_{k}(s)\setminus\mathcal{N}}(|\mathfrak{s}_{s}||\mathfrak{a}_{s}|+|\mathfrak{s}_{s}|^{2})\bigr]ds\le\sqrt{N}(\varepsilon_{1}+q_{0})\frac{\mathcal{Z}}{\sqrt{N}\varrho}+N(\varepsilon_{1}+q_{0})^{2}\frac{\mathcal{Z}}{N\varrho^{2}}=\mathcal{Z}\Bigl(\frac{\varepsilon_{1}+q_{0}}{\varrho}+\frac{(\varepsilon_{1}+q_{0})^{2}}{\varrho^{2}}\Bigr).$$

Since $\sum_{k}h_{k}=T$, the noise contributions of the $\mathcal{X}_{k}$ sum to at most $(C_{\mathcal{D}}+4(1+R^{2})(C_{\mathfrak{h}}+4lC_{Z}c_{e}))TNP(\mathcal{N})$. Hence $\sum_{k}\mathcal{X}_{k}$ is at most the sum of the first two lines of $\mathcal{B}_{N}$ restricted to their $\eta$-, $C_{\Psi}$- and noise-parts.

The covariance term. By clause (a) of the covariance lemma, $|\Theta^{\gamma\delta}(\Sigma_{s},\alpha_{s})-\Theta^{\star\gamma\delta}_{s}|\le c_{\Theta}N^{-1/2}|\mathfrak{z}_{s}|$ at every point; and $|\Theta^{\star\gamma\delta}_{s}|\le\bar{\Theta}$. Therefore, using $|Z^{\gamma\delta}_{s}|\le C_{Z}$ and summing $l^{2}$ index pairs,

$$\Bigl|\sum_{k}\int_{[t_{k},t_{k+1}]}\sum_{\gamma,\delta}Z^{\gamma\delta}_{s}\mathbb{E}[\mathbf{1}_{\mathcal{T}_{k}(s)}\Theta^{\gamma\delta}(\Sigma_{s},\alpha_{s})]ds-\int_{[0,T]}\sum_{\gamma,\delta}Z^{\gamma\delta}_{s}\Theta^{\star\gamma\delta}_{s}ds\Bigr|\ \le\ l^{2}C_{Z}\Bigl(c_{\Theta}N^{-1/2}\mathcal{I}_{1}+\bar{\Theta}\,\mathcal{I}_{2}\Bigr),$$

where $\mathcal{I}_{1}=\sum_{k}\int_{[t_{k},t_{k+1}]}\mathbb{E}[\mathbf{1}_{\mathcal{T}_{k}(s)}|\mathfrak{z}_{s}|]ds$ and $\mathcal{I}_{2}=\sum_{k}\int_{[t_{k},t_{k+1}]}\mathbb{E}[1-\mathbf{1}_{\mathcal{T}_{k}(s)}]ds$ (the letters $\mathcal{I}_{1},\mathcal{I}_{2}$ being local to this paragraph), the intervals $[t_{k},t_{k+1}]$ covering $[0,T]$ with overlaps of Lebesgue measure zero. Since $x\le\tfrac{1}{2}(1+x^{2})$ for every real $x\ge0$, which is $(1-x)^{2}\ge0$ rearranged, one has $\mathbf{1}_{\mathcal{T}_{k}(s)}|\mathfrak{z}_{s}|\le\tfrac{1}{2}\mathbf{1}_{\mathcal{T}_{k}(s)}(1+|\mathfrak{z}_{s}|^{2})$ pointwise, so $\mathcal{I}_{1}\le\tfrac{1}{2}(T+\mathcal{S}^{\mathrm{tr}}+\mathcal{Z})$ by claims 1(a) and 1(b) and the bound $\sum_{k}\int_{[t_{k},t_{k+1}]}\mathbb{E}[\mathbf{1}_{\mathcal{T}_{k}(s)}]ds\le T$; and $\mathcal{I}_{2}\le T\mathcal{P}$ by claim 1(d).

Assembling the three displays with claims 3, 4, 5 and 6, and recognizing the total noise coefficient $2TC_{\mathcal{D}}+2C_{\mathcal{D}G}+4lC_{Z}+4T(1+R^{2})(C_{\mathfrak{h}}+4lC_{Z}c_{e})=C_{\mathcal{N}}$, gives the first inequality of claim 7. The second follows because each summand $\int_{[t_{k},t_{k+1}]}\mathbb{E}[\mathbf{1}_{\mathcal{T}^{\mathrm{nr}}_{k}(s)}u_{s}\cdot R_{s}u_{s}]ds$ is nonnegative by claim 2. $\square$