Proof 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

Throughout, labels are written $c=(\sigma,\gamma)$ and $q=(c,j)$ denotes an element of $\mathsf{L}$. We use the following facts.

(F1) $|v_c|=\sqrt{2}$ for every label ($v_c=\delta_\gamma-\delta_\sigma$ has two coordinates $\pm1$ and the others $0$; claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$), and $|\sum_ia_iy_i|\le\sum_i|a_i|\,|y_i|$ for vectors $y_i\in\mathbb{R}^l$ and reals $a_i$ (claims 6 and 5 of that lemma, by induction). For reals $u,v$ and $\zeta>0$, $(u+v)^{2}\le(1+\zeta)u^{2}+(1+1/\zeta)v^{2}$, since $2|uv|\le\zeta u^{2}+v^{2}/\zeta$ by $(\sqrt{\zeta}|u|-|v|/\sqrt{\zeta})^{2}\ge0$.

(F2) The simplex $\Delta^l$ is convex (for $x,x'\in\Delta^l$ and $\tau\in[0,1]$ the point $x+\tau(x'-x)$ has nonnegative coordinates summing to $1$), and $\Delta^l\subset\tilde{U}$ by Twice Continuously Differentiable Extension of an Observation-Rate Family; so the segment between two points of $\Delta^l$ lies in $\Delta^l\subset\tilde{U}$, and the Euclidean distance between them is the norm of their difference.

(F3) Fix $\omega\in\Omega$ and $r\in\mathbf{R}$. By claims 2(a) and 2(b) 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 (applied through claim 3 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record to the clocks $\mathsf{P}^{(y)}$ and $\mathsf{P}^{\sharp}$), for each $y\in\mathbb{N}_0^{\mathsf{L}}$ the path $t\mapsto\bar\Sigma^{(y),r}_t(\omega)$ is constant on each of finitely many intervals partitioning $[0,T]$, and $t\mapsto\bar\Sigma^{(y),r}_{t-}(\omega)$ differs from it at only finitely many $t$; the same holds for $\bar\Sigma^{\sharp,r}(\omega)$. Taking a common refinement of the finitely many partitions belonging to $\bar\Sigma^{\sharp,r}(\omega)$ and to $\bar\Sigma^{(\mathsf{K}(\omega)-\mathsf{m}e_q),r}(\omega)$ for those $q\in\mathsf{L}$ with $\mathsf{K}_q(\omega)\ge\mathsf{m}$ (all of $\mathsf{L}$ when $\omega\in G^{\mathsf{m}}$), and splitting off each of the finitely many exceptional points as a degenerate interval $[u,u]$ (on which every function is constant), we obtain a finite family of pairwise disjoint intervals with union $[0,T]$ on each of which all these paths and all their left-limit versions are constant, with values in the finite set $\mathbb{G}_N\subseteq\Delta^l$. Consequently, if $\Phi$ is any real function of these finitely many path values, the map $t\mapsto\Phi(\cdots)$ is a finite sum of constants times indicators of intervals, hence bounded and measurable (intervals are Borel sets by Borel Sigma-Algebra on the Real Line; claims 1 and 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions). All maps of $t$ considered below are of this form, possibly multiplied by the components of $\hat\psi_t$ (themselves of this form, as linear combinations of differences of path values) or composed with measurable functions of $t$, and are bounded and measurable by claims 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. In particular, for fixed $(\omega,r)$ and $\upsilon$, the maps $t\mapsto\lambda^{\sharp,\omega,\upsilon}_t(r)=N\tilde{b}^\upsilon(\bar\Sigma^{\sharp,r}_{t-}(\omega))$ and $t\mapsto\lambda^{-q,\omega,\upsilon}_t(r)$ (equal to $N\tilde{b}^\upsilon(\bar\Sigma^{(\mathsf{K}(\omega)-\mathsf{m}e_q),r}_{t-}(\omega))$ or to $\lambda^{\sharp,\omega,\upsilon}_t(r)$; claim 3 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 and the definition of the effective removed intensities) are bounded and measurable, with values in $[N\underline{b},N\tilde{B}]$ for the former (claim 1 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 in $[0,N\tilde{B}]$ for the latter.

Step 1 (Claim 1). Fix $\omega\in G^{\mathsf{m}}$, $r\in\mathsf{T}_\omega$ and $q=(c,j)\in\mathsf{L}$; put $y=y^{(q)}=\mathsf{K}(\omega)-\mathsf{m}e_q$, which lies in $\mathbb{N}_0^{\mathsf{L}}$ because its $q$-coordinate is $y_q=\mathsf{K}_q(\omega)-\mathsf{m}\ge0$ (its other coordinates being those of $\mathsf{K}(\omega)$), and satisfies $y+\mathsf{m}e_q=\mathsf{K}(\omega)$. First, $G^{\mathsf{m}}=G_{L,D}\cap\bigcap_{q\in\mathsf{L}}\{\mathsf{K}_q\ge\mathsf{m}\}$ belongs to $\mathcal{F}$, since $G_{L,D}\in\mathcal{F}$ by claim 1 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 each $\mathsf{K}_q$ is a random variable by claim 1 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record. Since $G_{L,D}\subseteq\Omega^{U}_0$, claim 2 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record gives $\mathsf{P}^{\sharp,c'}(\omega)=\mathsf{P}^{(\mathsf{K}(\omega)),c'}(\omega)=\mathsf{P}^{(y),c'}(\omega)$ for $c'\neq c$ and

where the $\mathsf{m}$ inserted points $U^{q}_i(\omega)$, $y_q<i\le y_q+\mathsf{m}=\mathsf{K}_q(\omega)$, lie in $I_q\subseteq(0,R]$, are pairwise distinct, and differ from every point $U^{c,j'}_{i'}(\omega)$ with $1\le j'\le J_c$ and $1\le i'\le y_{c,j'}$. List the inserted points increasingly as $0<u_1<\dots<u_{\mathsf{m}}$. By the formula defining the deterministic-count clocks in The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record, on $\Omega^{U}_0$ one has $\mathsf{P}^{(y),c}_u(\omega)=\sum_{j'=1}^{J_c}\sum_{i'=1}^{y_{c,j'}}\mathbf{1}\{U^{c,j'}_{i'}(\omega)\le u\}$, whose value at $u$ exceeds its left limit $\sum_{j'}\sum_{i'\le y_{c,j'}}\mathbf{1}\{U^{c,j'}_{i'}(\omega)<u\}$ exactly when $u$ equals one of the points $U^{c,j'}_{i'}(\omega)$, $i'\le y_{c,j'}$; so the jump times of the counting path $\mathsf{P}^{(y),c}(\omega)$ are these points, and none of $u_1,\dots,u_{\mathsf{m}}$ is among them. Hence $\mathsf{P}^{\sharp}(\omega)$ is exactly the perturbed clock family $p^{+}$ of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect for $p=\mathsf{P}^{(y)}(\omega)$, the label $c_0=c$ and the insertion points $u_1,\dots,u_{\mathsf{m}}$.

Since $r\in\mathsf{T}_\omega$ and $\mathsf{K}_q(\omega)\ge\mathsf{m}$, the definition of the tracked records gives $(r,\omega)\in\mathsf{G}^{\sharp}$ and $(r,\omega)\in\mathsf{G}^{(y)}$, that is, the data $(\mathsf{P}^{\sharp}(\omega),a^{r},x_0)$ and $(\mathsf{P}^{(y)}(\omega),a^{r},x_0)$ are conflict-free (claim 1 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 as applied in claim 3 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record), and by claim 2(c) 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 the paths $\bar\Sigma^{\sharp,r}(\omega)$ and $\bar\Sigma^{(y),r}(\omega)$ are the open-loop aggregate solutions for these data. Thus, in the notation of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect, $\Sigma^{+}=\bar\Sigma^{\sharp,r}(\omega)$, $\Sigma=\bar\Sigma^{(y),r}(\omega)$, $Y=Y^{(q)}$, $\mathsf{C}^{+,c'}=\mathsf{C}^{\sharp,c'}$ and $\mathsf{C}^{c'}=\mathsf{C}^{(q),c'}$. The real numbers $R\ge NBT$, $L\ge0$, $D\ge0$ are those of the setting; the membership $\omega\in G_{L,D}$ states precisely hypothesis (D$^{+}$) of Removal Form of the Shared-Clock Insertion Response: Discrepancy on the Enlarged Clock Family, the Base-Clock Insertion Count, and the Linearisation Defect Along Either Path for $p^{+}=\mathsf{P}^{\sharp}(\omega)$; the constant $A_0^{+}$ of that lemma is $\sqrt{2}(\mathsf{m}+l(l-1)(D+\mathsf{m}))\exp(\sqrt{2}l(l-1)\Lambda_1T)=A_0$; and hypothesis (W) gives $\Lambda_1TA_0<L$. Therefore claims 1 and 3 of Removal Form of the Shared-Clock Insertion Response: Discrepancy on the Enlarged Clock Family, the Base-Clock Insertion Count, and the Linearisation Defect Along Either Path apply: $|Y^{(q)}_t|\le A_0$ and $|\mathsf{C}^{\sharp,c'}_t-\mathsf{C}^{(q),c'}_t|\le\Lambda_1TA_0$ for all $t$ and $c'$, and, since the base-clock insertion count of that lemma is $\#\{k:u_k\le\mathsf{C}^{(q),c}_t\}=\iota^{(q)}_t$, the first identity of its claim 3 (linearisation along $\Sigma^{+}=\bar\Sigma^{\sharp}$) is the displayed equation for $Y^{(q)}$ with the stated bound on $\mathsf{d}^{(q)}_t$. The bound $|Y^{(q)}_{t-}|\le A_0$ is claim 2 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 claim 1 of Removal Form of the Shared-Clock Insertion Response: Discrepancy on the Enlarged Clock Family, the Base-Clock Insertion Count, and the Linearisation Defect Along Either Path (which imports claim 1 of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect) the map $s\mapsto Y^{(q)}_s$ is bounded with measurable components, and by claim 3 of the former so is $s\mapsto\mathcal{E}(\bar\Sigma^{\sharp}_s,a^{r}_s)Y^{(q)}_s$.

Now $\hat\psi=\frac1N\sum_qw_qY^{(q)}$ and $\mathcal{E}(\bar\Sigma^{\sharp}_s,a^{r}_s)\hat\psi_s=\frac1N\sum_qw_q\mathcal{E}(\bar\Sigma^{\sharp}_s,a^{r}_s)Y^{(q)}_s$ (the matrix-vector product being linear in the vector) are finite linear combinations of bounded maps with measurable components, hence bounded with measurable components (claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions), and $|\hat\psi_t|\le\frac1N\sum_q|w_q|A_0$ by (F1). For $F$: $t\mapsto\mathsf{C}^{(q),c}_t$ is nondecreasing (claim 2(a) of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution) and $x\mapsto\#\{i:\mathsf{K}_q-\mathsf{m}<i\le\mathsf{K}_q,\ U^{q}_i\le x\}$ is nondecreasing, so $t\mapsto\iota^{(q)}_t$ is nondecreasing with values in $\{0,\dots,\mathsf{m}\}$; each level set $\{t:\iota^{(q)}_t\ge k\}$ is then an interval, so $\iota^{(q)}=\sum_{k=1}^{\mathsf{m}}\mathbf{1}\{\iota^{(q)}\ge k\}$ is a finite sum of indicators of intervals, hence measurable, and $F$ has bounded measurable components. Multiplying the equation for $Y^{(q)}$ by $w_q/N$, summing over $q$, and using linearity of the integral componentwise gives $\hat\psi_t=F_t+\int_{[0,t]}\mathcal{E}(\bar\Sigma^{\sharp}_s,a^{r}_s)\hat\psi_s\,ds+\hat{\mathsf{d}}_t$ with $\hat{\mathsf{d}}_t=\frac1N\sum_qw_q\mathsf{d}^{(q)}_t$, which is the defining identity of $\hat{\mathsf{d}}_t$; by (F1), $|\hat{\mathsf{d}}_t|\le\frac1N\sum_q|w_q|\,\sqrt{2}l(l-1)(D+\Lambda_2TA_0^{2}/N)$, as claimed.

Step 2 (Claim 2). Fix $\omega\in G^{\mathsf{m}}$, $r\in\mathsf{T}_\omega$ and $t\in[0,T]$. For $q=(c,j)$ define $n_q:[0,R]\to\{0,\dots,\mathsf{m}\}$ by $n_q(x)=\#\{i:\mathsf{K}_q(\omega)-\mathsf{m}<i\le\mathsf{K}_q(\omega),\ U^{q}_i(\omega)\le x\}$. The $\mathsf{m}$ points counted lie in $I_q=(b^{c}_{j-1},b^{c}_j]$ (Step 1), so $n_q(x)=0$ for $x\le b^{c}_{j-1}$ and $n_q(x)=\mathsf{m}$ for $x\ge b^{c}_j$: $n_q$ is a cell counter in the sense 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, and $\iota^{(q)}_t=n_q(x_q)$ with $x_q=\mathsf{C}^{(q),c}_t$. All consumed clock times lie in $[0,NBT]\subseteq[0,R]$ by claim 2(a) of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution. Put $x'_c=\mathsf{C}^{\sharp,c}_t\in[0,R]$ for each label $c$. The hypotheses 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 hold for the present $l$, $m$, $N$, $\mathsf{m}$, $T$, $B$, $R\ge NBT$, cells, $\phi_c$, the profile $\varpi$ (in the role of its $\lambda$) and $\Lambda$, so its claim 4 with the cell counters $(n_q)_q$ and the family $(x'_c)_c$ gives

It remains to bound $F_t-\frac1N\sum_cv_c\sum_jw_{c,j}n_{c,j}(x'_c)=\frac1N\sum_cv_c\sum_jw_{c,j}\bigl(n_{c,j}(x_{c,j})-n_{c,j}(x'_c)\bigr)$. Fix $c$ and put $\Xi=\Lambda_1TA_0$ (a real number, not to be confused with the simplex $\Delta^l$), so that $|x_{c,j}-x'_c|\le\Xi$ for every $j$ by claim 1. If $n_{c,j}(x_{c,j})\neq n_{c,j}(x'_c)$, then some counted point $U^{c,j}_i(\omega)$ satisfies $\min(x_{c,j},x'_c)<U^{c,j}_i(\omega)\le\max(x_{c,j},x'_c)$, so it lies in $[x'_c-\Xi,x'_c+\Xi]$, and since it also lies in $I_{c,j}$, the cell $I_{c,j}$ meets $[x'_c-\Xi,x'_c+\Xi]$. Let $\mathsf{J}_c$ be the set of $j\in\{1,\dots,J_c\}$ for which $I_{c,j}$ meets this interval; only $j\in\mathsf{J}_c$ contribute to the sum below, so if $\mathsf{J}_c$ is empty that sum vanishes and the bound is trivial. If $\mathsf{J}_c$ is nonempty, let $j_1$ and $j_2$ be its least and greatest elements, and pick $z_1\in I_{c,j_1}\cap[x'_c-\Xi,x'_c+\Xi]$ and $z_2\in I_{c,j_2}\cap[x'_c-\Xi,x'_c+\Xi]$. For $j_1<j<j_2$ one has $z_1\le b^{c}_{j_1}\le b^{c}_{j-1}<b^{c}_j\le b^{c}_{j_2-1}<z_2$, so $b^{c}_j\in I_{c,j}\cap[x'_c-\Xi,x'_c+\Xi]$; hence $\mathsf{J}_c=\{j_1,\dots,j_2\}$. Moreover $b^{c}_{j_1}\ge x'_c-\Delta$ (the right endpoint of $I_{c,j_1}$ is at least a point of $I_{c,j_1}\cap[x'_c-\Xi,x'_c+\Xi]$) and $b^{c}_{j_2-1}\le x'_c+\Delta$ (the left endpoint of $I_{c,j_2}$ is below such a point), so

(the middle bracket is at most $2\Xi$, and is negative when $j_1=j_2$). Since every cell has length at least $\mu_{\min}$, which is positive because all cell lengths are, the number of elements of $\mathsf{J}_c$ is at most $2(\Xi+\mu_{\max})/\mu_{\min}$. As $|n_{c,j}(x_{c,j})-n_{c,j}(x'_c)|\le\mathsf{m}$ and, by claim 2 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, $|w_{c,j}|\le\sqrt{2}\Lambda\mu_{c,j}/\mathsf{m}\le\sqrt{2}\Lambda\mu_{\max}/\mathsf{m}$,

Multiplying by $|v_c|/N=\sqrt{2}/N$, summing over the $l(l-1)$ labels with (F1), and adding the bound from claim 4 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 by the triangle inequality gives claim 2.

Step 3 (Claim 3). Fix $\omega\in\Omega$ and $r\in\mathbf{R}$. By claim 1 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, $\lambda^{\sharp,\omega}$ is a causal intensity with bound $N\tilde{B}$ and $\lambda^{\sharp,\omega,\upsilon}\ge N\underline{b}>0$ (hypothesis (OC)), each $\lambda^{-q,\omega}$ is a causal intensity with bound $N\tilde{B}$, and Pair Expansion of the Weighted Likelihood-Ratio Square: Pair Covariances of Causal-Intensity Likelihoods, the Exact Expansion, and Its First-Order Form applies with base $\lambda^{\sharp,\omega}$ and perturbed intensities $(\lambda^{-q,\omega})_q$; its pair exponents are

real numbers by its claim 1. By (F3) the maps $t\mapsto\lambda^{\sharp,\omega,\upsilon}_t(r)$ and $t\mapsto\lambda^{-q,\omega,\upsilon}_t(r)$ are bounded and measurable, the denominators lie in $[N\underline{b},N\tilde{B}]$, and reciprocals of such maps are bounded and measurable (compose with the sequentially continuous map $z\mapsto1/z$ on $[N\underline{b},\infty)$, Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable); so every integrand below is bounded and measurable, hence integrable on $[0,T]$. Multiplying by $w_qw_{q'}$, summing over $q,q'$, using linearity of the integral and the identity $\sum_{q}\sum_{q'}w_qw_{q'}a_qa_{q'}=(\sum_qw_qa_q)^{2}$ for real numbers $a_q$ gives the displayed formula for $\mathcal{Q}^{\omega}(r)$; its integrand is nonnegative, so $\mathcal{Q}^{\omega}(r)\ge0$ by monotonicity.

Now let $\omega\in G^{\mathsf{m}}$ and $r\in\mathsf{T}_\omega$, and let $\zeta>0$ be real. Fix $t\in[0,T]$, $\upsilon$ and $q=(c,j)$, and put $x=\bar\Sigma^{\sharp}_{t-}$ and $x'=\bar\Sigma^{(y^{(q)})}_{t-}$, both in $\mathbb{G}_N\subseteq\Delta^l$, with $x'-x=-Y^{(q)}_{t-}/N$. By claim 3 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 (through claim 3 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record) and the definition of the effective removed intensities (note $\mathsf{K}_q(\omega)\ge\mathsf{m}$), $\lambda^{\sharp,\upsilon}_t(r)=N\tilde{b}^\upsilon(x)=N\bar{\tilde{b}}^\upsilon(x)$ and $\lambda^{-q,\upsilon}_t(r)=N\tilde{b}^\upsilon(x')=N\bar{\tilde{b}}^\upsilon(x')$, using that $\bar{\tilde{b}}$ agrees with $\tilde{b}$ on $\Delta^l$ (claim (i) of Regularity and Derivative Bounds of the Extended Aggregate Observation Drift). Apply part (ii) of Multivariate Taylor Expansion with Uniform Second-Order Remainder on the open set $\tilde{U}\subseteq\mathbb{R}^l$ to $f=\bar{\tilde{b}}^\upsilon$, which is of class $C^2$ on $\tilde{U}$ by claim (i) of Regularity and Derivative Bounds of the Extended Aggregate Observation Drift, with the points $x$ and $x'$ (segment in $\Delta^l\subset\tilde{U}$ by (F2)) and $M_2=3\tilde{K}$ (claim (iii) of Regularity and Derivative Bounds of the Extended Aggregate Observation Drift bounds the second partial derivatives by $3\tilde{K}$ on $\Delta^l$, hence on the segment): with $\mathsf{h}=x'-x$ (coordinates $\mathsf{h}^{\gamma}$) and, the partial derivatives $\partial_i$ of the Taylor lemma being the present $\partial_\gamma$ ($n=l$), $\sum_{\gamma=1}^{l}\partial_\gamma\bar{\tilde{b}}^\upsilon(x)\mathsf{h}^{\gamma}=g_\upsilon(x)\cdot\mathsf{h}$,

Multiplying by $N$ and using $N\mathsf{h}=-Y^{(q)}_{t-}$ and $|Y^{(q)}_{t-}|\le A_0$ (claim 1), we get $\lambda^{-q,\upsilon}_t(r)-\lambda^{\sharp,\upsilon}_t(r)=-g_\upsilon(x)\cdot Y^{(q)}_{t-}+\mathrm{rem}^{(q),\upsilon}_t$, where $\mathrm{rem}^{(q),\upsilon}_t=N\bigl(\bar{\tilde{b}}^\upsilon(x')-\bar{\tilde{b}}^\upsilon(x)\bigr)+g_\upsilon(x)\cdot Y^{(q)}_{t-}$ satisfies $|\mathrm{rem}^{(q),\upsilon}_t|\le3l\tilde{K}A_0^{2}/(2N)$. Multiplying by $w_q$ and summing over $q$,

by the bilinearity of the dot product. By (F1) with $u=-Ng_\upsilon(x)\cdot\hat\psi_{t-}$ and $v=\mathrm{Rem}^{\upsilon}_t$, and since $\lambda^{\sharp,\upsilon}_t(r)=N\tilde{b}^\upsilon(x)\ge N\underline{b}$,

Summing over $\upsilon$ and using the identity $z\cdot(\tilde{D}(x)z)=\sum_{\gamma,\delta}\tilde{D}^{\gamma\delta}(x)z^{\gamma}z^{\delta}=\sum_\upsilon\bigl(\sum_\gamma\partial_\gamma\bar{\tilde{b}}^\upsilon(x)z^{\gamma}\bigr)^{2}/\tilde{b}^\upsilon(x)=\sum_\upsilon(g_\upsilon(x)\cdot z)^{2}/\tilde{b}^\upsilon(x)$ for $z\in\mathbb{R}^l$ (from the entries of $\tilde{D}$ and the matrix-vector product), the integrand of $\mathcal{Q}^{\omega}(r)$ at $t$ is at most $(1+\zeta)N\,\hat\psi_{t-}\cdot(\tilde{D}(\bar\Sigma^{\sharp}_{t-})\hat\psi_{t-})+(1+1/\zeta)\tilde{l}\mathsf{Z}^{2}/(N\underline{b})$. Both sides are bounded measurable functions of $t$ by (F3), so integrating over $[0,T]$ (monotonicity, the restricted Lebesgue measure of $[0,T]$ being $T$) gives

Finally, by (F3) the two functions $t\mapsto\hat\psi_{t-}\cdot(\tilde{D}(\bar\Sigma^{\sharp}_{t-})\hat\psi_{t-})$ and $t\mapsto\hat\psi_t\cdot(\tilde{D}(\bar\Sigma^{\sharp}_t)\hat\psi_t)$ are bounded and measurable and differ only at the finitely many exceptional points of (F3), a set $E_0$ of Lebesgue measure $0$ (a finite union of singletons $[u,u]$, claim 4 there); both functions are nonnegative (the identity above exhibits the quadratic form as a sum of squares over positive denominators), so by claim 6 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval with $D_0=[0,T]\setminus E_0$, on which they agree, their integrals over $[0,T]$ coincide. This yields the bound of claim 3 in the stated form.