Proof of 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

Tools. We use: from Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, claim 1 (partial derivatives of sums, scalar multiples and products of two functions), claim 2 (constants and coordinate functions are smooth, that is, by that definition, of class $C^{k}$ for every $k$) and claim 3 (sums, scalar multiples and products of $C^{k}$ functions are $C^{k}$); the definition of the class $C^{1}$ in C^k Maps on a Euclidean Open Set, by which a $C^{1}$ function has partial derivatives everywhere which are themselves continuous, and by which a $C^{2}$ function has partial derivatives of class $C^{1}$; Composition of Continuous Euclidean Maps (composition of maps continuous at a point and at its image is continuous at the point); claim 1 of Euclidean Continuity Agrees with Metric Continuity for Real-Valued Functions (for real-valued maps on a subset of a Euclidean space, Euclidean continuity at a point and metric continuity relative to the subset agree); claim 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space (sums, products and scalar multiples of continuous real-valued maps are continuous); claim 1 of Restriction Stability of Continuity and of the Derivative (the restriction of a continuous map to a subset is continuous relative to the subset); claim 1 of Componentwise Estimates, Transpose Identities, and Indefinite Riemann Integrals together with the identity $d(x,y)=|x-y|$ recorded in its setting, giving the estimate $d(x,y)\le\sum_{i}|x_{i}-y_{i}|$ for the Euclidean distance on $\mathbb{R}^{n}$ and in particular $|y_{i}|\le|y|\le\sum_{i}|y_{i}|$; claim 2 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous (class $C^{k}$ implies class $C^{1}$); claim 3 of Semicontinuity and Continuity Under Composition with a Continuous Map (the composition of a continuous map into a metric space with a continuous real function on its image set is continuous, in the metric sense); claim 2 of Jump Representation and Positive Semidefiniteness of the Aggregate Fluctuation Covariance (the quadratic form of $\Theta$ is $\sum_{(\sigma,\gamma')\in\mathcal{L}}\Sigma^{\sigma}\beta(\sigma,\gamma',\Sigma,\alpha)(x^{\gamma'}-x^{\sigma})^{2}$, the sum running over all ordered pairs of distinct states, hence nonnegative); Continuous Real-Valued Functions on a Compact Interval are Bounded (a continuous real function on a compact interval is bounded); claims 1, 3 and 4 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval ($\lambda_{[0,u]}([0,u])=u$; a continuous function on a compact interval is a square-integrable random variable on the normalised interval, in particular $\mathcal{B}_{[0,u]}$-measurable; and a measurable $f$ with $\int_{[0,u]}f^{2}\,d\lambda_{[0,u]}<\infty$ has $|f|$ integrable); Linearity and Monotonicity of the Lebesgue Integral (linearity and monotonicity of the integral); claims 2, 3 and 7 of Properties of Finite Sums (additivity, homogeneity, and a sum with a single possibly nonzero summand); claim 2 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers (the absolute value of a finite sum is at most the sum of the absolute values); Interchange of a Finite Double Sum (interchange of a finite double sum); claims 3 and 4 of Peeling, Splitting, and Interchange for Sums over a Finite Index Set (splitting a sum over a finite index set along a partition into two nonempty blocks, and dropping terms vanishing outside a subset); and claim 4 of Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum, $w\cdot(Mz)=\sum_{i}\sum_{j}M_{ij}w_{i}z_{j}$. Sums over the finite label set $\mathcal{L}$, or over subsets of it, are sums over a finite index set, independent of the enumeration by A Sum over a Finite Index Set Does Not Depend on the Enumeration, so that they may be computed along any enumeration as finite sums of Finite Sum Notation in a Field; iterating claims 1 and 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set along such an enumeration shows that a finite sum of $C^{k}$ functions with constant coefficients is $C^{k}$, with partial derivatives the corresponding sums of partial derivatives (finite linearity).

Proof of claim 1. Fix $c=(\sigma_{0},\gamma_{0})$. The set $U\times V$ is open in $\mathbb{R}^{l+m}$ (clause 2 of Twice Continuously Differentiable Extension of a Transition-Rate Family). The coordinate function $x\mapsto x_{\sigma_{0}}=\Sigma^{\sigma_{0}}$ is smooth on $U\times V$ (claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set), hence of class $C^{2}$ (Smooth Map on a Euclidean Open Set), and $\bar\beta(\sigma_{0},\gamma_{0},\cdot,\cdot)$ is of class $C^{2}$ on $U\times V$ (clause 2 of Twice Continuously Differentiable Extension of a Transition-Rate Family); so their product $\psi_{c}$ is of class $C^{2}$ (claim 3). By the definition of the class $C^{2}$, every partial derivative $\partial_{i}\psi_{c}$ exists at every point of $U\times V$ and is of class $C^{1}$ there; hence $g^{c}$ is defined everywhere with components $\partial_{1}\psi_{c},\dots,\partial_{l}\psi_{c}$ of class $C^{1}$, and $\mathcal{E}^{\gamma\eta}=\sum_{c}v_{c}^{\gamma}\partial_{\eta}\psi_{c}$ is defined everywhere and of class $C^{1}$ by finite linearity (the coefficients $v_{c}^{\gamma}\in\{-1,0,1\}$ being constants). A $C^{1}$ function is continuous at every point by the definition of the class $C^{1}$. For the identity, fix $\gamma,\eta$ and $x\in U\times V$. By Extended Aggregate State Drift,

since $\Sigma^{\sigma}\bar\beta(\sigma,\gamma,\Sigma,\alpha)=\psi_{(\sigma,\gamma)}(\Sigma,\alpha)$ and $\Sigma^{\gamma}\bar\beta(\gamma,\sigma,\Sigma,\alpha)=\psi_{(\gamma,\sigma)}(\Sigma,\alpha)$. By finite linearity, $\partial_{\eta}\bar{b}^{\gamma}(x)=\sum_{\sigma\ne\gamma}(\partial_{\eta}\psi_{(\sigma,\gamma)}(x)-\partial_{\eta}\psi_{(\gamma,\sigma)}(x))$. On the other hand, for a label $c=(\sigma',\gamma')$ the coefficient $v_{c}^{\gamma}$ is the $\gamma$-th component of $\delta_{\gamma'}-\delta_{\sigma'}$, which equals $1$ if $\gamma'=\gamma$ (then $\sigma'\ne\gamma$), equals $-1$ if $\sigma'=\gamma$ (then $\gamma'\ne\gamma$), and equals $0$ otherwise; the labels of the first kind form the set $\mathcal{L}_{+}=\{(\sigma,\gamma):\sigma\ne\gamma\}$, those of the second kind the set $\mathcal{L}_{-}=\{(\gamma,\sigma):\sigma\ne\gamma\}$, both nonempty (as $l\ge2$) and disjoint, and the terms of the third kind vanish. By claim 4 of Peeling, Splitting, and Interchange for Sums over a Finite Index Set (terms vanishing outside $\mathcal{L}_{+}\cup\mathcal{L}_{-}$) and then claim 3 (splitting along $\mathcal{L}_{+}$, $\mathcal{L}_{-}$), the sum $\mathcal{E}^{\gamma\eta}(x)=\sum_{c\in\mathcal{L}}v_{c}^{\gamma}\partial_{\eta}\psi_{c}(x)$ equals $\sum_{c\in\mathcal{L}_{+}}\partial_{\eta}\psi_{c}(x)+\sum_{c\in\mathcal{L}_{-}}(-\partial_{\eta}\psi_{c}(x))$. Rewriting the sums over $\mathcal{L}_{+}$ and $\mathcal{L}_{-}$ as sums over the index set $\{\sigma:\sigma\ne\gamma\}$ through the bijections $\sigma\mapsto(\sigma,\gamma)$ and $\sigma\mapsto(\gamma,\sigma)$ (if $\varphi$ enumerates $\{\sigma:\sigma\ne\gamma\}$ and $\chi$ is either bijection, then $\chi\circ\varphi$ enumerates the corresponding block, and the two sums over a finite index set, computed along $\varphi$ and along $\chi\circ\varphi$ respectively, are termwise identical) and using homogeneity for the sign, this equals $\sum_{\sigma\ne\gamma}\partial_{\eta}\psi_{(\sigma,\gamma)}(x)-\sum_{\sigma\ne\gamma}\partial_{\eta}\psi_{(\gamma,\sigma)}(x)$, which is $\partial_{\eta}\bar{b}^{\gamma}(x)$ by additivity of finite sums applied to the expression for $\partial_{\eta}\bar{b}^{\gamma}(x)$ above.

Proof of claim 2. By condition 1 of Mean-Field Trajectory Pair, every component $t\mapsto S^{\gamma}_{t}$ and $t\mapsto A^{j}_{t}$ is continuous on $[0,T]$. Fix $t\in[0,T]$ and $\varepsilon>0$. For each of the $l+m$ components choose $\vartheta>0$ such that the component varies by less than $\varepsilon/(l+m)$ on the points $t'\in[0,T]$ with $|t'-t|<\vartheta$, and let $\vartheta_{0}$ be the least of these $\vartheta$; then for $|t'-t|<\vartheta_{0}$ the Euclidean distance of $(S_{t'},A_{t'})$ from $(S_{t},A_{t})$ is at most $\sum_{\gamma}|S^{\gamma}_{t'}-S^{\gamma}_{t}|+\sum_{j}|A^{j}_{t'}-A^{j}_{t}|<\varepsilon$ (claim 1 of Componentwise Estimates, Transpose Identities, and Indefinite Riemann Integrals). Thus $t\mapsto(S_{t},A_{t})$ is continuous at $t$ in the sense of Continuity at a Point for Maps Between Euclidean Spaces; the same argument gives the continuity of $t\mapsto S_{t}$ into $\mathbb{R}^{l}$. The values lie in $\Delta^{l}\times\mathcal{A}$ and in $\Delta^{l}$ by Mean-Field Trajectory Pair, and $\Delta^{l}\times\mathcal{A}\subseteq U\times V$ by clause 1 of Twice Continuously Differentiable Extension of a Transition-Rate Family and $\Delta^{l}\subseteq\tilde{U}$ by clause 1 of Twice Continuously Differentiable Extension of an Observation-Rate Family.

Now $\mathcal{E}^{\gamma\eta}$ is continuous at every point of $U\times V$ (claim 1), so $t\mapsto\mathcal{E}^{\gamma\eta}(S_{t},A_{t})$ is continuous at every $t$ in the Euclidean sense by Composition of Continuous Euclidean Maps, hence continuous on $[0,T]$ in the metric sense by claim 1 of Euclidean Continuity Agrees with Metric Continuity for Real-Valued Functions; and $\mathcal{E}^{\gamma\eta}(S_{t},A_{t})=\partial_{\eta}\bar{b}^{\gamma}(S_{t},A_{t})=(\mathcal{E}_{t})_{\gamma\eta}$ by claim 1 and the definition of $\mathcal{E}_{t}$. For $\Theta$: let $\bar\Theta^{\gamma\delta}:U\times V\to\mathbb{R}$ be given by the formulas of Aggregate Fluctuation Covariance with $\bar\beta$ in place of $\beta$, that is, $\bar\Theta^{\gamma\gamma}=\sum_{\sigma\ne\gamma}(\psi_{(\sigma,\gamma)}+\psi_{(\gamma,\sigma)})$ and $\bar\Theta^{\gamma\delta}=-\psi_{(\gamma,\delta)}-\psi_{(\delta,\gamma)}$ for $\gamma\ne\delta$. Each $\bar\Theta^{\gamma\delta}$ is of class $C^{2}$ by claim 1 and finite linearity, hence continuous at every point of $U\times V$, and it agrees with $\Theta^{\gamma\delta}$ on $\Delta^{l}\times\mathcal{A}$ because $\bar\beta$ agrees with $\beta$ there (clause 1 of Twice Continuously Differentiable Extension of a Transition-Rate Family). Hence $t\mapsto\Theta^{\gamma\delta}(S_{t},A_{t})=\bar\Theta^{\gamma\delta}(S_{t},A_{t})$ is continuous on $[0,T]$ as before, and equals $(\Theta^{\star}_{t})_{\gamma\delta}$ by definition. For the observation data: by claim (i) of Regularity and Derivative Bounds of the Extended Aggregate Observation Drift, $\bar{\tilde{b}}^{\upsilon}$ is of class $C^{2}$ on the open set $\tilde{U}$, so $\bar{\tilde{b}}^{\upsilon}$ is of class $C^{1}$ (claim 2 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous) and each $\partial_{\gamma}\bar{\tilde{b}}^{\upsilon}$ is of class $C^{1}$ (definition of the class $C^{2}$), hence both are continuous at every point of $\tilde{U}$ (definition of the class $C^{1}$); composing with $t\mapsto S_{t}$ as before, $t\mapsto\partial_{\gamma}\bar{\tilde{b}}^{\upsilon}(S_{t})=(\tilde{\mathcal{E}}_{t})_{\upsilon\gamma}$ and $t\mapsto\bar{\tilde{b}}^{\upsilon}(S_{t})$ are continuous on $[0,T]$, and the latter equals $\tilde{b}^{\upsilon}(S_{t})$ because $\bar{\tilde{b}}$ agrees with $\tilde{b}$ on $\Delta^{l}$ (same claim), with values in $[\underline{b},\infty)$ by (OC). Restrictions to $[0,s]$ are continuous on $[0,s]$ by claim 1 of Restriction Stability of Continuity and of the Derivative; a continuous function on the compact interval $[0,s]$ is bounded by Continuous Real-Valued Functions on a Compact Interval are Bounded and measurable by claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval.

Proof of claim 3. By claim 2 the restrictions $u\mapsto\mathcal{E}^{\gamma\eta}(S_{u},A_{u})$ to $[0,s]$ are continuous on $[0,s]$; so the assignment $\mathcal{E}^{\star}$ has continuous entries, which is the hypothesis of the setting of Variation of Constants with Bounded Measurable Forcing and the Two-Parameter Fundamental Solution with $k=l$ and horizon $s$ in the role of its $T$. That setting furnishes a real number $\bar\Phi\ge0$ (its constant of the same name) with $\lVert\Phi^{\mathcal{E}}(t,u)\rVert_{\mathrm{rs}}\le\bar\Phi^{2}$ for all $t,u\in[0,s]$ (its claim 1), and $|My|\le\lVert M\rVert_{\mathrm{rs}}|y|$ for its row-sum vector norm (as recorded in its setting); hence $|\Phi^{\mathcal{E}}(t,u)y|\le\bar\Phi^{2}|y|$.

Proof of claim 4. Each component $\lambda^{\gamma}$ is continuous on $[0,T]$, hence bounded by some $C_{\gamma}\ge0$ (Continuous Real-Valued Functions on a Compact Interval are Bounded); with $\Lambda=\sum_{\gamma}C_{\gamma}$ we get $|\lambda(u)|\le\sum_{\gamma}|\lambda^{\gamma}(u)|\le\Lambda$ (claim 1 of Componentwise Estimates, Transpose Identities, and Indefinite Riemann Integrals). Consider the setting of Variation of Constants with Bounded Measurable Forcing and the Two-Parameter Fundamental Solution with $k=l$, horizon $T$ and the assignment $r\mapsto\mathcal{E}_{r}$ in the role of its coefficient matrix (written $A$ there), whose entries are continuous on $[0,T]$ by claim 2. The forcing $g(r)=\Theta^{\star}_{r}\lambda(r)$ has components $g^{\gamma}(r)=\sum_{\delta}(\Theta^{\star}_{r})_{\gamma\delta}\lambda^{\delta}(r)$ (Matrix-Vector Product), finite sums of products of functions continuous on $[0,T]$ (claim 2 and the hypothesis on $\lambda$), hence continuous on $[0,T]$ by claim 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space, hence 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 $g$ is bounded measurable in the sense of that lemma. With $\xi=0$, claim 2 of that lemma furnishes a map, which we call $\psi_{\lambda}$, with continuous, hence bounded measurable, components satisfying $\psi_{\lambda}(u)=\int_{[0,u]}(\mathcal{E}_{r}\psi_{\lambda}(r)+\Theta^{\star}_{r}\lambda(r))\,dr$ for all $u\in[0,T]$, and claim 3 of that lemma states that every bounded measurable $z$ satisfying the same equation equals $\psi_{\lambda}$; this is the existence and uniqueness asserted. The bound $\mathsf{M}$ is obtained from the continuous components exactly as $\Lambda$ was. The restrictions $\varpi$ and $\bar\psi$ have continuous components on $[0,s]$ (claim 1 of Restriction Stability of Continuity and of the Derivative), hence bounded measurable ones, with the same pointwise bounds. For $u\in[0,s]$ the displayed equation for $\psi_{\lambda}(u)$ involves only the values of $\psi_{\lambda}$, $\lambda$, $\mathcal{E}_{r}$ and $\Theta^{\star}_{r}$ for $r\in[0,u]\subseteq[0,s]$, where $\psi_{\lambda}(r)=\bar\psi_{r}$, $\lambda(r)=\varpi_{r}$, $\mathcal{E}_{r}=\mathcal{E}(S_{r},A_{r})$ (claim 2) and $\Theta^{\star}_{r}=\Theta(S_{r},A_{r})$ (definition); this is the displayed equation for $\bar\psi_{u}$. Finally, the setting of Variation of Constants with Bounded Measurable Forcing and the Two-Parameter Fundamental Solution is also instantiated with horizon $s$, coefficient matrix $\mathcal{E}^{\star}$ (claim 3) and the forcing $r\mapsto\Theta(S_{r},A_{r})\varpi_{r}$ on $[0,s]$, the restriction of $g$, which is bounded measurable on $[0,s]$ because the components of $g$ are continuous on $[0,T]$, so that their restrictions are continuous on $[0,s]$ (claim 1 of Restriction Stability of Continuity and of the Derivative), hence bounded and measurable there; $\bar\psi$ is bounded measurable and satisfies the equation of its claim 3 with $\xi=0$, and so does any other bounded measurable solution $z$ on $[0,s]$; by that claim both equal the map furnished by its claim 2, hence $z=\bar\psi$.

Proof of claim 5. For $t\in[0,T]$, $\upsilon\in\{1,\dots,\tilde{l}\}$ and $\gamma\in\{1,\dots,l\}$ write $\rho_{\upsilon}(t)=\tilde{b}^{\upsilon}(S_{t})\ge\underline{b}>0$ and $e_{\upsilon\gamma}(t)=\partial_{\gamma}\bar{\tilde{b}}^{\upsilon}(S_{t})$, functions on $[0,T]$ which are continuous by claim 2, so that $(\tilde{\Theta}^{\star}_{t})_{\upsilon\upsilon'}=\mathbf{1}_{\{\upsilon=\upsilon'\}}\rho_{\upsilon}(t)$ and $(\tilde{\mathcal{E}}_{t})_{\upsilon\gamma}=e_{\upsilon\gamma}(t)$. Fix $t$, and let $\tilde{\Theta}^{\flat}_{t}$ be the matrix with $\tilde{l}$ rows and columns and entries $(\tilde{\Theta}^{\flat}_{t})_{\upsilon\upsilon'}=\mathbf{1}_{\{\upsilon=\upsilon'\}}/\rho_{\upsilon}(t)$. By Product of Real Matrices, $(\tilde{\Theta}^{\star}_{t}\tilde{\Theta}^{\flat}_{t})_{\upsilon\upsilon''}=\sum_{\upsilon'}\mathbf{1}_{\{\upsilon=\upsilon'\}}\rho_{\upsilon}(t)\mathbf{1}_{\{\upsilon'=\upsilon''\}}/\rho_{\upsilon'}(t)$; the only possibly nonzero summand is the one with $\upsilon'=\upsilon$, equal to $\mathbf{1}_{\{\upsilon=\upsilon''\}}\rho_{\upsilon}(t)/\rho_{\upsilon}(t)=\mathbf{1}_{\{\upsilon=\upsilon''\}}$ (claim 7 of Properties of Finite Sums); so $\tilde{\Theta}^{\star}_{t}\tilde{\Theta}^{\flat}_{t}=I_{\tilde{l}}$, and symmetrically $\tilde{\Theta}^{\flat}_{t}\tilde{\Theta}^{\star}_{t}=I_{\tilde{l}}$. Hence $\tilde{\Theta}^{\star}_{t}$ is invertible with inverse $\tilde{\Theta}^{\flat}_{t}$ (Inverse Matrix and Invertible Real Square Matrix, the inverse being unique by Uniqueness of the Matrix Inverse). Next, $\tilde{\mathcal{E}}_{t}^{\top}$ has $l$ rows and $\tilde{l}$ columns with $(\tilde{\mathcal{E}}_{t}^{\top})_{\gamma\upsilon}=e_{\upsilon\gamma}(t)$ (Transpose of a Real Matrix), and $(\tilde{\Theta}^{\flat}_{t}\tilde{\mathcal{E}}_{t})_{\upsilon\delta}=\sum_{\upsilon'}\mathbf{1}_{\{\upsilon=\upsilon'\}}e_{\upsilon'\delta}(t)/\rho_{\upsilon}(t)=e_{\upsilon\delta}(t)/\rho_{\upsilon}(t)$ (claim 7 again); therefore $\tilde{D}_{t}=\tilde{\mathcal{E}}_{t}^{\top}(\tilde{\Theta}^{\flat}_{t}\tilde{\mathcal{E}}_{t})$ has $l$ rows and $l$ columns with

which is visibly symmetric in $(\gamma,\delta)$. For $z\in\mathbb{R}^{l}$, claim 4 of Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum gives $z\cdot(\tilde{D}_{t}z)=\sum_{\gamma}\sum_{\delta}(\tilde{D}_{t})_{\gamma\delta}z_{\gamma}z_{\delta}$; inserting the displayed formula, homogeneity moves the factor $z_{\gamma}z_{\delta}$ inside the sum over $\upsilon$, Interchange of a Finite Double Sum (applied first to the innermost pair of indices $(\delta,\upsilon)$ and then to the pair $(\gamma,\upsilon)$) brings the sum over $\upsilon$ outside, homogeneity (applied twice) extracts the factor $1/\rho_{\upsilon}(t)$ from the double sum over $(\gamma,\delta)$, and homogeneity again, applied twice, gives $\sum_{\gamma}\sum_{\delta}e_{\upsilon\gamma}(t)z_{\gamma}e_{\upsilon\delta}(t)z_{\delta}=\sum_{\gamma}e_{\upsilon\gamma}(t)z_{\gamma}\bigl(\sum_{\delta}e_{\upsilon\delta}(t)z_{\delta}\bigr)=(\sum_{\gamma}e_{\upsilon\gamma}(t)z_{\gamma})^{2}$; hence

each summand being nonnegative as $\rho_{\upsilon}(t)>0$. Finally, by claim (ii) of Regularity and Derivative Bounds of the Extended Aggregate Observation Drift, $|e_{\upsilon\gamma}(t)|\le\tilde{B}+\tilde{K}$ for all $\upsilon,\gamma,t$ (as $S_{t}\in\Delta^{l}$), so by claim 2 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers and $\rho_{\upsilon}(t)\ge\underline{b}$, $|(\tilde{D}_{t})_{\gamma\delta}|\le\tilde{l}(\tilde{B}+\tilde{K})^{2}/\underline{b}$. For continuity: the function $y\mapsto1/y$ on $E=[\underline{b},\infty)$ is continuous on $E$ relative to $E$, since $|1/y-1/y'|=|y'-y|/(yy')\le|y'-y|/\underline{b}^{2}$ for $y,y'\in E$ (so $\vartheta=\underline{b}^{2}\varepsilon$ serves for every $\varepsilon>0$); the map $t\mapsto\rho_{\upsilon}(t)$ is continuous on $[0,T]$ with values in $E$ (claim 2); so $t\mapsto1/\rho_{\upsilon}(t)$ is continuous on $[0,T]$ by claim 3 of Semicontinuity and Continuity Under Composition with a Continuous Map, and $t\mapsto(\tilde{D}_{t})_{\gamma\delta}=\sum_{\upsilon}e_{\upsilon\gamma}(t)e_{\upsilon\delta}(t)(1/\rho_{\upsilon}(t))$ is continuous on $[0,T]$ by claim 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space.

Proof of claim 6. By claim 4 of Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum, $\lambda(r)\cdot(\Theta^{\star}_{r}\lambda(r))=\sum_{\gamma}\sum_{\delta}(\Theta^{\star}_{r})_{\gamma\delta}\lambda^{\gamma}(r)\lambda^{\delta}(r)$, a finite sum of products of functions continuous on $[0,T]$ (claim 2 and the hypothesis on $\lambda$), hence continuous on $[0,T]$ (claim 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space); and it is nonnegative, since by claim 2 of Jump Representation and Positive Semidefiniteness of the Aggregate Fluctuation Covariance (applied at $(S_{r},A_{r})\in\Delta^{l}\times\mathcal{A}$) it equals $\sum_{(\sigma,\gamma')\in\mathcal{L}}S^{\sigma}_{r}\beta(\sigma,\gamma',S_{r},A_{r})(\lambda^{\gamma'}(r)-\lambda^{\sigma}(r))^{2}$ (a sum over all ordered pairs of distinct states), a finite sum of nonnegative terms ($S^{\sigma}_{r}\ge0$ as $S_{r}\in\Delta^{l}$, and $\beta\ge0$ by Transition-Rate Family). Likewise $\psi_{\lambda}(r)\cdot(\tilde{D}_{r}\psi_{\lambda}(r))=\sum_{\gamma}\sum_{\delta}(\tilde{D}_{r})_{\gamma\delta}\psi^{\gamma}_{\lambda}(r)\psi^{\delta}_{\lambda}(r)$ is a finite sum of products of functions continuous on $[0,T]$ (claims 4 and 5), hence continuous; by claim 5 it equals $\sum_{\upsilon}(g_{\upsilon}(S_{r})\cdot\psi_{\lambda}(r))^{2}/\rho_{\upsilon}(r)\ge0$, and since $|g_{\upsilon}(S_{r})\cdot\psi_{\lambda}(r)|=|\sum_{\gamma}e_{\upsilon\gamma}(r)\psi^{\gamma}_{\lambda}(r)|\le\sum_{\gamma}|e_{\upsilon\gamma}(r)||\psi^{\gamma}_{\lambda}(r)|\le l\,(\tilde{B}+\tilde{K})\,|\psi_{\lambda}(r)|\le l(\tilde{B}+\tilde{K})\mathsf{M}$ (claim 2 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers, $|\psi^{\gamma}_{\lambda}|\le|\psi_{\lambda}|$, claim (ii) of Regularity and Derivative Bounds of the Extended Aggregate Observation Drift and claim 4) and $\rho_{\upsilon}(r)\ge\underline{b}$, its values are at most $\tilde{l}\,l^{2}(\tilde{B}+\tilde{K})^{2}\mathsf{M}^{2}/\underline{b}$. The integrand of $\mathcal{A}_{s}(\lambda)$ is thus continuous and nonnegative on $[0,T]$; its restriction to $[0,s]$ is continuous on $[0,s]$ (claim 1 of Restriction Stability of Continuity and of the Derivative), hence bounded, say by $C\ge0$ (Continuous Real-Valued Functions on a Compact Interval are Bounded), and $\mathcal{B}_{[0,s]}$-measurable (claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval); writing $f$ for this restriction, $f^{2}\le C^{2}$ pointwise, so $\int_{[0,s]}f^{2}\,d\lambda_{[0,s]}\le C^{2}\lambda_{[0,s]}([0,s])=C^{2}s<\infty$ by monotonicity of the integral and claim 1 of that lemma, and $|f|=f$ (the integrand being nonnegative) is integrable with respect to $\lambda_{[0,s]}$ by claim 4 of that lemma; the same applies to each of its two summands separately. So $\mathcal{A}_{s}(\lambda)$ is a well-defined real number, equal by linearity of the integral to the sum of the integrals of the two summands, which are $\int_{[0,s]}\varpi_{r}\cdot(\Theta(S_{r},A_{r})\varpi_{r})\,dr$ and $\int_{[0,s]}\bar\psi_{r}\cdot(\tilde{D}(S_{r})\bar\psi_{r})\,dr$ by the identities $\varpi_{r}=\lambda(r)$, $\bar\psi_{r}=\psi_{\lambda}(r)$, $\Theta^{\star}_{r}=\Theta(S_{r},A_{r})$ and $\tilde{D}_{r}=\tilde{D}(S_{r})$ (claim 5) for $r\in[0,s]$; both integrals are nonnegative by monotonicity of the integral, their integrands being nonnegative, and so $\mathcal{A}_{s}(\lambda)\ge0$. $\qquad\blacksquare$