Proof of Pathwise Estimand Linearisation of an Open-Loop Aggregate Solution Along a Comparison Pair: Cell-Count Coefficients from the Fundamental Solution and the Three Error Terms

Throughout, $\mathbf{1}\{P\}$ denotes $1$ if the condition $P$ holds and $0$ otherwise, and claim numbers of Linearisation of a Perturbed Controlled Aggregate Flow Along a Comparison Pair: Exact Variation-of-Constants Identity and Residual Bound, Variation of Constants with Bounded Measurable Forcing and the Two-Parameter Fundamental Solution and Cell-Count Form of a Compensated Counting-Path Functional Along a Time Change: Window-Discrepancy Bounds for the Time-Change and Partial-Cell Errors are cited by name of the lemma. By the definition of the label rate in the setting of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect, $\psi_c(\Sigma,\alpha)=\Sigma^\sigma\bar{\beta}(\sigma,\gamma,\Sigma,\alpha)$ for $c=(\sigma,\gamma)$, and $\bar{\beta}=\beta$ on $\Delta^l\times\mathcal{A}$ by condition 1 of Twice Continuously Differentiable Extension of a Transition-Rate Family; hence $\psi_c(\Sigma,\alpha)=\Sigma^\sigma\beta(\sigma,\gamma,\Sigma,\alpha)$ on $\Delta^l\times\mathcal{A}$, and the consumed clock time of Open-Loop Aggregate Solution Driven by Aggregate Transition Clocks reads $\mathsf{C}^{c}_t=\int_{[0,t]}N\psi_c(\Sigma_u,a_u)\,du$, its integrand being measurable with values in $[0,NB]$ as recorded in that definition. For the comparison pair, $u\mapsto\psi_c(S_u,\mathsf{A}_u)=S^\sigma_u\beta(\sigma,\gamma,S_u,\mathsf{A}_u)$ is measurable: the map $(\Sigma,\alpha)\mapsto\beta(\sigma,\gamma,\Sigma,\alpha)$ is sequentially continuous on $\Delta^l\times\mathcal{A}$ for the Euclidean distance on $\mathbb{R}^{l+m}$ by condition 2 of Transition-Rate Family (the distances $d(\Sigma_n,\Sigma)$ and $d(\alpha_n,\alpha)$ being each at most $d((\Sigma_n,\alpha_n),(\Sigma,\alpha))$), so $u\mapsto\beta(\sigma,\gamma,S_u,\mathsf{A}_u)$ is measurable by Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable, and the product with the measurable coordinate $u\mapsto S^\sigma_u$ is measurable by claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions; its values lie in $[0,B]$ because $0\le S^\sigma_u\le1$ and $0\le\beta\le B$ (condition 1 of Transition-Rate Family). This proves the first assertion of claim 1.

Claim 1. By condition 1 of Open-Loop Aggregate Solution Driven by Aggregate Transition Clocks, $\Sigma$ takes values in $\mathbb{G}_N\subseteq\Delta^l$ and is constant on each of finitely many intervals partitioning $[0,T]$; each component is therefore a finite sum of constants times indicators of intervals, which are measurable by claims 1 and 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions (an interval in $[0,T]$ is a Borel set, being an intersection of $[0,T]$ with a Borel subset of the real line), so $\Sigma$ is bounded measurable; $y$ is measurable with values in $\Delta^l$ by hypothesis, so $e=\Sigma-y$ is bounded measurable, and $|e_t|\le2$ since $|z|\le1$ for $z\in\Delta^l$ (the coordinates being nonnegative with sum $1$, $|z|^{2}=\sum_\sigma(z^\sigma)^{2}\le(\sum_\sigma z^\sigma)^{2}=1$). For each $c$, claim 1 of Cumulative-Rate Time Change: Regularity, Substitution, and Crossing Times applied to the rate $u\mapsto N\psi_c(\Sigma_u,a_u)$, measurable with values in $[0,NB]$ (the cumulative rate there, an integral over $[0,T]$ against the indicator of $[0,t]$, agrees with the integral over the compact interval $[0,t]$ by claim 2 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval, both being the integral of the same zero extension over the real line), shows that $\mathsf{C}^{c}$ is nondecreasing and continuous on $[0,T]$ with $\mathsf{C}^{c}_0=0$ and $\mathsf{C}^{c}_T\le NBT\le R$; the same applies to $\bar{\mathsf{C}}^{c}$ with the rate $N\phi_c$, measurable with values in $[0,NB]$ by the first assertion of claim 1. Continuous functions on $[0,T]$ are measurable by claim 4 of Measurability of Countable Suprema, Bounded Pointwise Limits, Monotone Functions, and Continuous Functions.

Fix $q=(c,j)$ with $\bar{\mathsf{C}}^{c}_T\ge \mathsf{b}^{c}_j$ and let $Z=\{u\in[0,T]:\bar{\mathsf{C}}^{c}_u\ge \mathsf{b}^{c}_j\}$, nonempty since $T\in Z$, and let $\bar{\tau}$ be its greatest lower bound. Then $\bar{\mathsf{C}}^{c}_{\bar{\tau}}\ge \mathsf{b}^{c}_j$: otherwise, by continuity at $\bar{\tau}$ there is $\delta>0$ with $\bar{\mathsf{C}}^{c}_u<\mathsf{b}^{c}_j$ for all $u\in[0,T]$ with $|u-\bar{\tau}|<\delta$, whereas by the definition of the greatest lower bound some element of $Z$ lies in $[\bar{\tau},\bar{\tau}+\delta)$. Hence $\bar{\tau}\in Z$ is the least element of $Z$, and since $\bar{\mathsf{C}}^{c}$ is nondecreasing, $Z=[\bar{\tau},T]$.

By condition 2 of Open-Loop Aggregate Solution Driven by Aggregate Transition Clocks, $\Sigma_t=x_0+\frac{1}{N}\sum_cv_c\,p^{c}(\mathsf{C}^{c}_t)$. By linearity of the integral and the definition $b=\sum_cv_c\psi_c$ in Linearisation of a Perturbed Controlled Aggregate Flow Along a Comparison Pair: Exact Variation-of-Constants Identity and Residual Bound (its integrand below being bounded measurable by claim 1 there),

so $\Sigma_t=x_0+\int_{[0,t]}b(\Sigma_u,a_u)\,du+\frac{1}{N}\sum_cv_c\bigl(p^{c}(\mathsf{C}^{c}_t)-\mathsf{C}^{c}_t\bigr)=x_0+\int_{[0,t]}b(\Sigma_u,a_u)\,du+\mathfrak{m}_t$. Each $u\mapsto p^{c}(\mathsf{C}^{c}_u)$ is bounded measurable by claim 1 of Cell-Count Form of a Compensated Counting-Path Functional Along a Time Change: Window-Discrepancy Bounds for the Time-Change and Partial-Cell Errors ($\mathsf{C}^{c}$ being measurable with values in $[0,R]$), so $\mathfrak{m}$ is bounded measurable (claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions), and $\mathfrak{m}_0=0$ since $p^{c}(0)=0=\mathsf{C}^{c}_0$. The hypotheses of Linearisation of a Perturbed Controlled Aggregate Flow Along a Comparison Pair: Exact Variation-of-Constants Identity and Residual Bound on $x=\Sigma$, $y$, $a$ and $\mathfrak{m}$ are thus all met.

Claim 2. By the definition of $L_T$ and of $\mathfrak{m}$, using claims 1 and 2 of Linearity, Compatibility with the Matrix Product, and a Norm Bound for the Matrix-Vector Product (linearity of $z\mapsto\Phi^{\mathcal{E}}(T,u)(\mathcal{E}^{\star}_uz)$, and $\Phi^{\mathcal{E}}(T,u)(\mathcal{E}^{\star}_uv_c)=H^{c}_u$) and linearity of the integral over the finite sum,

where $H^{c}$ is bounded measurable by claim 2 of Variation of Constants with Bounded Measurable Forcing and the Two-Parameter Fundamental Solution, applied to $g_u=\mathcal{E}^{\star}_uv_c$, which is bounded measurable because the entries of $u\mapsto\mathcal{E}(S_u,\mathsf{A}_u)$ are (claim 1 of Linearisation of a Perturbed Controlled Aggregate Flow Along a Comparison Pair: Exact Variation-of-Constants Identity and Residual Bound) and its components are finite linear combinations of them (claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions), with $|H^{c}_u|\le\bar{\Phi}^{2}|\mathcal{E}^{\star}_uv_c|\le\bar{\Phi}^{2}\Lambda_{\mathcal{E}}\sqrt{2}$ by claim 1 of Linearisation of a Perturbed Controlled Aggregate Flow Along a Comparison Pair: Exact Variation-of-Constants Identity and Residual Bound, whence $\lVert H^{c}\rVert_1\le\sqrt{2}\bar{\Phi}^{2}\Lambda_{\mathcal{E}}T$ by monotonicity. Multiplying by $\sqrt{N}$ and taking the dot product with $\mathbf{c}$ gives the first identity.

Fix a label $c'$ and apply claim 4 of Cell-Count Form of a Compensated Counting-Path Functional Along a Time Change: Window-Discrepancy Bounds for the Time-Change and Partial-Cell Errors to $p=p^{c'}$, the cells $\mathsf{b}^{c'}_j$, $s=T$, $\mathsf{C}=\mathsf{C}^{c'}$, $\bar{\mathsf{C}}=\bar{\mathsf{C}}^{c'}$ (measurable with values in $[0,R]$ by claim 1, and satisfying $|\mathsf{C}^{c'}_u-\bar{\mathsf{C}}^{c'}_u|\le w_1$ by (CD)), $v=v_{c'}$ and $H=H^{c'}$: with the cell coefficients $\alpha^{c'}_j=v_{c'}\mathbf{1}\{\mathsf{b}^{c'}_j\le\bar{\mathsf{C}}^{c'}_T\}+\int_{[0,T]}\mathbf{1}\{\mathsf{b}^{c'}_j\le\bar{\mathsf{C}}^{c'}_u\}H^{c'}_u\,du$ of that lemma,

where the lemma's bound is stated with its own maximal cell length $\mu^{c'}_{\max}=\max_{1\le j\le J_{c'}}\mu_{c',j}\le\mu_{\max}$ and we have used $\mathrm{Disc}_{\mu^{c'}_{\max}}(p^{c'})\le\mathrm{Disc}_{\mu_{\max}}(p^{c'})$: for $0\le w\le w'$ the set whose least upper bound defines $\mathrm{Disc}_w(p^{c'})$ is contained in the one defining $\mathrm{Disc}_{w'}(p^{c'})$, both nonempty and bounded above by $p^{c'}(R)+R$, so the least upper bound is nondecreasing in $w$. Here $|v_{c'}|=\sqrt{2}$ because $v_{c'}=\delta_\gamma-\delta_\sigma$ with $\sigma\neq\gamma$ has exactly two nonzero coordinates, $1$ and $-1$. We identify $\mathbf{c}\cdot\alpha^{c'}_j$ with $\alpha_{(c',j)}$. If $\bar{\mathsf{C}}^{c'}_T<\mathsf{b}^{c'}_j$, both indicators vanish for every $u$ ($\bar{\mathsf{C}}^{c'}$ being nondecreasing), so $\alpha^{c'}_j=0=\alpha_{(c',j)}$. If $\bar{\mathsf{C}}^{c'}_T\ge \mathsf{b}^{c'}_j$, then by claim 1 the indicator $u\mapsto\mathbf{1}\{\mathsf{b}^{c'}_j\le\bar{\mathsf{C}}^{c'}_u\}$ is the indicator of $[\bar{\tau},T]$ with $\bar{\tau}=\bar{\tau}_{(c',j)}$, so, the integral over $[0,T]$ of a function vanishing off $[\bar{\tau},T]$ being its integral over $[\bar{\tau},T]$ (claim 2 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval, both being the integral of the same zero extension over the real line),

by claim 1 of Variation of Constants with Bounded Measurable Forcing and the Two-Parameter Fundamental Solution, whose integrand $\Phi^{\mathcal{E}}(T,u)(\mathcal{E}^{\star}_uv_{c'})$ equals $H^{c'}_u$ by claim 2 of Linearity, Compatibility with the Matrix Product, and a Norm Bound for the Matrix-Vector Product; hence $\mathbf{c}\cdot\alpha^{c'}_j=\alpha_{(c',j)}$. Summing over $c'$, dividing by $\sqrt{N}$, and using $|\mathbf{c}\cdot z|\le|\mathbf{c}||z|$ (Cauchy-Schwarz) together with the triangle inequality yields the bound $\mathsf{E}_{\mathrm{cell}}$.

Claim 3. By claim 1, claims 3 and 4 of Linearisation of a Perturbed Controlled Aggregate Flow Along a Comparison Pair: Exact Variation-of-Constants Identity and Residual Bound apply: $e_T=\Phi^{\mathcal{E}}(T,0)e_0+L_T(\mathfrak{m})+\int_{[0,T]}\Phi^{\mathcal{E}}(T,u)\rho_u\,du$ with $|\int_{[0,T]}\Phi^{\mathcal{E}}(T,u)\rho_u\,du|\le\bar{\Phi}^{2}\int_{[0,T]}|\rho_u|\,du$, and $e_0=x_0-y_0$ since $\Sigma_0=x_0$ (condition 2 of Open-Loop Aggregate Solution Driven by Aggregate Transition Clocks at $t=0$, where every $\mathsf{C}^{c}_0=0$ and $p^{c}(0)=0$). Taking the dot product with $\mathbf{c}$, multiplying by $\sqrt{N}$, subtracting $\frac{1}{\sqrt{N}}\sum_q\alpha_q(\mathsf{K}_q-\mu_q)$ and using claim 2, the triangle inequality, the Cauchy-Schwarz inequality and $|\Phi^{\mathcal{E}}(T,0)e_0|\le\bar{\Phi}^{2}|e_0|$ gives the displayed bound with $\mathsf{E}_{\mathrm{res}}=\sqrt{N}\int_{[0,T]}|\rho_u|\,du$. For the bound on $\mathsf{E}_{\mathrm{res}}$, claim 2 of Linearisation of a Perturbed Controlled Aggregate Flow Along a Comparison Pair: Exact Variation-of-Constants Identity and Residual Bound gives, for every $u$, with $|e_u|\le\bar{e}/\sqrt{N}$,

where $\bar{e}$, the least upper bound of the nonempty set $\{\sqrt{N}|e_u|:u\in[0,T]\}$ bounded above by $2\sqrt{N}$, is a real number. The maps $u\mapsto|y_u-S_u|$ and $u\mapsto\mathsf{d}_u$ are bounded measurable (Norm Bound for a Vector-Valued Lebesgue Integral over a Compact Interval for the norm of the bounded measurable $y-S$; claim 1 of Linearisation of a Perturbed Controlled Aggregate Flow Along a Comparison Pair: Exact Variation-of-Constants Identity and Residual Bound for $\mathsf{d}$), so integrating over $[0,T]$ by monotonicity and linearity and multiplying by $\sqrt{N}$ gives the stated bound.