Proof of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect

Throughout, $c=(\sigma,\gamma)$ denotes a label, $\mathbf{1}_{E}$ the indicator of a set $E$, and $\pi_\sigma$ the coordinate map $x\mapsto x_\sigma$ on $U\times V$. Two elementary facts are used repeatedly. (F1) The simplex is convex: if $x,y\in\Delta^l$ and $\tau\in[0,1]$ then $x+\tau(y-x)$ has nonnegative coordinates ($(1-\tau)x^\gamma+\tau y^\gamma\ge0$) summing to $(1-\tau)+\tau=1$, so it lies in $\Delta^l$; hence for $x,y\in\Delta^l$ and $\alpha\in\mathcal{A}$ the segment between $(x,\alpha)$ and $(y,\alpha)$ lies in $\Delta^l\times\mathcal{A}\subseteq U\times V$, and the Euclidean distance between these two points is $|y-x|$. (F2) Piecewise constant maps are measurable: by condition 1 of Open-Loop Aggregate Solution Driven by Aggregate Transition Clocks each component of $\Sigma$, of $\Sigma^{+}$ and hence of $Y$ is a finite sum of constants times indicators of subintervals of $[0,T]$, which is measurable by claims 1 and 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions and bounded because $\Sigma_s,\Sigma^{+}_s\in\mathbb{G}_N\subseteq\Delta^l$ have coordinates in $[0,1]$; and if $F$ is a function on $U\times V$ that is sequentially continuous on $\Delta^l\times\mathcal{A}$, then $s\mapsto F(\Sigma_s,a_s)$ and $s\mapsto F(\Sigma^{+}_s,a_s)$ are measurable by Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable, the components of $s\mapsto(\Sigma_s,a_s)$ and of $s\mapsto(\Sigma^{+}_s,a_s)$ being measurable; they are bounded whenever $F$ is bounded on $\Delta^l\times\mathcal{A}$.

Step 1 (Claim 1). The perturbed clock family. Put $\nu(u)=\#\{k:u_k\le u\}$, so $p^{+,c_0}=p^{c_0}+\nu$. The map $\nu$ takes values in $\{0,\dots,\mathsf{m}\}$, vanishes at $0$ (as every $u_k>0$), is nondecreasing, and is right-continuous: for $u\ge0$ let $\epsilon>0$ be smaller than every positive difference $u_k-u$; then $\nu(s)=\nu(u)$ for $s\in[u,u+\epsilon)$. Its left limit is $\nu(u-)=\#\{k:u_k<u\}$, so $\nu(u)-\nu(u-)=\#\{k:u_k=u\}\le1$, the $u_k$ being distinct. The sum of two nondecreasing right-continuous functions is nondecreasing and right-continuous (the greatest lower bound over $s>u$ of a sum of two nondecreasing functions is the sum of the greatest lower bounds), its left limits add, and $p^{+,c_0}(u)-p^{+,c_0}(u-)=(p^{c_0}(u)-p^{c_0}(u-))+(\nu(u)-\nu(u-))$: if $u=u_k$ for some $k$, then $u$ is not a jump time of $p^{c_0}$, so the first difference is $0$ and the sum is $1$; otherwise the second difference is $0$ and the sum is at most $1$. Also $p^{+,c_0}$ takes values in $\mathbb{N}_0$ as a sum of two such functions. Thus $p^{+,c_0}$ satisfies all four conditions of Counting Path and Its Jump Times, and since $p^{+,c}=p^{c}$ for $c\neq c_0$, $p^{+}$ is a clock family. We also record that $|v_c|=\sqrt{2}$ for every label: $v_c=\delta_\gamma-\delta_\sigma$ with $\gamma\neq\sigma$ has two coordinates equal to $\pm1$ and the others $0$, so $|v_c|^{2}=2$ by The Euclidean Norm on $\mathbb{R}^n$; and that for vectors $y_1,\dots,y_n\in\mathbb{R}^l$ and reals $\lambda_1,\dots,\lambda_n$ one has $|\sum_i\lambda_iy_i|\le\sum_i|\lambda_i||y_i|$, by induction on $n$ from claims 6 (triangle inequality) and 5 (absolute homogeneity) of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square. Also $p^{+,c_0}(0)=p^{c_0}(0)+\nu(0)=0$, and a maximum of finitely many continuous real functions is continuous, being obtained by iterating $\max(f,g)=\tfrac12(f+g+|f-g|)$.

Label rates. By claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set the coordinate map $\pi_\sigma$ is smooth on the open set $U\times V$, in particular of class $C^2$ there, and $\bar{\beta}(\sigma,\gamma,\cdot,\cdot)$ is of class $C^2$ there by condition 2 of Twice Continuously Differentiable Extension of a Transition-Rate Family; so $\psi_c=\pi_\sigma\bar{\beta}(\sigma,\gamma,\cdot,\cdot)$ is of class $C^2$ on $U\times V$ by claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set. Since $\partial_i\pi_\sigma=\mathbf{1}\{i=\sigma\}$ (the difference quotient of $\pi_\sigma$ in the $i$th variable is constantly $\mathbf{1}\{i=\sigma\}$, with $\mathbf{1}\{i=\sigma\}=1$ if $i=\sigma$ and $0$ otherwise), claim 1 of the same lemma gives, at every point $x=(\Sigma,\alpha)$ of $U\times V$,

$$\partial_i\psi_c(x)=\mathbf{1}\{i=\sigma\}\,\bar{\beta}(\sigma,\gamma,x)+\Sigma^{\sigma}\,\partial_i\bar{\beta}(\sigma,\gamma,x),\qquad \partial_j\partial_i\psi_c(x)=\mathbf{1}\{i=\sigma\}\,\partial_j\bar{\beta}(\sigma,\gamma,x)+\mathbf{1}\{j=\sigma\}\,\partial_i\bar{\beta}(\sigma,\gamma,x)+\Sigma^{\sigma}\,\partial_j\partial_i\bar{\beta}(\sigma,\gamma,x),$$

the second by applying claim 1 to the sum $\mathbf{1}\{i=\sigma\}\bar{\beta}+\pi_\sigma\,\partial_i\bar{\beta}$ (all partial derivatives involved existing, $\bar{\beta}$ being $C^2$). At points of $\Delta^l\times\mathcal{A}$ one has $0\le\Sigma^{\sigma}\le1$ and $0\le\bar{\beta}(\sigma,\gamma,x)=\beta(\sigma,\gamma,\Sigma,\alpha)\le B$ (conditions 1 of the extension and of Transition-Rate Family), and all partial derivatives of $\bar{\beta}$ of order one and two are bounded by $K$ (condition 3 of the extension), so $|\partial_i\psi_c|\le B+K$ and $|\partial_j\partial_i\psi_c|\le K+K+K=3K$ there. The identity $\psi_c=\Sigma^\sigma\beta(\sigma,\gamma,\Sigma,\alpha)$ on $\Delta^l\times\mathcal{A}$ is condition 1 of the extension, and the formula for $\mathsf{C}^{c}_t$ is then the definition of the consumed clock times, $\Sigma_s\in\mathbb{G}_N\subseteq\Delta^l$ and $a_s\in\mathcal{A}$.

Measurability. The map $s\mapsto Y_s$ is bounded and its components are measurable, by (F2). Each $\partial_\eta\psi_c$ is continuous on $U\times V$ (a $C^2$ map has continuous first partial derivatives, by clauses 1 and 2 of C^k Maps on a Euclidean Open Set), hence sequentially continuous on $\Delta^l\times\mathcal{A}$, so $s\mapsto\partial_\eta\psi_c(\Sigma_s,a_s)$ is measurable by (F2) and bounded by $B+K$; the components of $s\mapsto\mathcal{E}(\Sigma_s,a_s)Y_s$ are finite sums of products of such maps with the components of $Y$, hence bounded and measurable by claims 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions.

Exact identity. By condition 2 of Open-Loop Aggregate Solution Driven by Aggregate Transition Clocks for the two solutions, $\Sigma^{+}_t-\Sigma_t=\frac1N\sum_c v_c(\mathsf{N}^{+,c}_t-\mathsf{N}^{c}_t)$, with $\mathsf{N}^{c}_t=p^{c}(\mathsf{C}^{c}_t)$, $\mathsf{N}^{+,c}_t=p^{+,c}(\mathsf{C}^{+,c}_t)=p^{c}(\mathsf{C}^{+,c}_t)$ for $c\neq c_0$, and $\mathsf{N}^{+,c_0}_t=p^{c_0}(\mathsf{C}^{+,c_0}_t)+\nu(\mathsf{C}^{+,c_0}_t)=p^{c_0}(\mathsf{C}^{+,c_0}_t)+\iota_t$. Multiplying by $N$ gives the displayed identity, and $0\le\iota_t\le\mathsf{m}$ is clear.

Consumed-time difference. By the formula for the consumed clock times and linearity, $\mathsf{C}^{+,c}_t-\mathsf{C}^{c}_t=\int_{[0,t]}N\bigl(\psi_c(\Sigma^{+}_s,a_s)-\psi_c(\Sigma_s,a_s)\bigr)\,ds$. For each $s$, part (i) of Multivariate Taylor Expansion with Uniform Second-Order Remainder, applied on the open set $\mathcal{O}=U\times V\subseteq\mathbb{R}^{l+m}$ to $f=\psi_c$ (of class $C^1$ there) and the points $(\Sigma_s,a_s)$, $(\Sigma^{+}_s,a_s)$, whose segment lies in $\Delta^l\times\mathcal{A}$ by (F1) with $M_1=B+K$ there, gives $|\psi_c(\Sigma^{+}_s,a_s)-\psi_c(\Sigma_s,a_s)|\le\sqrt{l+m}\,(B+K)\,|\Sigma^{+}_s-\Sigma_s|=\Lambda_1|Y_s|/N$. The bound $|\mathsf{C}^{+,c}_t-\mathsf{C}^{c}_t|\le\Lambda_1\int_{[0,t]}|Y_s|\,ds$ follows from Norm Bound for a Vector-Valued Lebesgue Integral over a Compact Interval (with $n=1$) and monotonicity of the integral; the integrand $s\mapsto|Y_s|$ is measurable by Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable (the norm being continuous) and bounded.

Step 2 (Claim 2). Assume $\Lambda_1TA_0<L$; since $A_0>0$, this forces $L>0$. Put $W_t=\max_{c}|\mathsf{C}^{+,c}_t-\mathsf{C}^{c}_t|$ for $t\in[0,T]$; by claim 2(a) of Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution for both solutions, each $t\mapsto|\mathsf{C}^{+,c}_t-\mathsf{C}^{c}_t|$ is continuous on $[0,T]$ (the absolute value of a difference of continuous functions), hence so is $W$ (a maximum of finitely many continuous functions), $W_0=0$, and every level $\mathsf{C}^{c}_t$, $\mathsf{C}^{+,c}_t$ lies in $[0,NBT]\subseteq[0,R]$. Let $S=\{t\in[0,T]:W_s\le L\text{ for all }s\in[0,t]\}$ and let $t^{*}$ be its least upper bound; $0\in S$, $S$ is an interval containing $[0,t^{*})$, and $W_{t^{*}}\le L$ by continuity, so $t^{*}\in S$; moreover $t^{*}>0$ by continuity and $W_0=0<L$.

For $t\in[0,t^{*}]$ and every label $c$ the two levels $\mathsf{C}^{c}_t,\mathsf{C}^{+,c}_t$ bound a window in $[0,R]$ of length $W_t\le L$, so hypothesis (D) gives $|p^{c}(\mathsf{C}^{+,c}_t)-p^{c}(\mathsf{C}^{c}_t)|\le|\mathsf{C}^{+,c}_t-\mathsf{C}^{c}_t|+D$ (if $\mathsf{C}^{+,c}_t<\mathsf{C}^{c}_t$ apply (D) to the window $[\mathsf{C}^{+,c}_t,\mathsf{C}^{c}_t]$ and change signs). By the exact identity of claim 1, the triangle inequality for finite sums and $|v_c|=\sqrt{2}$ (both recorded in Step 1), $\iota_t\le\mathsf{m}$, and the consumed-time bound of claim 1,

$$|Y_t|\le\sqrt{2}\,\mathsf{m}+\sqrt{2}\sum_{c}\Bigl(\Lambda_1\int_{[0,t]}|Y_s|\,ds+D\Bigr)=\sqrt{2}\bigl(\mathsf{m}+l(l-1)D\bigr)+\sqrt{2}\,l(l-1)\Lambda_1\int_{[0,t]}|Y_s|\,ds\qquad(t\in[0,t^{*}]).$$

The function $u(t)=|Y_t|$ restricted to $[0,t^{*}]$ is bounded and measurable with respect to the trace Borel $\sigma$-algebra of $[0,t^{*}]$ (Step 1; the trace $\sigma$-algebra of $[0,t^{*}]$ consists of the sets $S'\cap[0,t^{*}]$ with $S'$ in the trace Borel $\sigma$-algebra of $[0,T]$), so Gronwall's Lemma for Bounded Measurable Functions, applied on $[0,t^{*}]$ (with $t^{*}>0$ in the role of $T$ there) with $a=\sqrt{2}(\mathsf{m}+l(l-1)D)$ and the constant $\sqrt{2}\,l(l-1)\Lambda_1\ge0$, yields $|Y_t|\le a\exp(\sqrt{2}\,l(l-1)\Lambda_1t)\le A_0$ for $t\in[0,t^{*}]$, $\exp$ being nondecreasing (claim 4 of Basic Properties of the Exponential Function). Consequently, by the consumed-time bound of claim 1 and monotonicity of the integral, $W_{t^{*}}\le\Lambda_1\int_{[0,t^{*}]}|Y_s|\,ds\le\Lambda_1t^{*}A_0\le\Lambda_1TA_0<L$. If $t^{*}<T$, then, since $W_{t^{*}}<L$ strictly, continuity of $W$ at $t^{*}$ would give $\epsilon>0$ with $t^{*}+\epsilon\le T$ and $W_s<L$ for all $s\in[t^{*},t^{*}+\epsilon]$; together with $W_s\le L$ on $[0,t^{*}]$ this yields $t^{*}+\epsilon\in S$, contradicting the definition of $t^{*}$. Hence $t^{*}=T$, and the two bounds of claim 2 hold on all of $[0,T]$ (the bound on $W_t$ by the same computation with $t$ in place of $t^{*}$).

Step 3 (Claim 3). Assume $\Lambda_1TA_0<L$, so that claim 2 applies. Fix $s\in[0,T]$ and a label $c$. Part (ii) of Multivariate Taylor Expansion with Uniform Second-Order Remainder, applied on $\mathcal{O}=U\times V$ to $f=\psi_c$ (of class $C^2$), the points $x=(\Sigma_s,a_s)$ and $y=(\Sigma^{+}_s,a_s)$, whose segment lies in $\Delta^l\times\mathcal{A}$ by (F1), and $M_2=3K$ (Step 1), gives, with $h=y-x=(\Sigma^{+}_s-\Sigma_s,0)$, whose coordinates $h_i$ vanish for $i>l$,

$$\Bigl|\psi_c(\Sigma^{+}_s,a_s)-\psi_c(\Sigma_s,a_s)-g^{c}(\Sigma_s,a_s)\cdot(\Sigma^{+}_s-\Sigma_s)\Bigr|\le\tfrac12(l+m)\,3K\,|\Sigma^{+}_s-\Sigma_s|^{2}=\frac{\Lambda_2|Y_s|^{2}}{N^{2}} .$$

Define $\rho^{c}_s=N\bigl(\psi_c(\Sigma^{+}_s,a_s)-\psi_c(\Sigma_s,a_s)\bigr)-g^{c}(\Sigma_s,a_s)\cdot Y_s$, a bounded measurable function of $s$ (Step 1 and (F2)) with $|\rho^{c}_s|\le\Lambda_2|Y_s|^{2}/N\le\Lambda_2A_0^{2}/N$ by claim 2. Integrating (linearity),

$$\mathsf{C}^{+,c}_t-\mathsf{C}^{c}_t=\int_{[0,t]}g^{c}(\Sigma_s,a_s)\cdot Y_s\,ds+\int_{[0,t]}\rho^{c}_s\,ds,\qquad\Bigl|\int_{[0,t]}\rho^{c}_s\,ds\Bigr|\le\frac{\Lambda_2TA_0^{2}}{N},$$

the last bound by Norm Bound for a Vector-Valued Lebesgue Integral over a Compact Interval, monotonicity, and the value $t\le T$ of the restricted Lebesgue measure of $[0,t]$. Next put $d^{c}_t=p^{c}(\mathsf{C}^{+,c}_t)-p^{c}(\mathsf{C}^{c}_t)-(\mathsf{C}^{+,c}_t-\mathsf{C}^{c}_t)$; by claim 2 the two levels bound a window in $[0,R]$ of length at most $\Lambda_1TA_0<L$, so hypothesis (D) gives $|d^{c}_t|\le D$ (when $\mathsf{C}^{+,c}_t<\mathsf{C}^{c}_t$, apply (D) to the window $[\mathsf{C}^{+,c}_t,\mathsf{C}^{c}_t]$ and note that $d^{c}_t$ changes sign). Substituting into the exact identity of claim 1,

$$Y_t=v_{c_0}\iota_t+\sum_{c}v_c\Bigl(\int_{[0,t]}g^{c}(\Sigma_s,a_s)\cdot Y_s\,ds+\int_{[0,t]}\rho^{c}_s\,ds+d^{c}_t\Bigr)=v_{c_0}\iota_t+\int_{[0,t]}\mathcal{E}(\Sigma_s,a_s)Y_s\,ds+\mathsf{d}_t,$$

where $\mathsf{d}_t=\sum_c v_c\bigl(\int_{[0,t]}\rho^{c}_s\,ds+d^{c}_t\bigr)$ and where we used, for each coordinate $\gamma$, linearity of the integral to write $\sum_c v_c^{\gamma}\int_{[0,t]}g^{c}(\Sigma_s,a_s)\cdot Y_s\,ds=\int_{[0,t]}\sum_c v_c^{\gamma}\,g^{c}(\Sigma_s,a_s)\cdot Y_s\,ds=\int_{[0,t]}\bigl(\mathcal{E}(\Sigma_s,a_s)Y_s\bigr)^{\gamma}\,ds$, by the definition of $\mathcal{E}$. Finally, by the triangle inequality for finite sums and $|v_c|=\sqrt{2}$ (Step 1), $|\mathsf{d}_t|\le\sqrt{2}\sum_c\bigl(\Lambda_2TA_0^{2}/N+D\bigr)=\sqrt{2}\,l(l-1)\bigl(D+\Lambda_2TA_0^{2}/N\bigr)$.