Proof 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

Throughout, $c=(\sigma,\gamma)$ denotes a label, and for $u\ge0$ we put $\nu(u)=\#\{k\in\{1,\dots,\mathsf{m}\}:u_k\le u\}$, so that $p^{+,c_0}=p^{c_0}+\nu$ and $p^{+,c}=p^{c}$ for $c\neq c_0$ by the definition of the perturbed clock family in Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect; $\nu$ is nondecreasing with values in $\{0,\dots,\mathsf{m}\}$, and $\iota_t=\nu(\mathsf{C}^{+,c_0}_t)$, $\iota^{-}_t=\nu(\mathsf{C}^{c_0}_t)$. Three elementary facts are used. (F1) $|v_c|=\sqrt{2}$ 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 claim 1 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square; and for vectors $y_1,\dots,y_n\in\mathbb{R}^l$ and real numbers $\lambda_1,\dots,\lambda_n$ one has $|\sum_i\lambda_iy_i|\le\sum_i|\lambda_i|\,|y_i|$, by induction on $n$ from claims 6 and 5 of the same lemma. (F2) The simplex is convex: for $x,y\in\Delta^l$ and $\tau\in[0,1]$ the point $x+\tau(y-x)$ has nonnegative coordinates $(1-\tau)x^\gamma+\tau y^\gamma$ summing to $1$, 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 $|y-x|$ (the control coordinates of the difference vanish). (F3) 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 $[0,T]$, so it is measurable for the trace Borel $\sigma$-algebra by claims 1 and 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, and bounded, the points $\Sigma_s,\Sigma^{+}_s\in\mathbb{G}_N\subseteq\Delta^l$ having coordinates in $[0,1]$; and if $F$ is a real 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 (the control path by the definition of the open-loop aggregate solution, the state paths as just noted), and they are bounded whenever $F$ is bounded on $\Delta^l\times\mathcal{A}$.

Step 1 (Claim 1). Let $c\in\mathcal{L}$ and let $0\le u\le u'\le R$ be real with $u'-u\le L$. If $c\neq c_0$ then $p^{c}=p^{+,c}$, and hypothesis (D$^{+}$) gives $|p^{c}(u')-p^{c}(u)-(u'-u)|\le D\le D+\mathsf{m}$. If $c=c_0$ then $p^{c_0}=p^{+,c_0}-\nu$, so

$$p^{c_0}(u')-p^{c_0}(u)-(u'-u)=\Bigl(p^{+,c_0}(u')-p^{+,c_0}(u)-(u'-u)\Bigr)-\bigl(\nu(u')-\nu(u)\bigr),$$

where the first bracket has absolute value at most $D$ by (D$^{+}$) and $0\le\nu(u')-\nu(u)\le\mathsf{m}$ because $\nu$ is nondecreasing with values in $\{0,\dots,\mathsf{m}\}$; the triangle inequality for absolute values gives $|p^{c_0}(u')-p^{c_0}(u)-(u'-u)|\le D+\mathsf{m}$. Thus $p$ satisfies hypothesis (D) of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect with tolerance $D+\mathsf{m}$. All other data of that lemma ($p$, $a$, $x_0$, $c_0$, $\mathsf{m}$, the insertion points, $R$, $L$, the conflict-freeness of $(p,a,x_0)$ and $(p^{+},a,x_0)$) are unchanged, so its claims 1, 2 and 3 apply with $D+\mathsf{m}$ in place of $D$; substituting $D+\mathsf{m}$ for $D$ in the formula for $A_0$ gives exactly $A_0^{+}$. The two bounds stated under the assumption $\Lambda_1TA_0^{+}<L$ are claim 2 of that lemma so applied.

Step 2 (Claim 2). Since $\iota^{-}_t=\nu(\mathsf{C}^{c_0}_t)$ and $\nu$ takes values in $\{0,\dots,\mathsf{m}\}$, $0\le\iota^{-}_t\le\mathsf{m}$. By claim 1 of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect (available by Step 1),

$$Y_t=v_{c_0}\,\iota_t+\sum_{c\in\mathcal{L}}v_c\Bigl(p^{c}\bigl(\mathsf{C}^{+,c}_t\bigr)-p^{c}\bigl(\mathsf{C}^{c}_t\bigr)\Bigr).$$

For $c\neq c_0$ the summand equals $v_c\bigl(p^{+,c}(\mathsf{C}^{+,c}_t)-p^{+,c}(\mathsf{C}^{c}_t)\bigr)$ because $p^{c}=p^{+,c}$. For $c=c_0$, the identity $p^{c_0}=p^{+,c_0}-\nu$ gives

$$p^{c_0}\bigl(\mathsf{C}^{+,c_0}_t\bigr)-p^{c_0}\bigl(\mathsf{C}^{c_0}_t\bigr)=p^{+,c_0}\bigl(\mathsf{C}^{+,c_0}_t\bigr)-p^{+,c_0}\bigl(\mathsf{C}^{c_0}_t\bigr)-\iota_t+\iota^{-}_t .$$

Substituting, the terms $v_{c_0}\iota_t$ and $-v_{c_0}\iota_t$ cancel and the removal-form identity follows.

Step 3 (Claim 3). Assume $\Lambda_1TA_0^{+}<L$, so that the bounds of claim 1 hold.

Measurability. By claim 1 of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect, $\psi_c$ is of class $C^2$ on $U\times V$ with $|\partial_i\psi_c|\le B+K$ and $|\partial_j\partial_i\psi_c|\le3K$ at every point of $\Delta^l\times\mathcal{A}$, and $\psi_c(x,\alpha)=x^{\sigma}\beta(\sigma,\gamma,x,\alpha)$ for all $(x,\alpha)\in\Delta^l\times\mathcal{A}$, so that $0\le\psi_c\le B$ there by the rate bound of Transition-Rate Family and $0\le x^\sigma\le1$. The map $\psi_c$ is continuous on $U\times V$ (clauses 1 and 2 of C^k Maps on a Euclidean Open Set), so $s\mapsto\psi_c(\Sigma_s,a_s)$ and $s\mapsto\psi_c(\Sigma^{+}_s,a_s)$ are bounded and measurable by (F3). 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 and bounded by $B+K$ by (F3), $\Sigma^{+}_s$ lying in $\Delta^l$; this is the $\eta$-th component of $s\mapsto g^{c}(\Sigma^{+}_s,a_s)$. 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 and (F3); and the map $s\mapsto\mathcal{E}(\Sigma_s,a_s)Y_s$ is bounded and measurable by claim 1 of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect.

Two Taylor expansions. Fix $s\in[0,T]$ and a label $c$. Apply part (ii) of Multivariate Taylor Expansion with Uniform Second-Order Remainder on the open set $W=U\times V\subseteq\mathbb{R}^{l+m}$ to $f=\psi_c$, with the points $x=(\Sigma^{+}_s,a_s)$ and $y=(\Sigma_s,a_s)$, whose segment lies in $\Delta^l\times\mathcal{A}$ by (F2), and with $M_2=3K$: here $h=y-x=(\Sigma_s-\Sigma^{+}_s,0)$ has coordinates $h_i=0$ for $i>l$, so $\sum_{i=1}^{l+m}\partial_i\psi_c(x)h_i=g^{c}(\Sigma^{+}_s,a_s)\cdot(\Sigma_s-\Sigma^{+}_s)$, and $|h|=|\Sigma_s-\Sigma^{+}_s|=|Y_s|/N$ by (F2) and claim 5 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square. This gives

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

Multiplying by $N$ (note $N(\Sigma_s-\Sigma^{+}_s)=-Y_s$) and rearranging, $N\bigl(\psi_c(\Sigma^{+}_s,a_s)-\psi_c(\Sigma_s,a_s)\bigr)=g^{c}(\Sigma^{+}_s,a_s)\cdot Y_s+\rho^{+,c}_s$, where $\rho^{+,c}_s=N\bigl(\psi_c(\Sigma^{+}_s,a_s)-\psi_c(\Sigma_s,a_s)\bigr)-g^{c}(\Sigma^{+}_s,a_s)\cdot Y_s$ satisfies $|\rho^{+,c}_s|\le\Lambda_2|Y_s|^{2}/N\le\Lambda_2(A_0^{+})^{2}/N$ by claim 1. Applying part (ii) instead with $x=(\Sigma_s,a_s)$ and $y=(\Sigma^{+}_s,a_s)$ (same segment, same $M_2$, now $h=(\Sigma^{+}_s-\Sigma_s,0)$ with $Nh=(Y_s,0)$) gives likewise $N\bigl(\psi_c(\Sigma^{+}_s,a_s)-\psi_c(\Sigma_s,a_s)\bigr)=g^{c}(\Sigma_s,a_s)\cdot Y_s+\rho^{-,c}_s$, where $\rho^{-,c}_s=N\bigl(\psi_c(\Sigma^{+}_s,a_s)-\psi_c(\Sigma_s,a_s)\bigr)-g^{c}(\Sigma_s,a_s)\cdot Y_s$ satisfies $|\rho^{-,c}_s|\le\Lambda_2(A_0^{+})^{2}/N$. Both $s\mapsto\rho^{\pm,c}_s$ are bounded (by these estimates) and measurable: $s\mapsto\psi_c(\Sigma_s,a_s)$ and $s\mapsto\psi_c(\Sigma^{+}_s,a_s)$ are measurable by the Measurability paragraph, and the dot products are finite sums of products of measurable functions (claims 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, using the measurability of $s\mapsto g^{c}(\Sigma_s,a_s)$ from claim 1 of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect and of $s\mapsto g^{c}(\Sigma^{+}_s,a_s)$ from the previous paragraph).

Consumed-time differences. By the formula $\mathsf{C}^{c}_t=\int_{[0,t]}N\psi_c(\Sigma_s,a_s)\,ds$ of claim 1 of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect, its analogue for $\Sigma^{+}$, and linearity of the integral,

$$\mathsf{C}^{+,c}_t-\mathsf{C}^{c}_t=\int_{[0,t]}g^{c}(\Sigma^{+}_s,a_s)\cdot Y_s\,ds+\int_{[0,t]}\rho^{+,c}_s\,ds=\int_{[0,t]}g^{c}(\Sigma_s,a_s)\cdot Y_s\,ds+\int_{[0,t]}\rho^{-,c}_s\,ds ,$$

and $\bigl|\int_{[0,t]}\rho^{\pm,c}_s\,ds\bigr|\le\Lambda_2T(A_0^{+})^{2}/N$ by Norm Bound for a Vector-Valued Lebesgue Integral over a Compact Interval (with $n=1$), monotonicity, and the value $t\le T$ of the restricted Lebesgue measure of $[0,t]$. Moreover, since $0\le\psi_c\le B$ on $\Delta^l\times\mathcal{A}$ (Measurability paragraph), monotonicity gives $0\le\mathsf{C}^{c}_t\le NBT\le R$ and likewise $0\le\mathsf{C}^{+,c}_t\le R$.

Discrepancy on the enlarged clock family. For a label $c$ put $d^{c}_t=p^{+,c}(\mathsf{C}^{+,c}_t)-p^{+,c}(\mathsf{C}^{c}_t)-(\mathsf{C}^{+,c}_t-\mathsf{C}^{c}_t)$. The two levels $\mathsf{C}^{c}_t$ and $\mathsf{C}^{+,c}_t$ lie in $[0,R]$ and, by claim 1, are at distance at most $\Lambda_1TA_0^{+}<L$, so hypothesis (D$^{+}$) applied to the window they bound 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 $d^{c}_t$ merely changes sign).

Assembly. Substituting the consumed-time differences into the identity of claim 2,

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

where $\mathsf{d}^{+}_t=\sum_cv_c\bigl(\int_{[0,t]}\rho^{+,c}_s\,ds+d^{c}_t\bigr)$ and where, for each coordinate $\gamma$, linearity of the integral gives $\sum_cv_c^{\gamma}\int_{[0,t]}g^{c}(\Sigma^{+}_s,a_s)\cdot Y_s\,ds=\int_{[0,t]}\sum_cv_c^{\gamma}\,g^{c}(\Sigma^{+}_s,a_s)\cdot Y_s\,ds=\int_{[0,t]}\bigl(\mathcal{E}(\Sigma^{+}_s,a_s)Y_s\bigr)^{\gamma}\,ds$ by the definition of $\mathcal{E}$ in Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect. The same substitution with $g^{c}(\Sigma_s,a_s)$ and $\rho^{-,c}$ gives the second identity with $\mathsf{d}^{-}_t=\sum_cv_c\bigl(\int_{[0,t]}\rho^{-,c}_s\,ds+d^{c}_t\bigr)$. Finally, by (F1) and the bounds just obtained, $|\mathsf{d}^{\pm}_t|\le\sqrt{2}\sum_c\bigl(\Lambda_2T(A_0^{+})^{2}/N+D\bigr)=\sqrt{2}\,l(l-1)\bigl(D+\Lambda_2T(A_0^{+})^{2}/N\bigr)$, the set $\mathcal{L}$ having $l(l-1)$ elements.