Proof of Closeness to a Mean-Field Pair on the Synthetic Copy: Clock Discrepancy, Gronwall Comparison of the Weighted Response with the Profile Response, and the Observation-Information Bound with Mean-Field Data

Throughout, $c=(\sigma,\gamma)$ denotes a label, and we use the following facts.

(F1) $|v_c|=\sqrt{2}$, $|x\cdot y|\le|x||y|$ (The Cauchy-Schwarz Inequality on $\mathbb{R}^n$) and $|\sum_ia_iy_i|\le\sum_i|a_i||y_i|$ for vectors $y_i$ and reals $a_i$ (claims 1, 5, 6 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square). Consequently, for $(\Sigma,\alpha)\in\Delta^l\times\mathcal{A}$ and $y\in\mathbb{R}^l$, since $\mathcal{E}(\Sigma,\alpha)y=\sum_cv_c(g^{c}(\Sigma,\alpha)\cdot y)$ 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, and $|g^{c}(\Sigma,\alpha)|\le\sqrt{l}(B+K)$ (its $l$ components are bounded by $B+K$ by claim 1 of that lemma),

$$|\mathcal{E}(\Sigma,\alpha)y|\le\sqrt{2}\,l(l-1)\sqrt{l}(B+K)\,|y|=\Lambda_{\mathcal{E}}|y|,\qquad |\mathcal{E}(\Sigma,\alpha)y-\mathcal{E}(\Sigma',\alpha')y|\le\sqrt{2}\sum_c|g^{c}(\Sigma,\alpha)-g^{c}(\Sigma',\alpha')|\,|y| .$$

(F2) Lipschitz bounds in the state. The simplex $\Delta^l$ is convex: for $\Sigma,\Sigma'\in\Delta^l$ and $\tau\in[0,1]$ the coordinates $(1-\tau)\Sigma^{\gamma}+\tau\Sigma'^{\gamma}$ of $\Sigma+\tau(\Sigma'-\Sigma)$ are nonnegative and sum to $1$. Hence for $\Sigma,\Sigma'\in\Delta^l$ and $\alpha\in\mathcal{A}$ the segment between $(\Sigma,\alpha)$ and $(\Sigma',\alpha)$ lies in $\Delta^l\times\mathcal{A}\subseteq U\times W_\beta$, with Euclidean distance $|\Sigma-\Sigma'|$ between its endpoints. 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 the open set $U\times W_\beta$ with $|\partial_i\psi_c|\le B+K$ and $|\partial_j\partial_i\psi_c|\le3K$ on $\Delta^l\times\mathcal{A}$; hence part (i) of Multivariate Taylor Expansion with Uniform Second-Order Remainder (with $n=l+m$) gives $|\psi_c(\Sigma,\alpha)-\psi_c(\Sigma',\alpha)|\le\Lambda_1|\Sigma-\Sigma'|$, and, applied to each $\partial_i\psi_c$, $i\in\{1,\dots,l+m\}$ (of class $C^1$ on $U\times W_\beta$ by clause 2 of C^k Maps on a Euclidean Open Set, with first partial derivatives $\partial_j\partial_i\psi_c$ bounded by $3K$ on the segment), $|\partial_i\psi_c(\Sigma,\alpha)-\partial_i\psi_c(\Sigma',\alpha)|\le\sqrt{l+m}\,3K|\Sigma-\Sigma'|$, so that $|g^{c}(\Sigma,\alpha)-g^{c}(\Sigma',\alpha)|\le\sqrt{l}\sqrt{l+m}\,3K|\Sigma-\Sigma'|=\Lambda_3|\Sigma-\Sigma'|$ (a vector of $\mathbb{R}^l$ whose components are bounded in absolute value by $M_1$ has norm at most $\sqrt{l}\,M_1$, 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). Likewise, for $x,x'\in\Delta^l$ and every $\upsilon$: $\Delta^l\subset\tilde{U}$ and the segment between $x$ and $x'$ lies in $\Delta^l$; by claims (i)--(iii) of Regularity and Derivative Bounds of the Extended Aggregate Observation Drift, $\bar{\tilde{b}}^\upsilon$ is of class $C^1$ on the open set $\tilde{U}$ and so is each $\partial_\gamma\bar{\tilde{b}}^\upsilon$ (the $C^1$ notion used there — coordinate functions and all first partial derivatives existing and continuous at every point — being clause 1 of C^k Maps on a Euclidean Open Set), so that $\bar{\tilde{b}}^\upsilon$ is of class $C^2$ in the sense of clause 2 of C^k Maps on a Euclidean Open Set; it agrees with $\tilde{b}^\upsilon$ on $\Delta^l$, has $|\partial_\gamma\bar{\tilde{b}}^\upsilon|\le\tilde{B}+\tilde{K}$ and $|\partial_\delta\partial_\gamma\bar{\tilde{b}}^\upsilon|\le3\tilde{K}$ on $\Delta^l$, and $|\tilde{b}^\upsilon(x)-\tilde{b}^\upsilon(x')|\le\Gamma|x-x'|$; hence $|g_\upsilon(x)|\le\Gamma$, and part (i) of Multivariate Taylor Expansion with Uniform Second-Order Remainder applied to each $\partial_\gamma\bar{\tilde{b}}^\upsilon$ gives $|g_\upsilon(x)-g_\upsilon(x')|\le\sqrt{l}\cdot\sqrt{l}\,3\tilde{K}|x-x'|=3l\tilde{K}|x-x'|$. Finally $\tilde{b}^\upsilon\ge\underline{b}>0$ on $\Delta^l$ by hypothesis (OC). All these bounds are applied below at points $(\bar\Sigma^{\sharp,r}_s(\omega),a^{r}_s)$ and $(S_s,\mathsf{A}_s)$, which lie in $\Delta^l\times\mathcal{A}$: for $r\in\mathsf{T}_\omega$ the path $\bar\Sigma^{\sharp,r}(\omega)$ is an open-loop aggregate solution (response data 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), which takes values in $\mathbb{G}_N\subseteq\Delta^l$ (Open-Loop Aggregate Solution Driven by Aggregate Transition Clocks), and $a^{r}_s\in\mathcal{A}$.

(F3) Measurability. The components of $t\mapsto(S_t,\mathsf{A}_t)$ are measurable and the map takes values in the nonempty set $\Delta^l\times\mathcal{A}$; $\psi_c$ and its partial derivatives $\partial_i\psi_c$ are of class $C^2$ and $C^1$ on the open set $U\times W_\beta$ (F2), hence continuous there by claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, hence sequentially continuous on $\Delta^l\times\mathcal{A}$ (an $\varepsilon$-$\delta$ continuous map preserves convergent sequences); so $t\mapsto\psi_c(S_t,\mathsf{A}_t)$ and $t\mapsto g^{c}(S_t,\mathsf{A}_t)$ have measurable components by Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable, bounded by $B$ and $\sqrt{l}(B+K)$. Each entry of $\Theta(\Sigma,\alpha)$ is a finite sum of products of coordinates of $\Sigma$ with values $\beta(\sigma,\gamma,\Sigma,\alpha)$ (Aggregate Fluctuation Covariance), and $\beta$ is jointly sequentially continuous on $\Delta^l\times\mathcal{A}$ by clause 2 of Transition-Rate Family, so $t\mapsto\Theta(S_t,\mathsf{A}_t)$ has measurable entries, bounded in absolute value by $(l-1)B$ (a diagonal entry is at most $B(1-\Sigma^{\gamma})+(l-1)B\Sigma^{\gamma}\le(l-1)B$, an off-diagonal one at most $(\Sigma^{\gamma}+\Sigma^{\delta})B\le B\le(l-1)B$, the coordinates of a point of $\Delta^l$ being nonnegative with sum $1$). The same applies with $a^{r}$ in place of $\mathsf{A}$, the record-frozen control path being a control path with measurable components (Open-Loop Aggregate Solution Driven by Aggregate Transition Clocks), and to $t\mapsto g_\upsilon(S_t)$ and $t\mapsto\tilde{b}^\upsilon(S_t)=\bar{\tilde{b}}^\upsilon(S_t)$ (continuous functions on $\tilde{U}$, by (F2) and the same lemma, composed with $S$). Products, sums and absolute values of bounded measurable functions are bounded and measurable (claims 2--4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions).

Step 1 (Claim 1). By (F3), $\phi_c=\psi_c(S_\cdot,\mathsf{A}_\cdot)$ is measurable, and $0\le\psi_c\le B$ on $\Delta^l\times\mathcal{A}$ (claim 1 of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect and the rate bound of Transition-Rate Family), hence $0\le\phi_c\le B$ on $[0,T]$, so 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 on the mean-field label rates hold. The integrand of $\mathsf{D}^{r}_{\mathrm{ctl}}$ and the maps $s\mapsto\mathcal{E}(S_s,\mathsf{A}_s)\bar\psi_s$, $s\mapsto\Theta(S_s,\mathsf{A}_s)\varpi_s$ are bounded and measurable by (F3), $\bar\psi$ and $\varpi$ being bounded with measurable components. By claim 1 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 with $y=\varpi_s$, $\sum_cv_c(v_c\cdot\varpi_s)\phi_c(s)=\sum_cv_c(v_c\cdot\varpi_s)S^{\sigma}_s\beta(\sigma,\gamma,S_s,\mathsf{A}_s)=\Theta(S_s,\mathsf{A}_s)\varpi_s$ (recall $\psi_c(\Sigma,\alpha)=\Sigma^{\sigma}\beta(\sigma,\gamma,\Sigma,\alpha)$ on $\Delta^l\times\mathcal{A}$ by claim 1 of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect), and $\sum_c(v_c\cdot\varpi_s)^{2}\phi_c(s)=\varpi_s\cdot(\Theta(S_s,\mathsf{A}_s)\varpi_s)$; integrating gives the formulas for $\bar{F}_t$ and $\mathcal{P}$.

Step 2 (Claim 2). Fix $\omega$, $r$, $\varepsilon_S$, $\varepsilon_{\mathrm{ctl}}$ as in claim 2 and drop them from the notation. For a label $c$ and $t\in[0,T]$, by claim 1 of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect (the path $\bar\Sigma^{\sharp}$ being the open-loop aggregate solution for $(\mathsf{P}^{\sharp}(\omega),a^{r},x_0)$, as in the response data 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) and the definition of $\bar{\mathsf{C}}^{c}$, $\mathsf{C}^{\sharp,c}_t-\bar{\mathsf{C}}^{c}_t=N\int_{[0,t]}\bigl(\psi_c(\bar\Sigma^{\sharp}_s,a^{r}_s)-\psi_c(S_s,\mathsf{A}_s)\bigr)\,ds$ by linearity. For every $s$, by (F2),

$$|\psi_c(\bar\Sigma^{\sharp}_s,a^{r}_s)-\psi_c(S_s,\mathsf{A}_s)|\le|\psi_c(\bar\Sigma^{\sharp}_s,a^{r}_s)-\psi_c(S_s,a^{r}_s)|+|\psi_c(S_s,a^{r}_s)-\psi_c(S_s,\mathsf{A}_s)|\le\Lambda_1\varepsilon_S+|\psi_c(S_s,a^{r}_s)-\psi_c(S_s,\mathsf{A}_s)| ,$$

so by monotonicity of the integral, $\mathbf{1}_{[0,t]}\le1$, and the definition of $\mathsf{D}^{r}_{\mathrm{ctl}}$ (whose integrand dominates the last summand), $|\mathsf{C}^{\sharp,c}_t-\bar{\mathsf{C}}^{c}_t|\le N(\Lambda_1\varepsilon_S\,T+\mathsf{D}^{r}_{\mathrm{ctl}})\le N(\Lambda_1T\varepsilon_S+\varepsilon_{\mathrm{ctl}})$, the restricted Lebesgue measure of $[0,t]$ being $t\le T$. Inserting this into claim 2 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 and summing over the $l(l-1)$ labels,

$$|F_t-\bar{F}_t|\le\frac{2\Lambda}{N}\,l(l-1)\bigl(N(\Lambda_1T\varepsilon_S+\varepsilon_{\mathrm{ctl}})+3\mu_{\max}\bigr)+\frac{4\Lambda\,l(l-1)\,\mu_{\max}(\Lambda_1TA_0+\mu_{\max})}{N\mu_{\min}}=\mathsf{e}_F .$$

Step 3 (Claim 3). Keep the data of Step 2 and put $e_t=\hat\psi_t-\bar\psi_t$, a bounded map with measurable components (claim 1 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 and the hypothesis on $\bar\psi$). Subtracting the profile response equation (with $\int_{[0,t]}\Theta(S_s,\mathsf{A}_s)\varpi_s\,ds=\bar{F}_t$ by claim 1) from the weighted response equation of claim 1 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 and using linearity of the integral,

$$e_t=(F_t-\bar{F}_t)+\hat{\mathsf{d}}_t+\int_{[0,t]}\mathcal{E}(\bar\Sigma^{\sharp}_s,a^{r}_s)\,e_s\,ds+\int_{[0,t]}\Bigl(\mathcal{E}(\bar\Sigma^{\sharp}_s,a^{r}_s)-\mathcal{E}(S_s,\mathsf{A}_s)\Bigr)\bar\psi_s\,ds ,$$

where we wrote $\mathcal{E}(\bar\Sigma^{\sharp}_s,a^{r}_s)\hat\psi_s-\mathcal{E}(S_s,\mathsf{A}_s)\bar\psi_s=\mathcal{E}(\bar\Sigma^{\sharp}_s,a^{r}_s)e_s+(\mathcal{E}(\bar\Sigma^{\sharp}_s,a^{r}_s)-\mathcal{E}(S_s,\mathsf{A}_s))\bar\psi_s$ (linearity of the matrix-vector product; all integrands bounded with measurable components by (F3) and claim 1 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). By (F1) and (F2), for every $s$,

$$\bigl|(\mathcal{E}(\bar\Sigma^{\sharp}_s,a^{r}_s)-\mathcal{E}(S_s,\mathsf{A}_s))\bar\psi_s\bigr|\le\sqrt{2}\,\mathsf{M}\sum_c\Bigl(|g^{c}(\bar\Sigma^{\sharp}_s,a^{r}_s)-g^{c}(S_s,a^{r}_s)|+|g^{c}(S_s,a^{r}_s)-g^{c}(S_s,\mathsf{A}_s)|\Bigr)\le\sqrt{2}\,\mathsf{M}\Bigl(l(l-1)\Lambda_3\varepsilon_S+\sum_c|g^{c}(S_s,a^{r}_s)-g^{c}(S_s,\mathsf{A}_s)|\Bigr),$$

so by Norm Bound for a Vector-Valued Lebesgue Integral over a Compact Interval, monotonicity and the definition of $\mathsf{D}^{r}_{\mathrm{ctl}}$, the last integral in the identity for $e_t$ has norm at most $\sqrt{2}\mathsf{M}(l(l-1)\Lambda_3T\varepsilon_S+\varepsilon_{\mathrm{ctl}})$. Also $|\mathcal{E}(\bar\Sigma^{\sharp}_s,a^{r}_s)e_s|\le\Lambda_{\mathcal{E}}|e_s|$ by (F1). Hence, by Norm Bound for a Vector-Valued Lebesgue Integral over a Compact Interval, claim 2 and the bound on $\hat{\mathsf{d}}_t$ from claim 1 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, the function $u(t)=|e_t|$ (bounded and measurable by Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable, the norm being continuous) satisfies

$$u(t)\le\mathsf{c}_\psi+\Lambda_{\mathcal{E}}\int_{[0,t]}u(s)\,ds\quad(t\in[0,T]),\qquad \mathsf{c}_\psi=\mathsf{e}_F+\frac{\sqrt{2}\,l(l-1)\lVert w\rVert_1}{N}\Bigl(D+\frac{\Lambda_2TA_0^{2}}{N}\Bigr)+\sqrt{2}\mathsf{M}\bigl(l(l-1)\Lambda_3T\varepsilon_S+\varepsilon_{\mathrm{ctl}}\bigr),$$

and Gronwall's Lemma for Bounded Measurable Functions gives $u(t)\le\mathsf{c}_\psi\exp(\Lambda_{\mathcal{E}}t)\le\mathsf{c}_\psi\exp(\Lambda_{\mathcal{E}}T)=\epsilon_\psi$, since $\mathsf{c}_\psi\ge0$ (every summand is nonnegative) and $\exp$ is nondecreasing (claim 4 of Basic Properties of the Exponential Function).

Step 4 (Claim 4). Keep the data of Step 2. By (F3) the map $t\mapsto\bar\psi_t\cdot(\tilde{D}(S_t)\bar\psi_t)=\sum_\upsilon(g_\upsilon(S_t)\cdot\bar\psi_t)^{2}/\tilde{b}^\upsilon(S_t)$ (the identity of the setting 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) is bounded and measurable, reciprocals of measurable functions with values in $[\underline{b},\infty)$ being measurable by Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable; it is nonnegative as a sum of squares over positive denominators. Fix $t\in[0,T]$ and $\upsilon$, and put $x=\bar\Sigma^{\sharp}_t$, $x'=S_t$, $z=\hat\psi_t$, $z'=\bar\psi_t$, $g=g_\upsilon(x)$, $g'=g_\upsilon(x')$, $b=\tilde{b}^\upsilon(x)$, $b'=\tilde{b}^\upsilon(x')$. Then $|x-x'|\le\varepsilon_S$, $|z'|\le\mathsf{M}$, $|z-z'|\le\epsilon_\psi$ (claim 3), $|z|\le\mathsf{M}+\epsilon_\psi$, $|g|,|g'|\le\Gamma$, $|g-g'|\le3l\tilde{K}\varepsilon_S$, $b,b'\ge\underline{b}$ and $|b-b'|\le\Gamma\varepsilon_S$ by (F2). Writing

$$\frac{(g\cdot z)^{2}}{b}-\frac{(g'\cdot z')^{2}}{b'}=\frac{(g\cdot z-g'\cdot z')(g\cdot z+g'\cdot z')}{b}+(g'\cdot z')^{2}\,\frac{b'-b}{b\,b'}$$

and using $|g\cdot z-g'\cdot z'|\le|(g-g')\cdot z|+|g'\cdot(z-z')|\le3l\tilde{K}\varepsilon_S(\mathsf{M}+\epsilon_\psi)+\Gamma\epsilon_\psi$, $|g\cdot z+g'\cdot z'|\le\Gamma(2\mathsf{M}+\epsilon_\psi)$ and $(g'\cdot z')^{2}\le\Gamma^{2}\mathsf{M}^{2}$ (all by (F1)), we obtain

$$\frac{(g_\upsilon(x)\cdot\hat\psi_t)^{2}}{\tilde{b}^\upsilon(x)}\le\frac{(g_\upsilon(S_t)\cdot\bar\psi_t)^{2}}{\tilde{b}^\upsilon(S_t)}+\kappa .$$

Summing over $\upsilon$ gives $\hat\psi_t\cdot(\tilde{D}(\bar\Sigma^{\sharp}_t)\hat\psi_t)\le\bar\psi_t\cdot(\tilde{D}(S_t)\bar\psi_t)+\tilde{l}\kappa$ for every $t$; both sides are bounded and measurable (claim 3 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 and the first paragraph of this step), so integrating over $[0,T]$ by monotonicity and multiplying by $(1+\zeta)N$, claim 3 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 yields claim 4.