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

Adopt the setting, hypotheses and notation 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 hence of Uniform Pair-Exponent Bound on the Synthetic Copy: Removed-Clock Intensities, the Insertion Response on the Clock-Good Event, and Bounds on the Pair Covariances, 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 and Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect): in particular the data $N$, $l$, $m$, $\tilde{l}$, $B$, $K$, $\tilde{B}$, $\tilde{K}$, $\underline{b}$, $T$, $D$, the control set $\mathcal{A}$, the transition-rate family $\beta$ with its aggregate fluctuation covariance $\Theta$, the label rates $\psi_c$, state gradients $g^{c}$ and drift Jacobian $\mathcal{E}$ on $\Delta^l\times\mathcal{A}$, the constants $\Lambda_1$, $\Lambda_2$, $\Gamma=\sqrt{l}(\tilde{B}+\tilde{K})$ and $A_0$, the cells with $\mu_{\max}$ and $\mu_{\min}$, the move size $\mathsf{m}$, the event $G^{\mathsf{m}}$, the tracked records $\mathsf{T}_\omega$, the regularised paths $\bar\Sigma^{\sharp,r}(\omega)$ with consumed clocks $\mathsf{C}^{\sharp,c,r}_t(\omega)$, the record-frozen control paths $a^{r}$, the profile $\varpi$ with bound $\Lambda$, the mean-field label rates $\phi_c$ and consumed clocks $\bar{\mathsf{C}}^{c}$, the injection weights $w$ with $\lVert w\rVert_1$, the profile injection $\bar{F}_t$ and profile energy $\mathcal{P}$, the weighted response $\hat\psi^{r}_t(\omega)$, the realized injection $F^{r}_t(\omega)$, the pair-exponent quadratic form $\mathcal{Q}^{\omega}(r)$, the observation gradients $g_\upsilon$ and the observation information matrix $\tilde{D}(x)$ ($x\in\Delta^l$). Integrals, measurability and norms are as there; $\exp$ is the exponential function.

Three notational reservations: the letter $\Theta$ denotes here the aggregate fluctuation covariance and never the parameter coordinate map of the copy; the bound $\mathsf{M}$ below is unrelated to the pathwise score map $\Psi$ of the adopted setting; and $\mathsf{D}^{r}_{\mathrm{ctl}}$ below is unrelated to the record coordinate map $\mathsf{D}$ of the copy.

Comparison data. Let $S:[0,T]\to\Delta^l$ and $\mathsf{A}:[0,T]\to\mathcal{A}$, written $t\mapsto S_t$ and $t\mapsto\mathsf{A}_t$, be maps with measurable components (a comparison pair; no dynamic relation between $S$ and $\mathsf{A}$ is assumed, so this is weaker than a mean-field trajectory pair), and take the mean-field label rates to be

$$\phi_c(t)=\psi_c(S_t,\mathsf{A}_t)\qquad(c\in\mathcal{L},\ t\in[0,T])$$

(admissible by claim 1). Let $\mathsf{M}\ge0$ be a real number and let $\bar\psi:[0,T]\to\mathbb{R}^l$, $t\mapsto\bar\psi_t$, be a map with measurable components, $|\bar\psi_t|\le\mathsf{M}$ for all $t$, satisfying the profile response equation

$$\bar\psi_u=\int_{[0,u]}\Bigl(\mathcal{E}(S_s,\mathsf{A}_s)\,\bar\psi_s+\Theta(S_s,\mathsf{A}_s)\,\varpi_s\Bigr)\,ds\qquad(u\in[0,T])$$

(the integrand being bounded with measurable components by claim 1). Put $\Lambda_3=3K\sqrt{l(l+m)}$ and $\Lambda_{\mathcal{E}}=\sqrt{2}\,l(l-1)\sqrt{l}\,(B+K)$. For $r\in\mathbf{R}$ define the control discrepancy of the record

$$\mathsf{D}^{r}_{\mathrm{ctl}}=\int_{[0,T]}\sum_{c\in\mathcal{L}}\Bigl(\bigl|\psi_c(S_t,a^{r}_t)-\psi_c(S_t,\mathsf{A}_t)\bigr|+\bigl|g^{c}(S_t,a^{r}_t)-g^{c}(S_t,\mathsf{A}_t)\bigr|\Bigr)\,dt$$

(its integrand being bounded and measurable by claim 1).

1. (Mean-field rates, injection and energy.) Each $\phi_c$ is measurable with values in $[0,B]$, so the objects 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 are defined; the integrand of $\mathsf{D}^{r}_{\mathrm{ctl}}$ is bounded and measurable for every $r$; the maps $s\mapsto\mathcal{E}(S_s,\mathsf{A}_s)\bar\psi_s$ and $s\mapsto\Theta(S_s,\mathsf{A}_s)\varpi_s$ are bounded with measurable components; and

$$\bar{F}_t=\int_{[0,t]}\Theta(S_s,\mathsf{A}_s)\,\varpi_s\,ds\quad(t\in[0,T]),\qquad \mathcal{P}=\int_{[0,T]}\varpi_s\cdot\bigl(\Theta(S_s,\mathsf{A}_s)\,\varpi_s\bigr)\,ds .$$

2. (Clock discrepancy and injection error on close records.) Let $\omega\in G^{\mathsf{m}}$, $r\in\mathsf{T}_\omega$, and let $\varepsilon_S\ge0$ and $\varepsilon_{\mathrm{ctl}}\ge0$ be real numbers with $|\bar\Sigma^{\sharp,r}_t(\omega)-S_t|\le\varepsilon_S$ for every $t\in[0,T]$ and $\mathsf{D}^{r}_{\mathrm{ctl}}\le\varepsilon_{\mathrm{ctl}}$. Then for every label $c$ and every $t\in[0,T]$, $|\mathsf{C}^{\sharp,c,r}_t(\omega)-\bar{\mathsf{C}}^{c}_t|\le N(\Lambda_1T\varepsilon_S+\varepsilon_{\mathrm{ctl}})$, and

$$|F^{r}_t(\omega)-\bar{F}_t|\le\mathsf{e}_F,\qquad \mathsf{e}_F=2\Lambda\,l(l-1)\,(\Lambda_1T\varepsilon_S+\varepsilon_{\mathrm{ctl}})+\frac{2\Lambda\,l(l-1)\,\mu_{\max}}{N}\Bigl(3+\frac{2(\Lambda_1TA_0+\mu_{\max})}{\mu_{\min}}\Bigr).$$

3. (Gronwall comparison with the profile response.) Under the assumptions of claim 2, for every $t\in[0,T]$,

$$|\hat\psi^{r}_t(\omega)-\bar\psi_t|\le\epsilon_\psi,\qquad \epsilon_\psi=\Bigl(\mathsf{e}_F+\frac{\sqrt{2}\,l(l-1)\,\lVert w\rVert_1}{N}\Bigl(D+\frac{\Lambda_2TA_0^{2}}{N}\Bigr)+\sqrt{2}\,\mathsf{M}\bigl(l(l-1)\Lambda_3T\varepsilon_S+\varepsilon_{\mathrm{ctl}}\bigr)\Bigr)\exp(\Lambda_{\mathcal{E}}T).$$

4. (Observation-information bound with mean-field data.) Under the assumptions of claim 2, the map $t\mapsto\bar\psi_t\cdot(\tilde{D}(S_t)\bar\psi_t)$ is bounded, measurable and nonnegative, and for every real $\zeta>0$

$$\mathcal{Q}^{\omega}(r)\le(1+\zeta)\,N\Bigl(\int_{[0,T]}\bar\psi_t\cdot\bigl(\tilde{D}(S_t)\bar\psi_t\bigr)\,dt+\tilde{l}\,T\,\kappa\Bigr)+\Bigl(1+\frac1\zeta\Bigr)\frac{9\,\tilde{l}\,l^{2}\tilde{K}^{2}T\lVert w\rVert_1^{2}A_0^{4}}{4N^{3}\underline{b}},$$

where

$$\kappa=\frac{\Gamma\,(2\mathsf{M}+\epsilon_\psi)\bigl(3l\tilde{K}\,\varepsilon_S\,(\mathsf{M}+\epsilon_\psi)+\Gamma\,\epsilon_\psi\bigr)}{\underline{b}}+\frac{\Gamma^{3}\mathsf{M}^{2}\,\varepsilon_S}{\underline{b}^{2}} .$$