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

Let $l\ge2$, $m\ge1$, $N\ge1$ and $\mathsf{m}\ge1$ be natural numbers ($\mathsf{m}$ is unrelated to $m$), and let $T>0$, $B\ge0$ and $R>0$ be real numbers with $R\ge NBT$. Let $\mathcal{L}$ be the set of transition labels $c=(\sigma,\gamma)$ on $l$ states, which has $l(l-1)$ elements, and for $c=(\sigma,\gamma)$ put $v_c=\delta_\gamma-\delta_\sigma$, where $\delta_1,\dots,\delta_l$ are the standard basis vectors of Euclidean space $\mathbb{R}^l$, as in Open-Loop Aggregate Solution Driven by Aggregate Transition Clocks; coordinates of points of $\mathbb{R}^l$ carry superscripts. Write $|\cdot|$ for the Euclidean norm and, for real numbers, for the absolute value, $x\cdot y$ for the dot product, $\sqrt{\cdot}$ for the nonnegative square root, and $\mathbf{1}\{P\}$ for $1$ if the condition $P$ holds and $0$ otherwise.

Cells. As in The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record, for every label $c$ let $J_c\ge1$ be a natural number and $0=b^{c}_0<b^{c}_1<\dots<b^{c}_{J_c}=R$ real numbers; the cells of the clock $c$ are the intervals $I_{c,j}=(b^{c}_{j-1},b^{c}_j]$ with lengths $\mu_{c,j}=b^{c}_j-b^{c}_{j-1}>0$ ($1\le j\le J_c$; written $|I_{c,j}|$ there); $\mathsf{L}$ is the set of pairs $q=(c,j)$ with $c\in\mathcal{L}$ and $1\le j\le J_c$; for $q=(c,j)\in\mathsf{L}$ we write $\mu_q=\mu_{c,j}$, and likewise $w_q=w_{c,j}$, $n_q=n_{c,j}$ and $\bar{J}_q=\bar{J}_{c,j}$ for the objects defined below; and $\mu_{\max}=\max_{q\in\mathsf{L}}\mu_q$.

Integration on $[0,T]$. Let $\mathcal{B}_{[0,T]}$ and $\mathrm{Leb}$ be the trace Borel $\sigma$-algebra and the restricted Lebesgue measure on $[0,T]$ (written $\lambda_{[0,T]}$ in that lemma). A real function on $[0,T]$ is measurable when it is measurable with respect to $\mathcal{B}_{[0,T]}$ and the Borel $\sigma$-algebra of the real line. For a bounded measurable $f:[0,T]\to\mathbb{R}$ and $J\in\mathcal{B}_{[0,T]}$ we write $\int_{J}f(s)\,ds=\int_{[0,T]}\mathbf{1}_{J}f\,d\mathrm{Leb}$, the Lebesgue integral of the integrable function $\mathbf{1}_{J}f$, with $\mathbf{1}_{J}$ the indicator of $J$; for maps into $\mathbb{R}^l$ with bounded measurable components the integral is taken componentwise.

Data. For every label $c$ let $\phi_c:[0,T]\to[0,B]$ be measurable (the mean-field label rates) and put

$$\bar{\mathsf{C}}^{c}_t=N\int_{[0,t]}\phi_c(s)\,ds\qquad(t\in[0,T])$$

(the mean-field consumed clocks). Let $\lambda:[0,T]\to\mathbb{R}^l$, written $t\mapsto\lambda_t$, be a map with measurable components and let $\Lambda\ge0$ be a real number with $|\lambda_t|\le\Lambda$ for every $t\in[0,T]$ (the profile and its bound). For every $c$ define the time cells

$$\bar{J}_{c,1}=\{s\in[0,T]:\bar{\mathsf{C}}^{c}_s\le b^{c}_1\},\qquad \bar{J}_{c,j}=\{s\in[0,T]:\bar{\mathsf{C}}^{c}_s\in I_{c,j}\}\quad(2\le j\le J_c),$$

the injection weights

$$w_{c,j}=\frac{N}{\mathsf{m}}\int_{\bar{J}_{c,j}}(v_c\cdot\lambda_s)\,\phi_c(s)\,ds\qquad((c,j)\in\mathsf{L})$$

(the time cells belong to $\mathcal{B}_{[0,T]}$ and all integrands below are bounded and measurable by claim 2, so these definitions make sense), the profile injection and the profile energy

$$\bar{F}_t=\int_{[0,t]}\sum_{c\in\mathcal{L}}v_c\,(v_c\cdot\lambda_s)\,\phi_c(s)\,ds\in\mathbb{R}^l\quad(t\in[0,T]),\qquad \mathcal{P}=\int_{[0,T]}\sum_{c\in\mathcal{L}}(v_c\cdot\lambda_s)^{2}\,\phi_c(s)\,ds .$$

Finally, a cell counter for $q=(c,j)\in\mathsf{L}$ is a map $n_q:[0,R]\to\{0,1,\dots,\mathsf{m}\}$ with $n_q(x)=0$ for $x\le b^{c}_{j-1}$ and $n_q(x)=\mathsf{m}$ for $x\ge b^{c}_j$.

1. (Label-rate form of the fluctuation covariance.) This claim involves only $l$, $m$, $B$ and the vectors $v_c$; none of the objects introduced in the three preceding paragraphs enters it. Let $\mathcal{A}$ be a nonempty subset of $\mathbb{R}^m$, let $\beta$ be a transition-rate family on $l$ states with control set $\mathcal{A}$ and rate bound $B$, and let $\Theta$ be its aggregate fluctuation covariance. Then for every $\Sigma$ in the probability simplex $\Delta^l$, every $\alpha\in\mathcal{A}$ and every $y\in\mathbb{R}^l$, with the matrix-vector product,

$$\Theta(\Sigma,\alpha)\,y=\sum_{c=(\sigma,\gamma)\in\mathcal{L}}v_c\,(v_c\cdot y)\,\Sigma^{\sigma}\beta(\sigma,\gamma,\Sigma,\alpha),\qquad y\cdot\bigl(\Theta(\Sigma,\alpha)\,y\bigr)=\sum_{c=(\sigma,\gamma)\in\mathcal{L}}(v_c\cdot y)^{2}\,\Sigma^{\sigma}\beta(\sigma,\gamma,\Sigma,\alpha).$$

2. (Clocks, time cells and weight bounds.) For every label $c$: $t\mapsto\bar{\mathsf{C}}^{c}_t$ is nondecreasing with $0\le\bar{\mathsf{C}}^{c}_t\le NBT\le R$; each time cell $\bar{J}_{c,j}$ is an interval (possibly empty) belonging to $\mathcal{B}_{[0,T]}$; the time cells $\bar{J}_{c,1},\dots,\bar{J}_{c,J_c}$ are pairwise disjoint with union $[0,T]$; and $N\int_{\bar{J}_{c,j}}\phi_c(s)\,ds\le\mu_{c,j}$. The integrands defining $w_{c,j}$, $\bar{F}_t$ and $\mathcal{P}$ are bounded and measurable, so these quantities are well defined, and

$$|w_{c,j}|\le\frac{\sqrt{2}\,\Lambda\,\mu_{c,j}}{\mathsf{m}},\qquad \sum_{q\in\mathsf{L}}|w_q|\le\frac{\sqrt{2}\,\Lambda\,l(l-1)\,R}{\mathsf{m}},\qquad |\bar{F}_t|\le2\Lambda BT\,l(l-1),\qquad 0\le\mathcal{P}\le2\Lambda^{2}BT\,l(l-1).$$

3. (Prior quadratic form.) Let $\kappa\ge0$ and, for every $q\in\mathsf{L}$, let $\mathsf{h}_q\ge0$ be real numbers with $\mu_q\,\mathsf{h}_q\le\kappa\,\mathsf{m}^{2}$. Then

$$\sum_{q\in\mathsf{L}}w_q^{2}\,\mathsf{h}_q\ \le\ \kappa\,N\,\mathcal{P}.$$

4. (Step-function injection.) Let $(n_q)_{q\in\mathsf{L}}$ be cell counters. Then for every $t\in[0,T]$ and every family $x=(x_c)_{c\in\mathcal{L}}$ of points of $[0,R]$,

$$\Bigl|\frac{1}{N}\sum_{c\in\mathcal{L}}v_c\sum_{j=1}^{J_c}w_{c,j}\,n_{c,j}(x_c)\ -\ \bar{F}_t\Bigr|\ \le\ \frac{2\Lambda}{N}\sum_{c\in\mathcal{L}}\bigl(|x_c-\bar{\mathsf{C}}^{c}_t|+3\mu_{\max}\bigr).$$