Regularity and Derivative Bounds of the Extended Aggregate State Drift

Let $l$ and $m$ be \reftext{def:natural-numbers-2026a}{natural numbers} with $l\ge2$ and $m\ge1$, let $\beta$ be a \reftext{def:transition-rate-family-2026a}{transition-rate family} on $l$ states with control dimension $m$ and rate bound $B$, let $(U,\bar{\beta})$ be a \reftext{def:c2-transition-rate-extension-2026a}{twice continuously differentiable extension} of $\beta$ with derivative bound $K$, and let $\bar{b}$ be the \reftext{def:extended-aggregate-state-drift-2026a}{extended aggregate state drift} of $(U,\bar{\beta})$. Adopt the coordinate and partial-derivative notation $\partial_i$, $\partial_j\partial_i$ of the extension definition, write $\Delta^l$ for the \reftext{def:probability-simplex-2026a}{probability simplex}, write $d$ for the \reftext{def:euclidean-distance-rn-2026a}{Euclidean distance}, and for $i\in\{1,\dots,l+m\}$ and $\sigma\in\{1,\dots,l\}$ set $\delta_{i\sigma}=1$ if $i=\sigma$ and $\delta_{i\sigma}=0$ otherwise. Then for every $\gamma\in\{1,\dots,l\}$:

\textbf{(i) (Regularity, derivative formulas, and restriction.)} $\bar{b}^\gamma$ is a \reftext{def:c1-map-euclidean-open-set-2026a}{$C^1$ map} on $U\times\mathbb{R}^m$, each $\partial_i\bar{b}^\gamma$ is again a $C^1$ map on $U\times\mathbb{R}^m$, and for all $i,j\in\{1,\dots,l+m\}$ and $x=(\Sigma,\alpha)\in U\times\mathbb{R}^m$:

Moreover $\bar{b}$ agrees on $\Delta^l\times\mathbb{R}^m$ with the \reftext{def:aggregate-state-drift-2026a}{aggregate state drift} of $\beta$.

\textbf{(ii) (First-order bounds on the simplex.)} $|\partial_i\bar{b}^\gamma(x)|\le l\,(B+K)$ for all $i\in\{1,\dots,l+m\}$ and all $x\in\Delta^l\times\mathbb{R}^m$, and

\textbf{(iii) (Second-order bounds and uniform continuity on the simplex.)} $|\partial_j\partial_i\bar{b}^\gamma(x)|\le 3\,l\,K$ for all $i,j\in\{1,\dots,l+m\}$ and $x\in\Delta^l\times\mathbb{R}^m$; and for every real $\varepsilon>0$ there is a real $\delta>0$ such that $|\partial_j\partial_i\bar{b}^\gamma(x)-\partial_j\partial_i\bar{b}^\gamma(y)|\le\varepsilon$ for all $i,j\in\{1,\dots,l+m\}$, all $\gamma\in\{1,\dots,l\}$, and all $x,y\in\Delta^l\times\mathbb{R}^m$ with $d(x,y)\le\delta$.