Regularity and Derivative Bounds of the Extended Aggregate Observation Drift

Let $l$ and $\tilde{l}$ be \reftext{def:natural-numbers-2026a}{natural numbers} with $l\ge2$ and $\tilde{l}\ge1$, let $\tilde{\beta}$ be an \reftext{def:observation-rate-family-2026a}{observation-rate family} on $l$ states with $\tilde{l}$ observation channels and rate bound $\tilde{B}$, let $(\tilde{U},\bar{\tilde{\beta}})$ be a \reftext{def:c2-observation-rate-extension-2026a}{twice continuously differentiable extension} of $\tilde{\beta}$ with derivative bound $\tilde{K}$, and let $\bar{\tilde{b}}$ be the \reftext{def:extended-aggregate-observation-drift-2026a}{extended aggregate observation drift} of $(\tilde{U},\bar{\tilde{\beta}})$. Adopt the coordinate and partial-derivative notation $\partial_\gamma$, $\partial_\delta\partial_\gamma$ of the extension definition, write $\Delta^l$ for the \reftext{def:probability-simplex-2026a}{probability simplex}, and write $d$ for the \reftext{def:euclidean-distance-rn-2026a}{Euclidean distance}. Then for every $\upsilon\in\{1,\dots,\tilde{l}\}$:

\textbf{(i) (Regularity, derivative formulas, and restriction.)} $\bar{\tilde{b}}^\upsilon$ is a \reftext{def:c1-map-euclidean-open-set-2026a}{$C^1$ map} on $\tilde{U}$, each $\partial_\gamma\bar{\tilde{b}}^\upsilon$ is again a $C^1$ map on $\tilde{U}$, and for all $\gamma,\delta\in\{1,\dots,l\}$ and $\Sigma\in\tilde{U}$:

Moreover $\bar{\tilde{b}}$ agrees on $\Delta^l$ with the \reftext{def:aggregate-observation-drift-2026a}{aggregate observation drift} of $\tilde{\beta}$.

\textbf{(ii) (First-order bounds on the simplex.)} $|\partial_\gamma\bar{\tilde{b}}^\upsilon(\Sigma)|\le\tilde{B}+\tilde{K}$ for all $\gamma\in\{1,\dots,l\}$ and all $\Sigma\in\Delta^l$, and

\textbf{(iii) (Second-order bounds and uniform continuity on the simplex.)} $|\partial_\delta\partial_\gamma\bar{\tilde{b}}^\upsilon(\Sigma)|\le3\,\tilde{K}$ for all $\gamma,\delta\in\{1,\dots,l\}$ and $\Sigma\in\Delta^l$; and for every real $\varepsilon>0$ there is a real $\delta^\circ>0$, which may be chosen independently of $\gamma$, $\delta$, and $\upsilon$, such that $|\partial_\delta\partial_\gamma\bar{\tilde{b}}^\upsilon(\Sigma)-\partial_\delta\partial_\gamma\bar{\tilde{b}}^\upsilon(\Sigma')|\le\varepsilon$ for all $\gamma,\delta\in\{1,\dots,l\}$, all $\upsilon\in\{1,\dots,\tilde{l}\}$, and all $\Sigma,\Sigma'\in\Delta^l$ with $d(\Sigma,\Sigma')\le\delta^\circ$.