Twice Continuously Differentiable Extension of an Observation-Rate Family

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 $\Delta^l\subset\mathbb{R}^l$ be the \reftext{def:probability-simplex-2026a}{probability simplex}, and let $\tilde{K}\ge0$ be a \reftext{def:real-numbers-c54-2026c}{real number}. Points of \reftext{def:euclidean-space-rn-2026a}{Euclidean space} $\mathbb{R}^{l}$ are written $\Sigma$, with coordinates $\Sigma^1,\dots,\Sigma^l$. For a real-valued function $f$ on an open subset of $\mathbb{R}^{l}$ we write $\partial_\gamma f$ for the \reftext{def:partial-derivative-coordinate-map-2026a}{partial derivative} of $f$ with respect to the coordinate $\Sigma^\gamma$, and $\partial_\delta\partial_\gamma f$ for $\partial_\delta$ applied to the function $\partial_\gamma f$.

A pair $(\tilde{U},\bar{\tilde{\beta}})$ is a \textbf{twice continuously differentiable extension of the observation-rate family} $\tilde{\beta}$ with \textbf{derivative bound} $\tilde{K}$ if it consists of an \reftext{def:open-subset-euclidean-space-2026a}{open} set $\tilde{U}\subseteq\mathbb{R}^l$ with $\Delta^l\subset\tilde{U}$ and a family of functions $\bar{\tilde{\beta}}(\sigma,\upsilon,\cdot):\tilde{U}\to\mathbb{R}$, indexed by the pairs $(\sigma,\upsilon)$ with $\sigma\in\{1,\dots,l\}$ and $\upsilon\in\{1,\dots,\tilde{l}\}$, such that for every such pair:

\textbf{1. (Extension.)} $\bar{\tilde{\beta}}(\sigma,\upsilon,\Sigma)=\tilde{\beta}(\sigma,\upsilon,\Sigma)$ for all $\Sigma\in\Delta^l$.

\textbf{2. (Regularity.)} $\bar{\tilde{\beta}}(\sigma,\upsilon,\cdot)$ is a \reftext{def:c1-map-euclidean-open-set-2026a}{$C^1$ map} on the open set $\tilde{U}$, and for every $\gamma\in\{1,\dots,l\}$ the partial derivative $\partial_\gamma\bar{\tilde{\beta}}(\sigma,\upsilon,\cdot)$ is again a $C^1$ map on $\tilde{U}$.

\textbf{3. (Derivative bounds.)} $|\partial_\gamma\bar{\tilde{\beta}}(\sigma,\upsilon,\Sigma)|\le\tilde{K}$ and $|\partial_\delta\partial_\gamma\bar{\tilde{\beta}}(\sigma,\upsilon,\Sigma)|\le\tilde{K}$ for all $\gamma,\delta\in\{1,\dots,l\}$ and all $\Sigma\in\tilde{U}$.

\textbf{4. (Uniform continuity of second derivatives.)} For every real $\varepsilon>0$ there is a real $\delta^\circ>0$ such that $|\partial_\delta\partial_\gamma\bar{\tilde{\beta}}(\sigma,\upsilon,\Sigma)-\partial_\delta\partial_\gamma\bar{\tilde{\beta}}(\sigma,\upsilon,\Sigma')|\le\varepsilon$ for all $\gamma,\delta\in\{1,\dots,l\}$, all pairs $(\sigma,\upsilon)$, and all $\Sigma,\Sigma'\in\tilde{U}$ whose \reftext{def:euclidean-distance-rn-2026a}{Euclidean distance} satisfies $d(\Sigma,\Sigma')\le\delta^\circ$.