The Gaussian Entropy Pair is a Noise Penalty Pair, with Nonnegative Penalty and Dense Score Domain

In the setting of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation, so that the reference measure is $\rho=\gamma_{c}$, let $\beta$ and $\kappa$ be positive real numbers with $c_{k}\le\kappa\,a_{k}$ for every $k\in\mathbb{N}$, and let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Gaussian entropy pair with temperature $\beta$, whose hypothesis holds with this $\kappa$. Noise penalty pairs on $\mathcal{P}^{a}_{\rho}$ are those of that definition.

1. (Nonnegative penalty)¶ $0\le\mathcal{E}(\mu)$ for every $\mu\in\mathcal{D}$.

2. (Density of the score domain)¶ For every $\mu\in\mathcal{D}$ and every positive $\varepsilon\in\mathbb{R}$ there is $\nu\in\mathcal{D}_{\Sigma}$ with $W_{a}(\nu,\mu)<\varepsilon$.

3. (Noise penalty pair)¶ $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ is a noise penalty pair on $\mathcal{P}^{a}_{\rho}$.