The Gaussian Entropy Pair Has Closed Score Along Noise Couplings

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 $\kappa$; it is a noise penalty pair on $\mathcal{P}^{a}_{\rho}$ by The Gaussian Entropy Pair is a Noise Penalty Pair, with Nonnegative Penalty and Dense Score Domain §pair.

(Closed score)¶ The pair has closed score along noise couplings.