Wall-Confined Free Energies with Closed Score

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, let $(\mathcal{D}_{0},\mathcal{E}_{0})$ be a free entropy penalty, let $R>0$ be real, and let $\mathcal{E}:\mathcal{D}\to\mathbb{R}$, on $\mathcal{D}=\mathcal{D}_{0}\cap\mathcal{D}_{R}$, be its wall-confined free energy with radius $R$, with score domain $\mathcal{D}_{\Xi}$ and score $\Xi$. The map $\kappa_{d}$ is the canonical map of Square-Integrable Noncommutative Laws: Standing Notation §laws. For $\mu\in\mathcal{D}_{\Xi}$, $\pi^{\Xi}_{\mu}\in\Sigma^{2}_{2d}$ is the score plan of $\mu$. The pairing $\langle\cdot,\cdot\rangle_{2}$ of $L^{2}$ $d$-tuples is that of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §pairing; differences, the $L^{2}$ norm $\lVert\cdot\rVert_{2}$, pairs and laws are those of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §tuples; convergence of real sequences is that of Limit of a Sequence of Real Numbers.

¶The wall-confined free energy $\mathcal{E}$ of $(\mathcal{D}_{0},\mathcal{E}_{0})$ with radius $R$ has closed score if the following holds for every tracial W*-probability space $(H,M,\Omega)$, every sequence $(\mu_{n})_{n\in\mathbb{N}}$ in $\mathcal{D}_{\Xi}$, every $\mu\in\mathcal{D}$, and all $L^{2}$ $d$-tuples $X_{n},Q_{n}$ $(n\in\mathbb{N})$ and $X$ of $(H,M,\Omega)$ such that $\mathrm{law}(X_{n},Q_{n})=\pi^{\Xi}_{\mu_{n}}$ for every $n$, $\mathrm{law}(X)=\kappa_{d}(\mu)$, $\lVert X_{n}-X\rVert_{2}\to0$, and there is a real $C$ with $\lVert Q_{n}\rVert_{2}\le C$ for every $n$: one has $\mu\in\mathcal{D}_{\Xi}$, and there is an $L^{2}$ $d$-tuple $Q$ of $(H,M,\Omega)$ with $\mathrm{law}(X,Q)=\pi^{\Xi}_{\mu}$ and $\langle Q_{n},Y\rangle_{2}\to\langle Q,Y\rangle_{2}$ for every $L^{2}$ $d$-tuple $Y$ of $(H,M,\Omega)$.