The Score Plan of a Law in the Score Domain of the Wall-Confined Free Energy

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 $\Xi$. For $\mu\in\Sigma_{d}$, $(\mathcal{H}_{\mu},\mathcal{M}_{\mu},\Omega_{\mu})$ is the tracial W*-probability space of The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §w-star, with conjugation $J_{\mu}$.

Positions and score. Let $\mu\in\mathcal{D}_{\Xi}$. The classes of the variables in $\mathcal{H}_{\mu}$,

$$X_{\mu}=(\widehat{x_{1}},\dots,\widehat{x_{d}}),$$

form an $L^{2}$ $d$-tuple of $(\mathcal{H}_{\mu},\mathcal{M}_{\mu},\Omega_{\mu})$, since $J_{\mu}\widehat{x_{i}}=\widehat{x_{i}^{*}}=\widehat{x_{i}}$ by Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §conjugation and Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint, and $\Xi(\mu)$ is an $L^{2}$ $d$-tuple of $(\mathcal{H}_{\mu},\mathcal{M}_{\mu},\Omega_{\mu})$ by Conjugate Variables, Wall Forces and Scores are Square-Integrable Tuples of the GNS Space §score. Hence the pair $(X_{\mu},\Xi(\mu))$ is an $L^{2}$ $2d$-tuple, with a law in $\Sigma^{2}_{2d}$.

¶Definition. The score plan of $\mu$ is the $L^{2}$ law

$$\pi^{\Xi}_{\mu}=\mathrm{law}\bigl(X_{\mu},\,\Xi(\mu)\bigr)\in\Sigma^{2}_{2d}.$$

The score plan $\pi^{\Xi}_{\mu}$ depends on $(\mathcal{D}_{0},\mathcal{E}_{0})$ and $R$ only through the score $\Xi(\mu)$.