Plan Jets at an Ekeland Point of the Doubled Difference with Logarithmic Confinement on Square-Integrable Noncommutative Laws

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, let $u,v:\Sigma^{2}_{d}\to\mathbb{R}$ and let $\varepsilon>0$, $\beta>0$ and $\delta\ge0$ be real. The plan superjets $J^{+}_{\delta}$ and plan subjets $J^{-}_{\delta}$ are those of Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §superjet and Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §subjet; sums, real multiples and the pairing of $L^{2}$ $d$-tuples are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §pairing, differences and the $L^{2}$ norm those of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §tuples; $\widehat{W}_{2}$ is the metric of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §metrics. Let $G:\Sigma^{2}_{d}\to\mathbb{R}$ be $G(\mu)=\log\bigl(1+\widehat{M}(\mu)\bigr)$, with the natural logarithm and the second moment $\widehat{M}$; it is defined since $\widehat{M}(\mu)\ge0$ by Calculus of Square-Integrable Noncommutative Laws: Agreement on Bounded Laws, Lipschitz Estimates, Functoriality of Push-Forwards, Moment Formulas, Positivity, the Cost and the Diagonal Coupling §moments. Let $\Phi:\Sigma^{2}_{2d}\to\mathbb{R}$ be

with the cost $\mathcal{I}$. Let $\hat{\gamma}\in\Sigma^{2}_{2d}$ satisfy $\Phi(\gamma)-\delta\,\widehat{W}_{2}(\gamma,\hat{\gamma})\le\Phi(\hat{\gamma})$ for every $\gamma\in\Sigma^{2}_{2d}$, and let $\hat{X},\hat{Y}$ be $L^{2}$ $d$-tuples of a tracial W*-probability space $(H,M,\Omega)$ with $\mathrm{law}(\hat{X},\hat{Y})=\hat{\gamma}$. Put

1. (Superjet of $u$)¶ $\mathrm{law}(\hat{X},p+q)\in J^{+}_{\delta}u(\mathrm{law}(\hat{X}))$.

2. (Subjet of $v$)¶ $\mathrm{law}(\hat{Y},p-q')\in J^{-}_{\delta}v(\mathrm{law}(\hat{Y}))$.

3. (Confinement terms)¶ $\lVert q\rVert_{2}\le\beta$, $\lVert q'\rVert_{2}\le\beta$, $(1+\lVert\hat{X}\rVert_{2})\lVert q\rVert_{2}\le3\beta$ and $(1+\lVert\hat{Y}\rVert_{2})\lVert q'\rVert_{2}\le3\beta$.

4. (Penalisation)¶ Suppose that $u$ and $v$ are bounded, let $\Phi'$ be defined as $\Phi$ with $2\varepsilon$ in place of $\varepsilon$, and let $\kappa\ge0$ be real with $\Phi(\hat{\gamma})\ge\sup\Phi-\kappa$. Then $\Phi$ and $\Phi'$ are bounded above, and