Scaling a Plan-Jet Viscosity Subsolution of an Equation with a Quadratic Hamiltonian Gives a Strict Subsolution

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, let $\rho>0$ be real, let $\mathcal{H}:\Sigma^{2}_{2d}\to\mathbb{R}$ be quadratic with a convex Lipschitz remainder, with a real $C$ as in Quadratic Hamiltonians with a Convex Remainder Lipschitz in the Momentum on Phase-Space Noncommutative Laws §bound, and let $u:\Sigma^{2}_{d}\to\mathbb{R}$ be a plan-jet viscosity subsolution of the discounted stationary Hamilton--Jacobi equation with discount rate $\rho$ and Hamiltonian $\mathcal{H}$. Let $\theta$ be real with $0<\theta<1$, and let $w=\theta u$, that is, $w(\mu)=\theta\,u(\mu)$. The plan superjets $J^{+}_{\delta}$, the lifts of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts, sums and the $L^{2}$ norm are as in Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation.

¶For every real $\delta\ge0$, every $\mu\in\Sigma^{2}_{d}$, every $\pi\in J^{+}_{\delta}w(\mu)$ and every real $\eta>0$, there are a tracial W*-probability space $(H,M,\Omega)$ and $L^{2}$ $d$-tuples $X,P,Q$ of $(H,M,\Omega)$ with $\mathrm{law}(X,P)=\pi$, $\lVert Q\rVert_{2}\le\delta$ and