Proof of The Bump Construction for Envelope Viscosity Subsolutions

Each result cited is universally quantified over the data in its own statement.

Conventions. The conventions of The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation are in force, with $\rho,\sigma,R$, $\mathcal{D}$, $\mathcal{E}$, $\mathcal{D}_{\Xi}$, $\Xi$ and $\mathcal{H}$ as fixed there; lifts $\mathcal{H}_{M}$ are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts. Sums, real multiples, the pairing $\langle\cdot,\cdot\rangle_{2}$ and the norm $\lVert\cdot\rVert_{2}$ of $L^{2}$ $d$-tuples are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §pairing, that is, the operations, inner product and norm of the Hilbert space $H^{d}$ (Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §operations, Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §pairing); so the Cauchy--Schwarz and triangle inequalities hold for them. By The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation §metric, $\mathcal{D}\subseteq\Sigma_{d,R}$ carries the metric $W_{2}$, and envelopes and semicontinuity on $\mathcal{D}$ refer to it.

Isometry. By The Metric Completion is a Complete Metric Space with a Dense Isometric Copy of the Space, and Maps Preserving Cauchy Sequences Extend to It §isometry, applied to $(\Sigma_{d},W_{2})$, whose completion is $\Sigma^{2}_{d}$ (Square-Integrable Noncommutative Laws: the Wasserstein Completion of the Laws, Affine Push-Forwards, Moments, Couplings and Cost §laws), $\widehat{W}_{2}(\kappa_{d}(\lambda),\kappa_{d}(\lambda'))=W_{2}(\lambda,\lambda')$ for $\lambda,\lambda'\in\Sigma_{d}$; in particular $\kappa_{d}$ is injective.

Law invariance. If $Z,Z'$ are $L^{2}$ tuples, possibly of different tracial W*-probability spaces, with $\mathrm{law}(Z)=\mathrm{law}(Z')$, and $T$ is an affine datum, then $\mathrm{law}(TZ)=\mathrm{law}(TZ')$ by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward, the inner products of components of $Z$ agree with those of $Z'$ by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §moments, and lifts take equal values by Lifts of Functions on Square-Integrable Noncommutative Laws to Square-Integrable Tuples §lift. Hence every norm, pairing and lifted value formed from real affine combinations of blocks of $Z$ equals the corresponding quantity for $Z'$.

Bounded realisations. Let $(H,M,\Omega)$ be a tracial W*-probability space and $k\in\mathbb{N}$. If $Z$ is an $L^{2}$ $k$-tuple of it with $\mathrm{law}(Z)=\kappa_{k}(\lambda)$ for some $\lambda\in\Sigma_{k,r}$, then by Square-Integrable Tuples with a Bounded Law are Vacuum Tuples of Bounded Self-Adjoint Operators §operator there is exactly one self-adjoint $k$-tuple $z$ in $M$ with $z\Omega=Z$, and it satisfies $\lambda_{z}=\lambda$ by Square-Integrable Tuples with a Bounded Law are Vacuum Tuples of Bounded Self-Adjoint Operators §law. Every law in $\Sigma_{k}$ lies in some $\Sigma_{k,r}$ (Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law). Conversely, for a self-adjoint $k$-tuple $z$ in $M$, $\lambda_{z}\in\Sigma_{k}$ and $\mathrm{law}(z\Omega)=\kappa_{k}(\lambda_{z})$ (Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §law, Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §bounded), and for self-adjoint tuples $z,z'$ in $M$ one has $\lambda_{(z,z')}\in\Pi(\lambda_{z},\lambda_{z'})$ by Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §coupling, so that $\lambda_{(z,z')}\circ\iota^{1}=\lambda_{z}$ by Couplings of Two Noncommutative Laws and Their Quadratic Cost §coupling. Moreover, $M$ is closed under sums, products and complex multiples (Tracial W*-Probability Spaces §space). For a bounded plan $\varpi$ at $\lambda\in\Sigma_{d}$ and any $L^{2}$ $d$-tuples $X,P$ with $\mathrm{law}(X,P)=\kappa_{2d}(\varpi)$ one has $\mathrm{law}(X)=\mathrm{pr}^{1}_{\#}\kappa_{2d}(\varpi)=\kappa_{d}(\lambda)$ by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward and The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §pairings.

Transport of the score. Let $\lambda\in\mathcal{D}_{\Xi}$, let $\varpi$ be a bounded plan at $\lambda$, and let $X,P,Q$ be $L^{2}$ $d$-tuples of a tracial W*-probability space $(H,M,\Omega)$ with $\mathrm{law}(X,P)=\kappa_{2d}(\varpi)$ and $\mathrm{law}(X,Q)=\pi^{\Xi}_{\lambda}$, the score plan. Then $\lambda\in\mathcal{D}\subseteq\Sigma_{d,R}$, as $\mathcal{D}_{\Xi}\subseteq\mathcal{D}$ (The Wall-Confined Free Energy and Its Score §score), and $\mathrm{law}(X)=\kappa_{d}(\lambda)$ (bounded realisations), and $\pi^{\Xi}_{\lambda}=\mathrm{law}(X_{\lambda},\Xi(\lambda))$, where $X_{\lambda}$, the tuple of classes of the variables in $\mathcal{H}_{\lambda}$ (The Score Plan of a Law in the Score Domain of the Wall-Confined Free Energy), is the tuple $X_{\lambda}$ of Realising a Square-Integrable Field of the GNS Space of a Bounded Law next to Its Positions: Uniqueness and the Joint Law with a Momentum, and $\Xi(\lambda)$ is an $L^{2}$ $d$-tuple of $(\mathcal{H}_{\lambda},\mathcal{M}_{\lambda},\Omega_{\lambda})$. Hence Realising a Square-Integrable Field of the GNS Space of a Bounded Law next to Its Positions: Uniqueness and the Joint Law with a Momentum, with $r=R$, this $\lambda$, $\zeta=\Xi(\lambda)$, the bounded plan $\varpi$ and the tuples $X,P,Q$, gives: $\mathrm{law}(X,P+tQ)=\varpi\oplus t\,\Xi(\lambda)$ for every real $t$ (Realising a Square-Integrable Field of the GNS Space of a Bounded Law next to Its Positions: Uniqueness and the Joint Law with a Momentum §shift), so that $\mathcal{H}_{M}(X,P+tQ)=\mathcal{H}(\varpi\oplus t\,\Xi(\lambda))$ by Lifts of Functions on Square-Integrable Noncommutative Laws to Square-Integrable Tuples §lift; $\langle Q,P\rangle_{2}=\mathcal{J}(\Xi(\lambda),\varpi)$ (Realising a Square-Integrable Field of the GNS Space of a Bounded Law next to Its Positions: Uniqueness and the Joint Law with a Momentum §pairing); and $\lVert Q\rVert_{2}=\lVert\Xi(\lambda)\rVert_{2}$ (Realising a Square-Integrable Field of the GNS Space of a Bounded Law next to Its Positions: Uniqueness and the Joint Law with a Momentum §norm).

Step 0 (the constant $\delta_{1}$ and the order of choices). By Shift Semicontinuity of a Hamiltonian on Phase-Space Noncommutative Laws §upper with $r=R$ fix $\delta_{\mathrm{u}}>0$ as there, and by Score Bounds at Envelope Test Inequalities from the Absorption Slack with $r=R$ (its hypotheses, absorption of shifts and boundedness at zero momentum, are assumed) fix $\delta_{\mathrm{s}}>0$ as there. Put $\delta_{1}=\min\{\delta_{\mathrm{u}},\delta_{\mathrm{s}}\}>0$. Let $0<\delta_{0}\le\delta_{1}$, and let $g$, $w$ be as in the statement, with $w$ not an envelope supersolution with shift range $\delta_{0}$. Fix a real $b\ge0$ with $|w|\le b$ and $|g|\le b$, and, by The Wall-Confined Free Energy: Norm Bound, Lower Bound, Weak-Star Compact Sublevel Sets and Displacement Monotonicity of the Score §bounds, a real $e$ with $e\le\mathcal{E}(\lambda)$ for all $\lambda\in\mathcal{D}$. The remaining objects are chosen in this order: $\delta,\varphi,\mu,\pi$, then $\beta$ and $r_{g}$ (Step 1); $a_{P}$, $r_{1}$, $m$, $h$, $\theta$ and $\Phi$ (Step 2); $r_{3}$ and $\varepsilon_{3}$ (Step 3); $r_{\Phi}$, $r_{2}$, $\varepsilon$ and $U$ (Step 4). In Step 5 the test data are arbitrary.

Envelope facts. For a bounded $v:\Sigma^{2}_{d}\to\mathbb{R}$ and real $\delta''>0$: $v\circ\kappa_{d}-\delta''\mathcal{E}$ is bounded above and $v\circ\kappa_{d}+\delta''\mathcal{E}$ bounded below near each point (The Penalty Envelopes of a Bounded Function on the Domain of the Wall-Confined Free Energy §upper, The Penalty Envelopes of a Bounded Function on the Domain of the Wall-Confined Free Energy §lower); $v(\kappa_{d}(\lambda))-\delta''\mathcal{E}(\lambda)\le v^{-}_{\delta''}(\lambda)$ and $v^{+}_{\delta''}(\lambda)\le v(\kappa_{d}(\lambda))+\delta''\mathcal{E}(\lambda)$ for $\lambda\in\mathcal{D}$ (Properties of the Upper Semicontinuous Envelope §bounds, Properties of the Lower Semicontinuous Envelope, by Duality §bounds); and $v^{+}_{\delta''}$ is lower semicontinuous on $\mathcal{D}$ (Properties of the Lower Semicontinuous Envelope, by Duality §lsc).

Step 1 (the failure and the gap). Negating Envelope Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §super for $w$, fix a real $\delta$ with $0<\delta\le\delta_{0}$, a function $\varphi:\Sigma^{2}_{d}\to\mathbb{R}$, $\mu\in\mathcal{D}$ with $w^{+}_{\delta}(\lambda)-\varphi(\kappa_{d}(\lambda))>w^{+}_{\delta}(\mu)-\varphi(\kappa_{d}(\mu))$ for every $\lambda\in\mathcal{D}$ with $\lambda\ne\mu$, and a bounded plan $\pi$ at $\mu$ with $\kappa_{2d}(\pi)\in J^{-}\varphi(\kappa_{d}(\mu))$, such that either $\mu\notin\mathcal{D}_{\Xi}$, or $\mu\in\mathcal{D}_{\Xi}$ and

$$F:=\rho\bigl(w^{+}_{\delta}(\mu)-\delta\,\mathcal{E}(\mu)\bigr)+\mathcal{H}\bigl(\pi\oplus(-\delta)\,\Xi(\mu)\bigr)+\frac{\sigma^{2}}{2}\Bigl(\mathcal{J}\bigl(\Xi(\mu),\pi\bigr)-\delta\,\lVert\Xi(\mu)\rVert_{2}^{2}\Bigr)<0.\tag{F}$$

Since $w\le g$, $w\circ\kappa_{d}+\delta\mathcal{E}\le g\circ\kappa_{d}+\delta\mathcal{E}$ on $\mathcal{D}$, so $w^{+}_{\delta}\le g^{+}_{\delta}$ by Properties of the Lower Semicontinuous Envelope, by Duality §monotone. Suppose $g^{+}_{\delta}(\mu)=w^{+}_{\delta}(\mu)$. Then for $\lambda\in\mathcal{D}$, $\lambda\ne\mu$, we get $g^{+}_{\delta}(\lambda)-\varphi(\kappa_{d}(\lambda))\ge w^{+}_{\delta}(\lambda)-\varphi(\kappa_{d}(\lambda))>g^{+}_{\delta}(\mu)-\varphi(\kappa_{d}(\mu))$, and since $g$ is a supersolution with shift range $\delta_{0}\ge\delta$, Envelope Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §super gives $\mu\in\mathcal{D}_{\Xi}$ and that the left side of (F), in which $w^{+}_{\delta}(\mu)=g^{+}_{\delta}(\mu)$, is nonnegative; this contradicts the choice of the failure. Hence $\beta=\frac12\bigl(g^{+}_{\delta}(\mu)-w^{+}_{\delta}(\mu)\bigr)>0$, and by lower semicontinuity of $g^{+}_{\delta}$ (Lower Semicontinuous Function on a Subset of a Metric Space) fix $r_{g}>0$ with

$$g^{+}_{\delta}(\lambda)>w^{+}_{\delta}(\mu)+\beta\qquad\text{for every }\lambda\in\mathcal{D}\text{ with }W_{2}(\lambda,\mu)<r_{g}.\tag{G}$$

Step 2 (gauge and test function). Let $a_{P}=|\pi|_{\mathrm{mom}}$. If $(X,P)$ is any realisation of $\kappa_{2d}(\pi)$, then $\lVert P\rVert_{2}=a_{P}$ and $\mathrm{law}(X)=\kappa_{d}(\mu)$: with the bounded realisation $(X,P)=(z,z')\Omega$, $\lambda_{(z,z')}=\pi$, Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §moments gives $\lVert P\rVert_{2}^{2}=\sum_{j}\pi(x_{d+j}x_{d+j})=a_{P}^{2}$ (Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §plans), and the second assertion is a convention above. $B$ and $C$ are the affine data of Couplings of a Square-Integrable Plan with a Law: Displacement and Momentum Pairing, the couplings $\mathcal{C}_{\kappa_{2d}(\pi)}(\cdot)$ are those of Couplings of a Square-Integrable Plan with a Law: Displacement and Momentum Pairing §couplings, and the displacement $s(\gamma)$ and momentum pairing $p(\gamma)$ are those of Couplings of a Square-Integrable Plan with a Law: Displacement and Momentum Pairing §displacement-pairing; for a realisation $(X,P,X')$ of $\gamma\in\Sigma^{2}_{3d}$ one has $\mathrm{law}(X,P)=B_{\#}\gamma$ and $\mathrm{law}(X')=C_{\#}\gamma$ by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward.

The modulus. For $\gamma\in\Sigma^{2}_{3d}$ with $B_{\#}\gamma=\kappa_{2d}(\pi)$ put $v(\gamma)=\varphi(\kappa_{d}(\mu))+p(\gamma)-\varphi(C_{\#}\gamma)$. For real $\eta>0$ let $r_{\eta}>0$ be as in Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §sub for $\varphi$, $\kappa_{2d}(\pi)$ and $\eta$; applying that inequality to a realisation of $\gamma$ (the lift of $\varphi$ being evaluated through Lifts of Functions on Square-Integrable Noncommutative Laws to Square-Integrable Tuples §lift) gives $v(\gamma)\le\eta\,s(\gamma)$ whenever $s(\gamma)<r_{\eta}$. Let $r_{1}=r_{1}^{\circ}/2$, where $r_{1}^{\circ}=r_{\eta}$ for $\eta=1$. For $s\in[0,r_{1}]$ let $m(s)$ be the least upper bound (The Real Numbers: Standing Notation and Background §bounds) of the set consisting of $0$ and of the numbers $v(\gamma)$ with $B_{\#}\gamma=\kappa_{2d}(\pi)$ and $s(\gamma)\le s$; this set is bounded above by $s$, so $0\le m(s)\le s\le r_{1}$. Thus $m(0)=0$, $m$ is bounded by $r_{1}$, and for $\eta>0$ one has $2m(s)\le\eta s$ whenever $s<r_{\eta/2}$. By A Nondecreasing Differentiable Majorant with Vanishing Derivative at Zero for a Little-o Modulus, applied to $2m$ on $[0,r_{1}]$, fix $h:[0,\infty)\to[0,\infty)$ as there, and $K_{h}$ the bound of A Nondecreasing Differentiable Majorant with Vanishing Derivative at Zero for a Little-o Modulus §derivative. Then $h$ is nondecreasing with $h(0)=0$ (A Nondecreasing Differentiable Majorant with Vanishing Derivative at Zero for a Little-o Modulus §monotone), $h(s)\ge2m(s)$ for $s\in[0,r_{1}]$ (A Nondecreasing Differentiable Majorant with Vanishing Derivative at Zero for a Little-o Modulus §majorant), $|h'|\le K_{h}$, $h'(0)=0$ and $h'(s)\to0$ as $s\to0$ (A Nondecreasing Differentiable Majorant with Vanishing Derivative at Zero for a Little-o Modulus §derivative); and since $\bar{h}$ is differentiable at every $s\ge0$ with $\bar{h}=h$ on $[0,\infty)$ (A Nondecreasing Differentiable Majorant with Vanishing Derivative at Zero for a Little-o Modulus §differentiable), the definition of the derivative (Single-Variable Calculus on an Interval §derivative) gives, for $s\ge0$ and $\eta>0$, a real $r>0$ with $|h(t)-h(s)-h'(s)(t-s)|\le\eta|t-s|$ for all real $t\ge0$ with $|t-s|<r$. So $h$ satisfies the gauge hypotheses of The Coupling Test Function of a Bounded Plan: Maximising Couplings, Continuity, Plan Subjets and Localisation with $L=K_{h}$.

The weight and $\Phi$. Put

$$\theta=\max\Bigl\{1,\ \frac{2a_{P}}{r_{1}},\ \frac{2\bigl(2b+\delta(\mathcal{E}(\mu)-e)+2\bigr)}{r_{1}^{2}}\Bigr\},$$

so $\theta>0$, $\theta r_{1}\ge2a_{P}$ and $\theta r_{1}^{2}/2\ge2b+\delta(\mathcal{E}(\mu)-e)+2$. Let $\Phi=\Phi^{h,\theta}_{\kappa_{2d}(\pi)}$ be the coupling test function; The Coupling Test Function of a Bounded Plan: Maximising Couplings, Continuity, Plan Subjets and Localisation applies with $r=R$, $\mu\in\Sigma_{d,R}$, $\pi$, $\theta$, $L=K_{h}$ and $h$, and "maximising coupling" refers to it.

(2a) Localisation hypothesis. If $\gamma\in\mathcal{C}_{\kappa_{2d}(\pi)}(\kappa_{d}(\mu))$ and $s(\gamma)<r_{1}$, then $C_{\#}\gamma=\kappa_{d}(\mu)$, so $p(\gamma)=v(\gamma)\le m(s(\gamma))\le\frac12h(s(\gamma))$. Thus the hypothesis of The Coupling Test Function of a Bounded Plan: Maximising Couplings, Continuity, Plan Subjets and Localisation §localisation and The Coupling Test Function of a Bounded Plan: Maximising Couplings, Continuity, Plan Subjets and Localisation §convergence holds with $r_{0}=r_{1}$, and $\Phi(\kappa_{d}(\mu))=0$ by The Coupling Test Function of a Bounded Plan: Maximising Couplings, Continuity, Plan Subjets and Localisation §localisation.

(2b) Upper bound. For $\zeta\in\Sigma^{2}_{d}$ and $\gamma\in\mathcal{C}_{\kappa_{2d}(\pi)}(\zeta)$ with $s=s(\gamma)$, Cauchy--Schwarz gives $p(\gamma)\le a_{P}s$, so, as $h\ge0$, $p(\gamma)-h(s)-\theta s^{2}\le a_{P}s-\theta s^{2}=\frac{a_{P}^{2}}{4\theta}-\theta\bigl(s-\frac{a_{P}}{2\theta}\bigr)^{2}\le\frac{a_{P}^{2}}{4\theta}$. Hence $\Phi(\zeta)\le a_{P}^{2}/(4\theta)$.

(2c) The bump functions. For real $\varepsilon>0$ define $\psi_{\varepsilon}:\mathcal{D}\to\mathbb{R}$ by $\psi_{\varepsilon}(\lambda)=w^{+}_{\delta}(\mu)+\varepsilon-\delta\,\mathcal{E}(\lambda)+\Phi(\kappa_{d}(\lambda))$. By (2b) and $e\le\mathcal{E}$, for $0<\varepsilon\le1$,

$$\psi_{\varepsilon}(\lambda)\le b_{\psi}:=w^{+}_{\delta}(\mu)+1-\delta e+\frac{a_{P}^{2}}{4\theta}\qquad(\lambda\in\mathcal{D}).\tag{B}$$

We claim that for every $\varepsilon>0$ and $\lambda\in\mathcal{D}$

$$\psi_{\varepsilon}(\lambda)\le w(\kappa_{d}(\lambda))+\varepsilon-\min\bigl\{\theta\,W_{2}(\lambda,\mu)^{2},1\bigr\}.\tag{K}$$

By The Coupling Test Function of a Bounded Plan: Maximising Couplings, Continuity, Plan Subjets and Localisation §attained fix a maximising coupling $\gamma$ for $\lambda$ and put $s=s(\gamma)$. For a realisation $(X,P,X')$ of $\gamma$, $\mathrm{law}(X)=\kappa_{d}(\mu)$ and $\mathrm{law}(X')=\kappa_{d}(\lambda)$, so $s=\lVert X'-X\rVert_{2}\ge W_{2}(\lambda,\mu)$ by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §lipschitz and the isometry. If $s\le r_{1}$: $v(\gamma)$ belongs to the set defining $m(s)$, so $p(\gamma)\le m(s)+\varphi(\kappa_{d}(\lambda))-\varphi(\kappa_{d}(\mu))$, and as $h(s)\ge2m(s)\ge m(s)$, $\Phi(\kappa_{d}(\lambda))\le\varphi(\kappa_{d}(\lambda))-\varphi(\kappa_{d}(\mu))-\theta s^{2}$. The minimum property of $\mu$ (with equality allowed at $\lambda=\mu$) and the envelope facts give $\varphi(\kappa_{d}(\lambda))-\varphi(\kappa_{d}(\mu))\le w^{+}_{\delta}(\lambda)-w^{+}_{\delta}(\mu)\le w(\kappa_{d}(\lambda))+\delta\mathcal{E}(\lambda)-w^{+}_{\delta}(\mu)$; hence $\psi_{\varepsilon}(\lambda)\le w(\kappa_{d}(\lambda))+\varepsilon-\theta s^{2}\le w(\kappa_{d}(\lambda))+\varepsilon-\theta W_{2}(\lambda,\mu)^{2}$. If $s>r_{1}$: $a_{P}\le\theta r_{1}/2<\theta s/2$, so $\Phi(\kappa_{d}(\lambda))\le a_{P}s-\theta s^{2}<-\theta s^{2}/2<-\theta r_{1}^{2}/2\le-(2b+\delta(\mathcal{E}(\mu)-e)+2)$; with $w^{+}_{\delta}(\mu)\le w(\kappa_{d}(\mu))+\delta\mathcal{E}(\mu)\le b+\delta\mathcal{E}(\mu)$, $-\delta\mathcal{E}(\lambda)\le-\delta e$ and $-b\le w(\kappa_{d}(\lambda))$ this gives $\psi_{\varepsilon}(\lambda)<-b+\varepsilon-2\le w(\kappa_{d}(\lambda))+\varepsilon-1$. In both cases (K) holds.

Step 3 (the test inequality near $\mu$). For $\nu\in\mathcal{D}$ and a maximising coupling $\gamma$ for $\nu$, a score quadruple for $(\nu,\gamma)$ is a quadruple $(X,P,X',Q)$ of $L^{2}$ $d$-tuples of one tracial W*-probability space with $\mathrm{law}(X,P,X')=\gamma$, $\nu\in\mathcal{D}_{\Xi}$ and $\mathrm{law}(X',Q)=\pi^{\Xi}_{\nu}$. With $S$ formed from $X,P,X'$ as in The Coupling Test Function of a Bounded Plan: Maximising Couplings, Continuity, Plan Subjets and Localisation §subjet and $G^{-}_{\delta}$ as in Shift Semicontinuity of a Hamiltonian on Phase-Space Noncommutative Laws, put, for $\varepsilon>0$,

$$\Lambda_{\varepsilon}=\rho\,\psi_{\varepsilon}(\nu)+G^{-}_{\delta}(X',S,Q)+\frac{\sigma^{2}}{2}\langle Q,S\rangle_{2}=\rho\,\psi_{\varepsilon}(\nu)+\mathcal{H}_{M}(X',S-\delta Q)+\frac{\sigma^{2}}{2}\bigl(\langle Q,S\rangle_{2}-\delta\lVert Q\rVert_{2}^{2}\bigr).$$

Since $S$ and $S-\delta Q$ are real affine combinations of $X,P,X',Q$ with coefficients fixed by $s(\gamma)$, $\Lambda_{\varepsilon}$ depends only on $\mathrm{law}(X,P,X',Q)$, by law invariance. Claim: there are reals $r_{3}>0$ and $\varepsilon_{3}\in(0,1]$ such that $\Lambda_{\varepsilon}\le0$ for every $\varepsilon\in(0,\varepsilon_{3}]$, every $\nu\in\mathcal{D}_{\Xi}$ with $W_{2}(\nu,\mu)<r_{3}$, every maximising coupling $\gamma$ for $\nu$ and every score quadruple for $(\nu,\gamma)$.

Suppose not. Then for every $n\in\mathbb{N}$ (taking $r_{3}=\varepsilon_{3}=1/n$) there are $\varepsilon_{n}\in(0,1/n]$, $\nu_{n}\in\mathcal{D}_{\Xi}$ with $W_{2}(\nu_{n},\mu)<1/n$, a maximising coupling $\gamma_{n}$ for $\nu_{n}$ and a score quadruple $(X_{n},P_{n},X'_{n},Q_{n})$ for $(\nu_{n},\gamma_{n})$ with $\Lambda_{n}>0$, where $\Lambda_{n}$ is $\Lambda_{\varepsilon_{n}}$ for these data.

(3a) One space. Let $\hat{\gamma}_{n}=\mathrm{law}(X_{n},P_{n},X'_{n},Q_{n})\in\Sigma^{2}_{4d}$; its push-forward to the first $2d$ variables is $\mathrm{law}(X_{n},P_{n})=B_{\#}\gamma_{n}=\kappa_{2d}(\pi)$. By Gluing Countably Many Square-Integrable Noncommutative Laws along a Common Marginal §glue, with $k=2d$, $m_{n}=2d$ and common marginal $\kappa_{2d}(\pi)$, fix a tracial W*-probability space $(H,M,\Omega)$, an $L^{2}$ $2d$-tuple $(X,P)$ and $L^{2}$ $2d$-tuples $(X'_{n},Q_{n})$ of it (we reuse the names) with $\mathrm{law}(X,P,X'_{n},Q_{n})=\hat{\gamma}_{n}$ for every $n$. By law invariance, $(X,P,X'_{n},Q_{n})$ is a score quadruple for $(\nu_{n},\gamma_{n})$ with the same value $\Lambda_{n}>0$; let $S_{n}$ be formed from $X,P,X'_{n}$. Also $\mathrm{law}(X,P)=\kappa_{2d}(\pi)$, so $\lVert P\rVert_{2}=a_{P}$ and $\mathrm{law}(X)=\kappa_{d}(\mu)$ (Step 2), and $s(\gamma_{n})=\lVert X'_{n}-X\rVert_{2}$.

(3b) Convergence. As $\nu_{n}\in\Sigma_{d,R}$, $W_{2}(\nu_{n},\mu)\to0$ and (2a) holds, The Coupling Test Function of a Bounded Plan: Maximising Couplings, Continuity, Plan Subjets and Localisation §convergence gives $s(\gamma_{n})\to0$, $\Phi(\kappa_{d}(\nu_{n}))\to0$ and $\lVert S_{n}-P\rVert_{2}\to0$. Fix $N_{0}$ with $\lVert S_{n}-P\rVert_{2}\le1$, hence $\lVert S_{n}\rVert_{2}\le a_{P}+1$, for $n\ge N_{0}$.

(3c) Score bound. Let $C_{\mathrm{s}}\ge0$ be the constant (called $C$ there) of Score Bounds at Envelope Test Inequalities from the Absorption Slack for $r=R$, the level $\delta_{\mathrm{s}}$ of Step 0, and $m_{\mathrm{s}}=a_{P}+1$ and $b_{\mathrm{s}}=|b_{\psi}|$ in the roles of its $m$ and $b$. For $n\ge N_{0}$, (B) and $\varepsilon_{n}\le1$ give $\rho\,\psi_{\varepsilon_{n}}(\nu_{n})\le\rho\,b_{\psi}$, so $\Lambda_{n}>0$ yields

$$\rho\,b_{\psi}+\mathcal{H}_{M}(X'_{n},S_{n}-\delta Q_{n})+\frac{\sigma^{2}}{2}\bigl(\langle Q_{n},S_{n}\rangle_{2}-\delta\lVert Q_{n}\rVert_{2}^{2}\bigr)\ge0.$$

Since $\mathrm{law}(X'_{n})=\kappa_{d}(\nu_{n})\in\kappa_{d}(\Sigma_{d,R})$ and $0<\delta\le\delta_{0}\le\delta_{\mathrm{s}}$, Score Bounds at Envelope Test Inequalities from the Absorption Slack §super, with $X'_{n},S_{n},Q_{n}$ and $a=b_{\psi}$, gives $\lVert Q_{n}\rVert_{2}\le C_{\mathrm{s}}/\delta$; and $\lVert\Xi(\nu_{n})\rVert_{2}=\lVert Q_{n}\rVert_{2}$ by law invariance, as $\mathrm{law}(X'_{n},Q_{n})=\pi^{\Xi}_{\nu_{n}}=\mathrm{law}(X_{\nu_{n}},\Xi(\nu_{n}))$. Replacing all sequences by their tails from $N_{0}$ on, we assume these bounds for every $n$.

(3d) Limits. By The Wall-Confined Free Energy Converges along Wasserstein-Convergent Sequences with Bounded Scores §convergence, $\mathcal{E}(\nu_{n})\to\mathcal{E}(\mu)$; with $\varepsilon_{n}\to0$ and (3b), $\psi_{\varepsilon_{n}}(\nu_{n})\to w^{+}_{\delta}(\mu)-\delta\mathcal{E}(\mu)$. As $\mathcal{E}$ has closed score, Wall-Confined Free Energies with Closed Score §closed, applied in $(H,M,\Omega)$ to $(\nu_{n})$, $\mu\in\mathcal{D}$, $X'_{n},Q_{n}$ and $X$ (note $\lVert X'_{n}-X\rVert_{2}=s(\gamma_{n})\to0$ and $\lVert Q_{n}\rVert_{2}\le C_{\mathrm{s}}/\delta$), gives $\mu\in\mathcal{D}_{\Xi}$ and an $L^{2}$ $d$-tuple $Q$ 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 $Y$. Since $\mu\in\mathcal{D}_{\Xi}$, Step 1 gives (F): $F<0$. Put $\varepsilon'=-F/4>0$.

(3e) Upper shift semicontinuity. $(X'_{n},S_{n},Q_{n})\to(X,P,Q)$ is a shift-convergent sequence at radius $R$ in the sense of Shift-Convergent Sequences of Positions, Momenta and Shifts in a Tracial W*-Probability Space §sequence: $\mathrm{law}(X'_{n})=\kappa_{d}(\nu_{n})$ and $\mathrm{law}(X)=\kappa_{d}(\mu)$ lie in $\kappa_{d}(\Sigma_{d,R})$ and $\lVert X'_{n}-X\rVert_{2}\to0$; $\lVert S_{n}-P\rVert_{2}\to0$; and $\lVert Q_{n}\rVert_{2}\le C_{\mathrm{s}}/\delta$ with the weak convergence of (3d). As $0<\delta\le\delta_{\mathrm{u}}$, Shift Semicontinuity of a Hamiltonian on Phase-Space Noncommutative Laws §upper gives $N_{1}$ with $G^{-}_{\delta}(X'_{n},S_{n},Q_{n})\le G^{-}_{\delta}(X,P,Q)+\varepsilon'$ for $n\ge N_{1}$. Moreover $|\langle Q_{n},S_{n}\rangle_{2}-\langle Q,P\rangle_{2}|\le\frac{C_{\mathrm{s}}}{\delta}\lVert S_{n}-P\rVert_{2}+|\langle Q_{n},P\rangle_{2}-\langle Q,P\rangle_{2}|\to0$.

(3f) Identification. The transport of the score (Realising a Square-Integrable Field of the GNS Space of a Bounded Law next to Its Positions: Uniqueness and the Joint Law with a Momentum §shift, Realising a Square-Integrable Field of the GNS Space of a Bounded Law next to Its Positions: Uniqueness and the Joint Law with a Momentum §pairing, Realising a Square-Integrable Field of the GNS Space of a Bounded Law next to Its Positions: Uniqueness and the Joint Law with a Momentum §norm), with $\lambda=\mu$, $\varpi=\pi$, the tuples $X,P,Q$ and $t=-\delta$ (so that $P+tQ=P-\delta Q$), gives $\mathcal{H}_{M}(X,P-\delta Q)=\mathcal{H}(\pi\oplus(-\delta)\,\Xi(\mu))$, $\langle Q,P\rangle_{2}=\mathcal{J}(\Xi(\mu),\pi)$ and $\lVert Q\rVert_{2}=\lVert\Xi(\mu)\rVert_{2}$. Hence

$$\rho\bigl(w^{+}_{\delta}(\mu)-\delta\mathcal{E}(\mu)\bigr)+G^{-}_{\delta}(X,P,Q)+\frac{\sigma^{2}}{2}\langle Q,P\rangle_{2}=F.$$

(3g) Contradiction. Fix $N_{2}$ such that for $n\ge N_{2}$ the numbers $\rho\,\psi_{\varepsilon_{n}}(\nu_{n})$ and $\frac{\sigma^{2}}{2}\langle Q_{n},S_{n}\rangle_{2}$ are within $\varepsilon'$ of $\rho(w^{+}_{\delta}(\mu)-\delta\mathcal{E}(\mu))$ and $\frac{\sigma^{2}}{2}\langle Q,P\rangle_{2}$ respectively. For $n\ge\max\{N_{1},N_{2}\}$, (3e) and (3f) give $\Lambda_{n}<F+3\varepsilon'=F/4<0$, contradicting $\Lambda_{n}>0$. This proves the claim; fix $r_{3}$ and $\varepsilon_{3}$.

Step 4 (the bump). By The Coupling Test Function of a Bounded Plan: Maximising Couplings, Continuity, Plan Subjets and Localisation §continuity and (2a) fix $r_{\Phi}>0$ with $|\Phi(\kappa_{d}(\lambda))|<\beta/2$ for every $\lambda\in\Sigma_{d,R}$ with $W_{2}(\lambda,\mu)<r_{\Phi}$. Put

$$r_{2}=\min\{r_{g},r_{\Phi},r_{3}\},\qquad\varepsilon=\min\Bigl\{\varepsilon_{3},\ \frac{\beta}{4},\ \frac{\theta r_{2}^{2}}{8},\ \frac12\Bigr\},\qquad\psi=\psi_{\varepsilon}.$$

Define $U:\Sigma^{2}_{d}\to\mathbb{R}$ as follows: if $\zeta=\kappa_{d}(\lambda)$ for some $\lambda\in\mathcal{D}$ (unique by injectivity of $\kappa_{d}$), let $U(\zeta)=\max\{w(\zeta),\psi(\lambda)\}$; otherwise $U(\zeta)=w(\zeta)$. So $U\ge w$, and $U(\kappa_{d}(\lambda))=\max\{w(\kappa_{d}(\lambda)),\psi(\lambda)\}$ for $\lambda\in\mathcal{D}$.

(4a) Far from $\mu$. If $\lambda\in\mathcal{D}$ and $W_{2}(\lambda,\mu)\ge r_{2}/2$, then $\min\{\theta W_{2}(\lambda,\mu)^{2},1\}\ge\min\{\theta r_{2}^{2}/4,1\}>\varepsilon$, so (K) gives $\psi(\lambda)<w(\kappa_{d}(\lambda))$.

(4b) $U\le g$. Let $\lambda\in\mathcal{D}$. If $W_{2}(\lambda,\mu)<r_{2}$, then by the choice of $r_{\Phi}$, (G) and the envelope facts,

$$\psi(\lambda)+\delta\mathcal{E}(\lambda)=w^{+}_{\delta}(\mu)+\varepsilon+\Phi(\kappa_{d}(\lambda))<w^{+}_{\delta}(\mu)+\tfrac{\beta}{4}+\tfrac{\beta}{2}<g^{+}_{\delta}(\lambda)\le g(\kappa_{d}(\lambda))+\delta\mathcal{E}(\lambda),$$

so $\psi(\lambda)<g(\kappa_{d}(\lambda))$; if $W_{2}(\lambda,\mu)\ge r_{2}$, then $\psi(\lambda)<w(\kappa_{d}(\lambda))\le g(\kappa_{d}(\lambda))$ by (4a). Together with $w\le g$ this gives $w\le U\le g$ on $\Sigma^{2}_{d}$, and $|U|\le b$, so $U$ is bounded.

(4c) $U$ exceeds $w$ somewhere. By Properties of the Lower Semicontinuous Envelope, by Duality §approximation, applied to $w\circ\kappa_{d}+\delta\mathcal{E}$ on $\mathcal{D}$, fix $\lambda_{k}\in\mathcal{D}$ with $W_{2}(\lambda_{k},\mu)\to0$ and $w(\kappa_{d}(\lambda_{k}))+\delta\mathcal{E}(\lambda_{k})\to w^{+}_{\delta}(\mu)$. By The Coupling Test Function of a Bounded Plan: Maximising Couplings, Continuity, Plan Subjets and Localisation §continuity and (2a), $\Phi(\kappa_{d}(\lambda_{k}))\to0$. Hence $\psi(\lambda_{k})-w(\kappa_{d}(\lambda_{k}))=w^{+}_{\delta}(\mu)-\bigl(w(\kappa_{d}(\lambda_{k}))+\delta\mathcal{E}(\lambda_{k})\bigr)+\varepsilon+\Phi(\kappa_{d}(\lambda_{k}))\to\varepsilon>0$, and for some $k$, $U(\kappa_{d}(\lambda_{k}))\ge\psi(\lambda_{k})>w(\kappa_{d}(\lambda_{k}))$ with $\lambda_{k}\in\mathcal{D}$.

Step 5 ($U$ is a subsolution with shift range $\delta_{0}$). Let $0<\delta'\le\delta_{0}$, $\varphi':\Sigma^{2}_{d}\to\mathbb{R}$, $\nu\in\mathcal{D}$ with $U^{-}_{\delta'}(\lambda)-\varphi'(\kappa_{d}(\lambda))<U^{-}_{\delta'}(\nu)-\varphi'(\kappa_{d}(\nu))$ for every $\lambda\in\mathcal{D}$ with $\lambda\ne\nu$, and let $\pi'$ be a bounded plan at $\nu$ with $\kappa_{2d}(\pi')\in J^{+}\varphi'(\kappa_{d}(\nu))$. We must show $\nu\in\mathcal{D}_{\Xi}$ and

$$\rho\bigl(U^{-}_{\delta'}(\nu)+\delta'\mathcal{E}(\nu)\bigr)+\mathcal{H}\bigl(\pi'\oplus\delta'\,\Xi(\nu)\bigr)+\frac{\sigma^{2}}{2}\Bigl(\mathcal{J}\bigl(\Xi(\nu),\pi'\bigr)+\delta'\lVert\Xi(\nu)\rVert_{2}^{2}\Bigr)\le0.\tag{S}$$

(5a) The envelope of $U$. On $\mathcal{D}$, $U\circ\kappa_{d}-\delta'\mathcal{E}$ is the pointwise maximum of $f_{1}=w\circ\kappa_{d}-\delta'\mathcal{E}$ and $f_{2}=\psi-\delta'\mathcal{E}$. Both are bounded above near each point: $f_{1}$ by the envelope facts, $f_{2}$ because $f_{2}\le b_{\psi}-\delta'e$ by (B) (Upper and Lower Semicontinuous Envelopes of a Real-Valued Function §near-bounds). Moreover $f_{2}=w^{+}_{\delta}(\mu)+\varepsilon+\Phi\circ\kappa_{d}-(\delta+\delta')\mathcal{E}$ is upper semicontinuous on $\mathcal{D}$: given $\lambda\in\mathcal{D}$ and $\eta>0$, the continuity of $\Phi\circ\kappa_{d}$ on $\Sigma_{d,R}\supseteq\mathcal{D}$ (The Coupling Test Function of a Bounded Plan: Maximising Couplings, Continuity, Plan Subjets and Localisation §continuity) and the lower semicontinuity of $\mathcal{E}$ on $\mathcal{D}$ (Sublevel Sets of the Wall-Confined Free Energy are Closed for the Wasserstein Distance §lsc, Lower Semicontinuous Function on a Subset of a Metric Space) give $r>0$ with $\Phi(\kappa_{d}(\lambda'))<\Phi(\kappa_{d}(\lambda))+\eta/2$ and $\mathcal{E}(\lambda')>\mathcal{E}(\lambda)-\eta/(2(\delta+\delta'))$ for $\lambda'\in\mathcal{D}$ with $W_{2}(\lambda',\lambda)<r$, whence $f_{2}(\lambda')<f_{2}(\lambda)+\eta$ (Upper Semicontinuous Function on a Subset of a Metric Space). So $f_{2}^{*}=f_{2}$ by Properties of the Upper Semicontinuous Envelope §fixed, and The Semicontinuous Envelopes of a Pointwise Maximum and of a Pointwise Minimum §upper gives

$$U^{-}_{\delta'}(\lambda)=\max\bigl\{w^{-}_{\delta'}(\lambda),\ \psi(\lambda)-\delta'\mathcal{E}(\lambda)\bigr\}\qquad(\lambda\in\mathcal{D}).$$

(5b) Case A: $U^{-}_{\delta'}(\nu)=w^{-}_{\delta'}(\nu)$. By (5a), $w^{-}_{\delta'}\le U^{-}_{\delta'}$ on $\mathcal{D}$, so for $\lambda\ne\nu$, $w^{-}_{\delta'}(\lambda)-\varphi'(\kappa_{d}(\lambda))\le U^{-}_{\delta'}(\lambda)-\varphi'(\kappa_{d}(\lambda))<U^{-}_{\delta'}(\nu)-\varphi'(\kappa_{d}(\nu))=w^{-}_{\delta'}(\nu)-\varphi'(\kappa_{d}(\nu))$. As $w$ is a subsolution with shift range $\delta_{0}\ge\delta'$, Envelope Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §sub gives $\nu\in\mathcal{D}_{\Xi}$ and (S) with $w^{-}_{\delta'}(\nu)$ in place of $U^{-}_{\delta'}(\nu)$, which is (S).

(5c) Case B: otherwise. Then $U^{-}_{\delta'}(\nu)=\psi(\nu)-\delta'\mathcal{E}(\nu)>w^{-}_{\delta'}(\nu)\ge w(\kappa_{d}(\nu))-\delta'\mathcal{E}(\nu)$, so $\psi(\nu)>w(\kappa_{d}(\nu))$ and $W_{2}(\nu,\mu)<r_{2}/2<r_{3}$ by (4a).

(i) A minimum of $\mathcal{E}-\chi$. Let $\chi:\Sigma^{2}_{d}\to\mathbb{R}$, $\chi=(\Phi-\varphi')/(\delta+\delta')$. For $\lambda\in\mathcal{D}$, (5a) and the maximum property give $\psi(\lambda)-\delta'\mathcal{E}(\lambda)-\varphi'(\kappa_{d}(\lambda))\le U^{-}_{\delta'}(\lambda)-\varphi'(\kappa_{d}(\lambda))\le\psi(\nu)-\delta'\mathcal{E}(\nu)-\varphi'(\kappa_{d}(\nu))$; inserting the definition of $\psi$ and dividing by $\delta+\delta'>0$, $\mathcal{E}(\lambda)-\chi(\kappa_{d}(\lambda))\ge\mathcal{E}(\nu)-\chi(\kappa_{d}(\nu))$ for every $\lambda\in\mathcal{D}$.

(ii) A common realisation. By The Coupling Test Function of a Bounded Plan: Maximising Couplings, Continuity, Plan Subjets and Localisation §attained fix a maximising coupling $\gamma$ for $\nu$, a realisation $(\tilde{X},\tilde{P},\tilde{X}')$ of it (Couplings of a Square-Integrable Plan with a Law: Displacement and Momentum Pairing §displacement-pairing) and, by Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs §law, $L^{2}$ $d$-tuples $(\tilde{Y},\tilde{P}')$ with law $\kappa_{2d}(\pi')$. Then $\mathrm{law}(\tilde{X}')=\kappa_{d}(\nu)=\mathrm{law}(\tilde{Y})$, so Gluing Two Square-Integrable Noncommutative Laws along a Common Marginal §glue (with $k=d$, $m=2d$, $n=d$, $\pi_{1}=\mathrm{law}(\tilde{X}',\tilde{X},\tilde{P})$, $\pi_{2}=\kappa_{2d}(\pi')$) gives a tracial W*-probability space $(H,M,\Omega)$ and $L^{2}$ $d$-tuples $X',X,P,P'$ of it with $\mathrm{law}(X',X,P)=\mathrm{law}(\tilde{X}',\tilde{X},\tilde{P})$ and $\mathrm{law}(X',P')=\kappa_{2d}(\pi')$; by law invariance (a permutation of blocks is an affine datum) $\mathrm{law}(X,P,X')=\gamma$. Let $s=s(\gamma)$ and $S$ be formed from $X,P,X'$ as in The Coupling Test Function of a Bounded Plan: Maximising Couplings, Continuity, Plan Subjets and Localisation §subjet, so that $\mathrm{law}(X',S)\in J^{-}\Phi(\kappa_{d}(\nu))$. Put $Q=\frac{1}{\delta+\delta'}(S-P')$.

(iii) Boundedness. By bounded realisations there are self-adjoint $d$-tuples $x,p,x',p'$ in $M$ with $(X,P)=(x,p)\Omega$, $X'=x'\Omega$ (as $\nu\in\Sigma_{d,R}$) and $(X',P')=(x'',p')\Omega$, and $x''=x'$ by uniqueness. With $c=h'(s)/s+2\theta$ if $s>0$ and $c=0$ if $s=0$, $S=(p-c(x'-x))\Omega$ and $Q=q\Omega$ for the self-adjoint $d$-tuple $q=\frac{1}{\delta+\delta'}(p-c(x'-x)-p')$ in $M$. Hence $\tilde{\pi}=\lambda_{(x',q)}\in\Sigma_{2d}$ satisfies $\tilde{\pi}\circ\iota^{1}=\lambda_{x'}=\nu$, so $\tilde{\pi}$ is a bounded plan at $\nu$, and $\kappa_{2d}(\tilde{\pi})=\mathrm{law}(X',Q)$.

(iv) Plan jets. First, $\mathrm{law}(X',-P')\in J^{-}(-\varphi')(\kappa_{d}(\nu))$: it is a plan at $\kappa_{d}(\nu)$ (Plans at a Square-Integrable Noncommutative Law §plan); given $\eta>0$, let $r>0$ be as in Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §super for $\varphi'$, $\kappa_{2d}(\pi')$ and $\eta$; if $\hat{X},\hat{P},\hat{X}'$ are $L^{2}$ $d$-tuples of any tracial W*-probability space with $\mathrm{law}(\hat{X},\hat{P})=\mathrm{law}(X',-P')$ and $\lVert\hat{X}'-\hat{X}\rVert_{2}<r$, then $\mathrm{law}(\hat{X},-\hat{P})=\kappa_{2d}(\pi')$ by law invariance, so the superjet inequality for $(\hat{X},-\hat{P},\hat{X}')$, multiplied by $-1$, is the subjet inequality of Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §sub for $-\varphi'$ at $(\hat{X},\hat{P},\hat{X}')$. Next, Plan Jets of a Sum along a Common Realisation §sub, with $\varphi_{1}=\Phi$, $\varphi_{2}=-\varphi'$, slacks $0$, the tuples $X',S,-P'$ and $\lambda=\kappa_{d}(\nu)$, gives $\mathrm{law}(X',S-P')\in J^{-}(\Phi-\varphi')(\kappa_{d}(\nu))$. Finally $\mathrm{law}(X',Q)\in J^{-}\chi(\kappa_{d}(\nu))$: with $c'=\delta+\delta'$, given $\eta>0$ let $r>0$ be as in Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §sub for $\Phi-\varphi'$, $\mathrm{law}(X',S-P')$ and $c'\eta$; if $\mathrm{law}(\hat{X},\hat{P})=\mathrm{law}(X',Q)$ and $\lVert\hat{X}'-\hat{X}\rVert_{2}<r$, then $\mathrm{law}(\hat{X},c'\hat{P})=\mathrm{law}(X',c'Q)=\mathrm{law}(X',S-P')$, and dividing the resulting subjet inequality for $\Phi-\varphi'$ by $c'$ gives the one for $\chi$ with $\hat{P}$ and $\eta$.

(v) Regular plan subgradients. By (i) (with any $r>0$), (iii) and (iv), Wall-Confined Free Energies with Regular Plan Subgradients §regular, applied to $\chi$, $\nu$ and the bounded plan $\tilde{\pi}$, gives $\nu\in\mathcal{D}_{\Xi}$ and $\mathrm{law}(X',Q)=\kappa_{2d}(\tilde{\pi})=\pi^{\Xi}_{\nu}$. Thus $(X,P,X',Q)$ is a score quadruple for $(\nu,\gamma)$.

(vi) The inequality. The transport of the score (Realising a Square-Integrable Field of the GNS Space of a Bounded Law next to Its Positions: Uniqueness and the Joint Law with a Momentum §shift, Realising a Square-Integrable Field of the GNS Space of a Bounded Law next to Its Positions: Uniqueness and the Joint Law with a Momentum §pairing, Realising a Square-Integrable Field of the GNS Space of a Bounded Law next to Its Positions: Uniqueness and the Joint Law with a Momentum §norm), with $\lambda=\nu$, $\varpi=\pi'$, the tuples $X',P',Q$ and $t=\delta'$, gives $\mathcal{H}(\pi'\oplus\delta'\,\Xi(\nu))=\mathcal{H}_{M}(X',P'+\delta'Q)$, $\mathcal{J}(\Xi(\nu),\pi')=\langle Q,P'\rangle_{2}$ and $\lVert\Xi(\nu)\rVert_{2}=\lVert Q\rVert_{2}$. As $(\delta+\delta')Q=S-P'$, we have $P'+\delta'Q=S-\delta Q$ and $\langle Q,P'\rangle_{2}+\delta'\lVert Q\rVert_{2}^{2}=\langle Q,S-\delta Q\rangle_{2}=\langle Q,S\rangle_{2}-\delta\lVert Q\rVert_{2}^{2}$; and $U^{-}_{\delta'}(\nu)+\delta'\mathcal{E}(\nu)=\psi(\nu)$. So the left side of (S) equals $\Lambda_{\varepsilon}$ for the score quadruple $(X,P,X',Q)$, which is at most $0$ by Step 3, since $0<\varepsilon\le\varepsilon_{3}$ and $W_{2}(\nu,\mu)<r_{3}$.

Conclusion. In both cases $\nu\in\mathcal{D}_{\Xi}$ and (S) holds, so $U$ is an envelope viscosity subsolution of $(\mathrm{E})$ with shift range $\delta_{0}$ (Envelope Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §sub). By Step 4, $U$ is bounded, $w\le U\le g$ on $\Sigma^{2}_{d}$, and $U(\kappa_{d}(\lambda_{k}))>w(\kappa_{d}(\lambda_{k}))$ for some $\lambda_{k}\in\mathcal{D}$. Since $\delta_{1}$ was fixed in Step 0 before $\delta_{0}$, $g$ and $w$, the lemma is proved.