Proof of The Supremum of a Bounded Family of Envelope Viscosity Subsolutions is an Envelope Viscosity Subsolution

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. Throughout, $\rho$, $\sigma$, $R$, $\mathcal{D}$, $\mathcal{E}$, $\mathcal{D}_{\Xi}$, $\Xi$ and $\mathcal{H}$ are those of The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation §data, and lifts $f_{M}$ of functions $f$ on $L^{2}$ laws are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts, so that $f_{M}(Z)=f(\mathrm{law}(Z))$ by Lifts of Functions on Square-Integrable Noncommutative Laws to Square-Integrable Tuples §lift. By The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation §metric, $\mathcal{D}\subseteq\Sigma_{d,R}$, and semicontinuity of real functions on subsets of $\mathcal{D}$ refers to the metric space $(\Sigma_{d,R},W_{2})$ of The Noncommutative Laws with a Norm Bound Form a Complete Bounded Metric Space with Interpolation Points; the envelopes of The Penalty Envelopes of a Bounded Function on the Domain of the Wall-Confined Free Energy are formed in this space, and Properties of the Upper Semicontinuous Envelope is applied there with $S=\mathcal{D}$. Sums, differences, 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; by Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §operations and Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §pairing they are the operations, the inner product (real on $L^{2}$ tuples) and the norm of the Hilbert space $H^{d}$, so the triangle and Cauchy--Schwarz inequalities hold for them (Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §background) and $\langle Y,Z\rangle_{2}=\langle Z,Y\rangle_{2}$. Pairs, triples, affine images $TZ$ and laws of $L^{2}$ tuples are those of Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws. For $k\in\mathbb{N}$, $\Sigma^{2}_{k}$ is the metric completion of $(\Sigma_{k},W_{2})$ with canonical map $\kappa_{k}$ (Square-Integrable Noncommutative Laws: the Wasserstein Completion of the Laws, Affine Push-Forwards, Moments, Couplings and Cost §laws), so 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 gives the isometry identity

$$\widehat{W}_{2}\bigl(\kappa_{k}(\lambda),\kappa_{k}(\lambda')\bigr)=W_{2}(\lambda,\lambda')\qquad(\lambda,\lambda'\in\Sigma_{k});$$

together with The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §separation and the fact that $\widehat{W}_{2}$ is a metric, it shows that $\kappa_{k}$ is injective. We write $N=(A^{N},0)$ for the affine datum from $2d$ to $2d$ variables with $A^{N}_{jj}=1$ and $A^{N}_{d+j,d+j}=-1$ for $j\in[d]$ and all other entries $0$; thus $N(Y,Z)=(Y,-Z)$ for $L^{2}$ $d$-tuples $Y,Z$ (Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations), and $N(N(Y,Z))=(Y,Z)$. Finally, $W_{2}(\lambda,\lambda')\le2R\sqrt{d}$ for all $\lambda,\lambda'\in\Sigma_{d,R}$, since $W_{2}(\lambda,\lambda')\ge0$ and $W_{2}(\lambda,\lambda')^{2}\le4dR^{2}$ by The Noncommutative Laws with a Norm Bound Form a Complete Bounded Metric Space with Interpolation Points §bounded.

Preliminary facts. (F1) Law invariance. Let $Z$ and $Z'$ be $L^{2}$ $k$-tuples of possibly different tracial W*-probability spaces with $\mathrm{law}(Z)=\mathrm{law}(Z')$, and let $T$ be an affine datum from $k$ to $n$ variables. By Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward, $\mathrm{law}(TZ)=T_{\#}\mathrm{law}(Z)=\mathrm{law}(TZ')$; hence $\lVert TZ\rVert_{2}=\lVert TZ'\rVert_{2}$, both squares being $\widehat{M}$ of this law by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §moments, and in particular $TZ'=0$ whenever $TZ=0$. If $U,V$ are $d$-blocks of $Z$, formed by the entries with indices $i_{1},\dots,i_{d}$ and $l_{1},\dots,l_{d}$, and $U',V'$ are the corresponding blocks of $Z'$, then by the same clause

$$\langle U,V\rangle_{2}=\sum_{j=1}^{d}\mathrm{m}_{i_{j}l_{j}}\bigl(\mathrm{law}(Z)\bigr)=\langle U',V'\rangle_{2}.$$

Blocks, and real linear combinations of blocks, are affine images of $Z$ (Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations). So a block has the law given by the coordinate push-forward, lifts of functions on laws take equal values at corresponding affine images of $Z$ and $Z'$, and every linear relation among blocks of $Z$ holds among the corresponding blocks of $Z'$. Every law in $\Sigma^{2}_{k}$ is the law of an $L^{2}$ $k$-tuple of some tracial W*-probability space by Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs §law.

(F2) GNS realisations. Let $n\in\mathbb{N}$ and $\lambda\in\Sigma_{n}$, and let $(\mathcal{H}_{\lambda},\mathcal{M}_{\lambda},\Omega_{\lambda})$ be 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, where $\mathcal{M}_{\lambda}=\mathcal{A}_{\lambda}''$. Each $L_{q}$ with $q\in\mathcal{P}_{n}$ lies in $\mathcal{A}_{\lambda}$, hence commutes with every element of $\mathcal{A}_{\lambda}'$ and lies in $\mathcal{M}_{\lambda}$ (The Commutant of a Set of Bounded Operators on a Complex Hilbert Space §commutant), and it is self-adjoint when $q\in\mathcal{P}_{n,\mathrm{sa}}$ (Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §adjoint). Thus $L^{\lambda}=(L_{x_{1}},\dots,L_{x_{n}})$ is a self-adjoint $n$-tuple in $\mathcal{M}_{\lambda}$, the variables being self-adjoint by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint; its law is $\lambda_{L^{\lambda}}=\lambda$ by Self-Adjoint Tuples in a Tracial W*-Probability Space and Their Laws §law and The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §law; and its vacuum tuple is $(\widehat{x_{1}},\dots,\widehat{x_{n}})$ 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 §vacuum. Hence $\mathrm{law}(\widehat{x_{1}},\dots,\widehat{x_{n}})=\kappa_{n}(\lambda)$ by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §bounded. In particular, for $\nu\in\mathcal{D}_{\Xi}$ the positions $X_{\nu}$ of The Score Plan of a Law in the Score Domain of the Wall-Confined Free Energy satisfy $\mathrm{law}(X_{\nu})=\kappa_{d}(\nu)$. For $\nu\in\Sigma_{d}$ and a bounded plan $\varpi$ at $\nu$, with $X_{\varpi},P_{\varpi}$ as in The Shift of a Bounded Plan by a Self-Adjoint Field §tuples, we get $\mathrm{law}(X_{\varpi},P_{\varpi})=\kappa_{2d}(\varpi)$; $\mathrm{law}(X_{\varpi})=\mathrm{pr}^{1}_{\#}\kappa_{2d}(\varpi)=\kappa_{d}(\nu)$ by (F1) and The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §pairings; $\mathrm{law}(P_{\varpi})\in\kappa_{d}(\Sigma_{d})$, since $P_{\varpi}$ is the vacuum tuple of the self-adjoint $d$-tuple $(L_{x_{d+1}},\dots,L_{x_{2d}})$, by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §bounded and Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §law; and

$$\lVert P_{\varpi}\rVert_{2}^{2}=\sum_{j=1}^{d}\lVert\widehat{x_{d+j}}\rVert^{2}=\sum_{j=1}^{d}\varpi(x_{d+j}x_{d+j})=|\varpi|_{\mathrm{mom}}^{2}$$

by Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples, Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §vacuum and Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §plans, so $\lVert P_{\varpi}\rVert_{2}=|\varpi|_{\mathrm{mom}}$.

(F3) Bounded plans from tuples. Let $(H,M,\Omega)$ be a tracial W*-probability space, let $Y$ be an $L^{2}$ $d$-tuple of it with $\mathrm{law}(Y)=\kappa_{d}(\nu)$ for some $\nu\in\Sigma_{d}$, let $Z_{1},\dots,Z_{n}$ be $L^{2}$ $d$-tuples of it with $\mathrm{law}(Z_{i})\in\kappa_{d}(\Sigma_{d})$, let $t_{1},\dots,t_{n}$ be real, and put $Z=\sum_{i=1}^{n}t_{i}Z_{i}$. By Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law each of these laws lies in $\kappa_{d}(\Sigma_{d,r})$ for some real $r>0$, so Square-Integrable Tuples with a Bounded Law are Vacuum Tuples of Bounded Self-Adjoint Operators §operator and Square-Integrable Tuples with a Bounded Law are Vacuum Tuples of Bounded Self-Adjoint Operators §law give self-adjoint $d$-tuples $y,z_{1},\dots,z_{n}$ in $M$ with $y\Omega=Y$, $z_{i}\Omega=Z_{i}$ and $\lambda_{y}=\nu$. The $d$-tuple $z=\sum_{i}t_{i}z_{i}$ consists of self-adjoint elements of $M$, and $z\Omega=Z$. By Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §coupling and Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §law, $(y,z)$ is a self-adjoint $2d$-tuple in $M$, and $\varpi=\lambda_{(y,z)}\in\Sigma_{2d}$ lies in $\Pi(\nu,\lambda_{z})$, so $\varpi\circ\iota^{1}=\nu$ (Couplings of Two Noncommutative Laws and Their Quadratic Cost §coupling) and $\varpi$ is a bounded plan at $\nu$ (Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §plans). Its vacuum tuple is the pair $(Y,Z)$, so $\kappa_{2d}(\varpi)=\mathrm{law}(Y,Z)$ by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §bounded; and $|\varpi|_{\mathrm{mom}}^{2}=\sum_{j}\varpi(x_{d+j}x_{d+j})=\sum_{j}\lVert z_{j}\Omega\rVert^{2}=\lVert Z\rVert_{2}^{2}$ by Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §moments, so $|\varpi|_{\mathrm{mom}}=\lVert Z\rVert_{2}$.

(F4) Shifted plans. Let $\nu\in\mathcal{D}_{\Xi}$ and let $\varpi$ be a bounded plan at $\nu$. By The Shift of a Bounded Plan by a Self-Adjoint Field §field and The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation §shifts, $V^{1}_{\varpi}\Xi(\nu)$ is an $L^{2}$ $d$-tuple of $(\mathcal{H}_{\varpi},\mathcal{M}_{\varpi},\Omega_{\varpi})$; put

$$\tau(\varpi)=\mathrm{law}\bigl(X_{\varpi},P_{\varpi},V^{1}_{\varpi}\Xi(\nu)\bigr)\in\Sigma^{2}_{3d}.$$

We claim: for every tracial W*-probability space $(H,M,\Omega)$, all $L^{2}$ $d$-tuples $\bar X,\bar P,\bar Q$ of it with $\mathrm{law}(\bar X,\bar P,\bar Q)=\tau(\varpi)$, and every real $t$,

$$\mathcal{H}\bigl(\varpi\oplus t\,\Xi(\nu)\bigr)=\mathcal{H}_{M}(\bar X,\bar P+t\bar Q),\qquad\mathcal{J}\bigl(\Xi(\nu),\varpi\bigr)=\langle\bar Q,\bar P\rangle_{2},\qquad\lVert\Xi(\nu)\rVert_{2}=\lVert\bar Q\rVert_{2},$$ $$\mathrm{law}(\bar X,\bar Q)=\pi^{\Xi}_{\nu},\qquad\mathrm{law}(\bar X,\bar P)=\kappa_{2d}(\varpi).$$

By (F1), applied with the affine data $(x,p,q)\mapsto(x,p+tq)$, $(x,p,q)\mapsto(x,q)$ and $(x,p,q)\mapsto(x,p)$ and to the blocks, it suffices to prove these for $(\bar X,\bar P,\bar Q)=(X_{\varpi},P_{\varpi},V^{1}_{\varpi}\Xi(\nu))$. The first identity is then The Shift of a Bounded Plan by a Self-Adjoint Field §shift together with Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts, and the last is (F2). Write $\Xi(\nu)=(\xi_{1},\dots,\xi_{d})$ and $V=V^{1}_{\varpi}$; by Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §isometries, $V$ is the isometry of Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §isometry for the law $\varpi$ and the tuple $(x_{1},\dots,x_{d})$, whose substitution is $\iota^{1}$ (Couplings of Two Noncommutative Laws and Their Quadratic Cost §marginals), so that the marginal law of that lemma is $\varpi\circ\iota^{1}=\nu$. Each $\langle V\xi_{j},\widehat{x_{d+j}}\rangle$ is real, both vectors being entries of $L^{2}$ tuples of $(\mathcal{H}_{\varpi},\mathcal{M}_{\varpi},\Omega_{\varpi})$ (Standard Form of a Tracial W*-Probability Space: the Commutation Theorem, Right-Bounded Vectors, Faithfulness and Self-Adjoint Vectors §self-adjoint), so Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §plan-pairing gives $\mathcal{J}(\Xi(\nu),\varpi)=\sum_{j}\langle V\xi_{j},\widehat{x_{d+j}}\rangle=\langle V\Xi(\nu),P_{\varpi}\rangle_{2}$. As $V^{*}V=I$, $\lVert V\xi_{j}\rVert=\lVert\xi_{j}\rVert$ for every $j$, whence $\lVert V\Xi(\nu)\rVert_{2}=\lVert\Xi(\nu)\rVert_{2}$. By Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §embedding and Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §homomorphism, the map of that lemma, which we denote $\pi_{V}:\mathcal{M}_{\nu}\to\mathcal{M}_{\varpi}$, is a trace-preserving embedding of $(\mathcal{H}_{\nu},\mathcal{M}_{\nu},\Omega_{\nu})$ into $(\mathcal{H}_{\varpi},\mathcal{M}_{\varpi},\Omega_{\varpi})$ in the sense of Trace-Preserving Embeddings of Tracial W*-Probability Spaces, Their Implementing Isometries and Conditional Expectations §embedding (the traces being $\tau_{\nu}$ and $\tau_{\varpi}$ by The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §w-star), and $\pi_{V}(S)\Omega_{\varpi}=VS\Omega_{\nu}$ for $S\in\mathcal{M}_{\nu}$; by the uniqueness in A Trace-Preserving Unital -Homomorphism between Tracial W-Probability Spaces is Implemented by an Isometry §isometry, $V$ is its implementing isometry (Trace-Preserving Embeddings of Tracial W*-Probability Spaces, Their Implementing Isometries and Conditional Expectations §isometry). Hence Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §embedding gives $\mathrm{law}(VX_{\nu},V\Xi(\nu))=\mathrm{law}(X_{\nu},\Xi(\nu))=\pi^{\Xi}_{\nu}$ (The Score Plan of a Law in the Score Domain of the Wall-Confined Free Energy §score-plan); and $VX_{\nu}=X_{\varpi}$, because $V\widehat{x_{i}}^{\,\nu}=\widehat{\iota^{1}(x_{i})}^{\,\varpi}=\widehat{x_{i}}^{\,\varpi}$ for $i\in[d]$ by Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §isometries and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values. This proves the claim.

(F5) Envelope bounds. By The Wall-Confined Free Energy: Norm Bound, Lower Bound, Weak-Star Compact Sublevel Sets and Displacement Monotonicity of the Score §bounds we fix, once and for all, a real $e$ with $e\le\mathcal{E}(\lambda)$ for every $\lambda\in\mathcal{D}$. Let $b'\ge0$ be real, let $w,w':\Sigma^{2}_{d}\to\mathbb{R}$ satisfy $|w|\le b'$ and $|w'|\le b'$, and let $\delta'>0$ be real. Then for every $\nu\in\mathcal{D}$

$$w\bigl(\kappa_{d}(\nu)\bigr)-\delta'\mathcal{E}(\nu)\le w^{-}_{\delta'}(\nu)\le b'-\delta'\mathcal{E}(\nu)\le b'-\delta'e,$$

and if $w\le w'$ on $\Sigma^{2}_{d}$, then $w^{-}_{\delta'}\le w'^{-}_{\delta'}$ on $\mathcal{D}$. Indeed, $w^{-}_{\delta'}$ is the upper semicontinuous envelope of $f=w\circ\kappa_{d}-\delta'\mathcal{E}$ on $\mathcal{D}$ (The Penalty Envelopes of a Bounded Function on the Domain of the Wall-Confined Free Energy §upper), so the first inequality is Properties of the Upper Semicontinuous Envelope §bounds. The function $g=b'-\delta'\mathcal{E}$ is upper semicontinuous on $\mathcal{D}$: given $\nu\in\mathcal{D}$ and a real $\eta>0$, as $\mathcal{E}$ is lower semicontinuous 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 with $\eta/\delta'$ gives a real $r>0$ with $\mathcal{E}(\nu)-\eta/\delta'<\mathcal{E}(\lambda)$ for all $\lambda\in\mathcal{D}$ with $W_{2}(\nu,\lambda)<r$, and then $g(\lambda)<g(\nu)+\eta$, which is Upper Semicontinuous Function on a Subset of a Metric Space. As $f\le g$ (because $w\le b'$), Properties of the Upper Semicontinuous Envelope §least gives the second inequality, and the third holds as $e\le\mathcal{E}(\nu)$ and $\delta'>0$. Finally, $w\circ\kappa_{d}-\delta'\mathcal{E}$ and $w'\circ\kappa_{d}-\delta'\mathcal{E}$ are bounded above near each point of $\mathcal{D}$ (The Penalty Envelopes of a Bounded Function on the Domain of the Wall-Confined Free Energy §upper), so the monotonicity is Properties of the Upper Semicontinuous Envelope §monotone.

Step 0 (The constant $\delta_{1}$). Since $\mathcal{H}$ is lower shift-semicontinuous at noise level $\sigma$, Shift Semicontinuity of a Hamiltonian on Phase-Space Noncommutative Laws §lower with $r=R$ gives a real $\delta_{\mathrm{ls}}>0$ as there. Since $\mathcal{H}$ absorbs shifts at noise level $\sigma$ and is bounded at zero momentum at bounded positions, Score Bounds at Envelope Test Inequalities from the Absorption Slack with $r=R$ gives a real $\delta_{\mathrm{sb}}>0$ as there. Put $\delta_{1}=\min\{\delta_{\mathrm{ls}},\delta_{\mathrm{sb}}\}>0$; it depends only on the data of the setting. Now let $\delta_{0}$ be real with $0<\delta_{0}\le\delta_{1}$, and let $b$, $\mathcal{F}$ and $W$ be as in the statement.

Part 1 (Bound). Let $\lambda\in\Sigma^{2}_{d}$. The set $\{v(\lambda):v\in\mathcal{F}\}$ is nonempty and bounded above by $b$, so its least upper bound $W(\lambda)$ satisfies $W(\lambda)\le b$ (The Real Numbers: Standing Notation and Background §bounds); and for any $v\in\mathcal{F}$, $W(\lambda)\ge v(\lambda)\ge-b$. Hence $|W(\lambda)|\le b$. In particular $W$ is bounded, and its penalty envelopes are defined.

Part 2 (Subsolution): data and order of choices. Let $\delta$ be real with $0<\delta\le\delta_{0}$, let $\varphi:\Sigma^{2}_{d}\to\mathbb{R}$, let $\mu\in\mathcal{D}$ satisfy $W^{-}_{\delta}(\nu)-\varphi(\kappa_{d}(\nu))<W^{-}_{\delta}(\mu)-\varphi(\kappa_{d}(\mu))$ for every $\nu\in\mathcal{D}$ with $\nu\ne\mu$, and let $\pi$ be a bounded plan at $\mu$ with $\kappa_{2d}(\pi)\in J^{+}\varphi(\kappa_{d}(\mu))$. By Envelope Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §sub we must show that $\mu\in\mathcal{D}_{\Xi}$ and

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

By hypothesis,

$$W^{-}_{\delta}(\nu)-W^{-}_{\delta}(\mu)\le\varphi\bigl(\kappa_{d}(\nu)\bigr)-\varphi\bigl(\kappa_{d}(\mu)\bigr)\qquad(\nu\in\mathcal{D}).\tag{SM}$$

For $v\in\mathcal{F}$ write $v^{-}_{\delta}$ for its upper penalty envelope; as $v\le W$ and $|v|,|W|\le b$ (Part 1), (F5) gives

$$v^{-}_{\delta}\le W^{-}_{\delta}\ \text{ on }\mathcal{D},\qquad W^{-}_{\delta}(\nu)\le b-\delta e\ \ (\nu\in\mathcal{D}),\qquad W^{-}_{\delta}(\mu)\ge W(\kappa_{d}(\mu))-\delta\mathcal{E}(\mu)\ge-b-\delta\mathcal{E}(\mu).\tag{MON}$$

The objects below are chosen in this order: $e$ (F5); $a_{P}$, $\pi^{-}$, $r'$, $r_{1}$, $\omega$, $h$, $K$, $A_{0}$, $\theta$, $\Phi$, $T$, $\ell$, $m_{*}$ and $C_{*}$ (Step 1), none of which depends on $\mathcal{F}$ beyond $W$; then, for every $j\in\mathbb{N}$, $\nu^{0}_{j}$, $v_{j}$ and $\varepsilon_{j}$, and then $j_{0}$ (Step 3); then, for every $j$, the weights $w^{j}_{k}$, the point $\nu_{j}$ and the centres $x^{j}_{k}$ (Step 3), the maximising coupling $\gamma_{j}$ and its realisation (Step 4), the spaces and tuples of Step 5 and the plan $\varpi_{j}$, and the glued law $\Gamma_{j}$ (Step 8); finally the common space and the tuple $Q$ (Step 8).

Step 1 (Constants and the test function $T$). By (F2) applied to $\pi$, $X_{\pi},P_{\pi}$ are $L^{2}$ $d$-tuples of $(\mathcal{H}_{\pi},\mathcal{M}_{\pi},\Omega_{\pi})$ with $\mathrm{law}(X_{\pi},P_{\pi})=\kappa_{2d}(\pi)$, $\mathrm{law}(X_{\pi})=\kappa_{d}(\mu)$, $\mathrm{law}(P_{\pi})\in\kappa_{d}(\Sigma_{d})$ and $\lVert P_{\pi}\rVert_{2}=|\pi|_{\mathrm{mom}}$; put $a_{P}=|\pi|_{\mathrm{mom}}$. By (F3) with $Y=X_{\pi}$, $n=1$, $Z_{1}=P_{\pi}$ and $t_{1}=-1$, there is a bounded plan $\pi^{-}$ at $\mu$ with $\kappa_{2d}(\pi^{-})=\mathrm{law}(X_{\pi},-P_{\pi})$ and $|\pi^{-}|_{\mathrm{mom}}=a_{P}$. As $(X_{\pi},-P_{\pi})=N(X_{\pi},P_{\pi})$ and $N(N(Y,Z))=(Y,Z)$, (F1) gives the following fact (R): if $\bar X,\bar P_{-}$ are $L^{2}$ $d$-tuples of a tracial W*-probability space with $\mathrm{law}(\bar X,\bar P_{-})=\kappa_{2d}(\pi^{-})$, then $\mathrm{law}(\bar X,-\bar P_{-})=\kappa_{2d}(\pi)$, $\mathrm{law}(\bar X)=\kappa_{d}(\mu)$ and $\lVert\bar P_{-}\rVert_{2}=a_{P}$.

The modulus. Let $B$ and $C$ be the affine data of Couplings of a Square-Integrable Plan with a Law: Displacement and Momentum Pairing, and $s(\gamma)$ and $p(\gamma)$ the displacement and momentum pairing of $\gamma\in\Sigma^{2}_{3d}$ (Couplings of a Square-Integrable Plan with a Law: Displacement and Momentum Pairing §displacement-pairing). Since $\kappa_{2d}(\pi)\in J^{+}\varphi(\kappa_{d}(\mu))$, Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §super with slack $0$ gives, for every real $\eta>0$, a real $r_{\eta}>0$ such that

$$\varphi\bigl(\mathrm{law}(\bar X')\bigr)\le\varphi\bigl(\kappa_{d}(\mu)\bigr)+\langle\bar P,\bar X'-\bar X\rangle_{2}+\eta\lVert\bar X'-\bar X\rVert_{2}$$

for all $L^{2}$ $d$-tuples $\bar X,\bar P,\bar X'$ of any tracial W*-probability space with $\mathrm{law}(\bar X,\bar P)=\kappa_{2d}(\pi)$ and $\lVert\bar X'-\bar X\rVert_{2}<r_{\eta}$. Let $r'$ be the radius $r_{\eta}$ for $\eta=1$, and put $r_{1}=r'/2$. For $s\in[0,r_{1}]$ let

$$A(s)=\{0\}\cup\bigl\{\varphi(C_{\#}\gamma)-\varphi(\kappa_{d}(\mu))-p(\gamma):\ \gamma\in\Sigma^{2}_{3d},\ B_{\#}\gamma=\kappa_{2d}(\pi),\ s(\gamma)\le s\bigr\}.$$

If $\gamma$ is as in the braces and $(\bar X,\bar P,\bar X')$ realises it, then $\mathrm{law}(\bar X,\bar P)=B_{\#}\gamma=\kappa_{2d}(\pi)$, $\mathrm{law}(\bar X')=C_{\#}\gamma$ by (F1), $p(\gamma)=\langle\bar P,\bar X'-\bar X\rangle_{2}$ and $\lVert\bar X'-\bar X\rVert_{2}=s(\gamma)\le s<r'$; so the element is at most $s(\gamma)\le s$, and, for every $\eta>0$, at most $\eta s$ if $s<r_{\eta}$. Thus $A(s)$ is nonempty and bounded above by $s$, and $\omega(s)=\sup A(s)$ (The Real Numbers: Standing Notation and Background §bounds) satisfies $0\le\omega(s)\le s\le r_{1}$, $\omega(0)=0$, and $\omega(s)\le\eta s$ for $s\in[0,r_{1}]$ with $s<r_{\eta}$ (as also $0\le\eta s$). Applied to $\gamma=\mathrm{law}(\bar X,\bar P,\bar X')$, the definition gives the following fact ($\Omega$): if $\bar X,\bar P,\bar X'$ are $L^{2}$ $d$-tuples of a tracial W*-probability space with $\mathrm{law}(\bar X,\bar P)=\kappa_{2d}(\pi)$, $\mathrm{law}(\bar X')=\kappa_{d}(\nu)$ for some $\nu\in\Sigma_{d}$ and $\lVert\bar X'-\bar X\rVert_{2}\le s\le r_{1}$, then $\varphi(\kappa_{d}(\nu))-\varphi(\kappa_{d}(\mu))-\langle\bar P,\bar X'-\bar X\rangle_{2}\le\omega(s)$.

The gauge. The function $2\omega:[0,r_{1}]\to[0,\infty)$ vanishes at $0$, is bounded by $2r_{1}$, and for every $\eta>0$ satisfies $2\omega(s)\le\eta s$ for $s<r_{\eta/2}$. So A Nondecreasing Differentiable Majorant with Vanishing Derivative at Zero for a Little-o Modulus, with $r=r_{1}$ and $2\omega$ in the role of $m$, gives $h:[0,\infty)\to[0,\infty)$ and a real $K$ such that: $h$ is nondecreasing with $h(0)=0$ (A Nondecreasing Differentiable Majorant with Vanishing Derivative at Zero for a Little-o Modulus §monotone); $h'(0)=0$, $|h'(s)|\le K$ for $s\ge0$, and $h'(s)\to0$ as $s\to0$ (A Nondecreasing Differentiable Majorant with Vanishing Derivative at Zero for a Little-o Modulus §derivative); and $h\ge2\omega$ on $[0,r_{1}]$ (A Nondecreasing Differentiable Majorant with Vanishing Derivative at Zero for a Little-o Modulus §majorant). For $s\ge0$ the extension $\bar h$ is differentiable at the interior point $s$ of $\mathbb{R}$ (A Nondecreasing Differentiable Majorant with Vanishing Derivative at Zero for a Little-o Modulus §differentiable), so by Single-Variable Calculus on an Interval §derivative, for every $\eta>0$ there is a real $r>0$ with $|h(t)-h(s)-h'(s)(t-s)|\le\eta|t-s|$ for every real $t\ge0$ with $|t-s|<r$ (for $t\ne s$ apply the difference quotient bound with increment $t-s$; for $t=s$ both sides vanish). Hence $h$ satisfies the hypotheses of The Coupling Test Function of a Bounded Plan: Maximising Couplings, Continuity, Plan Subjets and Localisation with $L=K$ ($K\ge|h'(0)|=0$).

The weight and $T$. Put $A_{0}=2b+\delta(\mathcal{E}(\mu)-e)+1\ge1$ and

$$\theta=\frac{4a_{P}+1}{r_{1}}+\frac{4A_{0}}{r_{1}^{2}}>0,$$

so that $\theta r_{1}\ge4a_{P}+1$ (in particular $\theta r_{1}\ge2|\pi^{-}|_{\mathrm{mom}}$) and $\theta r_{1}^{2}\ge4A_{0}$. Let $\Phi=\Phi^{h,\theta}_{\kappa_{2d}(\pi^{-})}$ be the coupling test function of $\kappa_{2d}(\pi^{-})$ with gauge $h$ and weight $\theta$, and $T=-\Phi:\Sigma^{2}_{d}\to\mathbb{R}$. We apply The Coupling Test Function of a Bounded Plan: Maximising Couplings, Continuity, Plan Subjets and Localisation with $r=R$, the law $\mu\in\Sigma_{d,R}$, the bounded plan $\pi^{-}$, this $\theta$, $L=K$ and $h$; maximising couplings are those of that lemma, and $\mathcal{C}(\lambda)=\mathcal{C}_{\kappa_{2d}(\pi^{-})}(\lambda)$ (Couplings of a Square-Integrable Plan with a Law: Displacement and Momentum Pairing §couplings).

(T0) Let $\nu\in\Sigma_{d,R}$, let $\gamma$ be a maximising coupling for $\nu$ (one exists by The Coupling Test Function of a Bounded Plan: Maximising Couplings, Continuity, Plan Subjets and Localisation §attained), let $(\bar X,\bar P_{-},\bar X')$ realise $\gamma$, and put $\bar P=-\bar P_{-}$ and $s=s(\gamma)$. As $\gamma\in\mathcal{C}(\kappa_{d}(\nu))$, (F1) and (R) give $\mathrm{law}(\bar X,\bar P)=\kappa_{2d}(\pi)$, $\mathrm{law}(\bar X)=\kappa_{d}(\mu)$, $\mathrm{law}(\bar X')=\kappa_{d}(\nu)$ and $\lVert\bar P\rVert_{2}=a_{P}$; moreover $s=\lVert\bar X'-\bar X\rVert_{2}$ and $p(\gamma)=\langle\bar P_{-},\bar X'-\bar X\rangle_{2}=-\langle\bar P,\bar X'-\bar X\rangle_{2}$. Hence, by the definition of a maximising coupling, the Cauchy--Schwarz inequality, $h\ge0$ and $\theta(s-a_{P}/(2\theta))^{2}\ge0$,

$$T\bigl(\kappa_{d}(\nu)\bigr)=\langle\bar P,\bar X'-\bar X\rangle_{2}+h(s)+\theta s^{2}\ge-a_{P}s+\theta s^{2}\ge-\frac{a_{P}^{2}}{4\theta};$$

and $W_{2}(\nu,\mu)\le s$ by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §lipschitz, the isometry identity and The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §symmetry.

The level and the score constant. Put

$$\ell=\frac{1}{\delta}\Bigl(b+\frac{a_{P}^{2}}{4\theta}-W^{-}_{\delta}(\mu)+2\Bigr),\qquad\mathcal{D}_{\ell}=\{\nu\in\mathcal{D}:\mathcal{E}(\nu)\le\ell\},\qquad m_{*}=a_{P}+K+2\theta r_{1}+4R\sqrt{d},$$

and let $C_{*}\ge0$ be the constant given by Score Bounds at Envelope Test Inequalities from the Absorption Slack for $r=R$ (with the $\delta_{\mathrm{sb}}$ of Step 0), $m=m_{*}$ and the given $b$.

Step 2 (The key bound). Let $\nu\in\mathcal{D}$ and let $\gamma,\bar X,\bar P,\bar X',s$ be as in (T0). We claim

$$W^{-}_{\delta}(\nu)-T\bigl(\kappa_{d}(\nu)\bigr)\le W^{-}_{\delta}(\mu)-\frac{\theta}{2}s^{2},\qquad\text{and}\qquad W^{-}_{\delta}(\nu)-T\bigl(\kappa_{d}(\nu)\bigr)\le W^{-}_{\delta}(\mu)-1\ \text{ if }s\ge r_{1}.\tag{K}$$

If $s<r_{1}$, then (SM) and ($\Omega$) (applicable by (T0)) give $W^{-}_{\delta}(\nu)\le W^{-}_{\delta}(\mu)+\langle\bar P,\bar X'-\bar X\rangle_{2}+\omega(s)$, and $\omega(s)\le h(s)/2$; so by (T0), $W^{-}_{\delta}(\nu)-T(\kappa_{d}(\nu))\le W^{-}_{\delta}(\mu)-h(s)/2-\theta s^{2}\le W^{-}_{\delta}(\mu)-\theta s^{2}$, which gives the first inequality. If $s\ge r_{1}$, then by (MON), (T0) and the definition of $A_{0}$,

$$W^{-}_{\delta}(\nu)-T\bigl(\kappa_{d}(\nu)\bigr)\le b-\delta e+a_{P}s-\theta s^{2}\le W^{-}_{\delta}(\mu)+A_{0}-1+a_{P}s-\theta s^{2}.$$

Since $s\ge r_{1}$, $\theta s/4\ge\theta r_{1}/4>a_{P}$, so $a_{P}s-\theta s^{2}/2\le-\theta s^{2}/4\le-\theta r_{1}^{2}/4\le-A_{0}$, and the right side is at most $W^{-}_{\delta}(\mu)-1-\theta s^{2}/2$; both inequalities of (K) follow.

Localisation at $\mu$. Let $\gamma\in\mathcal{C}(\kappa_{d}(\mu))$ with $s(\gamma)<r_{1}$, realised by $(\bar X,\bar P_{-},\bar X')$, and put $\bar P=-\bar P_{-}$. As in (T0), $\mathrm{law}(\bar X,\bar P)=\kappa_{2d}(\pi)$, $\mathrm{law}(\bar X')=\kappa_{d}(\mu)$, $\lVert\bar X'-\bar X\rVert_{2}=s(\gamma)$ and $p(\gamma)=-\langle\bar P,\bar X'-\bar X\rangle_{2}$, so ($\Omega$) with $\nu=\mu$ gives $p(\gamma)\le\omega(s(\gamma))\le\frac{1}{2}h(s(\gamma))$. As $\theta r_{1}\ge2|\pi^{-}|_{\mathrm{mom}}$, the hypotheses of The Coupling Test Function of a Bounded Plan: Maximising Couplings, Continuity, Plan Subjets and Localisation §localisation and of The Coupling Test Function of a Bounded Plan: Maximising Couplings, Continuity, Plan Subjets and Localisation §convergence hold with $r_{0}=r_{1}$; in particular $T(\kappa_{d}(\mu))=-\Phi(\kappa_{d}(\mu))=0$.

Step 3 (Almost maximisers and the variational principle). Two bounds. Let $v\in\mathcal{F}$ and $\nu\in\mathcal{D}$. By (MON) and (K), applied with a maximising coupling for $\nu$,

$$v^{-}_{\delta}(\nu)-T\bigl(\kappa_{d}(\nu)\bigr)\le W^{-}_{\delta}(\nu)-T\bigl(\kappa_{d}(\nu)\bigr)\le W^{-}_{\delta}(\mu).\tag{I}$$

If moreover $\nu\notin\mathcal{D}_{\ell}$, that is $\mathcal{E}(\nu)>\ell$, then (F5) (with $w=v$, $b'=b$) and (T0) give, as $\delta>0$,

$$v^{-}_{\delta}(\nu)-T\bigl(\kappa_{d}(\nu)\bigr)\le b-\delta\mathcal{E}(\nu)+\frac{a_{P}^{2}}{4\theta}<b-\delta\ell+\frac{a_{P}^{2}}{4\theta}=W^{-}_{\delta}(\mu)-2.\tag{O}$$

Completeness. With the restriction of $W_{2}$, $\mathcal{D}_{\ell}$ is a metric space (Metric Space). It is complete: a Cauchy sequence $(\lambda_{m})_{m}$ in it converges in the complete space $(\Sigma_{d,R},W_{2})$ (The Noncommutative Laws with a Norm Bound Form a Complete Bounded Metric Space with Interpolation Points §complete) to some $\lambda$, that is $W_{2}(\lambda_{m},\lambda)\to0$ (Convergent Sequence in a Metric Space), and Sublevel Sets of the Wall-Confined Free Energy are Closed for the Wasserstein Distance §closed gives $\lambda\in\mathcal{D}$ and $\mathcal{E}(\lambda)\le\ell$, so $\lambda\in\mathcal{D}_{\ell}$ and the sequence converges in $\mathcal{D}_{\ell}$ (Complete Metric Space).

Almost maximisers. Let $j\in\mathbb{N}$. By Properties of the Upper Semicontinuous Envelope §approximation, applied to $W\circ\kappa_{d}-\delta\mathcal{E}$ (whose upper semicontinuous envelope is $W^{-}_{\delta}$) at $\mu$ with $\varepsilon=1/j$, choose $\nu^{0}_{j}\in\mathcal{D}$ with $W_{2}(\nu^{0}_{j},\mu)\le1/j$ and $W(\kappa_{d}(\nu^{0}_{j}))-\delta\mathcal{E}(\nu^{0}_{j})>W^{-}_{\delta}(\mu)-1/j$; as $W(\kappa_{d}(\nu^{0}_{j}))$ is the least upper bound of $\{v(\kappa_{d}(\nu^{0}_{j})):v\in\mathcal{F}\}$, choose $v_{j}\in\mathcal{F}$ with $v_{j}(\kappa_{d}(\nu^{0}_{j}))>W(\kappa_{d}(\nu^{0}_{j}))-1/j$. Then $\delta\mathcal{E}(\nu^{0}_{j})<W(\kappa_{d}(\nu^{0}_{j}))-W^{-}_{\delta}(\mu)+1\le b-W^{-}_{\delta}(\mu)+1<\delta\ell$, so $\nu^{0}_{j}\in\mathcal{D}_{\ell}$. Write $v^{-}_{j,\delta}=(v_{j})^{-}_{\delta}$. By (F5),

$$v^{-}_{j,\delta}(\nu^{0}_{j})-T\bigl(\kappa_{d}(\nu^{0}_{j})\bigr)\ge v_{j}\bigl(\kappa_{d}(\nu^{0}_{j})\bigr)-\delta\mathcal{E}(\nu^{0}_{j})-T\bigl(\kappa_{d}(\nu^{0}_{j})\bigr)>W^{-}_{\delta}(\mu)-\varepsilon_{j},\qquad\varepsilon_{j}=\frac{2}{j}+\bigl|T\bigl(\kappa_{d}(\nu^{0}_{j})\bigr)\bigr|.$$

Since $W_{2}(\nu^{0}_{j},\mu)\to0$, the function $T\circ\kappa_{d}$ is continuous on $\Sigma_{d,R}$ (The Coupling Test Function of a Bounded Plan: Maximising Couplings, Continuity, Plan Subjets and Localisation §continuity) and $T(\kappa_{d}(\mu))=0$ (Step 2), we have $\varepsilon_{j}\to0$. Fix $j_{0}\in\mathbb{N}$ with $\varepsilon_{j}<1$ for all $j\ge j_{0}$, and replace the sequences $(\nu^{0}_{j})$, $(v_{j})$ and $(\varepsilon_{j})$ by $(\nu^{0}_{j+j_{0}-1})$, $(v_{j+j_{0}-1})$ and $(\varepsilon_{j+j_{0}-1})$. All properties just proved persist, $W_{2}(\nu^{0}_{j},\mu)\to0$ and $\varepsilon_{j}\to0$ still hold, and now $0<\varepsilon_{j}<1$ for every $j\in\mathbb{N}$.

The variational principle. Fix $j\in\mathbb{N}$ and put $w^{j}_{k}=\frac{1}{j}\bigl(\frac{1}{2}\bigr)^{k}>0$ for $k\in\mathbb{N}$; by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §geometric and Elementary Properties of Series of Real Numbers §linearity, $\sum_{k}w^{j}_{k}$ converges with sum $1/j$. We apply A Smooth Variational Principle of Borwein-Preiss Type with a Gauge on a Complete Metric Space on the nonempty complete metric space $(\mathcal{D}_{\ell},W_{2})$ with: $f=f_{j}$, the restriction to $\mathcal{D}_{\ell}$ of $v^{-}_{j,\delta}-T\circ\kappa_{d}$; $g(\lambda,\lambda')=W_{2}(\lambda,\lambda')^{2}$ and $G=4dR^{2}$; the weights $(w^{j}_{k})_{k}$; $\varepsilon=\varepsilon_{j}$; and $x_{1}=\nu^{0}_{j}$. Its hypotheses hold. By (I), $f_{j}\le W^{-}_{\delta}(\mu)$. The function $f_{j}$ is upper semicontinuous on $\mathcal{D}_{\ell}$: given $\lambda\in\mathcal{D}_{\ell}$ and $\eta>0$, Properties of the Upper Semicontinuous Envelope §usc and Upper Semicontinuous Function on a Subset of a Metric Space give $r_{a}>0$ with $v^{-}_{j,\delta}(\lambda')<v^{-}_{j,\delta}(\lambda)+\eta/2$ for $\lambda'\in\mathcal{D}$ with $W_{2}(\lambda,\lambda')<r_{a}$, and The Coupling Test Function of a Bounded Plan: Maximising Couplings, Continuity, Plan Subjets and Localisation §continuity with Continuous Map Between Metric Spaces gives $r_{b}>0$ with $|T(\kappa_{d}(\lambda'))-T(\kappa_{d}(\lambda))|<\eta/2$ for $\lambda'\in\Sigma_{d,R}$ with $W_{2}(\lambda,\lambda')<r_{b}$; so $f_{j}(\lambda')<f_{j}(\lambda)+\eta$ for $\lambda'\in\mathcal{D}_{\ell}$ with $W_{2}(\lambda,\lambda')<\min\{r_{a},r_{b}\}$. Next, $g(\lambda,\lambda)=0$ and $0\le g\le G$ (Metric Space, The Noncommutative Laws with a Norm Bound Form a Complete Bounded Metric Space with Interpolation Points §bounded); by the triangle inequality (The Noncommutative Wasserstein Distance Satisfies the Triangle Inequality and is a Metric on Noncommutative Laws §triangle), symmetry and the bound $W_{2}\le2R\sqrt{d}$ of the conventions, $|g(\lambda_{1},\lambda')-g(\lambda_{2},\lambda')|\le4R\sqrt{d}\,W_{2}(\lambda_{1},\lambda_{2})$, so each $g(\cdot,\lambda')$ is continuous, hence lower semicontinuous (Lower Semicontinuous Function on a Subset of a Metric Space); and for $\eta>0$ the number $\beta=\eta^{2}/4$ works, as $g(\lambda,\lambda')\le\beta$ gives $W_{2}(\lambda,\lambda')\le\eta/2<\eta$. Finally $\nu^{0}_{j}\in\mathcal{D}_{\ell}$ and $f_{j}(\nu^{0}_{j})>W^{-}_{\delta}(\mu)-\varepsilon_{j}\ge\sup f_{j}-\varepsilon_{j}$. The theorem gives $\nu_{j}\in\mathcal{D}_{\ell}$ and a sequence $(x^{j}_{k})_{k\in\mathbb{N}}$ in $\mathcal{D}_{\ell}$ with $x^{j}_{1}=\nu^{0}_{j}$. For $\nu\in\mathcal{D}$ the series $\sum_{k}w^{j}_{k}W_{2}(\nu,x^{j}_{k})^{2}$ converges by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §comparison, its terms lying in $[0,4dR^{2}w^{j}_{k}]$, and we put

$$\Psi_{j}(\nu)=v^{-}_{j,\delta}(\nu)-T\bigl(\kappa_{d}(\nu)\bigr)-\sum_{k=1}^{\infty}w^{j}_{k}W_{2}(\nu,x^{j}_{k})^{2}\qquad(\nu\in\mathcal{D});$$

on $\mathcal{D}_{\ell}$ this is the function called $\Phi$ in that theorem. By A Smooth Variational Principle of Borwein-Preiss Type with a Gauge on a Complete Metric Space §value and A Smooth Variational Principle of Borwein-Preiss Type with a Gauge on a Complete Metric Space §maximum, $\Psi_{j}(\nu_{j})\ge f_{j}(\nu^{0}_{j})>W^{-}_{\delta}(\mu)-\varepsilon_{j}$, and $\Psi_{j}(\nu)<\Psi_{j}(\nu_{j})$ for $\nu\in\mathcal{D}_{\ell}$ with $\nu\ne\nu_{j}$. For $\nu\in\mathcal{D}\setminus\mathcal{D}_{\ell}$, (O) and the nonnegativity of the series give $\Psi_{j}(\nu)<W^{-}_{\delta}(\mu)-2<W^{-}_{\delta}(\mu)-\varepsilon_{j}<\Psi_{j}(\nu_{j})$. Hence

$$\Psi_{j}(\nu)<\Psi_{j}(\nu_{j})\quad(\nu\in\mathcal{D},\ \nu\ne\nu_{j}),\qquad v^{-}_{j,\delta}(\nu_{j})-T\bigl(\kappa_{d}(\nu_{j})\bigr)\ge\Psi_{j}(\nu_{j})>W^{-}_{\delta}(\mu)-\varepsilon_{j}.\tag{BP}$$

Step 4 (Localisation of the maximisers). For every $j\in\mathbb{N}$ fix, by The Coupling Test Function of a Bounded Plan: Maximising Couplings, Continuity, Plan Subjets and Localisation §attained, a maximising coupling $\gamma_{j}$ for $\nu_{j}$, and a realisation $(X^{j},P^{j}_{-},X'^{j})$ of it in a tracial W*-probability space $(H_{j},M_{j},\Omega_{j})$ (Couplings of a Square-Integrable Plan with a Law: Displacement and Momentum Pairing §displacement-pairing); put $s_{j}=s(\gamma_{j})=\lVert X'^{j}-X^{j}\rVert_{2}$ and $\tilde P^{j}=-P^{j}_{-}$. By (MON) and (BP), $W^{-}_{\delta}(\nu_{j})-T(\kappa_{d}(\nu_{j}))>W^{-}_{\delta}(\mu)-\varepsilon_{j}>W^{-}_{\delta}(\mu)-1$, so (K) forces $s_{j}<r_{1}$ and $\frac{\theta}{2}s_{j}^{2}<\varepsilon_{j}$. As $\varepsilon_{j}\to0$, $s_{j}\to0$; and $W_{2}(\nu_{j},\mu)\le s_{j}$ by (T0), so $W_{2}(\nu_{j},\mu)\to0$. Let $\alpha_{j}=h'(s_{j})/s_{j}+2\theta$ if $s_{j}>0$ and $\alpha_{j}=0$ if $s_{j}=0$, so that the tuple formed from $(X^{j},P^{j}_{-},X'^{j})$ in The Coupling Test Function of a Bounded Plan: Maximising Couplings, Continuity, Plan Subjets and Localisation §subjet is $S^{j}_{-}=P^{j}_{-}-\alpha_{j}(X'^{j}-X^{j})$. By The Coupling Test Function of a Bounded Plan: Maximising Couplings, Continuity, Plan Subjets and Localisation §convergence (its hypotheses were verified in Step 2), $T(\kappa_{d}(\nu_{j}))=-\Phi(\kappa_{d}(\nu_{j}))\to0$ and $\lVert S^{j}_{-}-P^{j}_{-}\rVert_{2}\to0$. Put $S^{j}=-S^{j}_{-}=\tilde P^{j}+\alpha_{j}(X'^{j}-X^{j})$. By The Coupling Test Function of a Bounded Plan: Maximising Couplings, Continuity, Plan Subjets and Localisation §subjet, $|h'|\le K$ and $s_{j}<r_{1}$,

$$\lVert S^{j}-\tilde P^{j}\rVert_{2}=\lVert S^{j}_{-}-P^{j}_{-}\rVert_{2}\le h'(s_{j})+2\theta s_{j}\le K+2\theta r_{1}.$$

By (T0), $\mathrm{law}(X^{j},\tilde P^{j})=\kappa_{2d}(\pi)$, $\mathrm{law}(X^{j})=\kappa_{d}(\mu)$, $\mathrm{law}(X'^{j})=\kappa_{d}(\nu_{j})$ and $\lVert\tilde P^{j}\rVert_{2}=a_{P}$. Finally, $v^{-}_{j,\delta}(\nu_{j})\to W^{-}_{\delta}(\mu)$: by (BP), $v^{-}_{j,\delta}(\nu_{j})>W^{-}_{\delta}(\mu)-\varepsilon_{j}+T(\kappa_{d}(\nu_{j}))$, and the right side tends to $W^{-}_{\delta}(\mu)$; and by (MON), $v^{-}_{j,\delta}(\nu_{j})\le W^{-}_{\delta}(\nu_{j})$, while $W^{-}_{\delta}$ is upper semicontinuous at $\mu$ (Properties of the Upper Semicontinuous Envelope §usc), so for every $\eta>0$, $W^{-}_{\delta}(\nu_{j})<W^{-}_{\delta}(\mu)+\eta$ for all large $j$.

Step 5 (A plan superjet for $v_{j}$ and its subsolution inequality). Fix $j\in\mathbb{N}$.

A superjet of $T$. By The Coupling Test Function of a Bounded Plan: Maximising Couplings, Continuity, Plan Subjets and Localisation §subjet, $\mathrm{law}(X'^{j},S^{j}_{-})\in J^{-}\Phi(\kappa_{d}(\nu_{j}))$. We claim $\mathrm{law}(X'^{j},S^{j})\in J^{+}T(\kappa_{d}(\nu_{j}))$. It is a plan at $\kappa_{d}(\nu_{j})$, its first marginal being $\mathrm{law}(X'^{j})$ by (F1). Let $\eta>0$, let $r>0$ be given for $\eta$ by Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §sub (slack $0$) for $\Phi$ at $\kappa_{d}(\nu_{j})$ and the plan $\mathrm{law}(X'^{j},S^{j}_{-})$, and let $Y,S',Y'$ be $L^{2}$ $d$-tuples of a tracial W*-probability space with $\mathrm{law}(Y,S')=\mathrm{law}(X'^{j},S^{j})$ and $\lVert Y'-Y\rVert_{2}<r$. By (F1), $\mathrm{law}(Y,-S')=N_{\#}\mathrm{law}(X'^{j},S^{j})=\mathrm{law}(X'^{j},S^{j}_{-})$, so the subdifferential inequality for $Y,-S',Y'$ gives $\Phi(\mathrm{law}(Y'))\ge\Phi(\kappa_{d}(\nu_{j}))-\langle S',Y'-Y\rangle_{2}-\eta\lVert Y'-Y\rVert_{2}$; multiplying by $-1$,

$$T\bigl(\mathrm{law}(Y')\bigr)\le T\bigl(\kappa_{d}(\nu_{j})\bigr)+\langle S',Y'-Y\rangle_{2}+\eta\lVert Y'-Y\rVert_{2},$$

which is Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §super with slack $0$.

A superjet of the distance series. By The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §attained fix optimal couplings $\gamma^{j}_{k}\in\Pi(\nu_{j},x^{j}_{k})$ ($k\in\mathbb{N}$); all these laws lie in $\mathcal{D}_{\ell}\subseteq\Sigma_{d,R}$. By Gluing Countably Many Noncommutative Couplings with a Common First Marginal in One Tracial W*-Probability Space §glue, applied with $R$, $\nu_{j}$, $(x^{j}_{k})_{k}$ and $(\gamma^{j}_{k})_{k}$, fix a tracial W*-probability space $(H^{j},M^{j},\Omega^{j})$ and self-adjoint $d$-tuples $y^{j}$ and $t^{j,k}$ ($k\in\mathbb{N}$) in $M^{j}$ with the properties listed there, in particular $\lambda_{y^{j}}=\nu_{j}$. Apply Plan Jets of a Series of Squared Wasserstein Distances at Optimal Couplings of Bounded Noncommutative Laws with $R$, $\nu_{j}$, $(x^{j}_{k})$, $(\gamma^{j}_{k})$, $(w^{j}_{k})$, $(H^{j},M^{j},\Omega^{j})$, $y^{j}$ and $(t^{j,k})$ in the roles of $R$, $\mu$, $(\nu_{k})$, $(\gamma_{k})$, $(c_{k})$, $(H,M,\Omega)$, $s$ and $(t^{k})$, and let $g^{j}$, $\pi^{j}=\lambda_{(y^{j},g^{j})}$ and $\varphi_{j}$ be the objects called $P$, $\pi$ and $\varphi$ there. By Plan Jets of a Series of Squared Wasserstein Distances at Optimal Couplings of Bounded Noncommutative Laws §plans and Plan Jets of a Series of Squared Wasserstein Distances at Optimal Couplings of Bounded Noncommutative Laws §superjet, $\pi^{j}$ is a bounded plan at $\nu_{j}$ and $\kappa_{2d}(\pi^{j})\in J^{+}\varphi_{j}(\kappa_{d}(\nu_{j}))$, and by the same clause and the isometry identity $\varphi_{j}(\kappa_{d}(\nu))=\sum_{k}w^{j}_{k}W_{2}(\nu,x^{j}_{k})^{2}$ for $\nu\in\Sigma_{d,R}$. Put $G_{j}=g^{j}\Omega^{j}$. By Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §coupling and Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §bounded, $\kappa_{2d}(\pi^{j})=\mathrm{law}(y^{j}\Omega^{j},G_{j})$, the vacuum tuple of $(y^{j},g^{j})$ being this pair; also $\mathrm{law}(y^{j}\Omega^{j})=\kappa_{d}(\nu_{j})$ and $\mathrm{law}(G_{j})=\kappa_{d}(\lambda_{g^{j}})\in\kappa_{d}(\Sigma_{d})$ by the same clause and Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §law. By Plan Jets of a Series of Squared Wasserstein Distances at Optimal Couplings of Bounded Noncommutative Laws §momentum, the bound $W_{2}\le2R\sqrt{d}$, Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §comparison and Elementary Properties of Series of Real Numbers §linearity,

$$\lVert G_{j}\rVert_{2}\le2\sum_{k=1}^{\infty}w^{j}_{k}W_{2}(\nu_{j},x^{j}_{k})\le4R\sqrt{d}\sum_{k=1}^{\infty}w^{j}_{k}=\frac{4R\sqrt{d}}{j}.$$

The test function. Let $\psi_{j}=T+\varphi_{j}:\Sigma^{2}_{d}\to\mathbb{R}$. For $\nu\in\mathcal{D}$, $v^{-}_{j,\delta}(\nu)-\psi_{j}(\kappa_{d}(\nu))=\Psi_{j}(\nu)$, so (BP) gives

$$v^{-}_{j,\delta}(\nu)-\psi_{j}\bigl(\kappa_{d}(\nu)\bigr)<v^{-}_{j,\delta}(\nu_{j})-\psi_{j}\bigl(\kappa_{d}(\nu_{j})\bigr)\qquad(\nu\in\mathcal{D},\ \nu\ne\nu_{j}).$$

Adding the superjets. The laws $\mathrm{law}(X'^{j},S^{j},X^{j},\tilde P^{j})\in\Sigma^{2}_{4d}$ and $\kappa_{2d}(\pi^{j})\in\Sigma^{2}_{2d}$ have the same law $\kappa_{d}(\nu_{j})$ of their first $d$ variables, by (F1). By Gluing Two Square-Integrable Noncommutative Laws along a Common Marginal §glue with $k=d$, $m=3d$ and $n=d$, fix a tracial W*-probability space $(H'_{j},M'_{j},\Omega'_{j})$ and $L^{2}$ $d$-tuples $Y_{j},\check S_{j},\check X_{j},\check P_{j},\check G_{j}$ of it with

$$\mathrm{law}(Y_{j},\check S_{j},\check X_{j},\check P_{j})=\mathrm{law}(X'^{j},S^{j},X^{j},\tilde P^{j}),\qquad\mathrm{law}(Y_{j},\check G_{j})=\kappa_{2d}(\pi^{j}).$$

By (F1) and Step 4: $\mathrm{law}(Y_{j})=\kappa_{d}(\nu_{j})$; $\mathrm{law}(Y_{j},\check S_{j})=\mathrm{law}(X'^{j},S^{j})\in J^{+}T(\kappa_{d}(\nu_{j}))$; $\mathrm{law}(\check X_{j},\check P_{j})=\kappa_{2d}(\pi)$, hence $\mathrm{law}(\check X_{j})=\kappa_{d}(\mu)$ and $\mathrm{law}(\check P_{j})=\mathrm{law}(P_{\pi})\in\kappa_{d}(\Sigma_{d})$ (Step 1); $\check S_{j}=\check P_{j}+\alpha_{j}(Y_{j}-\check X_{j})$; $\lVert\check P_{j}\rVert_{2}=a_{P}$, $\lVert Y_{j}-\check X_{j}\rVert_{2}=s_{j}$ and $\lVert\check S_{j}-\check P_{j}\rVert_{2}=\lVert S^{j}-\tilde P^{j}\rVert_{2}$; and $\mathrm{law}(\check G_{j})=\mathrm{law}(G_{j})\in\kappa_{d}(\Sigma_{d})$ with $\lVert\check G_{j}\rVert_{2}=\lVert G_{j}\rVert_{2}$. By Plan Jets of a Sum along a Common Realisation §super, with $T$ and $\varphi_{j}$ in the roles of $\varphi_{1}$ and $\varphi_{2}$, $\lambda=\kappa_{d}(\nu_{j})$, both slacks $0$, $X=Y_{j}$, $P_{1}=\check S_{j}$ and $P_{2}=\check G_{j}$,

$$\mathrm{law}(Y_{j},\check\Sigma_{j})\in J^{+}\psi_{j}\bigl(\kappa_{d}(\nu_{j})\bigr),\qquad\check\Sigma_{j}=\check S_{j}+\check G_{j}=\check P_{j}+\alpha_{j}Y_{j}-\alpha_{j}\check X_{j}+\check G_{j}.$$

The bounded plan. By (F3), with $Y=Y_{j}$ and $\check\Sigma_{j}$ written as the above combination of $\check P_{j},Y_{j},\check X_{j},\check G_{j}$, there is a bounded plan $\varpi_{j}$ at $\nu_{j}$ with $\kappa_{2d}(\varpi_{j})=\mathrm{law}(Y_{j},\check\Sigma_{j})$ and

$$|\varpi_{j}|_{\mathrm{mom}}=\lVert\check\Sigma_{j}\rVert_{2}\le\lVert\check P_{j}\rVert_{2}+\lVert\check S_{j}-\check P_{j}\rVert_{2}+\lVert\check G_{j}\rVert_{2}\le a_{P}+K+2\theta r_{1}+4R\sqrt{d}=m_{*}.$$

The subsolution inequality. By Envelope Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §sub, applied to the envelope viscosity subsolution $v_{j}$ with shift range $\delta_{0}$ (bounded, as $|v_{j}|\le b$) with the level $\delta$, the test function $\psi_{j}$, the law $\nu_{j}\in\mathcal{D}$ and the bounded plan $\varpi_{j}$, we get $\nu_{j}\in\mathcal{D}_{\Xi}$ and, with $a_{j}=v^{-}_{j,\delta}(\nu_{j})+\delta\mathcal{E}(\nu_{j})$,

$$\rho a_{j}+\mathcal{H}\bigl(\varpi_{j}\oplus\delta\,\Xi(\nu_{j})\bigr)+\frac{\sigma^{2}}{2}\Bigl(\mathcal{J}\bigl(\Xi(\nu_{j}),\varpi_{j}\bigr)+\delta\,\lVert\Xi(\nu_{j})\rVert_{2}^{2}\Bigr)\le0.\tag{S}$$

Step 6 (Score bound). By (F5) with $w=v_{j}$ and $b'=b$, $-b-\delta\mathcal{E}(\nu_{j})\le v^{-}_{j,\delta}(\nu_{j})\le b-\delta\mathcal{E}(\nu_{j})$, that is $|a_{j}|\le b$. By (F4) with $t=\delta$, applied in $(\mathcal{H}_{\varpi_{j}},\mathcal{M}_{\varpi_{j}},\Omega_{\varpi_{j}})$ to $(X_{\varpi_{j}},P_{\varpi_{j}},V^{1}_{\varpi_{j}}\Xi(\nu_{j}))$, the inequality (S) is the hypothesis of Score Bounds at Envelope Test Inequalities from the Absorption Slack §sub with $X=X_{\varpi_{j}}$, $P=P_{\varpi_{j}}$, $Q=V^{1}_{\varpi_{j}}\Xi(\nu_{j})$ and $a=a_{j}$; there $\mathrm{law}(X_{\varpi_{j}})=\kappa_{d}(\nu_{j})\in\kappa_{d}(\Sigma_{d,R})$ and $\lVert P_{\varpi_{j}}\rVert_{2}=|\varpi_{j}|_{\mathrm{mom}}\le m_{*}$ by (F2), $|a_{j}|\le b$, and $0<\delta\le\delta_{0}\le\delta_{\mathrm{sb}}$. Hence, using (F4) once more,

$$\lVert\Xi(\nu_{j})\rVert_{2}=\lVert V^{1}_{\varpi_{j}}\Xi(\nu_{j})\rVert_{2}\le\frac{C_{*}}{\delta}\qquad(j\in\mathbb{N}).$$

Step 7 (Convergence of the energies). By Steps 4, 5 and 6, $(\nu_{j})_{j}$ is a sequence in $\mathcal{D}_{\Xi}$ with $W_{2}(\nu_{j},\mu)\to0$ and $\lVert\Xi(\nu_{j})\rVert_{2}\le C_{*}/\delta$, and $\mu\in\mathcal{D}$; so The Wall-Confined Free Energy Converges along Wasserstein-Convergent Sequences with Bounded Scores §convergence gives $\mathcal{E}(\nu_{j})\to\mathcal{E}(\mu)$. Together with Step 4, $a_{j}\to W^{-}_{\delta}(\mu)+\delta\mathcal{E}(\mu)$.

Step 8 (All tuples in one space, and closed score). Fix $j$. As $\nu_{j}\in\mathcal{D}_{\Xi}$, the law $\tau(\varpi_{j})$ of (F4) is defined, and by (F1) and (F2) the laws $\mathrm{law}(Y_{j},\check\Sigma_{j},\check X_{j},\check P_{j})\in\Sigma^{2}_{4d}$ and $\tau(\varpi_{j})\in\Sigma^{2}_{3d}$ have the same law $\kappa_{2d}(\varpi_{j})$ of their first $2d$ variables. By Gluing Two Square-Integrable Noncommutative Laws along a Common Marginal §glue with $k=2d$, $m=2d$ and $n=d$, fix a tracial W*-probability space and $L^{2}$ $d$-tuples $Y'_{j},\Sigma'_{j},X''_{j},P''_{j},Q'_{j}$ of it with

$$\mathrm{law}(Y'_{j},\Sigma'_{j},X''_{j},P''_{j})=\mathrm{law}(Y_{j},\check\Sigma_{j},\check X_{j},\check P_{j}),\qquad\mathrm{law}(Y'_{j},\Sigma'_{j},Q'_{j})=\tau(\varpi_{j}),$$

and put $\Gamma_{j}=\mathrm{law}(X''_{j},P''_{j},Y'_{j},\Sigma'_{j},Q'_{j})\in\Sigma^{2}_{5d}$. The law of its first $2d$ variables is $\mathrm{law}(X''_{j},P''_{j})=\mathrm{law}(\check X_{j},\check P_{j})=\kappa_{2d}(\pi)$ by (F1). Now Gluing Countably Many Square-Integrable Noncommutative Laws along a Common Marginal §glue, with $k=2d$, the law $\kappa_{2d}(\pi)$ in the role of $\pi$, $m_{j}=3d$ and $\gamma_{j}=\Gamma_{j}$, gives a tracial W*-probability space $(H,M,\Omega)$, an $L^{2}$ $2d$-tuple of it, written as a pair $(X,P)$ of $L^{2}$ $d$-tuples, and for every $j$ an $L^{2}$ $3d$-tuple, written as a triple $(\bar X_{j},\bar\Sigma_{j},\bar Q_{j})$, with $\mathrm{law}(X,P,\bar X_{j},\bar\Sigma_{j},\bar Q_{j})=\Gamma_{j}$ for every $j\in\mathbb{N}$. By (F1), Steps 4 and 5 and the constructions above, for every $j$:

$$\mathrm{law}(X,P)=\kappa_{2d}(\pi),\qquad\mathrm{law}(X)=\kappa_{d}(\mu),\qquad\mathrm{law}(\bar X_{j})=\kappa_{d}(\nu_{j}),\qquad\lVert\bar X_{j}-X\rVert_{2}=\lVert Y_{j}-\check X_{j}\rVert_{2}=s_{j},$$ $$\lVert\bar\Sigma_{j}-P\rVert_{2}=\lVert\check\Sigma_{j}-\check P_{j}\rVert_{2}\le\lVert\check S_{j}-\check P_{j}\rVert_{2}+\lVert\check G_{j}\rVert_{2}\le\lVert S^{j}_{-}-P^{j}_{-}\rVert_{2}+\frac{4R\sqrt{d}}{j},\qquad\mathrm{law}(\bar X_{j},\bar\Sigma_{j},\bar Q_{j})=\tau(\varpi_{j}).$$

Hence, by Step 4, $\lVert\bar X_{j}-X\rVert_{2}\to0$ and $\lVert\bar\Sigma_{j}-P\rVert_{2}\to0$. By (F4) with $t=\delta$ and Step 6, $\mathrm{law}(\bar X_{j},\bar Q_{j})=\pi^{\Xi}_{\nu_{j}}$ and $\lVert\bar Q_{j}\rVert_{2}=\lVert\Xi(\nu_{j})\rVert_{2}\le C_{*}/\delta$, and (S) becomes, with $G^{+}_{\delta}$ as in Shift Semicontinuity of a Hamiltonian on Phase-Space Noncommutative Laws,

$$\rho a_{j}+G^{+}_{\delta}(\bar X_{j},\bar\Sigma_{j},\bar Q_{j})+\frac{\sigma^{2}}{2}\langle\bar Q_{j},\bar\Sigma_{j}\rangle_{2}\le0\qquad(j\in\mathbb{N}).\tag{S'}$$

Since $\mathcal{E}$ has closed score, Wall-Confined Free Energies with Closed Score §closed, applied in $(H,M,\Omega)$ with the sequence $(\nu_{j})$ in $\mathcal{D}_{\Xi}$, the law $\mu\in\mathcal{D}$, the tuples $\bar X_{j},\bar Q_{j}$ and $X$, and the bound $C_{*}/\delta$, gives $\mu\in\mathcal{D}_{\Xi}$ and an $L^{2}$ $d$-tuple $Q$ of $(H,M,\Omega)$ with $\mathrm{law}(X,Q)=\pi^{\Xi}_{\mu}$ and $\langle\bar Q_{j},Y\rangle_{2}\to\langle Q,Y\rangle_{2}$ for every $L^{2}$ $d$-tuple $Y$ of $(H,M,\Omega)$.

Step 9 (Passage to the limit). By Step 8, the tuples $\bar X_{j},\bar\Sigma_{j},\bar Q_{j}$ ($j\in\mathbb{N}$) and $X,P,Q$ form a shift-convergent sequence at radius $R$ in $(H,M,\Omega)$ (Shift-Convergent Sequences of Positions, Momenta and Shifts in a Tracial W*-Probability Space §sequence): the laws of $\bar X_{j}$ and $X$ lie in $\kappa_{d}(\Sigma_{d,R})$ as $\nu_{j},\mu\in\mathcal{D}\subseteq\Sigma_{d,R}$, $\lVert\bar X_{j}-X\rVert_{2}\to0$, $\lVert\bar\Sigma_{j}-P\rVert_{2}\to0$, $\lVert\bar Q_{j}\rVert_{2}\le C_{*}/\delta$, and $\bar Q_{j}$ converges weakly to $Q$. Since $0<\delta\le\delta_{0}\le\delta_{\mathrm{ls}}$, Shift Semicontinuity of a Hamiltonian on Phase-Space Noncommutative Laws §lower (with $r=R$ and the $\delta_{\mathrm{ls}}$ of Step 0) gives, for every $\varepsilon>0$, an $N\in\mathbb{N}$ with $G^{+}_{\delta}(X,P,Q)\le G^{+}_{\delta}(\bar X_{j},\bar\Sigma_{j},\bar Q_{j})+\varepsilon$ for all $j\ge N$. Moreover, by the Cauchy--Schwarz inequality, the bound on $\lVert\bar Q_{j}\rVert_{2}$ and the weak convergence with $Y=P$,

$$\bigl|\langle\bar Q_{j},\bar\Sigma_{j}\rangle_{2}-\langle Q,P\rangle_{2}\bigr|\le\frac{C_{*}}{\delta}\lVert\bar\Sigma_{j}-P\rVert_{2}+\bigl|\langle\bar Q_{j},P\rangle_{2}-\langle Q,P\rangle_{2}\bigr|\to0,$$

and $a_{j}\to W^{-}_{\delta}(\mu)+\delta\mathcal{E}(\mu)$ by Step 7. Let $\varepsilon>0$. For all large $j$ the following three one-sided bounds hold: $\rho\bigl(W^{-}_{\delta}(\mu)+\delta\mathcal{E}(\mu)\bigr)\le\rho a_{j}+\varepsilon$, since $\rho a_{j}\to\rho\bigl(W^{-}_{\delta}(\mu)+\delta\mathcal{E}(\mu)\bigr)$; $G^{+}_{\delta}(X,P,Q)\le G^{+}_{\delta}(\bar X_{j},\bar\Sigma_{j},\bar Q_{j})+\varepsilon$, for $j\ge N$ with the $N$ above; and $\frac{\sigma^{2}}{2}\langle Q,P\rangle_{2}\le\frac{\sigma^{2}}{2}\langle\bar Q_{j},\bar\Sigma_{j}\rangle_{2}+\varepsilon$, since $\frac{\sigma^{2}}{2}\langle\bar Q_{j},\bar\Sigma_{j}\rangle_{2}\to\frac{\sigma^{2}}{2}\langle Q,P\rangle_{2}$ by the display above. Adding them and using (S') yields

$$\rho\bigl(W^{-}_{\delta}(\mu)+\delta\mathcal{E}(\mu)\bigr)+G^{+}_{\delta}(X,P,Q)+\frac{\sigma^{2}}{2}\langle Q,P\rangle_{2}\le\rho a_{j}+G^{+}_{\delta}(\bar X_{j},\bar\Sigma_{j},\bar Q_{j})+\frac{\sigma^{2}}{2}\langle\bar Q_{j},\bar\Sigma_{j}\rangle_{2}+3\varepsilon\le3\varepsilon.$$

As $\varepsilon>0$ was arbitrary, the left side is at most $0$. Finally, $\mu\in\mathcal{D}_{\Xi}$, $\pi$ is a bounded plan at $\mu$, $\mathrm{law}(X,P)=\kappa_{2d}(\pi)$ and $\mathrm{law}(X,Q)=\pi^{\Xi}_{\mu}$; also $\mathrm{law}(X)=\kappa_{d}(\mu)$ with $\mu\in\mathcal{D}\subseteq\Sigma_{d,R}$, and $\pi^{\Xi}_{\mu}=\mathrm{law}(X_{\mu},\Xi(\mu))$ (The Score Plan of a Law in the Score Domain of the Wall-Confined Free Energy §score-plan), where $X_{\mu}$ is the tuple $X_{\lambda}$ of classes of the variables in 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 for $\lambda=\mu$ and $\Xi(\mu)$ is an $L^{2}$ $d$-tuple of $(\mathcal{H}_{\mu},\mathcal{M}_{\mu},\Omega_{\mu})$. So 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 §joint, with $r=R$, $\lambda=\mu$, $\zeta=\Xi(\mu)$, the bounded plan $\pi$ and the tuples $X,P,Q$ of $(H,M,\Omega)$, gives $\mathrm{law}(X,P,Q)=\mathrm{law}(X_{\pi},P_{\pi},V^{1}_{\pi}\Xi(\mu))=\tau(\pi)$, and then (F4) with $t=\delta$ 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 $\lVert Q\rVert_{2}=\lVert\Xi(\mu)\rVert_{2}$ (these three identities are also 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 with $t=\delta$ together with Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts, 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 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 the same data). By the definition of $G^{+}_{\delta}$,

$$G^{+}_{\delta}(X,P,Q)+\frac{\sigma^{2}}{2}\langle Q,P\rangle_{2}=\mathcal{H}\bigl(\pi\oplus\delta\,\Xi(\mu)\bigr)+\frac{\sigma^{2}}{2}\Bigl(\mathcal{J}\bigl(\Xi(\mu),\pi\bigr)+\delta\,\lVert\Xi(\mu)\rVert_{2}^{2}\Bigr),$$

so the left side above is the left side of (Goal), and (Goal) holds. As $\delta$, $\varphi$, $\mu$ and $\pi$ were arbitrary, $W$ is an envelope viscosity subsolution of $(\mathrm{E})$ with shift range $\delta_{0}$, which is claim 2; claim 1 is Part 1, and $\delta_{1}$ was fixed in Step 0 before $\delta_{0}$, $b$ and $\mathcal{F}$.