Proof of Comparison at a Fixed Level for Envelope Viscosity Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall

Each result cited below is universally quantified over the data in its own statement. The conventions of The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation are in force. Throughout, $\rho$, $\sigma$ and $R$ are as in The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §parameters; $\mathcal{D}$, $\mathcal{E}$, $\mathcal{D}_{\Xi}$ and $\Xi$ as in The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §free-energy; and $\mathcal{H}$ as in The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §hamiltonian, with lifts $\mathcal{H}_{M}$ as in Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts. The results The Wall-Confined Free Energy: Norm Bound, Lower Bound, Weak-Star Compact Sublevel Sets and Displacement Monotonicity of the Score, Tangent Inequalities for the Wall Energy along Couplings and for the Wall-Confined Free Energy along Optimal Couplings, Sublevel Sets of the Wall-Confined Free Energy are Closed for the Wasserstein Distance and Perturbed Maximisers of the Doubled Difference Penalised by the Wall-Confined Free Energy are applied with the free entropy penalty $(\mathcal{D}_{0},\mathcal{E}_{0})$ and the radius $R$ of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §free-energy, and the penalty envelopes are those of The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation §envelopes. 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 $\mathcal{D}$ 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; this is also the metric space relative to which the envelopes of The Penalty Envelopes of a Bounded Function on the Domain of the Wall-Confined Free Energy are formed and in which Properties of the Upper Semicontinuous Envelope and Properties of the Lower Semicontinuous Envelope, by Duality are applied below, with $S=\mathcal{D}$. By Square-Integrable Noncommutative Laws: the Wasserstein Completion of the Laws, Affine Push-Forwards, Moments, Couplings and Cost §laws, $\Sigma^{2}_{d}$ is the metric completion of $(\Sigma_{d},W_{2})$ with metric $\widehat{W}_{2}$ and canonical map $\kappa_{d}$, so 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 we have the isometry identity

$$\widehat{W}_{2}\bigl(\kappa_{d}(\lambda),\kappa_{d}(\lambda')\bigr)=W_{2}(\lambda,\lambda')\qquad\text{for all }\lambda,\lambda'\in\Sigma_{d}.$$

Tracial W*-probability spaces are written $(H_{1},M_{1},\Omega_{1})$ and $(H_{2},M_{2},\Omega_{2})$ below, to avoid clashes with $\Psi_{\delta,\alpha}$ (Step 2), with the GNS spaces $(\mathcal{H}_{\pi},\mathcal{M}_{\pi},\Omega_{\pi})$ of (F6) and with the commutant notation of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §spaces. Sums, differences, real multiples and the $L^{2}$ 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 Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples and Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §operations they are componentwise, and $\lVert\cdot\rVert_{2}$ is the norm of the Hilbert space $H^{d}$ by Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §pairing, so $\lVert tZ\rVert_{2}=|t|\,\lVert Z\rVert_{2}$ for real $t$.

Preliminary facts. (F1) For all $\lambda,\lambda'\in\Sigma_{d,R}$, $0\le W_{2}(\lambda,\lambda')\le2R\sqrt{d}$: $W_{2}$ is nonnegative by The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §distance, and $W_{2}(\lambda,\lambda')^{2}\le4dR^{2}=(2R\sqrt{d})^{2}$ by The Noncommutative Laws with a Norm Bound Form a Complete Bounded Metric Space with Interpolation Points §bounded; as $0\le2R\sqrt{d}$, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $W_{2}(\lambda,\lambda')\le2R\sqrt{d}$.

(F2) For a real $\alpha>0$ let $T_{\alpha}=(A,0)$ be the affine datum from $2d$ to $2d$ variables with zero shift and with $A_{jj}=1$, $A_{d+j,j}=-\alpha$ and $A_{d+j,d+j}=\alpha$ for $j\in[d]$, all other entries of $A$ being $0$. Let $i\in\{1,2\}$ and let $a,b$ be self-adjoint $d$-tuples in $M_{i}$. By Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §coupling, $(a,b)$ is a self-adjoint $2d$-tuple in $M_{i}$, $\lambda_{(a,b)}\in\Pi(\lambda_{a},\lambda_{b})$ and $I(\lambda_{(a,b)})=\sum_{j=1}^{d}\lVert a_{j}\Omega_{i}-b_{j}\Omega_{i}\rVert^{2}$; by Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples this sum equals both $\lVert a\Omega_{i}-b\Omega_{i}\rVert_{2}^{2}$ and $\lVert b\Omega_{i}-a\Omega_{i}\rVert_{2}^{2}$. By Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §affine, applied to $(a,b)$ and $T_{\alpha}$, the $2d$-tuple $w$ with $w_{j}=a_{j}$ and $w_{d+j}=\alpha b_{j}-\alpha a_{j}$ ($j\in[d]$) is a self-adjoint $2d$-tuple in $M_{i}$ whose vacuum tuple is the pair $(a\Omega_{i},\alpha(b\Omega_{i}-a\Omega_{i}))$ of Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations, and $\lambda_{w}=\lambda_{(a,b)}\circ\sigma_{T_{\alpha}}$. Hence, by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §bounded and Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts,

$$\mathcal{H}_{M_{i}}\bigl(a\Omega_{i},\alpha(b\Omega_{i}-a\Omega_{i})\bigr)=\mathcal{H}\bigl(\kappa_{2d}(\lambda_{(a,b)}\circ\sigma_{T_{\alpha}})\bigr),$$

which depends only on $\alpha$ and on the law $\lambda_{(a,b)}$.

(F3) Let $i\in\{1,2\}$ and let $a,p$ be self-adjoint $d$-tuples in $M_{i}$. By Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §coupling, $(a,p)$ is a self-adjoint $2d$-tuple in $M_{i}$; its vacuum tuple is the pair $(a\Omega_{i},p\Omega_{i})$, so $\mathcal{H}(\kappa_{2d}(\lambda_{(a,p)}))=\mathcal{H}_{M_{i}}(a\Omega_{i},p\Omega_{i})$ by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §bounded and Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts.

(F4) The canonical map $\kappa_{d}$ is injective 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 the metric space $(\Sigma_{d},W_{2})$ whose metric completion is $\Sigma^{2}_{d}$ (Square-Integrable Noncommutative Laws: the Wasserstein Completion of the Laws, Affine Push-Forwards, Moments, Couplings and Cost §laws). Hence, as $\mathcal{D}\subseteq\Sigma_{d,R}$, for every $\zeta\in\Sigma^{2}_{d}$ there is at most one $\lambda\in\mathcal{D}$ with $\kappa_{d}(\lambda)=\zeta$.

(F5) Let $w:\Sigma^{2}_{d}\to\mathbb{R}$ with $|w(\zeta)|\le b$ for every $\zeta\in\Sigma^{2}_{d}$, let $\delta>0$ be real and let $e$ be a real lower bound of $\mathcal{E}$ on $\mathcal{D}$. Then, for every $\mu\in\mathcal{D}$,

$$w\bigl(\kappa_{d}(\mu)\bigr)-\delta\mathcal{E}(\mu)\le w^{-}_{\delta}(\mu)\le b-\delta\mathcal{E}(\mu)\le b-\delta e,\qquad -b+\delta e\le-b+\delta\mathcal{E}(\mu)\le w^{+}_{\delta}(\mu)\le w\bigl(\kappa_{d}(\mu)\bigr)+\delta\mathcal{E}(\mu).$$

Indeed, $w^{-}_{\delta}$ is the upper semicontinuous envelope of $f=w\circ\kappa_{d}-\delta\mathcal{E}$ on $\mathcal{D}$ by 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. Let $g=b-\delta\mathcal{E}$ on $\mathcal{D}$. It is upper semicontinuous on $\mathcal{D}$: given $\mu\in\mathcal{D}$ and a real $\eta>0$, since $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ by 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 $\mathcal{E}(\mu)-\eta/\delta<\mathcal{E}(\lambda)$ for every $\lambda\in\mathcal{D}$ with $W_{2}(\mu,\lambda)<r$, and multiplying by $-\delta<0$ and adding $b$ gives $g(\lambda)<g(\mu)+\eta$ for these $\lambda$, which is Upper Semicontinuous Function on a Subset of a Metric Space at $\mu$. As $f\le g$ on $\mathcal{D}$ (because $w\le b$), Properties of the Upper Semicontinuous Envelope §least gives $w^{-}_{\delta}\le g$, the second inequality; the third holds as $e\le\mathcal{E}(\mu)$ and $\delta>0$. Symmetrically, $w^{+}_{\delta}$ is the lower semicontinuous envelope of $w\circ\kappa_{d}+\delta\mathcal{E}$ by The Penalty Envelopes of a Bounded Function on the Domain of the Wall-Confined Free Energy §lower; the last inequality is Properties of the Lower Semicontinuous Envelope, by Duality §bounds; $-g=-b+\delta\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ by the same argument (now $-b+\delta\mathcal{E}(\mu)-\eta<-b+\delta\mathcal{E}(\lambda)$), and $-b+\delta\mathcal{E}\le w\circ\kappa_{d}+\delta\mathcal{E}$ on $\mathcal{D}$ (because $-b\le w$), so Properties of the Lower Semicontinuous Envelope, by Duality §greatest gives the second inequality; the first holds as $e\le\mathcal{E}(\mu)$.

(F6) Since $\mathcal{H}$ absorbs shifts at noise level $\sigma$, applying this with $r=R$ we fix a real $\delta_{\mathrm{a}}>0$ and a nondecreasing function $\omega_{\mathrm{a}}:[0,\infty)\to[0,\infty)$ as there. We claim: for every $\mu\in\mathcal{D}_{\Xi}$, every bounded plan $\pi$ at $\mu$ and every real $t$ with $0<t\le\delta_{\mathrm{a}}$,

$$\mathcal{H}\bigl(\pi\oplus t\,\Xi(\mu)\bigr)\ge\mathcal{H}\bigl(\kappa_{2d}(\pi)\bigr)-t\,\omega_{\mathrm{a}}\bigl(|\pi|_{\mathrm{mom}}\bigr)-\tfrac{\sigma^{2}t}{4}\lVert\Xi(\mu)\rVert_{2}^{2},\qquad\mathcal{H}\bigl(\pi\oplus(-t)\,\Xi(\mu)\bigr)\le\mathcal{H}\bigl(\kappa_{2d}(\pi)\bigr)+t\,\omega_{\mathrm{a}}\bigl(|\pi|_{\mathrm{mom}}\bigr)+\tfrac{\sigma^{2}t}{4}\lVert\Xi(\mu)\rVert_{2}^{2}.$$

Indeed, $\pi\in\Sigma_{2d}$ by Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §plans, so $\pi\in\Sigma_{2d,r'}$ for some real $r'>0$ by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law, and $(\mathcal{H}_{\pi},\mathcal{M}_{\pi},\Omega_{\pi})$ is a tracial W*-probability space by The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §w-star, with $\mathcal{M}_{\pi}=\mathcal{A}_{\pi}''$. Let $L=(L_{x_{1}},\dots,L_{x_{2d}})$ be the multiplication operators of Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §multiplication on $\mathcal{H}_{\pi}$. Each $L_{x_{i}}$ lies in $\mathcal{A}_{\pi}$, hence commutes with every element of $\mathcal{A}_{\pi}'$ and so lies in $\mathcal{A}_{\pi}''=\mathcal{M}_{\pi}$ (The Commutant of a Set of Bounded Operators on a Complex Hilbert Space §commutant), and it is self-adjoint 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 §adjoint, as $x_{i}^{*}=x_{i}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint; so $L$ is a self-adjoint $2d$-tuple in $\mathcal{M}_{\pi}$, whose law is $\lambda_{L}=\pi$ by The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §law, and whose vacuum tuple is $(\widehat{x_{1}},\dots,\widehat{x_{2d}})$ 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, that is, the pair $(X_{\pi},P_{\pi})$ of Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations formed from the tuples of The Shift of a Bounded Plan by a Self-Adjoint Field §tuples. Hence $\mathrm{law}(X_{\pi},P_{\pi})=\kappa_{2d}(\pi)$ by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §bounded. The marginal datum $\mathrm{pr}^{1}$ of Affine Data and Affine Substitutions of Noncommutative Polynomials §coordinate has zero shift and $P^{1}_{ij}=1$ exactly when $j=i$, so $\mathrm{pr}^{1}(X_{\pi},P_{\pi})=X_{\pi}$ by Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations; thus, by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward, Calculus of Square-Integrable Noncommutative Laws: Agreement on Bounded Laws, Lipschitz Estimates, Functoriality of Push-Forwards, Moment Formulas, Positivity, the Cost and the Diagonal Coupling §bounded, Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §coordinate and Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §plans, $\mathrm{law}(X_{\pi})=\mathrm{pr}^{1}_{\#}\kappa_{2d}(\pi)=\kappa_{d}(\pi\circ\iota^{1})=\kappa_{d}(\mu)\in\kappa_{d}(\Sigma_{d,R})$, as $\mu\in\mathcal{D}_{\Xi}\subseteq\mathcal{D}\subseteq\Sigma_{d,R}$ by The Wall-Confined Free Energy and Its Score §score and The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation §metric. Next, by Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples and 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, $\lVert P_{\pi}\rVert_{2}^{2}=\sum_{j=1}^{d}\lVert\widehat{x_{d+j}}\rVert^{2}=\sum_{j=1}^{d}\pi(x_{d+j}x_{d+j})=|\pi|_{\mathrm{mom}}^{2}$, so $\lVert P_{\pi}\rVert_{2}=|\pi|_{\mathrm{mom}}$ by the uniqueness in Existence and Uniqueness of the Nonnegative Square Root (Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §plans). Let $Q=V^{1}_{\pi}\Xi(\mu)$, an $L^{2}$ $d$-tuple of $(\mathcal{H}_{\pi},\mathcal{M}_{\pi},\Omega_{\pi})$ 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. As $(V^{1}_{\pi})^{*}V^{1}_{\pi}=I$ by Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §isometries and Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §isometry, $\lVert V^{1}_{\pi}\zeta_{j}\rVert^{2}=\langle\zeta_{j},(V^{1}_{\pi})^{*}V^{1}_{\pi}\zeta_{j}\rangle=\lVert\zeta_{j}\rVert^{2}$ for the components $\zeta_{j}$ of $\Xi(\mu)$, so $\lVert Q\rVert_{2}=\lVert-Q\rVert_{2}=\lVert\Xi(\mu)\rVert_{2}$. By The Shift of a Bounded Plan by a Self-Adjoint Field §shift and Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts, $\mathcal{H}(\pi\oplus t\,\Xi(\mu))=\mathcal{H}_{\mathcal{M}_{\pi}}(X_{\pi},P_{\pi}+tQ)$, $\mathcal{H}(\pi\oplus(-t)\,\Xi(\mu))=\mathcal{H}_{\mathcal{M}_{\pi}}(X_{\pi},P_{\pi}+t(-Q))$ (componentwise $(-t)Q=t(-Q)$) and $\mathcal{H}(\kappa_{2d}(\pi))=\mathcal{H}_{\mathcal{M}_{\pi}}(X_{\pi},P_{\pi})$. The absorption inequality of Absorption of Shifts by a Hamiltonian on Phase-Space Noncommutative Laws §absorption, applied in $(\mathcal{H}_{\pi},\mathcal{M}_{\pi},\Omega_{\pi})$ with $X_{\pi}$, $P_{\pi}$, the level $t$ and $Q$, respectively $-Q$, in the roles of $X$, $P$, $\delta$ and $Q$, now gives both inequalities of the claim.

Order of choices. Before $\delta$, $u$ and $v$ are given, the following are fixed: $e$, $C$ and $\delta_{*}$, and, for every $j\in\mathbb{N}$, $r_{j}$, $\tau_{j}$, $N_{j}$ and $R'_{j}$ (Step 1), together with $(\delta_{\mathrm{a}},\omega_{\mathrm{a}})$ of (F6); they determine $\epsilon$ (Step 1), which therefore depends on $b$, $\delta_{0}$ and the data of the setting only. Then $\delta$, $u$, $v$ and a law $\mu_{0}$ are given (Step 2), and an index $j$ is fixed (Step 2); then the index $\ell$ and the strength $\alpha$ (Step 3); $r_{\mathrm{m}}$ (Step 4); $c$, then the tolerance $\theta$, then $(\hat{\mu},\hat{\nu})$, $(\mu_{k})$, $(\nu_{k})$ and $(\varepsilon_{k})$ (Step 5); then $\gamma$, $(\gamma_{k})$ and $(H_{1},M_{1},\Omega_{1})$, $s$, $(t^{k})$ (Step 7); finally $(\gamma'_{k})$ and $(H_{2},M_{2},\Omega_{2})$, $s'$, $(t'^{k})$ (Step 8).

Step 1 (The function $\epsilon$). By The Wall-Confined Free Energy: Norm Bound, Lower Bound, Weak-Star Compact Sublevel Sets and Displacement Monotonicity of the Score §bounds fix a real $e$ with $e\le\mathcal{E}(\lambda)$ for every $\lambda\in\mathcal{D}$. Put $C=2b+2\delta_{0}|e|$, a real number with $C\ge0$, and $\delta_{*}=\min\{\delta_{0},\delta_{\mathrm{a}}\}>0$, with $(\delta_{\mathrm{a}},\omega_{\mathrm{a}})$ as in (F6). For every $j\in\mathbb{N}$, the structure condition at bounded positions, applied with the radius $R$ and with $\eta=1/j>0$, gives a real $r_{j}>0$, which we fix, such that, for every tracial W*-probability space $(H_{1},M_{1},\Omega_{1})$, all $L^{2}$ $d$-tuples $X,Y$ of it with $\mathrm{law}(X)\in\kappa_{d}(\Sigma_{d,R})$ and $\mathrm{law}(Y)\in\kappa_{d}(\Sigma_{d,R})$, and every real $\alpha>0$ with $\alpha\lVert X-Y\rVert_{2}^{2}+\lVert X-Y\rVert_{2}<r_{j}$,

$$\mathcal{H}_{M_{1}}\bigl(Y,\alpha(X-Y)\bigr)-\mathcal{H}_{M_{1}}\bigl(X,\alpha(X-Y)\bigr)<\tfrac{1}{j}.$$

Put $\tau_{j}=\min\{r_{j}/8,\,r_{j}^{2}/16\}>0$; by claim 1 (unboundedness) of The Archimedean Property of the Real Numbers fix $N_{j}\in\mathbb{N}$ with $N_{j}>2C/\tau_{j}$, so that $C/N_{j}<\tau_{j}/2$; and put $R'_{j}=R+2(2^{N_{j}+1}+2)R\sqrt{d}$. Now define $\epsilon$ on $(0,\delta_{0}]$ as follows. For $0<\delta\le\delta_{*}$ let

$$A_{\delta}=\Bigl\{\frac{5}{\rho j}+\frac{2\delta}{\rho}\,\omega_{\mathrm{a}}(R'_{j})\ :\ j\in\mathbb{N}\Bigr\},\qquad\epsilon(\delta)=2\delta|e|+\inf A_{\delta};$$

$A_{\delta}$ is nonempty and each of its elements is nonnegative (as $\rho>0$, $\delta>0$ and $\omega_{\mathrm{a}}\ge0$), so $\inf A_{\delta}$ exists by Existence of the Infimum of a Nonempty Subset of $\mathbb{R}$ Bounded Below, and $\inf A_{\delta}\ge0$ because $0$ is a lower bound of $A_{\delta}$ and $\inf A_{\delta}$ is the greatest one. For $\delta_{*}<\delta\le\delta_{0}$ let $\epsilon(\delta)=2b+2\delta|e|$. In both cases $\epsilon(\delta)\ge0$. By construction $\epsilon$ is determined by $b$, $\delta_{0}$, $e$, $(\delta_{\mathrm{a}},\omega_{\mathrm{a}})$, $\rho$, $R$, $d$ and the numbers $r_{j}$, none of which depends on the functions $u$ and $v$ of the statement.

The limit. Let $\eta>0$ be real. By claim 1 of The Archimedean Property of the Real Numbers choose $j\in\mathbb{N}$ with $j>15/(\rho\eta)$, so that $5/(\rho j)<\eta/3$, and let

$$\delta_{1}=\min\Bigl\{\delta_{*},\ \frac{\rho\eta}{6(\omega_{\mathrm{a}}(R'_{j})+1)},\ \frac{\eta}{6(|e|+1)}\Bigr\}>0.$$

Let $\delta$ lie in $(0,\delta_{0}]$ with $\delta<\delta_{1}$. Then $\delta<\delta_{*}$, so, as $\inf A_{\delta}$ is at most the element of $A_{\delta}$ indexed by $j$, $\epsilon(\delta)\le2\delta|e|+5/(\rho j)+(2\delta/\rho)\omega_{\mathrm{a}}(R'_{j})$; here $2\delta|e|\le\eta|e|/(3(|e|+1))<\eta/3$ and $(2\delta/\rho)\omega_{\mathrm{a}}(R'_{j})\le\eta\,\omega_{\mathrm{a}}(R'_{j})/(3(\omega_{\mathrm{a}}(R'_{j})+1))<\eta/3$. Hence $\epsilon(\delta)<\eta$, which is the required limit property. It remains to prove that, with this $\epsilon$, the displayed inequality of the statement holds for every $\delta$, $u$, $v$ and $\mu$ as there.

Step 2 (Reduction and the modified pair). Let $\delta$ be real with $0<\delta\le\delta_{0}$, let $u$ be an envelope viscosity subsolution and $v$ an envelope viscosity supersolution of $(\mathrm{E})$ with shift range $\delta_{0}$ and $|u|\le b$, $|v|\le b$ on $\Sigma^{2}_{d}$, let $\mu_{0}\in\mathcal{D}$, and put $m_{0}=u^{-}_{\delta}(\mu_{0})-v^{+}_{\delta}(\mu_{0})$. If $m_{0}\le0$, then $u^{-}_{\delta}(\mu_{0})\le v^{+}_{\delta}(\mu_{0})+\epsilon(\delta)$ as $\epsilon(\delta)\ge0$. By (F5), applied to $u$ and to $v$ with this $\delta$ and $e$, $m_{0}\le(b-\delta e)-(-b+\delta e)=2b-2\delta e\le2b+2\delta|e|$, since $-e\le|-e|=|e|$ by claims 3 and 2 of Properties of the Absolute Value in an Ordered Field; so the inequality also holds at $\mu_{0}$ if $\delta_{*}<\delta$. Assume from now on that $m_{0}>0$ and $\delta\le\delta_{*}$, so that $\delta\le\delta_{0}$ and $\delta\le\delta_{\mathrm{a}}$. We shall prove, for every $j\in\mathbb{N}$, the target inequality

$$m_{0}<2\delta|e|+\frac{5}{\rho j}+\frac{2\delta}{\rho}\,\omega_{\mathrm{a}}(R'_{j}).\tag{T}$$

This suffices: by (T), $m_{0}-2\delta|e|$ is a lower bound of $A_{\delta}$, hence at most $\inf A_{\delta}$, that is, $u^{-}_{\delta}(\mu_{0})\le v^{+}_{\delta}(\mu_{0})+\epsilon(\delta)$.

The modified pair. For $\zeta\in\Sigma^{2}_{d}$, if there is $\lambda\in\mathcal{D}$ with $\kappa_{d}(\lambda)=\zeta$ (such $\lambda$ is unique by (F4)) put $\tilde{u}(\zeta)=u^{-}_{\delta}(\lambda)+\delta\mathcal{E}(\lambda)$ and $\tilde{v}(\zeta)=v^{+}_{\delta}(\lambda)-\delta\mathcal{E}(\lambda)$; otherwise put $\tilde{u}(\zeta)=u(\zeta)$ and $\tilde{v}(\zeta)=v(\zeta)$. Thus, for every $\lambda\in\mathcal{D}$,

$$\tilde{u}\bigl(\kappa_{d}(\lambda)\bigr)-\delta\mathcal{E}(\lambda)=u^{-}_{\delta}(\lambda),\qquad\tilde{v}\bigl(\kappa_{d}(\lambda)\bigr)+\delta\mathcal{E}(\lambda)=v^{+}_{\delta}(\lambda).\tag{M}$$

By (F5) with $w=u$, $-b\le u(\kappa_{d}(\lambda))\le\tilde{u}(\kappa_{d}(\lambda))\le b$, and by (F5) with $w=v$, $-b\le\tilde{v}(\kappa_{d}(\lambda))\le v(\kappa_{d}(\lambda))\le b$, for every $\lambda\in\mathcal{D}$; off $\kappa_{d}(\mathcal{D})$, $\tilde{u}=u$ and $\tilde{v}=v$. Hence $|\tilde{u}|\le b$ and $|\tilde{v}|\le b$ on $\Sigma^{2}_{d}$, and the hypotheses of Perturbed Maximisers of the Doubled Difference Penalised by the Wall-Confined Free Energy hold with $\tilde{u}$, $\tilde{v}$, $b$ and $e$ in the roles of $u$, $v$, $b$ and $e$; let $\Psi_{\delta',\alpha}$ and $S(\delta',\alpha)$ be as there for this pair. By (M), for all $\mu,\nu\in\mathcal{D}$ and real $\alpha>0$,

$$\Psi_{\delta,\alpha}(\mu,\nu)=u^{-}_{\delta}(\mu)-v^{+}_{\delta}(\nu)-\tfrac{\alpha}{2}W_{2}(\mu,\nu)^{2}.$$

As $W_{2}(\mu_{0},\mu_{0})=0$ by Metric Space, $\Psi_{\delta,\alpha}(\mu_{0},\mu_{0})=m_{0}$, so $S(\delta,\alpha)\ge m_{0}$ for every real $\alpha>0$, $S(\delta,\alpha)$ being the supremum of the values of $\Psi_{\delta,\alpha}$ (a real number by Perturbed Maximisers of the Doubled Difference Penalised by the Wall-Confined Free Energy §bounds). Finally, by (M), the function $\mu\mapsto\tilde{u}(\kappa_{d}(\mu))-\delta\mathcal{E}(\mu)$ on $\mathcal{D}$ is $u^{-}_{\delta}$, which is upper semicontinuous on $\mathcal{D}$ by Properties of the Upper Semicontinuous Envelope §usc, and $\nu\mapsto\tilde{v}(\kappa_{d}(\nu))+\delta\mathcal{E}(\nu)$ is $v^{+}_{\delta}$, which is lower semicontinuous on $\mathcal{D}$ by Properties of the Lower Semicontinuous Envelope, by Duality §lsc; so Perturbed Maximisers of the Doubled Difference Penalised by the Wall-Confined Free Energy §perturbed applies to the pair $(\tilde{u},\tilde{v})$ at this $\delta$. For the rest of the proof fix $j\in\mathbb{N}$; we prove (T).

Step 3 (Choice of the strength by pigeonhole). Write $N=N_{j}$, $r_{\mathrm{s}}=r_{j}$, $\tau=\tau_{j}$ and $R'=R'_{j}$ (Step 1). For $i\in[N+1]$ let $q_{i}=S(\delta,2^{i})$, a real number by Perturbed Maximisers of the Doubled Difference Penalised by the Wall-Confined Free Energy §bounds. By the same clause every value of $\Psi_{\delta,2}$ is at most $2b-2\delta e$, so $q_{1}\le2b-2\delta e\le2b+2\delta_{0}|e|=C$ (as $-e\le|e|$, Step 2, and $0<\delta\le\delta_{0}$); and $q_{N+1}\ge m_{0}>0$ by Step 2. Hence the telescoping sum satisfies

$$\sum_{i=1}^{N}(q_{i}-q_{i+1})=q_{1}-q_{N+1}\le C.$$

If $q_{i}-q_{i+1}>C/N$ held for every $i\in[N]$, the sum would exceed $N\cdot(C/N)=C$; so we may fix $\ell\in[N]$ with $q_{\ell}-q_{\ell+1}\le C/N<\tau/2$ (Step 1). Put $\alpha=2^{\ell+1}$. Then $\alpha/2=2^{\ell}$, so $q_{\ell}=S(\delta,\alpha/2)$ and $q_{\ell+1}=S(\delta,\alpha)$; and, as $1\le\ell\le N$ and $i\mapsto2^{i}$ is nondecreasing on $\mathbb{N}$ (since $2^{i+1}=2^{i}+2^{i}\ge2^{i}$ for every $i\in\mathbb{N}$), $\alpha=2^{\ell+1}\ge2^{2}=4$ and $\alpha\le2^{N+1}$. Together with Step 2,

$$S(\delta,\alpha/2)-S(\delta,\alpha)<\tfrac{\tau}{2},\qquad S(\delta,\alpha)\ge m_{0},\qquad2\le\alpha\le2^{N+1}.$$

Step 4 (The momentum radius and the choice of $r_{\mathrm{m}}$). Recall $R'=R'_{j}=R+2(2^{N+1}+2)R\sqrt{d}$; as $\alpha\le2^{N+1}$ (Step 3), $R\le R'$ and $2\alpha R\sqrt{d}+4R\sqrt{d}\le R'$. Since $\mathcal{H}$ is uniformly continuous in the momentum at bounded positions, applying this with the radius $R'$ and with $\eta=1/j>0$ gives a real $r_{\mathrm{m}}>0$ such that, for $i\in\{1,2\}$, every tracial W*-probability space $(H_{i},M_{i},\Omega_{i})$, every $L^{2}$ $d$-tuple $X$ of it with $\mathrm{law}(X)\in\kappa_{d}(\Sigma_{d,R'})$, and all $L^{2}$ $d$-tuples $P_{\flat},P_{\sharp}$ of it with $\lVert P_{\flat}\rVert_{2}\le R'$, $\lVert P_{\sharp}\rVert_{2}\le R'$ and $\lVert P_{\flat}-P_{\sharp}\rVert_{2}<r_{\mathrm{m}}$,

$$\bigl|\mathcal{H}_{M_{i}}(X,P_{\flat})-\mathcal{H}_{M_{i}}(X,P_{\sharp})\bigr|<\tfrac{1}{j}.$$

Step 5 (Choice of the tolerance and the perturbed pair). By Perturbed Maximisers of the Doubled Difference Penalised by the Wall-Confined Free Energy §perturbed, applied to the pair $(\tilde{u},\tilde{v})$ at the level $\delta$ (its hypotheses hold by Step 2), fix a real $c$ as there. Let

$$\theta=\min\Bigl\{\tfrac{1}{2},\ \tfrac{\tau}{2},\ \frac{1}{\rho j},\ \frac{1}{2j(\sigma^{2}+1)(|c-e|+1)},\ \frac{r_{\mathrm{m}}}{8R\sqrt{d}}\Bigr\}.$$

Then $0<\theta<1$, $\theta\le\tau/2$, $\rho\theta\le1/j$, $2\sigma^{2}|c-e|\,\theta\le1/j$ and $4R\sqrt{d}\,\theta<r_{\mathrm{m}}$. By the same clause, applied with $\alpha$ and $\theta$, fix $(\hat{\mu},\hat{\nu})\in\mathcal{D}\times\mathcal{D}$, sequences $(\mu_{k})_{k\in\mathbb{N}}$, $(\nu_{k})_{k\in\mathbb{N}}$ in $\mathcal{D}$ and positive reals $\varepsilon_{k}$ as there, and let $\Phi$ be as there; thus $\sum_{k=1}^{\infty}\varepsilon_{k}\le\theta$, all of $\mathcal{E}(\hat{\mu})$, $\mathcal{E}(\hat{\nu})$, $\mathcal{E}(\mu_{k})$, $\mathcal{E}(\nu_{k})$ are at most $c$, $\Psi_{\delta,\alpha}(\hat{\mu},\hat{\nu})\ge S(\delta,\alpha)-\theta$, and $\Phi(\mu,\nu)<\Phi(\hat{\mu},\hat{\nu})$ for every $(\mu,\nu)\in\mathcal{D}\times\mathcal{D}$ other than $(\hat{\mu},\hat{\nu})$. Put $W=W_{2}(\hat{\mu},\hat{\nu})$. All these laws lie in $\mathcal{D}\subseteq\Sigma_{d,R}$. Since $e\le\mathcal{E}(\hat{\mu})\le c$, we have $c-e\ge0$, hence $2\sigma^{2}(c-e)\theta\le1/j$.

For $(\mu,\nu)\in\mathcal{D}\times\mathcal{D}$ the series $\sum_{k}\varepsilon_{k}W_{2}(\mu,\mu_{k})^{2}$ and $\sum_{k}\varepsilon_{k}W_{2}(\nu,\nu_{k})^{2}$ have terms in $[0,4dR^{2}\varepsilon_{k}]$ by (F1), so they converge by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §comparison, compared with $\sum_{k}4dR^{2}\varepsilon_{k}$, which converges by the scalar-multiple part of Elementary Properties of Series of Real Numbers §linearity (applied with both of its series taken to be the convergent $\sum_{k}\varepsilon_{k}$); and, as the sum of two convergent series is the series of the termwise sums by Elementary Properties of Series of Real Numbers §linearity,

$$\Phi(\mu,\nu)=\Psi_{\delta,\alpha}(\mu,\nu)-\sum_{k=1}^{\infty}\varepsilon_{k}W_{2}(\mu,\mu_{k})^{2}-\sum_{k=1}^{\infty}\varepsilon_{k}W_{2}(\nu,\nu_{k})^{2}.$$

Likewise, by (F1) and Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §comparison, compared with $\sum_{k}2R\sqrt{d}\,\varepsilon_{k}$ (convergent by Elementary Properties of Series of Real Numbers §linearity as before), the series $\sum_{k}\varepsilon_{k}W_{2}(\hat{\mu},\mu_{k})$ and $\sum_{k}\varepsilon_{k}W_{2}(\hat{\nu},\nu_{k})$ converge; and by Elementary Properties of Series of Real Numbers §order together with the scalar-multiple identity of Elementary Properties of Series of Real Numbers §linearity, both sums are at most $\sum_{k}2R\sqrt{d}\,\varepsilon_{k}=2R\sqrt{d}\sum_{k}\varepsilon_{k}\le2R\sqrt{d}\,\theta$, the last step by claim 5 of Elementary Arithmetic in an Ordered Field as $0\le2R\sqrt{d}$.

Step 6 (Size of $W$). By Perturbed Maximisers of the Doubled Difference Penalised by the Wall-Confined Free Energy §strength, applied with $\delta$, $\theta$, $\alpha$, $\alpha'=\alpha/2$ and the pair $(\hat{\mu},\hat{\nu})$ (recall $\Psi_{\delta,\alpha}(\hat{\mu},\hat{\nu})\ge S(\delta,\alpha)-\theta$), and as $(\alpha-\alpha/2)/2=\alpha/4$,

$$S(\delta,\alpha)-\theta+\tfrac{\alpha}{4}W^{2}\le S(\delta,\alpha/2).$$

Hence, by Step 3 and Step 5, $\frac{\alpha}{4}W^{2}\le S(\delta,\alpha/2)-S(\delta,\alpha)+\theta<\tau/2+\tau/2=\tau$. Consequently $\alpha W^{2}<4\tau\le r_{\mathrm{s}}/2$, and, as $\alpha\ge2$, $W^{2}<2\tau\le r_{\mathrm{s}}^{2}/8<r_{\mathrm{s}}^{2}/4$, so $W<r_{\mathrm{s}}/2$ since $W\ge0$. Hence

$$\alpha W^{2}+W<r_{\mathrm{s}},\qquad 0\le W\le2R\sqrt{d},$$

the second by (F1).

Step 7 (Subsolution test at $\hat{\mu}$). By The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §attained (with radius $R$) fix an optimal coupling $\gamma\in\Pi(\hat{\mu},\hat{\nu})$ and, for every $k\in\mathbb{N}$, an optimal coupling $\gamma_{k}\in\Pi(\hat{\mu},\mu_{k})$. Put $\chi^{1}=\hat{\nu}$, $\gamma^{1}=\gamma$, $a_{1}=\alpha/2$, and $\chi^{k+1}=\mu_{k}$, $\gamma^{k+1}=\gamma_{k}$, $a_{k+1}=\varepsilon_{k}$ for $k\in\mathbb{N}$. Then each $\chi^{k}\in\Sigma_{d,R}$, each $\gamma^{k}\in\Pi(\hat{\mu},\chi^{k})$ is optimal, and $(a_{k})$ is a sequence of nonnegative reals whose series converges, with $\sum_{k}a_{k}=\alpha/2+\sum_{k}\varepsilon_{k}$ by Shifting the Index of a Series of Real Numbers §shift (as $a_{k+1}=\varepsilon_{k}$). By Gluing Countably Many Noncommutative Couplings with a Common First Marginal in One Tracial W*-Probability Space §glue, applied with $R$, $\hat{\mu}$, $(\chi^{k})$ and $(\gamma^{k})$ in the roles of $R$, $\mu$, $(\nu_{k})$ and $(\gamma_{k})$, fix a tracial W*-probability space $(H_{1},M_{1},\Omega_{1})$ and self-adjoint $d$-tuples $s$ and $t^{k}$ ($k\in\mathbb{N}$) in $M_{1}$ with $\lVert s_{i}\rVert_{\mathrm{op}}\le R$ and $\lVert t^{k}_{i}\rVert_{\mathrm{op}}\le R$ for all $i\in[d]$ and $k\in\mathbb{N}$, $\lambda_{s}=\hat{\mu}$, $\lambda_{(s,t^{k})}=\gamma^{k}$ and $\lambda_{t^{k}}=\chi^{k}$. Apply Plan Jets of a Series of Squared Wasserstein Distances at Optimal Couplings of Bounded Noncommutative Laws with $R$, $\hat{\mu}$, $(\chi^{k})$, $(\gamma^{k})$, $(a_{k})$, $(H_{1},M_{1},\Omega_{1})$, $s$ and $(t^{k})$ in the roles of $R$, $\mu$, $(\nu_{k})$, $(\gamma_{k})$, $(c_{k})$, $(H,M,\Omega)$, $s$ and $(t^{k})$, and let $P$, $\pi=\lambda_{(s,P)}$ and $\varphi$ be as there.

Strict maximum. Let $\nu\in\mathcal{D}$. By Plan Jets of a Series of Squared Wasserstein Distances at Optimal Couplings of Bounded Noncommutative Laws §superjet, the isometry identity and Shifting the Index of a Series of Real Numbers §shift, $\varphi(\kappa_{d}(\nu))=\frac{\alpha}{2}W_{2}(\nu,\hat{\nu})^{2}+\sum_{k}\varepsilon_{k}W_{2}(\nu,\mu_{k})^{2}$. Hence, by Step 5 and (M), with the real constant $C_{1}=\tilde{v}(\kappa_{d}(\hat{\nu}))+\delta\mathcal{E}(\hat{\nu})+\sum_{k}\varepsilon_{k}W_{2}(\hat{\nu},\nu_{k})^{2}$,

$$u^{-}_{\delta}(\nu)-\varphi\bigl(\kappa_{d}(\nu)\bigr)=\tilde{u}\bigl(\kappa_{d}(\nu)\bigr)-\varphi\bigl(\kappa_{d}(\nu)\bigr)-\delta\mathcal{E}(\nu)=\Phi(\nu,\hat{\nu})+C_{1}.$$

If $\nu\ne\hat{\mu}$ then $(\nu,\hat{\nu})\ne(\hat{\mu},\hat{\nu})$, so $\Phi(\nu,\hat{\nu})<\Phi(\hat{\mu},\hat{\nu})$; thus $u^{-}_{\delta}(\nu)-\varphi(\kappa_{d}(\nu))<u^{-}_{\delta}(\hat{\mu})-\varphi(\kappa_{d}(\hat{\mu}))$ for every $\nu\in\mathcal{D}$ with $\nu\ne\hat{\mu}$.

Momentum bounds. By Plan Jets of a Series of Squared Wasserstein Distances at Optimal Couplings of Bounded Noncommutative Laws §momentum, Shifting the Index of a Series of Real Numbers §shift, (F1), Step 5 and Step 4 (as $\theta<1$),

$$\lVert P\Omega_{1}\rVert_{2}\le2\sum_{k=1}^{\infty}a_{k}W_{2}(\hat{\mu},\chi^{k})=\alpha W+2\sum_{k=1}^{\infty}\varepsilon_{k}W_{2}(\hat{\mu},\mu_{k})\le2\alpha R\sqrt{d}+4R\sqrt{d}\,\theta\le R'.$$

Put $X=s\Omega_{1}$ and $Y=t^{1}\Omega_{1}$, $L^{2}$ $d$-tuples of $(H_{1},M_{1},\Omega_{1})$ with $\mathrm{law}(X)=\kappa_{d}(\hat{\mu})$ and $\mathrm{law}(Y)=\kappa_{d}(\hat{\nu})$ by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §bounded. By (F2) with $(a,b)=(s,t^{1})$, $\lVert X-Y\rVert_{2}^{2}=I(\gamma)=W^{2}$ as $\gamma$ is optimal (The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §optimal), so $\lVert X-Y\rVert_{2}=W$ and $\lVert\alpha(X-Y)\rVert_{2}=\alpha W\le2\alpha R\sqrt{d}\le R'$. Since $2a_{1}(s\Omega_{1}-t^{1}\Omega_{1})=\alpha(X-Y)$, the tail bound of Plan Jets of a Series of Squared Wasserstein Distances at Optimal Couplings of Bounded Noncommutative Laws §momentum with $n=1$ and Step 5 give

$$\bigl\lVert P\Omega_{1}-\alpha(X-Y)\bigr\rVert_{2}\le2\sum_{k=1}^{\infty}\varepsilon_{k}W_{2}(\hat{\mu},\mu_{k})\le4R\sqrt{d}\,\theta<r_{\mathrm{m}}.$$

As $\lVert s_{i}\rVert_{\mathrm{op}}\le R\le R'$ for every $i\in[d]$, Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §law gives $\hat{\mu}=\lambda_{s}\in\Sigma_{d,R'}$, so $\mathrm{law}(X)\in\kappa_{d}(\Sigma_{d,R'})$. By (F3) with $(a,p)=(s,P)$ and the choice of $r_{\mathrm{m}}$ in Step 4 (with $i=1$),

$$\mathcal{H}\bigl(\kappa_{2d}(\pi)\bigr)=\mathcal{H}_{M_{1}}(X,P\Omega_{1})>\mathcal{H}_{M_{1}}\bigl(X,\alpha(X-Y)\bigr)-\tfrac{1}{j}.$$

The subsolution inequality. By Plan Jets of a Series of Squared Wasserstein Distances at Optimal Couplings of Bounded Noncommutative Laws §plans, $\pi$ is a bounded plan at $\hat{\mu}$ with $|\pi|_{\mathrm{mom}}=\lVert P\Omega_{1}\rVert_{2}\le R'$, and by Plan Jets of a Series of Squared Wasserstein Distances at Optimal Couplings of Bounded Noncommutative Laws §superjet, $\kappa_{2d}(\pi)\in J^{+}\varphi(\kappa_{d}(\hat{\mu}))$. Now Envelope Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §sub, applied to $u$ with the level $\delta$ (recall $0<\delta\le\delta_{0}$), the test function $\varphi$, the law $\hat{\mu}$ (a strict maximum point of $u^{-}_{\delta}-\varphi\circ\kappa_{d}$ on $\mathcal{D}$ by the strict maximum paragraph) and the plan $\pi$, gives $\hat{\mu}\in\mathcal{D}_{\Xi}$ and

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

By (F6) with $t=\delta$ (recall $0<\delta\le\delta_{\mathrm{a}}$), $\mathcal{H}(\pi\oplus\delta\,\Xi(\hat{\mu}))\ge\mathcal{H}(\kappa_{2d}(\pi))-\delta\,\omega_{\mathrm{a}}(|\pi|_{\mathrm{mom}})-\frac{\sigma^{2}\delta}{4}\lVert\Xi(\hat{\mu})\rVert_{2}^{2}$, and $\omega_{\mathrm{a}}(|\pi|_{\mathrm{mom}})\le\omega_{\mathrm{a}}(R')$ as $\omega_{\mathrm{a}}$ is nondecreasing and $|\pi|_{\mathrm{mom}}\le R'$. Substituting, using $u^{-}_{\delta}(\hat{\mu})+\delta\mathcal{E}(\hat{\mu})=\tilde{u}(\kappa_{d}(\hat{\mu}))$ from (M), and discarding the term $(\frac{\sigma^{2}\delta}{2}-\frac{\sigma^{2}\delta}{4})\lVert\Xi(\hat{\mu})\rVert_{2}^{2}=\frac{\sigma^{2}\delta}{4}\lVert\Xi(\hat{\mu})\rVert_{2}^{2}\ge0$, we obtain the absorbed subsolution inequality

$$\rho\,\tilde{u}\bigl(\kappa_{d}(\hat{\mu})\bigr)+\mathcal{H}\bigl(\kappa_{2d}(\pi)\bigr)+\tfrac{\sigma^{2}}{2}\mathcal{J}\bigl(\Xi(\hat{\mu}),\pi\bigr)\le\delta\,\omega_{\mathrm{a}}(R').$$

By Plan Jets of a Series of Squared Wasserstein Distances at Optimal Couplings of Bounded Noncommutative Laws §pairing, applied to $\zeta=\Xi(\hat{\mu})$, which lies in $\mathcal{H}_{\hat{\mu}}^{d}$ by The Wall-Confined Free Energy and Its Score §score as $\hat{\mu}\in\mathcal{D}_{\Xi}$, and by Shifting the Index of a Series of Real Numbers §shift, $\mathcal{J}(\Xi(\hat{\mu}),\pi)=\alpha\mathcal{J}^{1}_{\gamma}(\Xi(\hat{\mu}))+\sum_{k}2\varepsilon_{k}\mathcal{J}^{1}_{\gamma_{k}}(\Xi(\hat{\mu}))$. For each $k$, Tangent Inequalities for the Wall Energy along Couplings and for the Wall-Confined Free Energy along Optimal Couplings §tangent (with $\hat{\mu}\in\mathcal{D}_{\Xi}$, $\mu_{k}\in\mathcal{D}$ and $\gamma_{k}$ optimal) gives $\mathcal{J}^{1}_{\gamma_{k}}(\Xi(\hat{\mu}))\ge\mathcal{E}(\hat{\mu})-\mathcal{E}(\mu_{k})\ge e-c$. Hence, as $\varepsilon_{k}>0$, the convergent series $\sum_{k}2\varepsilon_{k}\mathcal{J}^{1}_{\gamma_{k}}(\Xi(\hat{\mu}))$ dominates termwise the series $\sum_{k}2(e-c)\varepsilon_{k}$, which converges with sum $2(e-c)\sum_{k}\varepsilon_{k}$ by Elementary Properties of Series of Real Numbers §linearity; so Elementary Properties of Series of Real Numbers §order and, as $e-c\le0$ and $\sum_{k}\varepsilon_{k}\le\theta$, claim 5 of Elementary Arithmetic in an Ordered Field give

$$\mathcal{J}\bigl(\Xi(\hat{\mu}),\pi\bigr)\ge\alpha\mathcal{J}^{1}_{\gamma}\bigl(\Xi(\hat{\mu})\bigr)+2(e-c)\sum_{k=1}^{\infty}\varepsilon_{k}\ge\alpha\mathcal{J}^{1}_{\gamma}\bigl(\Xi(\hat{\mu})\bigr)-2(c-e)\theta.$$

Step 8 (Supersolution test at $\hat{\nu}$). Let $\gamma'=\lambda_{(t^{1},s)}$. By (F2) with $(a,b)=(t^{1},s)$, $\gamma'\in\Pi(\lambda_{t^{1}},\lambda_{s})=\Pi(\hat{\nu},\hat{\mu})$ and $I(\gamma')=\lVert s\Omega_{1}-t^{1}\Omega_{1}\rVert_{2}^{2}=W^{2}=W_{2}(\hat{\nu},\hat{\mu})^{2}$ (Step 7 and The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §symmetry), so $\gamma'$ is optimal. By The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §attained fix, for every $k\in\mathbb{N}$, an optimal coupling $\gamma'_{k}\in\Pi(\hat{\nu},\nu_{k})$. Put $\chi'^{1}=\hat{\mu}$, $\gamma'^{1}=\gamma'$, and $\chi'^{k+1}=\nu_{k}$, $\gamma'^{k+1}=\gamma'_{k}$ for $k\in\mathbb{N}$, and keep the weights $(a_{k})$ of Step 7. By Gluing Countably Many Noncommutative Couplings with a Common First Marginal in One Tracial W*-Probability Space §glue, applied with $R$, $\hat{\nu}$, $(\chi'^{k})$ and $(\gamma'^{k})$, fix a tracial W*-probability space $(H_{2},M_{2},\Omega_{2})$ and self-adjoint $d$-tuples $s'$ and $t'^{k}$ ($k\in\mathbb{N}$) in $M_{2}$ with $\lVert s'_{i}\rVert_{\mathrm{op}}\le R$ and $\lVert t'^{k}_{i}\rVert_{\mathrm{op}}\le R$ for all $i\in[d]$ and $k\in\mathbb{N}$, $\lambda_{s'}=\hat{\nu}$, $\lambda_{(s',t'^{k})}=\gamma'^{k}$ and $\lambda_{t'^{k}}=\chi'^{k}$. Apply Plan Jets of a Series of Squared Wasserstein Distances at Optimal Couplings of Bounded Noncommutative Laws with $R$, $\hat{\nu}$, $(\chi'^{k})$, $(\gamma'^{k})$, $(a_{k})$, $(H_{2},M_{2},\Omega_{2})$, $s'$ and $(t'^{k})$ in the roles of $R$, $\mu$, $(\nu_{k})$, $(\gamma_{k})$, $(c_{k})$, $(H,M,\Omega)$, $s$ and $(t^{k})$, and let $P'$, $\pi'=\lambda_{(s',P')}$, $\pi'^{-}=\lambda_{(s',-P')}$ and $\varphi'$ be the objects called $P$, $\pi$, $\pi^{-}$ and $\varphi$ there.

Strict minimum. Let $\nu\in\mathcal{D}$. As in Step 7, now also using The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §symmetry, $\varphi'(\kappa_{d}(\nu))=\frac{\alpha}{2}W_{2}(\hat{\mu},\nu)^{2}+\sum_{k}\varepsilon_{k}W_{2}(\nu,\nu_{k})^{2}$, so, with the real constant $C_{2}=\tilde{u}(\kappa_{d}(\hat{\mu}))-\delta\mathcal{E}(\hat{\mu})-\sum_{k}\varepsilon_{k}W_{2}(\hat{\mu},\mu_{k})^{2}$, Step 5 and (M),

$$v^{+}_{\delta}(\nu)-(-\varphi')\bigl(\kappa_{d}(\nu)\bigr)=\tilde{v}\bigl(\kappa_{d}(\nu)\bigr)-(-\varphi')\bigl(\kappa_{d}(\nu)\bigr)+\delta\mathcal{E}(\nu)=C_{2}-\Phi(\hat{\mu},\nu).$$

If $\nu\ne\hat{\nu}$ then $\Phi(\hat{\mu},\nu)<\Phi(\hat{\mu},\hat{\nu})$; thus $v^{+}_{\delta}(\nu)-(-\varphi')(\kappa_{d}(\nu))>v^{+}_{\delta}(\hat{\nu})-(-\varphi')(\kappa_{d}(\hat{\nu}))$ for every $\nu\in\mathcal{D}$ with $\nu\ne\hat{\nu}$.

Momentum bounds. As in Step 7, by Plan Jets of a Series of Squared Wasserstein Distances at Optimal Couplings of Bounded Noncommutative Laws §momentum, (F1), The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §symmetry and Step 5, $\lVert P'\Omega_{2}\rVert_{2}\le\alpha W+2\sum_{k}\varepsilon_{k}W_{2}(\hat{\nu},\nu_{k})\le R'$, and $\lVert(-P')\Omega_{2}\rVert_{2}=\lVert P'\Omega_{2}\rVert_{2}$ since $(-P')\Omega_{2}=-(P'\Omega_{2})$ componentwise. Put $X'=s'\Omega_{2}$ and $Q'=\alpha(t'^{1}\Omega_{2}-s'\Omega_{2})$. By (F2) with $(a,b)=(s',t'^{1})$, $\lVert t'^{1}\Omega_{2}-s'\Omega_{2}\rVert_{2}^{2}=I(\gamma')=W^{2}$, so $\lVert Q'\rVert_{2}=\alpha W\le R'$. Componentwise, $(-P')\Omega_{2}-Q'=-\bigl(P'\Omega_{2}-2a_{1}(s'\Omega_{2}-t'^{1}\Omega_{2})\bigr)$, so the tail bound of Plan Jets of a Series of Squared Wasserstein Distances at Optimal Couplings of Bounded Noncommutative Laws §momentum with $n=1$ and Step 5 give $\lVert(-P')\Omega_{2}-Q'\rVert_{2}\le2\sum_{k}\varepsilon_{k}W_{2}(\hat{\nu},\nu_{k})\le4R\sqrt{d}\,\theta<r_{\mathrm{m}}$. As in Step 7, $\mathrm{law}(X')=\kappa_{d}(\hat{\nu})\in\kappa_{d}(\Sigma_{d,R'})$. By (F3) with $(a,p)=(s',-P')$ and Step 4 (with $i=2$), $\mathcal{H}(\kappa_{2d}(\pi'^{-}))=\mathcal{H}_{M_{2}}(X',(-P')\Omega_{2})<\mathcal{H}_{M_{2}}(X',Q')+1/j$. Finally, $\lambda_{(s',t'^{1})}=\gamma'=\lambda_{(t^{1},s)}$, so (F2), applied in $(H_{2},M_{2},\Omega_{2})$ with $(a,b)=(s',t'^{1})$ and in $(H_{1},M_{1},\Omega_{1})$ with $(a,b)=(t^{1},s)$, gives $\mathcal{H}_{M_{2}}(X',Q')=\mathcal{H}(\kappa_{2d}(\gamma'\circ\sigma_{T_{\alpha}}))=\mathcal{H}_{M_{1}}(Y,\alpha(X-Y))$, with $X,Y$ as in Step 7. Hence

$$\mathcal{H}\bigl(\kappa_{2d}(\pi'^{-})\bigr)<\mathcal{H}_{M_{1}}\bigl(Y,\alpha(X-Y)\bigr)+\tfrac{1}{j}.$$

The supersolution inequality. By Plan Jets of a Series of Squared Wasserstein Distances at Optimal Couplings of Bounded Noncommutative Laws §plans, $\pi'^{-}$ is a bounded plan at $\hat{\nu}$ with $|\pi'^{-}|_{\mathrm{mom}}=\lVert P'\Omega_{2}\rVert_{2}\le R'$, and by Plan Jets of a Series of Squared Wasserstein Distances at Optimal Couplings of Bounded Noncommutative Laws §subjet, $\kappa_{2d}(\pi'^{-})\in J^{-}(-\varphi')(\kappa_{d}(\hat{\nu}))$. Now Envelope Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §super, applied to $v$ with the level $\delta$ (recall $0<\delta\le\delta_{0}$), the test function $-\varphi'$, the law $\hat{\nu}$ (a strict minimum point of $v^{+}_{\delta}-(-\varphi')\circ\kappa_{d}$ on $\mathcal{D}$ by the strict minimum paragraph) and the plan $\pi'^{-}$, gives $\hat{\nu}\in\mathcal{D}_{\Xi}$ and

$$\rho\bigl(v^{+}_{\delta}(\hat{\nu})-\delta\,\mathcal{E}(\hat{\nu})\bigr)+\mathcal{H}\bigl(\pi'^{-}\oplus(-\delta)\,\Xi(\hat{\nu})\bigr)+\tfrac{\sigma^{2}}{2}\Bigl(\mathcal{J}\bigl(\Xi(\hat{\nu}),\pi'^{-}\bigr)-\delta\,\lVert\Xi(\hat{\nu})\rVert_{2}^{2}\Bigr)\ge0.$$

By (F6) with $t=\delta$, $\mathcal{H}(\pi'^{-}\oplus(-\delta)\,\Xi(\hat{\nu}))\le\mathcal{H}(\kappa_{2d}(\pi'^{-}))+\delta\,\omega_{\mathrm{a}}(|\pi'^{-}|_{\mathrm{mom}})+\frac{\sigma^{2}\delta}{4}\lVert\Xi(\hat{\nu})\rVert_{2}^{2}$, and $\omega_{\mathrm{a}}(|\pi'^{-}|_{\mathrm{mom}})\le\omega_{\mathrm{a}}(R')$ as $|\pi'^{-}|_{\mathrm{mom}}\le R'$. Substituting, using $v^{+}_{\delta}(\hat{\nu})-\delta\mathcal{E}(\hat{\nu})=\tilde{v}(\kappa_{d}(\hat{\nu}))$ from (M), and $\frac{\sigma^{2}\delta}{2}-\frac{\sigma^{2}\delta}{4}=\frac{\sigma^{2}\delta}{4}$, we obtain the absorbed supersolution inequality (the last step as $\frac{\sigma^{2}\delta}{4}\lVert\Xi(\hat{\nu})\rVert_{2}^{2}\ge0$)

$$\rho\,\tilde{v}\bigl(\kappa_{d}(\hat{\nu})\bigr)+\mathcal{H}\bigl(\kappa_{2d}(\pi'^{-})\bigr)+\tfrac{\sigma^{2}}{2}\mathcal{J}\bigl(\Xi(\hat{\nu}),\pi'^{-}\bigr)\ge-\delta\,\omega_{\mathrm{a}}(R')+\tfrac{\sigma^{2}\delta}{4}\lVert\Xi(\hat{\nu})\rVert_{2}^{2}\ge-\delta\,\omega_{\mathrm{a}}(R').$$

By Plan Jets of a Series of Squared Wasserstein Distances at Optimal Couplings of Bounded Noncommutative Laws §pairing, applied to $\zeta=\Xi(\hat{\nu})$, which lies in $\mathcal{H}_{\hat{\nu}}^{d}$ by The Wall-Confined Free Energy and Its Score §score as $\hat{\nu}\in\mathcal{D}_{\Xi}$, and by Shifting the Index of a Series of Real Numbers §shift, $\mathcal{J}(\Xi(\hat{\nu}),\pi'^{-})=-\alpha\mathcal{J}^{1}_{\gamma'}(\Xi(\hat{\nu}))-\sum_{k}2\varepsilon_{k}\mathcal{J}^{1}_{\gamma'_{k}}(\Xi(\hat{\nu}))$, and, exactly as in Step 7, Tangent Inequalities for the Wall Energy along Couplings and for the Wall-Confined Free Energy along Optimal Couplings §tangent (with $\hat{\nu}\in\mathcal{D}_{\Xi}$, $\nu_{k}\in\mathcal{D}$ and $\gamma'_{k}$ optimal) gives $\mathcal{J}^{1}_{\gamma'_{k}}(\Xi(\hat{\nu}))\ge e-c$, so, by the same use of Elementary Properties of Series of Real Numbers §order and Elementary Properties of Series of Real Numbers §linearity as in Step 7,

$$\mathcal{J}\bigl(\Xi(\hat{\nu}),\pi'^{-}\bigr)\le-\alpha\mathcal{J}^{1}_{\gamma'}\bigl(\Xi(\hat{\nu})\bigr)+2(c-e)\theta.$$

Step 9 (Upper bound). Subtracting the absorbed supersolution inequality of Step 8 from the absorbed subsolution inequality of Step 7,

$$\rho\bigl(\tilde{u}(\kappa_{d}(\hat{\mu}))-\tilde{v}(\kappa_{d}(\hat{\nu}))\bigr)\le\mathcal{H}\bigl(\kappa_{2d}(\pi'^{-})\bigr)-\mathcal{H}\bigl(\kappa_{2d}(\pi)\bigr)-\tfrac{\sigma^{2}}{2}\Bigl(\mathcal{J}\bigl(\Xi(\hat{\mu}),\pi\bigr)-\mathcal{J}\bigl(\Xi(\hat{\nu}),\pi'^{-}\bigr)\Bigr)+2\delta\,\omega_{\mathrm{a}}(R').$$

By the pairing estimates of Steps 7 and 8, the bracket is at least $\alpha\bigl(\mathcal{J}^{1}_{\gamma}(\Xi(\hat{\mu}))+\mathcal{J}^{1}_{\gamma'}(\Xi(\hat{\nu}))\bigr)-4(c-e)\theta$. By Tangent Inequalities for the Wall Energy along Couplings and for the Wall-Confined Free Energy along Optimal Couplings §tangent, applied to $(\hat{\mu},\hat{\nu},\gamma)$ and to $(\hat{\nu},\hat{\mu},\gamma')$ (both base points lie in $\mathcal{D}_{\Xi}$ by Steps 7 and 8, and $\gamma,\gamma'$ are optimal), $\mathcal{J}^{1}_{\gamma}(\Xi(\hat{\mu}))\ge\mathcal{E}(\hat{\mu})-\mathcal{E}(\hat{\nu})$ and $\mathcal{J}^{1}_{\gamma'}(\Xi(\hat{\nu}))\ge\mathcal{E}(\hat{\nu})-\mathcal{E}(\hat{\mu})$, so the sum of the two is nonnegative. Hence, as $\sigma^{2}\ge0$ and $2\sigma^{2}(c-e)\theta\le1/j$ (Step 5), the pairing term $-\frac{\sigma^{2}}{2}(\cdots)$ is at most $2\sigma^{2}(c-e)\theta\le1/j$. By the Hamiltonian estimates of Steps 7 and 8,

$$\mathcal{H}\bigl(\kappa_{2d}(\pi'^{-})\bigr)-\mathcal{H}\bigl(\kappa_{2d}(\pi)\bigr)<\mathcal{H}_{M_{1}}\bigl(Y,\alpha(X-Y)\bigr)-\mathcal{H}_{M_{1}}\bigl(X,\alpha(X-Y)\bigr)+\tfrac{2}{j}<\tfrac{1}{j}+\tfrac{2}{j},$$

the last by the choice of $r_{\mathrm{s}}=r_{j}$ in Step 1, since $\mathrm{law}(X)=\kappa_{d}(\hat{\mu})$ and $\mathrm{law}(Y)=\kappa_{d}(\hat{\nu})$ lie in $\kappa_{d}(\Sigma_{d,R})$ and $\alpha\lVert X-Y\rVert_{2}^{2}+\lVert X-Y\rVert_{2}=\alpha W^{2}+W<r_{\mathrm{s}}$ by Steps 7 and 6. Therefore

$$\rho\bigl(\tilde{u}(\kappa_{d}(\hat{\mu}))-\tilde{v}(\kappa_{d}(\hat{\nu}))\bigr)<\tfrac{4}{j}+2\delta\,\omega_{\mathrm{a}}(R').$$

Step 10 (Lower bound and conclusion). Here $-e\le|e|$ (Step 2), so, as $0\le2\delta$, claim 5 of Elementary Arithmetic in an Ordered Field and claim 2 of Zero Products and Elementary Identities in a Field give $2\delta e=-(2\delta(-e))\ge-2\delta|e|$. Since $\Psi_{\delta,\alpha}(\hat{\mu},\hat{\nu})\ge S(\delta,\alpha)-\theta$ (Step 5), $W^{2}\ge0$, $\mathcal{E}(\hat{\mu}),\mathcal{E}(\hat{\nu})\ge e$ (Step 1) and $S(\delta,\alpha)\ge m_{0}$ (Step 3),

$$\tilde{u}\bigl(\kappa_{d}(\hat{\mu})\bigr)-\tilde{v}\bigl(\kappa_{d}(\hat{\nu})\bigr)=\Psi_{\delta,\alpha}(\hat{\mu},\hat{\nu})+\tfrac{\alpha}{2}W^{2}+\delta\mathcal{E}(\hat{\mu})+\delta\mathcal{E}(\hat{\nu})\ge S(\delta,\alpha)-\theta+2\delta e\ge m_{0}-\theta-2\delta|e|.$$

Multiplying by $\rho>0$, combining with Step 9 and using $\rho\theta\le1/j$ (Step 5),

$$\rho m_{0}<\rho\theta+2\rho\delta|e|+\tfrac{4}{j}+2\delta\,\omega_{\mathrm{a}}(R')\le\tfrac{5}{j}+2\rho\delta|e|+2\delta\,\omega_{\mathrm{a}}(R'_{j}),$$

and dividing by $\rho>0$ gives (T). As $j\in\mathbb{N}$ was arbitrary, Step 2 yields $u^{-}_{\delta}(\mu_{0})\le v^{+}_{\delta}(\mu_{0})+\epsilon(\delta)$; as $\mu_{0}\in\mathcal{D}$, $\delta\in(0,\delta_{0}]$ and $u$, $v$ were arbitrary, and $\epsilon$ was fixed in Step 1 before them, the lemma is proved.