Proof of Perturbed Maximisers of the Doubled Difference Penalised by the Wall-Confined Free Energy

Each result cited is universally quantified over the data in its own statement. Throughout, $R>0$ is the wall radius of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §parameters and $\mathbb{N}=\{1,2,3,\dots\}$ as in Natural Numbers. For reals $\delta\ge0$ and $\alpha>0$ let $V(\delta,\alpha)=\{\Psi_{\delta,\alpha}(\mu,\nu):\ (\mu,\nu)\in\mathcal{D}\times\mathcal{D}\}$, so that $S(\delta,\alpha)=\sup V(\delta,\alpha)$.

Step 1 (Preliminaries). (a) By The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §free-energy, $\mathcal{D}$ and $\mathcal{E}$ are the domain and the energy of The Wall-Confined Free Energy: Norm Bound, Lower Bound, Weak-Star Compact Sublevel Sets and Displacement Monotonicity of the Score for the free entropy penalty $(\mathcal{D}_{0},\mathcal{E}_{0})$ and the radius $R$. Hence $\mathcal{D}\subseteq\Sigma_{d,R}$ by The Wall-Confined Free Energy: Norm Bound, Lower Bound, Weak-Star Compact Sublevel Sets and Displacement Monotonicity of the Score §bounds, and $\Sigma_{d,R}\subseteq\Sigma_{d}$ by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law.

(b) For $\mu,\nu\in\mathcal{D}$ the number $W_{2}(\mu,\nu)$ is a nonnegative square root by The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §distance, so $W_{2}(\mu,\nu)\ge0$, and then $0=W_{2}(\mu,\nu)\cdot0\le W_{2}(\mu,\nu)\cdot W_{2}(\mu,\nu)=W_{2}(\mu,\nu)^{2}$ by claim 1 of Zero Products and Elementary Identities in a Field and claim 5 of Elementary Arithmetic in an Ordered Field. In the same way, whenever $0\le a$ and $0\le x$ are reals, $0=a\cdot0\le ax$ by these two claims; we use this below for products of nonnegative factors.

(c) For $\delta\ge0$ and $\lambda\in\mathcal{D}$ we have $\delta e\le\delta\,\mathcal{E}(\lambda)$ by the choice of $e$ and claim 5 of Elementary Arithmetic in an Ordered Field; and for $\lambda,\lambda'\in\Sigma^{2}_{d}$ we have $u(\lambda)\le|u(\lambda)|\le b$ by claim 3 of Properties of the Absolute Value in an Ordered Field, and $-v(\lambda')\le|-v(\lambda')|=|v(\lambda')|\le b$ by claim 3 of Properties of the Absolute Value in an Ordered Field applied to $-v(\lambda')$ and claim 2 of Properties of the Absolute Value in an Ordered Field. We also record: $0<2$ by claim 8 of Elementary Order Arithmetic in an Ordered Field, so for every real $\alpha>0$ the number $\tfrac{\alpha}{2}=\alpha\cdot2^{-1}$ is positive by that claim 8, and in particular $0\le\tfrac{\alpha}{2}$.

(d) (Squares.) Let $s,s',t$ be reals with $0\le s$, $0\le s'$, $0\le t$ and $s\le s'+t$. Then $s^{2}\le s'^{2}+2st$. Indeed, $0\le2$ by claim 8 of Elementary Order Arithmetic in an Ordered Field, so $0\le2st$ and $0\le2s$ by the product rule of (b). If $s\le s'$, then claim 5 of Elementary Arithmetic in an Ordered Field, applied with the nonnegative factors $s$ and then $s'$, gives $ss\le ss'\le s's'$, so $s^{2}\le s'^{2}\le s'^{2}+2st$. Otherwise $s'<s$, the order being total; then $0\le s-s'$ by claim 3 of Elementary Arithmetic in an Ordered Field, while $s-s'\le t$ and $s+s'\le s+s=2s$ by the compatibility of the order with addition in Ordered Field; so claim 5 of Elementary Arithmetic in an Ordered Field, applied with the nonnegative factors $s-s'$ and then $2s$, gives $s^{2}-s'^{2}=(s-s')(s+s')\le(s-s')\,2s\le t\cdot2s$, that is, $s^{2}\le s'^{2}+2st$.

(e) (A criterion for lower semicontinuity.) Let $(Y,d_{Y})$ be a metric space, $A\subseteq Y$, $\varphi:A\to\mathbb{R}$ and $y\in A$, and suppose that there is a real $a\ge0$ with $\varphi(y)\le\varphi(y')+a\,d_{Y}(y,y')$ for every $y'\in A$. Then $\varphi$ is lower semicontinuous at $y$ relative to $A$ in the sense of Lower Semicontinuous Function on a Subset of a Metric Space. Indeed, let $\varepsilon>0$. Since $0<1$ by claim 6 of Elementary Order Arithmetic in an Ordered Field and $1=0+1\le a+1$ by the compatibility of the order with addition, $0<a+1$ by claim 2 there, so $\eta=\varepsilon(a+1)^{-1}$ is positive by claims 7 and 5 there. Then $a\eta+\eta=\varepsilon$, so $a\eta=\varepsilon-\eta<\varepsilon$ by claims 4 and 1 there. For $y'\in A$ with $d_{Y}(y,y')<\eta$ we get $a\,d_{Y}(y,y')\le a\eta$ by claim 5 of Elementary Arithmetic in an Ordered Field, hence $\varphi(y)-\varepsilon\le\varphi(y')+a\eta-\varepsilon<\varphi(y')$ by claims 1 and 2 of Elementary Order Arithmetic in an Ordered Field.

Step 2 (Clause bounds). Let $\delta\ge0$, $\alpha>0$, $\mu,\nu\in\mathcal{D}$. By Step 1(c), $u(\kappa_{d}(\mu))\le b$, $-v(\kappa_{d}(\nu))\le b$, $-\delta\mathcal{E}(\mu)\le-\delta e$ and $-\delta\mathcal{E}(\nu)\le-\delta e$; by Step 1(b), the positivity of $\tfrac{\alpha}{2}$ recorded in Step 1(c), and the product rule for nonnegative factors of Step 1(b), $\tfrac{\alpha}{2}W_{2}(\mu,\nu)^{2}\ge0$. Adding, $\Psi_{\delta,\alpha}(\mu,\nu)\le2b-2\delta e$. Since $\mathcal{D}\ne\varnothing$, $V(\delta,\alpha)$ is nonempty, and $2b-2\delta e$ is an upper bound for it in the sense of Upper Bound and Least Upper Bound, so $V(\delta,\alpha)$ is bounded above; hence its least upper bound $S(\delta,\alpha)$ exists as a real number, since $\mathbb{R}$ is Dedekind complete by The Real Numbers and Standard Notation. Moreover, for $\mu,\nu\in\mathcal{D}$,

$$\Psi_{\delta,\alpha}(\mu,\nu)=\Psi_{0,\alpha}(\mu,\nu)-\delta\mathcal{E}(\mu)-\delta\mathcal{E}(\nu)\le S(0,\alpha)-2\delta e,$$

because $S(0,\alpha)$ is an upper bound of $V(0,\alpha)$ and by Step 1(c). Thus $S(0,\alpha)-2\delta e$ is an upper bound of $V(\delta,\alpha)$, and the least upper bound satisfies $S(\delta,\alpha)\le S(0,\alpha)-2\delta e$.

Step 3 (Clause diagonal). Let $\alpha>0$ and $\lambda\in\mathcal{D}$. By Step 1(a), $\lambda\in\Sigma_{d,R}$, so $W_{2}(\lambda,\lambda)=0$ by The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §separation. Hence $\Psi_{0,\alpha}(\lambda,\lambda)=u(\kappa_{d}(\lambda))-v(\kappa_{d}(\lambda))$, and this value is at most $S(0,\alpha)$ since it lies in $V(0,\alpha)$.

Step 4 (Clause monotone). Let $0<\alpha'\le\alpha$ and $\mu,\nu\in\mathcal{D}$. Since $\alpha'\le\alpha$, claim 3 of Elementary Arithmetic in an Ordered Field gives $0\le\alpha-\alpha'$; and $0<2^{-1}$ by claims 8 and 7 of Elementary Order Arithmetic in an Ordered Field. Hence $\tfrac{\alpha-\alpha'}{2}=2^{-1}(\alpha-\alpha')\ge0$ by the product rule for nonnegative factors of Step 1(b). As $W_{2}(\mu,\nu)^{2}\ge0$ (Step 1(b)), the same rule gives $\tfrac{\alpha-\alpha'}{2}W_{2}(\mu,\nu)^{2}\ge0$, so

$$\Psi_{0,\alpha}(\mu,\nu)=\Psi_{0,\alpha'}(\mu,\nu)-\tfrac{\alpha-\alpha'}{2}W_{2}(\mu,\nu)^{2}\le\Psi_{0,\alpha'}(\mu,\nu)\le S(0,\alpha').$$

So $S(0,\alpha')$ is an upper bound of $V(0,\alpha)$, whence $S(0,\alpha)\le S(0,\alpha')$.

Step 5 (Clause strength). Let $\delta\ge0$, $\theta\ge0$ and $0<\alpha'<\alpha$, and let $(\mu,\nu)\in\mathcal{D}\times\mathcal{D}$ satisfy $\Psi_{\delta,\alpha}(\mu,\nu)\ge S(\delta,\alpha)-\theta$. Directly from the definition,

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

Adding $\tfrac{\alpha-\alpha'}{2}W_{2}(\mu,\nu)^{2}$ to both sides of the hypothesis, which preserves the order by the compatibility of the order with addition in Ordered Field, gives $S(\delta,\alpha)-\theta+\tfrac{\alpha-\alpha'}{2}W_{2}(\mu,\nu)^{2}\le\Psi_{\delta,\alpha'}(\mu,\nu)$; and $\Psi_{\delta,\alpha'}(\mu,\nu)\le S(\delta,\alpha')$ because this value lies in $V(\delta,\alpha')$, of which $S(\delta,\alpha')$ is an upper bound (Step 2).

Step 6 (Clause weight). Let $\alpha>0$ and $\varepsilon>0$ be given; the choices are made in the order $(\mu_{0},\nu_{0})$, then $K$, then $\delta_{1}$, and $\delta$ is arbitrary afterwards. By claim 8 of Elementary Order Arithmetic in an Ordered Field, $\varepsilon/2>0$, so claim 3 of Approximation Property of the Supremum and the Infimum in $\mathbb{R}$ gives $(\mu_{0},\nu_{0})\in\mathcal{D}\times\mathcal{D}$ with $\Psi_{0,\alpha}(\mu_{0},\nu_{0})>S(0,\alpha)-\varepsilon/2$. Put $K=|\mathcal{E}(\mu_{0})|+|\mathcal{E}(\nu_{0})|+1$, which is positive: $0\le|\mathcal{E}(\mu_{0})|+|\mathcal{E}(\nu_{0})|$ by claim 1 of Properties of the Absolute Value in an Ordered Field and claim 2 of Elementary Arithmetic in an Ordered Field, so $1=0+1\le K$ by the compatibility of the order with addition in Ordered Field, and $0<1$ by claim 6 of Elementary Order Arithmetic in an Ordered Field, whence $0<K$ by claim 2 there; and $\delta_{1}=\tfrac{\varepsilon}{2}K^{-1}$, which is positive by claims 5 and 7 of Elementary Order Arithmetic in an Ordered Field. Let $0<\delta\le\delta_{1}$. By claim 3 of Properties of the Absolute Value in an Ordered Field and claim 5 of Elementary Arithmetic in an Ordered Field,

$$\delta\bigl(\mathcal{E}(\mu_{0})+\mathcal{E}(\nu_{0})\bigr)\le\delta K\le\delta_{1}K=\tfrac{\varepsilon}{2}.$$

Therefore, since $\Psi_{\delta,\alpha}(\mu_{0},\nu_{0})\in V(\delta,\alpha)$,

$$S(\delta,\alpha)\ge\Psi_{\delta,\alpha}(\mu_{0},\nu_{0})=\Psi_{0,\alpha}(\mu_{0},\nu_{0})-\delta\bigl(\mathcal{E}(\mu_{0})+\mathcal{E}(\nu_{0})\bigr)>S(0,\alpha)-\tfrac{\varepsilon}{2}-\tfrac{\varepsilon}{2}=S(0,\alpha)-\varepsilon.$$

Step 7 (Clause perturbed: the localising set). Fix a real $\delta>0$ and, using $\mathcal{D}\ne\varnothing$, a point $\lambda_{0}\in\mathcal{D}$. The inverse $\delta^{-1}$ exists and is positive by claim 7 of Elementary Order Arithmetic in an Ordered Field. Put

$$K=2\,\mathcal{E}(\lambda_{0})+\delta^{-1}(4b+1),\qquad c=K-e;$$

these depend only on $\delta$ and on the data $b$, $e$, $\lambda_{0}$, all fixed before $\alpha$ and $\theta$. Now let $\alpha>0$ and $\theta$ be reals with $0<\theta<1$, and write $\Psi=\Psi_{\delta,\alpha}$ and $S=S(\delta,\alpha)$, a real number by Step 2.

(a) $S\ge-2b-2\delta\mathcal{E}(\lambda_{0})$. Indeed, $W_{2}(\lambda_{0},\lambda_{0})=0$ as in Step 3. Put $\lambda=\kappa_{d}(\lambda_{0})$. By claim 3 of Properties of the Absolute Value in an Ordered Field, the hypothesis on $b$ and the sign reversal of claim 4 of Elementary Order Arithmetic in an Ordered Field, $-b\le-|u(\lambda)|\le u(\lambda)$, and from $v(\lambda)\le|v(\lambda)|\le b$ also $-b\le-v(\lambda)$. Hence $S\ge\Psi(\lambda_{0},\lambda_{0})=u(\lambda)-v(\lambda)-2\delta\mathcal{E}(\lambda_{0})\ge-2b-2\delta\mathcal{E}(\lambda_{0})$, the first inequality because $\Psi(\lambda_{0},\lambda_{0})\in V(\delta,\alpha)$.

(b) If $(\mu,\nu)\in\mathcal{D}\times\mathcal{D}$ satisfies $K<\mathcal{E}(\mu)+\mathcal{E}(\nu)$, then $\Psi(\mu,\nu)<S-1$. Indeed, the bounds $u(\kappa_{d}(\mu))\le b$, $-v(\kappa_{d}(\nu))\le b$ and $\tfrac{\alpha}{2}W_{2}(\mu,\nu)^{2}\ge0$ used in Step 2 give $\Psi(\mu,\nu)\le2b-\delta(\mathcal{E}(\mu)+\mathcal{E}(\nu))$. By claim 10 of Elementary Order Arithmetic in an Ordered Field with the positive factor $\delta$, $\delta K<\delta(\mathcal{E}(\mu)+\mathcal{E}(\nu))$, where $\delta K=2\delta\mathcal{E}(\lambda_{0})+4b+1$ since $\delta\delta^{-1}=1$. By claims 4, 1 and 2 there and (a),

$$\Psi(\mu,\nu)\le2b-\delta\bigl(\mathcal{E}(\mu)+\mathcal{E}(\nu)\bigr)<2b-\delta K=-2b-2\delta\mathcal{E}(\lambda_{0})-1\le S-1.$$

(c) Let $X=\{(\mu,\nu)\in\mathcal{D}\times\mathcal{D}:\ \mathcal{E}(\mu)+\mathcal{E}(\nu)\le K\}$. For $(\mu,\nu)\in X$, since $e\le\mathcal{E}(\nu)$ and $e\le\mathcal{E}(\mu)$, the compatibility of the order with addition in Ordered Field and claim 4 of Elementary Order Arithmetic in an Ordered Field give

$$\mathcal{E}(\mu)\le K-\mathcal{E}(\nu)\le K-e=c,\qquad \mathcal{E}(\nu)\le K-\mathcal{E}(\mu)\le K-e=c.\tag{7.1}$$

Step 8 (The complete metric space $X$). By The Noncommutative Wasserstein Distance Satisfies the Triangle Inequality and is a Metric on Noncommutative Laws §metric, $W_{2}$ restricts to a metric on $\Sigma_{d,R}$; this is 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. Let $d_{\times}$ be the product metric on $\Sigma_{d,R}\times\Sigma_{d,R}$ built from two copies of it, a metric by claim 1 of The Product Metric is a Metric. By Step 1(a), $X\subseteq\mathcal{D}\times\mathcal{D}\subseteq\Sigma_{d,R}\times\Sigma_{d,R}$; let $d_{X}$ be the restriction of $d_{\times}$ to $X\times X$, a metric on $X$ by claim 1 of The Restriction of a Metric to a Subset Induces the Subspace Topology. For $x=(\mu,\nu)$ and $x'=(\mu',\nu')$ in $\Sigma_{d,R}\times\Sigma_{d,R}$, claim 2 of The Product Metric is a Metric gives $W_{2}(\mu,\mu')\le d_{\times}(x,x')$ and $W_{2}(\nu,\nu')\le d_{\times}(x,x')$.

(a) Estimates. Let $x=(\mu,\nu)$, $x'=(\mu',\nu')$ and $y=(\mu'',\nu'')$ lie in $\mathcal{D}\times\mathcal{D}$ and put $\rho=d_{\times}(x,x')$, so that $0\le\rho$ by Metric Space. By The Noncommutative Wasserstein Distance Satisfies the Triangle Inequality and is a Metric on Noncommutative Laws §triangle, The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §symmetry (all points lie in $\Sigma_{d,R}$ by Step 1(a)) and the domination just recorded, combined by the compatibility of the order with addition,

$$W_{2}(\mu,\nu)\le W_{2}(\mu,\mu')+W_{2}(\mu',\nu')+W_{2}(\nu',\nu)\le W_{2}(\mu',\nu')+2\rho,$$ $$W_{2}(\mu,\mu'')\le W_{2}(\mu,\mu')+W_{2}(\mu',\mu'')\le W_{2}(\mu',\mu'')+\rho,\qquad W_{2}(\nu,\nu'')\le W_{2}(\nu',\nu'')+\rho.\tag{8.1}$$

(b) Completeness. Let $(x_{m})_{m\in\mathbb{N}}$ be a Cauchy sequence in $(X,d_{X})$ and write $x_{m}=(\sigma_{m},\tau_{m})$. Given $\varepsilon>0$ and $N$ as in that definition, $W_{2}(\sigma_{m},\sigma_{l})\le d_{X}(x_{m},x_{l})<\varepsilon$ for all $m,l\ge N$, so $W_{2}(\sigma_{m},\sigma_{l})<\varepsilon$ by claim 2 of Elementary Order Arithmetic in an Ordered Field, and likewise for $(\tau_{m})$. Thus both are Cauchy sequences in $(\Sigma_{d,R},W_{2})$, and by The Noncommutative Laws with a Norm Bound Form a Complete Bounded Metric Space with Interpolation Points §complete they converge to points $\sigma,\tau\in\Sigma_{d,R}$. By claim 1 of Coordinatewise Convergence, Sequential Compactness and Density in a Product Metric Space, $(x_{m})$ converges to $(\sigma,\tau)$ in $(\Sigma_{d,R}\times\Sigma_{d,R},d_{\times})$; once $(\sigma,\tau)\in X$ is shown, the same holds in $(X,d_{X})$ because $d_{X}$ is the restriction of $d_{\times}$, and then $(X,d_{X})$ is complete. Since $0\le W_{2}(\sigma_{m},\sigma)$ by Metric Space, $|W_{2}(\sigma_{m},\sigma)-0|=W_{2}(\sigma_{m},\sigma)$ by Absolute Value in an Ordered Field, so the real sequence $(W_{2}(\sigma_{m},\sigma))_{m\in\mathbb{N}}$ converges to $0$ in the sense of Limit of a Sequence of Real Numbers; as $\mathcal{E}(\sigma_{m})\le c$ for every $m$ by (7.1), Sublevel Sets of the Wall-Confined Free Energy are Closed for the Wasserstein Distance §closed gives $\sigma\in\mathcal{D}$, and in the same way $\tau\in\mathcal{D}$. Finally let $\varepsilon>0$; then $\varepsilon'=\varepsilon2^{-1}>0$ by claim 8 of Elementary Order Arithmetic in an Ordered Field. By Sublevel Sets of the Wall-Confined Free Energy are Closed for the Wasserstein Distance §lsc and Lower Semicontinuous Function on a Subset of a Metric Space there are reals $\eta_{1},\eta_{2}>0$ with $\mathcal{E}(\sigma)-\varepsilon'<\mathcal{E}(\lambda)$ for every $\lambda\in\mathcal{D}$ with $W_{2}(\sigma,\lambda)<\eta_{1}$, and $\mathcal{E}(\tau)-\varepsilon'<\mathcal{E}(\lambda)$ for every $\lambda\in\mathcal{D}$ with $W_{2}(\tau,\lambda)<\eta_{2}$. Convergence gives $N_{1},N_{2}\in\mathbb{N}$ with $W_{2}(\sigma_{m},\sigma)<\eta_{1}$ for $m\ge N_{1}$ and $W_{2}(\tau_{m},\tau)<\eta_{2}$ for $m\ge N_{2}$; for $m=N_{1}+N_{2}$ both hold, since $N_{1}<N_{1}+N_{2}$ and $N_{2}<N_{2}+N_{1}=N_{1}+N_{2}$ by claim 6 of Properties of the Order on the Natural Numbers and the commutativity of claim 4 of Arithmetic of Addition on the Natural Numbers, whence $N_{1}\le m$ and $N_{2}\le m$ by claim 1 of Properties of the Order on the Natural Numbers; and by The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §symmetry we get $\mathcal{E}(\sigma)-\varepsilon'<\mathcal{E}(\sigma_{m})$ and $\mathcal{E}(\tau)-\varepsilon'<\mathcal{E}(\tau_{m})$. Adding by claim 3 of Elementary Order Arithmetic in an Ordered Field, and using $\varepsilon'+\varepsilon'=\varepsilon$ (claim 8 there) and $x_{m}\in X$, we obtain $\mathcal{E}(\sigma)+\mathcal{E}(\tau)-\varepsilon<\mathcal{E}(\sigma_{m})+\mathcal{E}(\tau_{m})\le K$, whence $\mathcal{E}(\sigma)+\mathcal{E}(\tau)\le K+\varepsilon$. By Comparison of Real Numbers with Arbitrary Positive Slack §slack-above, $\mathcal{E}(\sigma)+\mathcal{E}(\tau)\le K$, that is, $(\sigma,\tau)\in X$.

Step 9 (Semicontinuity and the gauge). Semicontinuity on $\mathcal{D}$ refers to $(\Sigma_{d,R},W_{2})$, and on subsets of $\mathcal{D}\times\mathcal{D}$ to $(\Sigma_{d,R}\times\Sigma_{d,R},d_{\times})$ unless $(X,d_{X})$ is named; in each case it is meant at every point of the set, so that the pointwise claims of the lemmas cited apply at each point.

(a) The function $f:X\to\mathbb{R}$, $f(x)=\Psi(x)$, is bounded above by $2b-2\delta e$ (Step 2) and is upper semicontinuous on $X$ in $(X,d_{X})$. (i) By the hypotheses of clause perturbed for the fixed $\delta$, the function $\mu\mapsto u(\kappa_{d}(\mu))-\delta\mathcal{E}(\mu)$ is upper semicontinuous on $\mathcal{D}$ and the function $\nu\mapsto v(\kappa_{d}(\nu))+\delta\mathcal{E}(\nu)$ is lower semicontinuous on $\mathcal{D}$. By claim 4 of Negation, Restriction, and Separated Differences of Semicontinuous Functions, with both metric spaces $(\Sigma_{d,R},W_{2})$ and $A=B=\mathcal{D}$, applied to these two functions, $h_{1}(\mu,\nu)=\bigl[u(\kappa_{d}(\mu))-\delta\mathcal{E}(\mu)\bigr]-\bigl[v(\kappa_{d}(\nu))+\delta\mathcal{E}(\nu)\bigr]$ is upper semicontinuous on $\mathcal{D}\times\mathcal{D}$. (ii) Let $q(\mu,\nu)=\tfrac{\alpha}{2}W_{2}(\mu,\nu)^{2}$ for $(\mu,\nu)\in\mathcal{D}\times\mathcal{D}$. For $x=(\mu,\nu)$ and $x'=(\mu',\nu')$ in $\mathcal{D}\times\mathcal{D}$ and $\rho=d_{\times}(x,x')$, (8.1) and Step 1(d) with $s=W_{2}(\mu,\nu)$, $s'=W_{2}(\mu',\nu')$ and $t=2\rho$ (nonnegative by Step 1(b)) give $W_{2}(\mu,\nu)^{2}\le W_{2}(\mu',\nu')^{2}+4\rho\,W_{2}(\mu,\nu)$; multiplying by $\tfrac{\alpha}{2}\ge0$ (claim 5 of Elementary Arithmetic in an Ordered Field) gives $q(x)\le q(x')+a\rho$ with $a=2\alpha W_{2}(\mu,\nu)\ge0$ (Step 1(b)). By Step 1(e), $q$ is lower semicontinuous on $\mathcal{D}\times\mathcal{D}$, so $-q$ is upper semicontinuous there by claim 1 of Negation, Restriction, and Separated Differences of Semicontinuous Functions. (iii) On $\mathcal{D}\times\mathcal{D}$ we have $\Psi=h_{1}+(-q)$ directly from the definition of $\Psi_{\delta,\alpha}$, which is upper semicontinuous by claim 1 of Sums and Nonnegative Multiples of Semicontinuous Functions; by claim 2 of Negation, Restriction, and Separated Differences of Semicontinuous Functions, its restriction $f$ is upper semicontinuous on $X$ relative to $X$ in $(\Sigma_{d,R}\times\Sigma_{d,R},d_{\times})$. As $d_{X}(x,y)=d_{\times}(x,y)$ for $x,y\in X$, the condition of Upper Semicontinuous Function on a Subset of a Metric Space reads the same in $(X,d_{X})$, so $f$ is upper semicontinuous on $X$ there.

(b) The gauge. For $x=(\mu,\nu)$ and $y=(\mu',\nu')$ in $\mathcal{D}\times\mathcal{D}$ put $g(x,y)=W_{2}(\mu,\mu')^{2}+W_{2}(\nu,\nu')^{2}$, and let $G=8dR^{2}$ with $d$ read as a real number. (i) $g(x,x)=0$ by The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §separation, and $0\le g(x,y)\le G$ by Step 1(b), claim 2 of Elementary Arithmetic in an Ordered Field and The Noncommutative Laws with a Norm Bound Form a Complete Bounded Metric Space with Interpolation Points §bounded (each square is at most $4dR^{2}$); in particular $0\le G$. (ii) For fixed $y=(\mu'',\nu'')\in X$, the function $g(\cdot,y)$ is lower semicontinuous on $X$ in $(X,d_{X})$: for $x=(\mu,\nu)$ and $x'=(\mu',\nu')$ in $X$ and $\rho=d_{X}(x,x')$, (8.1) and Step 1(d) with $t=\rho$ give $W_{2}(\mu,\mu'')^{2}\le W_{2}(\mu',\mu'')^{2}+2\rho\,W_{2}(\mu,\mu'')$ and the same inequality for $\nu,\nu',\nu''$; adding, $g(x,y)\le g(x',y)+a\rho$ with $a=2\bigl(W_{2}(\mu,\mu'')+W_{2}(\nu,\nu'')\bigr)\ge0$, and Step 1(e) applies. (iii) Let $\eta>0$ and $\beta=(\eta2^{-1})(\eta2^{-1})$, positive by claims 8 and 5 of Elementary Order Arithmetic in an Ordered Field. Let $x=(\mu,\nu)$ and $y=(\mu',\nu')$ in $X$ satisfy $g(x,y)\le\beta$, and write $w=W_{2}(\mu,\mu')$. Then $w^{2}\le g(x,y)\le\beta$ because $W_{2}(\nu,\nu')^{2}\ge0$. If $\eta2^{-1}<w$, then $0<w$ by claim 2 there, and claim 10 there, applied with the positive factors $\eta2^{-1}$ and then $w$, gives $\beta<(\eta2^{-1})w<ww$, contradicting $w^{2}\le\beta$. Hence $w\le\eta2^{-1}<\eta$ by totality and claim 8 there, so $w<\eta$ by claim 2 there; likewise $W_{2}(\nu,\nu')<\eta$, and $d_{X}(x,y)<\eta$ by claim 3 of The Product Metric is a Metric.

Step 10 (Weights and the starting point). (a) By claims 8 and 7 of Elementary Order Arithmetic in an Ordered Field, $\tfrac{1}{2}=2^{-1}$ is positive; so for $k\in\mathbb{N}$ the natural power $(\tfrac{1}{2})^{k}$ is nonnegative and nonzero by claims 5 and 4 of Properties of Natural Number Powers in a Field, hence positive, and $\varepsilon_{k}=\theta(\tfrac{1}{2})^{k}$ is positive by claim 5 of Elementary Order Arithmetic in an Ordered Field. By Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §geometric, $\sum_{k=1}^{\infty}(\tfrac{1}{2})^{k}$ converges with sum $1$, so by Elementary Properties of Series of Real Numbers §linearity, applied with both of its convergent series taken to be $\sum_{k=1}^{\infty}(\tfrac{1}{2})^{k}$ and with $\lambda=\theta$, the series $\sum_{k=1}^{\infty}\varepsilon_{k}$ converges with sum $\theta\cdot1=\theta$; in particular $\sum_{k=1}^{\infty}\varepsilon_{k}\le\theta$.

(b) $V(\delta,\alpha)$ is nonempty and bounded above (Step 2), so claim 3 of Approximation Property of the Supremum and the Infimum in $\mathbb{R}$ with $\varepsilon=\theta$ gives $x_{1}=(\mu_{1},\nu_{1})\in\mathcal{D}\times\mathcal{D}$ with $S-\theta<\Psi(x_{1})$. As $\theta<1$, claims 4 and 1 of Elementary Order Arithmetic in an Ordered Field give $S-1<S-\theta$. If $K<\mathcal{E}(\mu_{1})+\mathcal{E}(\nu_{1})$ held, Step 7(b) and claim 2 there would give $\Psi(x_{1})<S-\theta$, which is impossible; since the order is total, $x_{1}\in X$. In particular $X\ne\varnothing$, and $f(X)$ is nonempty and bounded above by $S$ (as $X\subseteq\mathcal{D}\times\mathcal{D}$), so $\sup_{x\in X}f(x)$ exists by The Real Numbers: Standing Notation and Background §bounds and is at most $S$. Hence $f(x_{1})>S-\theta\ge\sup_{x\in X}f(x)-\theta$.

Step 11 (The variational principle; conclusion of clause perturbed). Apply A Smooth Variational Principle of Borwein-Preiss Type with a Gauge on a Complete Metric Space to the complete metric space $(X,d_{X})$ (Step 8(b)), which is nonempty (Step 10(b)), the function $f$ (Step 9(a)), the bound $G$ and the restriction of $g$ to $X\times X$ (Step 9(b)), the weights $c_{k}=\varepsilon_{k}$ (Step 10(a)), its $\varepsilon$ taken to be $\theta$, and the point $x_{1}$ (Step 10(b)). It yields a point $(\hat{\mu},\hat{\nu})\in X$ and a sequence $(x_{k})_{k\in\mathbb{N}}$ in $X$ whose first term is $x_{1}$; write $x_{k}=(\mu_{k},\nu_{k})$, so that $(\mu_{k})_{k\in\mathbb{N}}$ and $(\nu_{k})_{k\in\mathbb{N}}$ are sequences in $\mathcal{D}$. The function $x\mapsto f(x)-\sum_{k=1}^{\infty}\varepsilon_{k}\,g(x,x_{k})$ on $X$ of that theorem is the restriction to $X$ of the function $\Phi$ of clause perturbed. We verify the clause with these data and the $c$ of Step 7.

The weights are those of Step 10(a). Since $(\hat{\mu},\hat{\nu})$ and every $x_{k}$ lie in $X$, (7.1) gives $\mathcal{E}(\hat{\mu})\le c$, $\mathcal{E}(\hat{\nu})\le c$, $\mathcal{E}(\mu_{k})\le c$ and $\mathcal{E}(\nu_{k})\le c$ for every $k$.

Let $(\mu,\nu)\in\mathcal{D}\times\mathcal{D}$ and $w_{k}=g((\mu,\nu),x_{k})$, so $0\le w_{k}\le G$ by Step 9(b)(i). Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §tail-bound, with its nonnegative convergent series taken to be $\sum_{k=1}^{\infty}\varepsilon_{k}$ and $M=G$, shows that $\sum_{k=1}^{\infty}\varepsilon_{k}w_{k}=\sum_{k=1}^{\infty}\varepsilon_{k}\bigl(W_{2}(\mu,\mu_{k})^{2}+W_{2}(\nu,\nu_{k})^{2}\bigr)$ converges. Its terms are nonnegative by the product rule of Step 1(b), so its sum is at least its first partial sum, which is nonnegative, by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates. Hence

$$\Phi(\mu,\nu)\le\Psi(\mu,\nu)\qquad\text{for every }(\mu,\nu)\in\mathcal{D}\times\mathcal{D}.\tag{11.1}$$

By A Smooth Variational Principle of Borwein-Preiss Type with a Gauge on a Complete Metric Space §value and Step 10(b), $\Phi(\hat{\mu},\hat{\nu})\ge f(x_{1})>S-\theta$, so $\Psi(\hat{\mu},\hat{\nu})\ge\Phi(\hat{\mu},\hat{\nu})>S-\theta$ by (11.1) and claim 2 of Elementary Order Arithmetic in an Ordered Field; in particular $\Psi_{\delta,\alpha}(\hat{\mu},\hat{\nu})\ge S(\delta,\alpha)-\theta$.

Finally let $(\mu,\nu)\in\mathcal{D}\times\mathcal{D}$ with $(\mu,\nu)\ne(\hat{\mu},\hat{\nu})$. If $(\mu,\nu)\in X$, then $\Phi(\mu,\nu)<\Phi(\hat{\mu},\hat{\nu})$ by A Smooth Variational Principle of Borwein-Preiss Type with a Gauge on a Complete Metric Space §maximum. Otherwise $K<\mathcal{E}(\mu)+\mathcal{E}(\nu)$, the order being total, and (11.1), Step 7(b), Step 10(b) and the preceding paragraph give

$$\Phi(\mu,\nu)\le\Psi(\mu,\nu)<S-1<S-\theta<\Phi(\hat{\mu},\hat{\nu}),$$

combined by claim 2 of Elementary Order Arithmetic in an Ordered Field. Since $c$ was fixed in Step 7 before $\alpha$ and $\theta$, clause perturbed holds. $\blacksquare$