Proof of Comparison and Uniqueness under Wasserstein Semicontinuity for Free-Energy-Penalised Viscosity Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall

Each result cited is universally quantified over the data in its own statement. 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 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. 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 1) 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.

Proof of clause 1 (comparison). The objects are chosen in the following order: $\beta$, $b$, $e$, $(\delta_{u},\omega_{u})$, $(\delta_{v},\omega_{v})$ (Step 1); $r_{\mathrm{s}}$, then $\tau$, $K$, $m$ and the strength $\alpha$ (Step 3); $R'$ and $r_{\mathrm{m}}$ (Step 4); $\delta_{1}$, then the weight $\delta$, then $c$, 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). In particular $\alpha$ is fixed before $\delta$, and $\theta$ after $\delta$ and $c$.

Step 1 (Setup). If $\mathcal{D}=\varnothing$ there is nothing to prove, so assume $\mathcal{D}\ne\varnothing$. Suppose, for a contradiction, that there is $\mu_{0}\in\mathcal{D}$ with $u(\kappa_{d}(\mu_{0}))-v(\kappa_{d}(\mu_{0}))>0$, and let $\beta>0$ be half of this difference. Since $u$ and $v$ are bounded, taking the larger of the two bounds given by Bounded Real-Valued Function on a Set (claim 1 of Elementary Properties of the Maximum of Two Elements), fix a real $b\ge0$ with $|u(\lambda)|\le b$ and $|v(\lambda)|\le b$ for every $\lambda\in\Sigma^{2}_{d}$. By The Wall-Confined Free Energy: Norm Bound, Lower Bound, Weak-Star Compact Sublevel Sets and Displacement Monotonicity of the Score §bounds, $\mathcal{D}\subseteq\Sigma_{d,R}$, and we fix a real $e$ with $e\le\mathcal{E}(\lambda)$ for every $\lambda\in\mathcal{D}$. Since $u$ is a free-energy-penalised viscosity subsolution, fix a real $\delta_{u}>0$ and a nondecreasing $\omega_{u}:[0,\infty)\to[0,\infty)$ as there; since $v$ is a free-energy-penalised viscosity supersolution, fix a real $\delta_{v}>0$ and a nondecreasing $\omega_{v}:[0,\infty)\to[0,\infty)$ as there. The hypotheses of Perturbed Maximisers of the Doubled Difference Penalised by the Wall-Confined Free Energy now hold with these $u$, $v$, $b$ and $e$ (the semicontinuity of $u$ and $v$ on $\mathcal{D}$ is assumed in clause 1); let $\Psi_{\delta,\alpha}$ and $S(\delta,\alpha)$ be as there.

Step 2 (The strength sequence). For $k\in\mathbb{N}$ let $q_{k}=S(0,2^{k})$, a real number by Perturbed Maximisers of the Doubled Difference Penalised by the Wall-Confined Free Energy §bounds. By Perturbed Maximisers of the Doubled Difference Penalised by the Wall-Confined Free Energy §diagonal with $\lambda=\mu_{0}$, $q_{k}\ge u(\kappa_{d}(\mu_{0}))-v(\kappa_{d}(\mu_{0}))=2\beta$ for every $k$. By Perturbed Maximisers of the Doubled Difference Penalised by the Wall-Confined Free Energy §monotone, with $2^{k}$ in the role of $\alpha'$ and $2^{k+1}$ in the role of $\alpha$, $q_{k+1}\le q_{k}$ for every $k$. Thus $(q_{k})_{k\in\mathbb{N}}$ is nonincreasing and its set of terms is bounded below by $2\beta$, so by A Bounded Monotone Sequence of Real Numbers Converges §nonincreasing it converges to a real number $q$.

Step 3 (Choice of $r_{\mathrm{s}}$ and of the strength). Since $\mathcal{H}$ satisfies the structure condition at bounded positions, applying it with the radius $R$ and with $\eta=\rho\beta/2>0$ gives a real $r_{\mathrm{s}}>0$ 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 $\alpha\lVert X-Y\rVert_{2}^{2}+\lVert X-Y\rVert_{2}<r_{\mathrm{s}}$,

$$\mathcal{H}_{M_{1}}\bigl(Y,\alpha(X-Y)\bigr)-\mathcal{H}_{M_{1}}\bigl(X,\alpha(X-Y)\bigr)<\tfrac{\rho\beta}{2}.$$

Let $\tau=\min\{r_{\mathrm{s}}/8,\,r_{\mathrm{s}}^{2}/16\}>0$. By Step 2 there is $K\in\mathbb{N}$ with $|q_{j}-q|<\tau/4$ for every $j\ge K$. By claim 1 (unboundedness) of The Archimedean Property of the Real Numbers choose $n_{0}\in\mathbb{N}$ with $n_{0}>\max\{4/\tau,\ (2\rho+1)/(\rho\beta)\}$, and let $m\in\mathbb{N}$ be the larger of $n_{0}$ and $K$, so that $m\ge K$ and $m\ge n_{0}$ by claim 1 of Elementary Properties of the Maximum of Two Elements, hence $m>\max\{4/\tau,\ (2\rho+1)/(\rho\beta)\}$ (natural numbers being read in $\mathbb{R}$, where their order is preserved by claim 6 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field); put $\alpha=2^{m+1}$. Then $\alpha\ge m+2>m$, since $2^{n}\ge n+1$ for every $n\in\mathbb{N}$ by Principle of Induction for the Natural Numbers (as $2^{1}=2$ and $2^{n+1}=2^{n}+2^{n}\ge(n+1)+1$); also $\alpha\ge2$ and $\alpha/2=2^{m}$, so $q_{m}=S(0,\alpha/2)$ and $q_{m+1}=S(0,\alpha)$. Put $\xi=q_{m}-q_{m+1}+2/\alpha$. Since $m,m+1\ge K$, $q_{m}-q_{m+1}\le|q_{m}-q|+|q_{m+1}-q|<\tau/2$, and $2/\alpha<2/m<\tau/2$; together with Step 2,

$$0\le\xi<\tau,\qquad\frac{2\rho+1}{\alpha}<\rho\beta.$$

Step 4 (Choice of the momentum radius and of $r_{\mathrm{m}}$). Let $R'=R+2(\alpha+2)R\sqrt{d}$, so that $R\le R'$ 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 $\eta=\rho\beta/8>0$ 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 $\lVert P_{\flat}-P_{\sharp}\rVert_{2}<r_{\mathrm{m}}$,

$$\bigl|\mathcal{H}_{M_{i}}(X,P_{\flat})-\mathcal{H}_{M_{i}}(X,P_{\sharp})\bigr|<\tfrac{\rho\beta}{8}.$$

Step 5 (Choice of the weight, the tolerance and the perturbed pair). By Perturbed Maximisers of the Doubled Difference Penalised by the Wall-Confined Free Energy §weight, applied with this $\alpha$ and with $1/\alpha$ in the role of $\varepsilon$, fix a real $\delta_{1}>0$ with $S(\delta',\alpha)\ge S(0,\alpha)-1/\alpha$ for every real $\delta'$ with $0<\delta'\le\delta_{1}$. Let

$$\delta=\min\Bigl\{\delta_{u},\ \delta_{v},\ \delta_{1},\ \frac{1}{\alpha\,(2|e|+\omega_{u}(R')+\omega_{v}(R')+1)}\Bigr\}.$$

Then $\delta>0$, $\delta\le\delta_{u}$, $\delta\le\delta_{v}$, and

$$S(\delta,\alpha)\ge q_{m+1}-\tfrac{1}{\alpha},\qquad 2\delta|e|\le\tfrac{1}{\alpha},\qquad\delta\bigl(\omega_{u}(R')+\omega_{v}(R')\bigr)\le\tfrac{1}{\alpha}.$$

By Perturbed Maximisers of the Doubled Difference Penalised by the Wall-Confined Free Energy §perturbed, applied with this $\delta$, fix a real $c$ as there. Let

$$\theta=\min\Bigl\{\tfrac{1}{2},\ \tfrac{\tau-\xi}{2},\ \tfrac{\beta}{8},\ \frac{\rho\beta}{16(\sigma^{2}+1)(|c-e|+1)},\ \frac{r_{\mathrm{m}}}{8R\sqrt{d}}\Bigr\}.$$

Then $0<\theta<1$, $\xi+\theta<\tau$, $\theta\le\beta/8$, $2\sigma^{2}|c-e|\,\theta\le\rho\beta/8$ 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 $\mathcal{D}\subseteq\Sigma_{d,R}$ (Step 1). Since $e\le\mathcal{E}(\hat{\mu})\le c$, we have $c-e\ge0$, hence $2\sigma^{2}(c-e)\theta\le\rho\beta/8$.

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 $\Psi_{\delta,\alpha}(\hat{\mu},\hat{\nu})\ge S(\delta,\alpha)-\theta$),

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

Here $-e\le|-e|=|e|$ by claims 3 and 2 of Properties of the Absolute Value in an Ordered Field, 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 $-2\delta e=2\delta(-e)\le2\delta|e|$, and $2\delta|e|\le1/\alpha$ by Step 5. Hence, by Perturbed Maximisers of the Doubled Difference Penalised by the Wall-Confined Free Energy §bounds, $S(\delta,\alpha/2)\le S(0,\alpha/2)-2\delta e\le q_{m}+1/\alpha$, while $S(\delta,\alpha)\ge q_{m+1}-1/\alpha$ by Step 5. Therefore $\frac{\alpha}{4}W^{2}\le\xi+\theta<\tau$. 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 all $\lVert s_{j}\rVert_{\mathrm{op}}\le R$ and $\lVert t^{k}_{j}\rVert_{\mathrm{op}}\le R$, $\lambda_{s}=\hat{\mu}$, $\lambda_{(s,t^{k})}=\gamma^{k}$ 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, with the real constant $C=v(\kappa_{d}(\hat{\nu}))+\delta\mathcal{E}(\hat{\nu})+\sum_{k}\varepsilon_{k}W_{2}(\hat{\nu},\nu_{k})^{2}$,

$$u\bigl(\kappa_{d}(\nu)\bigr)-\varphi\bigl(\kappa_{d}(\nu)\bigr)-\delta\mathcal{E}(\nu)=\Phi(\nu,\hat{\nu})+C.$$

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 the left-hand side at $\nu$ is strictly smaller than at $\hat{\mu}$.

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 $\lVert s_{j}\rVert_{\mathrm{op}}\le R\le R'$, 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 $i=1$),

$$\mathcal{H}\bigl(\kappa_{2d}(\pi)\bigr)=\mathcal{H}_{M_{1}}(X,P\Omega_{1})>\mathcal{H}_{M_{1}}\bigl(X,\alpha(X-Y)\bigr)-\tfrac{\rho\beta}{8}.$$

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 Free-Energy-Penalised Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §sub, applied to $u$ with $(\delta_{u},\omega_{u})$, the test function $\varphi$, the weight $\delta$ (recall $0<\delta\le\delta_{u}$), the law $\hat{\mu}$ and the plan $\pi$, gives $\hat{\mu}\in\mathcal{D}_{\Xi}$ and, as $\omega_{u}$ is nondecreasing,

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

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 all $\lVert s'_{j}\rVert_{\mathrm{op}}\le R$ and $\lVert t'^{k}_{j}\rVert_{\mathrm{op}}\le R$, $\lambda_{s'}=\hat{\nu}$, $\lambda_{(s',t'^{k})}=\gamma'^{k}$ 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 $C'=u(\kappa_{d}(\hat{\mu}))-\delta\mathcal{E}(\hat{\mu})-\sum_{k}\varepsilon_{k}W_{2}(\hat{\mu},\mu_{k})^{2}$ and Step 5,

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

If $\nu\ne\hat{\nu}$ then $\Phi(\hat{\mu},\nu)<\Phi(\hat{\mu},\hat{\nu})$, so the left-hand side at $\nu$ is strictly larger than at $\hat{\nu}$.

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 $i=2$), $\mathcal{H}(\kappa_{2d}(\pi'^{-}))=\mathcal{H}_{M_{2}}(X',(-P')\Omega_{2})<\mathcal{H}_{M_{2}}(X',Q')+\rho\beta/8$. 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

$$\mathcal{H}\bigl(\kappa_{2d}(\pi'^{-})\bigr)<\mathcal{H}_{M_{1}}\bigl(Y,\alpha(X-Y)\bigr)+\tfrac{\rho\beta}{8}.$$

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 Free-Energy-Penalised Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §super, applied to $v$ with $(\delta_{v},\omega_{v})$, the test function $-\varphi'$, the weight $\delta$ (recall $0<\delta\le\delta_{v}$), the law $\hat{\nu}$ and the plan $\pi'^{-}$, gives $\hat{\nu}\in\mathcal{D}_{\Xi}$ and, as $\omega_{v}$ is nondecreasing,

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

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 supersolution inequality of Step 8 from the subsolution inequality of Step 7 and using the last inequality of Step 5,

$$\rho\bigl(u(\kappa_{d}(\hat{\mu}))-v(\kappa_{d}(\hat{\nu}))\bigr)\le\mathcal{H}\bigl(\kappa_{2d}(\pi'^{-})\bigr)-\mathcal{H}\bigl(\kappa_{2d}(\pi)\bigr)-\tfrac{\sigma^{2}}{2}\Bigl(\mathcal{J}\bigl(\Xi(\hat{\mu}),\pi\bigr)-\mathcal{J}\bigl(\Xi(\hat{\nu}),\pi'^{-}\bigr)\Bigr)+\tfrac{1}{\alpha}.$$

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 $2\sigma^{2}(c-e)\theta\le\rho\beta/8$ (Step 5), the pairing term contributes at most $\rho\beta/8$. By the Hamiltonian estimates of Steps 7 and 8,

$$\mathcal{H}\bigl(\kappa_{2d}(\pi'^{-})\bigr)-\mathcal{H}\bigl(\kappa_{2d}(\pi)\bigr)<\mathcal{H}_{M_{1}}\bigl(Y,\alpha(X-Y)\bigr)-\mathcal{H}_{M_{1}}\bigl(X,\alpha(X-Y)\bigr)+\tfrac{\rho\beta}{4}<\tfrac{\rho\beta}{2}+\tfrac{\rho\beta}{4},$$

the last by the choice of $r_{\mathrm{s}}$ in Step 3, 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

$$\rho\bigl(u(\kappa_{d}(\hat{\mu}))-v(\kappa_{d}(\hat{\nu}))\bigr)<\tfrac{7}{8}\rho\beta+\tfrac{1}{\alpha}.$$

Step 10 (Lower bound and contradiction). Since $\Psi_{\delta,\alpha}(\hat{\mu},\hat{\nu})\ge S(\delta,\alpha)-\theta$, $W^{2}\ge0$, $\mathcal{E}(\hat{\mu}),\mathcal{E}(\hat{\nu})\ge e$ (Step 1) and $2\delta e\ge-2\delta|e|$ (Step 6),

$$u\bigl(\kappa_{d}(\hat{\mu})\bigr)-v\bigl(\kappa_{d}(\hat{\nu})\bigr)=\Psi_{\delta,\alpha}(\hat{\mu},\hat{\nu})+\tfrac{\alpha}{2}W^{2}+\delta\mathcal{E}(\hat{\mu})+\delta\mathcal{E}(\hat{\nu})\ge S(\delta,\alpha)-\theta-2\delta|e|\ge q_{m+1}-\tfrac{2}{\alpha}-\theta\ge2\beta-\tfrac{2}{\alpha}-\tfrac{\beta}{8},$$

by Step 5 and Step 2. Multiplying by $\rho>0$ and combining with Step 9 gives $2\rho\beta-\frac{2\rho}{\alpha}-\frac{\rho\beta}{8}<\frac{7}{8}\rho\beta+\frac{1}{\alpha}$, that is, $\rho\beta<\frac{2\rho+1}{\alpha}$, which contradicts Step 3. Hence $u(\kappa_{d}(\mu))\le v(\kappa_{d}(\mu))$ for every $\mu\in\mathcal{D}$.

Proof of clause 2 (uniqueness). Step 11. By Free-Energy-Penalised Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §solution, each of $u$ and $v$ is both a free-energy-penalised viscosity subsolution and a free-energy-penalised viscosity supersolution of $(\mathrm{E})$, and by hypothesis the functions $\mu\mapsto u(\kappa_{d}(\mu))$ and $\mu\mapsto v(\kappa_{d}(\mu))$ on $\mathcal{D}$ are both upper and lower semicontinuous on $\mathcal{D}$. Clause 1 applied to the pair $(u,v)$ gives $u(\kappa_{d}(\mu))\le v(\kappa_{d}(\mu))$, and clause 1 applied to the pair $(v,u)$ gives $v(\kappa_{d}(\mu))\le u(\kappa_{d}(\mu))$, for every $\mu\in\mathcal{D}$. Hence $u(\kappa_{d}(\mu))=v(\kappa_{d}(\mu))$ for every $\mu\in\mathcal{D}$.