Proof of Plan Jets of Squared Wasserstein Distances at an Optimal Coupling of Bounded Noncommutative Laws

Each result cited is applied with the data of its own statement. Write $W=W_{2}(\mu,\nu)$, and for $k,l\in[2d]$ let $\delta_{kl}$ be $1$ if $k=l$ and $0$ otherwise. Throughout, $j$ ranges over $[d]$ and $l$ over $[2d]$.

Step 1 (The data and their composites). By Affine Data and Affine Substitutions of Noncommutative Polynomials §tuple, the tuple of a datum $(A,c)$ from $2d$ to $2d$ variables has components $c_{i}1+\sum_{l=1}^{2d}A_{il}x_{l}$; since $1,x_{1},\dots,x_{2d}$ are distinct monomials, these components determine the numbers $c_{i}$ and $A_{il}$, so a datum is determined by its tuple. Reading off the coefficients of the given tuples, $T=(A,0)$ and $T'=(A',0)$ with

$$A_{jl}=\delta_{jl},\qquad A'_{jl}=\delta_{d+j,l},\qquad A_{d+j,l}=A'_{d+j,l}=\alpha\delta_{jl}-\alpha\delta_{d+j,l}.$$

In particular, with $h_{j}=x_{j}-x_{d+j}$ as in Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws, $a^{T}_{d+j}=a^{T'}_{d+j}=\alpha x_{j}-\alpha x_{d+j}=\alpha h_{j}$. As $\gamma\in\Pi(\mu,\nu)$, $\gamma$ is a tracial state on $\mathcal{P}_{2d}$ with $\gamma\circ\iota^{1}=\mu$ and $\gamma\circ\iota^{2}=\nu$ by Couplings of Two Noncommutative Laws and Their Quadratic Cost §coupling, By Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law there are real $R_{1},R_{2}>0$ with $\mu\in\Sigma_{d,R_{1}}$ and $\nu\in\Sigma_{d,R_{2}}$; with $R$ the larger of $R_{1}$ and $R_{2}$, Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §monotone gives $\mu,\nu\in\Sigma_{d,R}$, and then $\gamma\in\Pi(\mu,\nu)\subseteq\Sigma_{2d,R}\subseteq\Sigma_{2d}$ by Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §bound and Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law. By Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §self-adjoint, the tuples $a^{T}$ and $a^{T'}$ are $2d$-tuples in $\mathcal{P}_{2d,\mathrm{sa}}$, and $\pi,\pi'\in\Sigma_{2d}$. Let $S=(B,0)$ and $S'=(B',0)$ be the affine data from $2d$ to $2d$ variables with

$$B_{jl}=\delta_{jl},\qquad B_{d+j,l}=\delta_{jl}-\alpha^{-1}\delta_{d+j,l},\qquad B'_{jl}=\delta_{jl}+\alpha^{-1}\delta_{d+j,l},\qquad B'_{d+j,l}=\delta_{jl}.$$

By Affine Data and Affine Substitutions of Noncommutative Polynomials §composite, all composites below have zero shift, and their matrices are computed from $(P^{1}A)_{jl}=A_{jl}=\delta_{jl}=P^{1}_{jl}$, $(P^{1}A')_{jl}=A'_{jl}=\delta_{d+j,l}=P^{2}_{jl}$, where $P^{1},P^{2}$ are the matrices of the marginal data, and

$$(BA)_{jl}=A_{jl}=\delta_{jl},\quad(BA)_{d+j,l}=A_{jl}-\alpha^{-1}A_{d+j,l}=\delta_{d+j,l},\quad(B'A')_{jl}=A'_{jl}+\alpha^{-1}A'_{d+j,l}=\delta_{jl},\quad(B'A')_{d+j,l}=A'_{jl}=\delta_{d+j,l}.$$

Hence, with the identity datum of Affine Data and Affine Substitutions of Noncommutative Polynomials §identity,

$$\mathrm{pr}^{1}\circ T=\mathrm{pr}^{1},\qquad\mathrm{pr}^{1}\circ T'=\mathrm{pr}^{2},\qquad S\circ T=S'\circ T'=\mathrm{id}_{2d}.\tag{1}$$

Since $\sigma_{\mathrm{pr}^{1}}=\iota^{1}$ and $\sigma_{\mathrm{pr}^{2}}=\iota^{2}$ by Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §coordinate, (1) and Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §composition give

$$\sigma_{T}\circ\iota^{1}=\sigma_{\mathrm{pr}^{1}\circ T}=\iota^{1},\qquad\sigma_{T'}\circ\iota^{1}=\sigma_{\mathrm{pr}^{1}\circ T'}=\iota^{2},\qquad\pi\circ\sigma_{S}=\gamma\circ\sigma_{S\circ T}=\gamma,\qquad\pi'\circ\sigma_{S'}=\gamma\circ\sigma_{S'\circ T'}=\gamma.\tag{2}$$

Step 2 (Clause 1). By (2), $\pi\circ\iota^{1}=\gamma\circ(\sigma_{T}\circ\iota^{1})=\gamma\circ\iota^{1}=\mu$ and $\pi'\circ\iota^{1}=\gamma\circ\iota^{2}=\nu$; as $\pi,\pi'\in\Sigma_{2d}$, $\pi$ and $\pi'$ are bounded plans at $\mu$ and at $\nu$ by Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §plans. For the momentum, the moment formula of Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §moments for the tracial state $\gamma$, the datum $T$ (zero shift) and $i=k=d+j$, together with the symmetry $\mathrm{m}_{j,d+j}(\gamma)=\mathrm{m}_{d+j,j}(\gamma)$ of the same clause, gives

$$\pi(x_{d+j}x_{d+j})=\sum_{l=1}^{2d}\sum_{l'=1}^{2d}A_{d+j,l}A_{d+j,l'}\,\mathrm{m}_{ll'}(\gamma)=\alpha^{2}\bigl(\mathrm{m}_{jj}(\gamma)+\mathrm{m}_{d+j,d+j}(\gamma)-2\,\mathrm{m}_{j,d+j}(\gamma)\bigr),$$

and the same value for $\pi'(x_{d+j}x_{d+j})$, since the rows $d+j$ of $A$ and $A'$ agree. The same formula for the data $\mathrm{pr}^{1},\mathrm{pr}^{2}$ (zero shift), whose substitutions are $\iota^{1},\iota^{2}$, gives $\mathrm{m}_{jj}(\gamma\circ\iota^{1})=\mathrm{m}_{jj}(\gamma)$ and $\mathrm{m}_{jj}(\gamma\circ\iota^{2})=\mathrm{m}_{d+j,d+j}(\gamma)$. Summing over $j$ and using Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §cost-identity, Couplings of Two Noncommutative Laws and Their Quadratic Cost §cost and the optimality of $\gamma$ (The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §optimal),

$$\sum_{j=1}^{d}\pi(x_{d+j}x_{d+j})=\alpha^{2}\Bigl(M(\gamma\circ\iota^{1})+M(\gamma\circ\iota^{2})-2\sum_{j=1}^{d}\mathrm{m}_{j,d+j}(\gamma)\Bigr)=\alpha^{2}\gamma(\Delta_{d})=\alpha^{2}I(\gamma)=\alpha^{2}W^{2},$$

and likewise for $\pi'$. Since $\alpha W\ge0$ has square $\alpha^{2}W^{2}$, Existence and Uniqueness of the Nonnegative Square Root gives $|\pi|_{\mathrm{mom}}=|\pi'|_{\mathrm{mom}}=\alpha W$. Finally, by 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 for the datum $\mathrm{pr}^{1}$,

$$\mathrm{pr}^{1}_{\#}\kappa_{2d}(\pi)=\kappa_{d}(\pi\circ\iota^{1})=\kappa_{d}(\mu),\qquad\mathrm{pr}^{1}_{\#}\kappa_{2d}(\pi')=\kappa_{d}(\pi'\circ\iota^{1})=\kappa_{d}(\nu),$$

so $\kappa_{2d}(\pi)\in\Sigma^{2}_{2d}$ is a plan at $\kappa_{d}(\mu)$ and $\kappa_{2d}(\pi')$ one at $\kappa_{d}(\nu)$ by Plans at a Square-Integrable Noncommutative Law §plan.

Step 3 (Tuples of law $\kappa_{2d}(\gamma)$). Let $(H,M,\Omega)$ be a tracial W*-probability space and $X,Y$ $L^{2}$ $d$-tuples with $\mathrm{law}(X,Y)=\kappa_{2d}(\gamma)$. We claim $\mathrm{law}(X)=\kappa_{d}(\mu)$, $\mathrm{law}(Y)=\kappa_{d}(\nu)$ and $\lVert X-Y\rVert_{2}=W$. By Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §coupling, $\mathrm{law}(X,Y)\in\Pi^{2}(\mathrm{law}(X),\mathrm{law}(Y))$, so by Square-Integrable Noncommutative Laws: the Wasserstein Completion of the Laws, Affine Push-Forwards, Moments, Couplings and Cost §couplings and 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,

$$\mathrm{law}(X)=\mathrm{pr}^{1}_{\#}\kappa_{2d}(\gamma)=\kappa_{d}(\gamma\circ\iota^{1})=\kappa_{d}(\mu),\qquad\mathrm{law}(Y)=\mathrm{pr}^{2}_{\#}\kappa_{2d}(\gamma)=\kappa_{d}(\gamma\circ\iota^{2})=\kappa_{d}(\nu).$$

By the same clause of Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §coupling, the second part of 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 (as $\gamma\in\Pi(\mu,\nu)$) and optimality, $\lVert X-Y\rVert_{2}^{2}=\mathcal{I}(\kappa_{2d}(\gamma))=I(\gamma)=W^{2}$; both $\lVert X-Y\rVert_{2}$ and $W$ are nonnegative, so they are equal. Moreover $\widehat{W}_{2}(\kappa_{d}(\mu),\kappa_{d}(\nu))=W$ 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, and $\widehat{W}_{2}(\lambda,\lambda)=0$ for all $\lambda$ 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 §metric; hence

$$\varphi(\kappa_{d}(\mu))=\tfrac{\alpha}{2}W^{2},\qquad\psi(\kappa_{d}(\nu))=-\tfrac{\alpha}{2}W^{2}.\tag{3}$$

We also use that, by Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §pairing, $\langle\cdot,\cdot\rangle_{2}$ is the inner product of $H^{d}$, real on $L^{2}$ $d$-tuples, and $\lVert\cdot\rVert_{2}$ its norm, while sums, real multiples and differences are computed componentwise (Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §operations, Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples); so for $L^{2}$ $d$-tuples $U,V$ and real $t$, conjugate symmetry and the reality of the pairing give

$$\lVert U+tV\rVert_{2}^{2}=\lVert U\rVert_{2}^{2}+2t\langle U,V\rangle_{2}+t^{2}\lVert V\rVert_{2}^{2},\qquad\langle U,V\rangle_{2}=\langle V,U\rangle_{2}.\tag{4}$$

Step 4 (Clause 2). Let $\eta>0$ and put $r=2\eta/(\alpha+\beta)>0$. Let $(H,M,\Omega)$ be a tracial W*-probability space and $X,P,X'$ $L^{2}$ $d$-tuples with $\mathrm{law}(X,P)=\kappa_{2d}(\pi)$ and $\lVert X'-X\rVert_{2}<r$. Let $Y=X-\alpha^{-1}P$. By Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations, $S(X,P)$ has components $X_{j}$ and $X_{j}-\alpha^{-1}P_{j}$ in the positions $j$ and $d+j$, so $S(X,P)=(X,Y)$. 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 and (2),

$$\mathrm{law}(X,Y)=S_{\#}\kappa_{2d}(\pi)=\kappa_{2d}(\pi\circ\sigma_{S})=\kappa_{2d}(\gamma).$$

By Step 3, $\mathrm{law}(X)=\kappa_{d}(\mu)$, $\mathrm{law}(Y)=\kappa_{d}(\nu)$ and $\lVert X-Y\rVert_{2}=W$; also $X-Y=\alpha^{-1}P$ componentwise. By Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §lipschitz, $\widehat{W}_{2}(\mathrm{law}(X'),\kappa_{d}(\nu))\le\lVert X'-Y\rVert_{2}$ and $\widehat{W}_{2}(\mathrm{law}(X'),\kappa_{d}(\mu))\le\lVert X'-X\rVert_{2}$, and both sides are nonnegative. Since $X-Y=\alpha^{-1}P$, one has $X'-Y=(X'-X)+(X-Y)=(X'-X)+\alpha^{-1}P$ and $\lVert\alpha^{-1}P\rVert_{2}^{2}=\lVert X-Y\rVert_{2}^{2}=W^{2}$; so (4) with $U=X'-X$, $V=P$ and $t=\alpha^{-1}$, together with the symmetry in (4), gives $\lVert X'-Y\rVert_{2}^{2}=\lVert X'-X\rVert_{2}^{2}+2\alpha^{-1}\langle P,X'-X\rangle_{2}+\alpha^{-2}\lVert P\rVert_{2}^{2}=\lVert X'-X\rVert_{2}^{2}+2\alpha^{-1}\langle P,X'-X\rangle_{2}+W^{2}$, where $\alpha^{-2}\lVert P\rVert_{2}^{2}=\lVert\alpha^{-1}P\rVert_{2}^{2}$ by the real homogeneity of the pairing (the case $U=0$ of (4)). By Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts and Lifts of Functions on Square-Integrable Noncommutative Laws to Square-Integrable Tuples §lift, $\varphi_{M}(X')=\varphi(\mathrm{law}(X'))$, so

$$\varphi_{M}(X')\le\tfrac{\alpha}{2}\lVert X'-Y\rVert_{2}^{2}+\tfrac{\beta}{2}\lVert X'-X\rVert_{2}^{2}=\tfrac{\alpha}{2}W^{2}+\langle P,X'-X\rangle_{2}+\tfrac{\alpha+\beta}{2}\lVert X'-X\rVert_{2}^{2}.$$

As $0\le\lVert X'-X\rVert_{2}<r$, $\tfrac{\alpha+\beta}{2}\lVert X'-X\rVert_{2}^{2}\le\tfrac{\alpha+\beta}{2}r\lVert X'-X\rVert_{2}=\eta\lVert X'-X\rVert_{2}$. With (3),

$$\varphi_{M}(X')\le\varphi(\kappa_{d}(\mu))+\langle P,X'-X\rangle_{2}+(0+\eta)\lVert X'-X\rVert_{2}.$$

So the plan $\kappa_{2d}(\pi)$ at $\kappa_{d}(\mu)$ (Step 2) is a plan superdifferential of $\varphi$ at $\kappa_{d}(\mu)$ with slack $0$ by Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §super, and $\kappa_{2d}(\pi)\in J^{+}\varphi(\kappa_{d}(\mu))$ by Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §superjet.

Step 5 (Clause 3). Let $\eta>0$ and $r=2\eta/(\alpha+\beta)$. Let $(H,M,\Omega)$ be a tracial W*-probability space and $Y,P,Y'$ $L^{2}$ $d$-tuples with $\mathrm{law}(Y,P)=\kappa_{2d}(\pi')$ and $\lVert Y'-Y\rVert_{2}<r$. Let $X=Y+\alpha^{-1}P$. By Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations, $S'(Y,P)$ has components $Y_{j}+\alpha^{-1}P_{j}$ and $Y_{j}$ in the positions $j$ and $d+j$, so $S'(Y,P)=(X,Y)$, and as in Step 4, by (2), $\mathrm{law}(X,Y)=S'_{\#}\kappa_{2d}(\pi')=\kappa_{2d}(\pi'\circ\sigma_{S'})=\kappa_{2d}(\gamma)$. By Step 3, $\mathrm{law}(X)=\kappa_{d}(\mu)$, $\mathrm{law}(Y)=\kappa_{d}(\nu)$ and $\lVert X-Y\rVert_{2}=W$, and $X-Y=\alpha^{-1}P$ componentwise. Since $X-Y'=(X-Y)-(Y'-Y)$, (4) gives $\lVert X-Y'\rVert_{2}^{2}=W^{2}-2\alpha^{-1}\langle P,Y'-Y\rangle_{2}+\lVert Y'-Y\rVert_{2}^{2}$. By Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §lipschitz, $\widehat{W}_{2}(\kappa_{d}(\mu),\mathrm{law}(Y'))\le\lVert X-Y'\rVert_{2}$ and $\widehat{W}_{2}(\mathrm{law}(Y'),\kappa_{d}(\nu))\le\lVert Y'-Y\rVert_{2}$. With the lift $\psi_{M}(Y')=\psi(\mathrm{law}(Y'))$ of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts and (3),

$$\psi_{M}(Y')\ge-\tfrac{\alpha}{2}\lVert X-Y'\rVert_{2}^{2}-\tfrac{\beta}{2}\lVert Y'-Y\rVert_{2}^{2}=\psi(\kappa_{d}(\nu))+\langle P,Y'-Y\rangle_{2}-\tfrac{\alpha+\beta}{2}\lVert Y'-Y\rVert_{2}^{2}\ge\psi(\kappa_{d}(\nu))+\langle P,Y'-Y\rangle_{2}-(0+\eta)\lVert Y'-Y\rVert_{2},$$

the last step as in Step 4. Since $\kappa_{2d}(\pi')$ is a plan at $\kappa_{d}(\nu)$ (Step 2), Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §sub and Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §subjet give $\kappa_{2d}(\pi')\in J^{-}\psi(\kappa_{d}(\nu))$.

Step 6 (Clause 4). Apply Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §isometry to $\gamma\in\Sigma_{2d}$ and the $2d$-tuple $a^{T}$ in $\mathcal{P}_{2d,\mathrm{sa}}$ (Step 1), whose marginal law is $\gamma\circ\sigma_{T}=\pi$: there is $U\in\mathcal{L}(\mathcal{H}_{\pi},\mathcal{H}_{\gamma})$ with $U\widehat{p}^{\,\pi}=\widehat{\sigma_{T}(p)}^{\,\gamma}$ for all $p\in\mathcal{P}_{2d}$ and $U^{*}U=I$, so that $\langle U\xi,U\xi'\rangle=\langle\xi,U^{*}U\xi'\rangle=\langle\xi,\xi'\rangle$ for $\xi,\xi'\in\mathcal{H}_{\pi}$. Since $\pi\circ\iota^{1}=\mu$, Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §isometries gives $V^{1}_{\pi}\in\mathcal{L}(\mathcal{H}_{\mu},\mathcal{H}_{\pi})$ with $V^{1}_{\pi}\widehat{q}^{\,\mu}=\widehat{\iota^{1}(q)}^{\,\pi}$, so $UV^{1}_{\pi}\in\mathcal{L}(\mathcal{H}_{\mu},\mathcal{H}_{\gamma})$ and, by (2), $UV^{1}_{\pi}\widehat{q}^{\,\mu}=\widehat{\sigma_{T}(\iota^{1}(q))}^{\,\gamma}=\widehat{\iota^{1}(q)}^{\,\gamma}$ for every $q\in\mathcal{P}_{d}$. As $\gamma\circ\iota^{1}=\mu$, $V^{1}_{\gamma}$ is by Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §isometries the operator of Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §isometry for $\gamma$ and $(x_{1},\dots,x_{d})$, the only element of $\mathcal{L}(\mathcal{H}_{\mu},\mathcal{H}_{\gamma})$ with $\widehat{q}^{\,\mu}\mapsto\widehat{\iota^{1}(q)}^{\,\gamma}$ for all $q$; hence $UV^{1}_{\pi}=V^{1}_{\gamma}$. By Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values, $\sigma_{T}(x_{d+j})=a^{T}_{d+j}=\alpha h_{j}$, and 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 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 §algebra for $\gamma\in\Sigma_{2d,R}$, $\widehat{\alpha h_{j}}=L_{\alpha h_{j}}\Omega_{\gamma}=\alpha L_{h_{j}}\Omega_{\gamma}=\alpha\widehat{h_{j}}$. Hence $U\widehat{x_{d+j}}^{\,\pi}=\alpha\widehat{h_{j}}^{\,\gamma}$, and for $\zeta\in\mathcal{H}_{\mu}^{d}$, by linearity of the inner product in its second argument,

$$\bigl\langle V^{1}_{\pi}\zeta_{j},\widehat{x_{d+j}}\bigr\rangle_{\mathcal{H}_{\pi}}=\bigl\langle UV^{1}_{\pi}\zeta_{j},U\widehat{x_{d+j}}\bigr\rangle_{\mathcal{H}_{\gamma}}=\alpha\bigl\langle V^{1}_{\gamma}\zeta_{j},\widehat{h_{j}}\bigr\rangle_{\mathcal{H}_{\gamma}}.$$

Taking real parts ($\alpha$ is real) and summing over $j$, Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §plan-pairing and Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §coupling-pairing give $\mathcal{J}(\zeta,\pi)=\alpha\mathcal{J}^{1}_{\gamma}(\zeta)$. For $\pi'$ the same argument with $a^{T'}$ gives $U'\in\mathcal{L}(\mathcal{H}_{\pi'},\mathcal{H}_{\gamma})$ with $U'^{*}U'=I$; now $\pi'\circ\iota^{1}=\nu=\gamma\circ\iota^{2}$ and, by (2), $U'V^{1}_{\pi'}\widehat{q}^{\,\nu}=\widehat{\sigma_{T'}(\iota^{1}(q))}^{\,\gamma}=\widehat{\iota^{2}(q)}^{\,\gamma}$, so the uniqueness in Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §isometry for $\gamma$ and $(x_{d+1},\dots,x_{2d})$ gives $U'V^{1}_{\pi'}=V^{2}_{\gamma}$; and $U'\widehat{x_{d+j}}^{\,\pi'}=\alpha\widehat{h_{j}}^{\,\gamma}$ since $a^{T'}_{d+j}=\alpha h_{j}$. Hence $\mathcal{J}(\zeta',\pi')=\alpha\mathcal{J}^{2}_{\gamma}(\zeta')$ for $\zeta'\in\mathcal{H}_{\nu}^{d}$.

Step 7 (Clause 5). Let $\mathcal{A}_{\gamma}=\{L_{p}:\ p\in\mathcal{P}_{2d}\}$ as in The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications. By The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §w-star for $\gamma\in\Sigma_{2d}$, $(K,N,\Psi)=(\mathcal{H}_{\gamma},\mathcal{M}_{\gamma},\Omega_{\gamma})$ is a tracial W*-probability space with $\mathcal{M}_{\gamma}=\mathcal{A}_{\gamma}''$. Let $s=(L_{x_{1}},\dots,L_{x_{2d}})$. Each $L_{x_{i}}$ lies in $\mathcal{A}_{\gamma}\subseteq\mathcal{A}_{\gamma}''=\mathcal{M}_{\gamma}$ by Basic Properties of Commutants: Unital Algebras Closed under Weak Limits, Order Reversal, the Triple Commutant, and Conjugation §order, and 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, since $x_{i}^{*}=x_{i}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint. So $s$ is a self-adjoint $2d$-tuple in $N$ (Self-Adjoint Tuples in a Tracial W*-Probability Space and Their Laws §tuple), and its law is $\lambda_{s}=\gamma$ by Self-Adjoint Tuples in a Tracial W*-Probability Space and Their Laws §law and The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §law. By Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §bounded, $s\Psi$ is an $L^{2}$ $2d$-tuple of $(K,N,\Psi)$ with $\mathrm{law}(s\Psi)=\kappa_{2d}(\gamma)$. Let $X=(L_{x_{1}}\Psi,\dots,L_{x_{d}}\Psi)$ and $Y=(L_{x_{d+1}}\Psi,\dots,L_{x_{2d}}\Psi)$, $L^{2}$ $d$-tuples of $(K,N,\Psi)$ with $(X,Y)=s\Psi$. By Step 3, $\lVert X-Y\rVert_{2}=W$, and $\mathrm{law}(X)=\kappa_{d}(\mu)$, $\mathrm{law}(Y)=\kappa_{d}(\nu)$; so by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §moments and 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, $\lVert X\rVert_{2}^{2}=\widehat{M}(\kappa_{d}(\mu))=M(\mu)$ and $\lVert Y\rVert_{2}^{2}=\widehat{M}(\kappa_{d}(\nu))=M(\nu)$, where $M(\mu)=\sum_{j=1}^{d}\mu(x_{j}x_{j})$ is the second moment of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §laws, as in the statement. By Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations, $T(X,Y)$ has components $X_{j}$ and $\alpha X_{j}-\alpha Y_{j}$ in the positions $j$ and $d+j$, so $T(X,Y)=(X,\alpha(X-Y))$, and likewise $T'(X,Y)=(Y,\alpha(X-Y))$. By Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward and 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,

$$\mathrm{law}(X,\alpha(X-Y))=T_{\#}\kappa_{2d}(\gamma)=\kappa_{2d}(\gamma\circ\sigma_{T})=\kappa_{2d}(\pi),\qquad\mathrm{law}(Y,\alpha(X-Y))=T'_{\#}\kappa_{2d}(\gamma)=\kappa_{2d}(\pi').$$

This proves clause 5.