Proof of The Free Heat Flow: Realisation, Norm Bound, Moments, Distance to the Initial Law, Wasserstein Contraction, Semigroup Property and Continuity

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

Conventions and elementary facts. For $m,n\in\mathbb{N}$ write $\iota^{1}_{m,n}=\sigma_{(x_{1},\dots,x_{m})}:\mathcal{P}_{m}\to\mathcal{P}_{m+n}$ and $\iota^{2}_{m,n}=\sigma_{(x_{m+1},\dots,x_{m+n})}:\mathcal{P}_{n}\to\mathcal{P}_{m+n}$; these are the maps $\iota^{1},\iota^{2}$ of Freeness of Two Groups of Variables under a Noncommutative Law §free for the split of $m+n$ variables into the first $m$ and the last $n$, and a free product $\alpha\star\beta$ for that split is taken as in The Free Product of Two Noncommutative Laws: Existence, Uniqueness, Norm Bound, Substitutions and Weak-Star Continuity §product. For $m=n=d$ they are the marginal substitutions $\iota^{1}=\iota^{1}_{d,d}$ and $\iota^{2}=\iota^{2}_{d,d}$. Every tuple of variables is self-adjoint by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §self-adjoint, and the tuple of an affine datum is self-adjoint by Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §self-adjoint; the same clause shows that $\omega\circ\sigma_{T}$ is a tracial state (a law if $\omega$ is a law) for every tracial state $\omega$ and affine datum $T$.

(C) Composition rule. Let $a$ be an $n$-tuple in $\mathcal{P}_{m}$ and $b$ an $m$-tuple in $\mathcal{P}_{l}$. By Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §composition, $\sigma_{b}\circ\sigma_{a}=\sigma_{c}$ with $c_{j}=\sigma_{b}(a_{j})$, so $(\omega\circ\sigma_{b})\circ\sigma_{a}=\omega\circ\sigma_{c}$ for every linear $\omega:\mathcal{P}_{l}\to\mathbb{C}$. If $a_{j}=\sum_{k}A_{jk}x_{k}$ with real $A_{jk}$, then $c_{j}=\sum_{k}A_{jk}b_{k}$, because $\sigma_{b}$ is linear (Substitution of Noncommutative Polynomials into the Variables §substitution) with $\sigma_{b}(x_{k})=b_{k}$ (Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values). Substitutions are moreover multiplicative (Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §homomorphism) and $\sigma_{b}(1)=1$ (Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values).

(Q) If $a,b\ge0$ are real with $a^{2}\le b^{2}$, then $a\le b$: otherwise $0\le b<a$ gives $b^{2}\le ab<a^{2}$. (S) For real $u,v\ge0$, $\sqrt{u}\sqrt{v}=\sqrt{uv}$ and $\sqrt{0}=0$, since $\sqrt{u}\sqrt{v}\ge0$ and $(\sqrt{u}\sqrt{v})^{2}=uv$, and $0^{2}=0$, by the uniqueness in Existence and Uniqueness of the Nonnegative Square Root. (B) If $0<R\le R'$ then $\Sigma_{n,R}\subseteq\Sigma_{n,R'}$, because $R^{k}\le R'^{k}$ for every $k\in\mathbb{N}$ (Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §norm-bound).

(SC) The semicircular law. $\mathrm{sc}_{d}$ satisfies the Schwinger-Dyson equation (The Semicircular Law and Its Scalings §standard), so by The Vacuum Law of the Semicircular Operators is the Unique Noncommutative Law Satisfying the Schwinger-Dyson Equation §unique it equals the vacuum law of that theorem, which lies in $\Sigma_{d,2}$ by The Vacuum Law of the Semicircular Operators is the Unique Noncommutative Law Satisfying the Schwinger-Dyson Equation §vacuum-law; thus $\mathrm{sc}_{d}\in\Sigma_{d,2}$. By the Schwinger-Dyson equation and the definition of the free difference quotient (with $x_{\varnothing}=1$ and $\mathrm{sc}_{d}(1)=1$), for $i,k\in[d]$: $\mathrm{sc}_{d}(x_{i})=\mathrm{sc}_{d}(x_{i}1)=\partial^{\mathrm{sc}_{d}}_{i}(1)=0$ and $\mathrm{sc}_{d}(x_{i}x_{k})=\partial^{\mathrm{sc}_{d}}_{i}(x_{k})=\delta_{ik}$. The tuple of $T_{t}$ of The Semicircular Law and Its Scalings §scaled is $(\sqrt{t}\,x_{1},\dots,\sqrt{t}\,x_{d})$, and $\mathrm{sc}_{d,t}=\mathrm{sc}_{d}\circ\sigma_{T_{t}}$.

For $\lambda\in\Sigma_{d}$ put $\gamma=\lambda\star\mathrm{sc}_{d}\in\Sigma_{2d}$ (split $d+d$). By The Free Product of Two Noncommutative Laws: Existence, Uniqueness, Norm Bound, Substitutions and Weak-Star Continuity §product, $\gamma$ is free, $\gamma\circ\iota^{1}=\lambda$ and $\gamma\circ\iota^{2}=\mathrm{sc}_{d}$. Write $g^{t}=(x_{1}+\sqrt{t}\,x_{d+1},\dots,x_{d}+\sqrt{t}\,x_{2d})$ for the tuple of $\Gamma_{t}$, so $\sigma_{\Gamma_{t}}=\sigma_{g^{t}}$ by Affine Data and Affine Substitutions of Noncommutative Polynomials §substitution.

Clause 1 (Realisation). By Free Additive Convolution of Two Noncommutative Laws §convolution, $\lambda\boxplus\mathrm{sc}_{d,t}=(\lambda\star\mathrm{sc}_{d,t})\circ\sigma_{\mathrm{add}}$, where $\mathrm{add}$ has tuple $(x_{1}+x_{d+1},\dots,x_{d}+x_{2d})$. Apply The Free Product of Two Noncommutative Laws: Existence, Uniqueness, Norm Bound, Substitutions and Weak-Star Continuity §substitution with $m=n=m'=n'=d$, $a=(x_{1},\dots,x_{d})$ in $\mathcal{P}_{d}$ and $b=(\sqrt{t}\,x_{1},\dots,\sqrt{t}\,x_{d})$, the tuple of $T_{t}$. By (C), $\iota^{1}(x_{j})=x_{j}$ and $\iota^{2}(\sqrt{t}\,x_{j})=\sqrt{t}\,x_{d+j}$, so the tuple of that clause is $c=(x_{1},\dots,x_{d},\sqrt{t}\,x_{d+1},\dots,\sqrt{t}\,x_{2d})$, and

$$\gamma\circ\sigma_{c}=(\lambda\circ\sigma_{a})\star(\mathrm{sc}_{d}\circ\sigma_{T_{t}})=\lambda\star\mathrm{sc}_{d,t},$$

using $\sigma_{a}(p)=p$ from Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §identity. By (C), $(\gamma\circ\sigma_{c})\circ\sigma_{\mathrm{add}}=\gamma\circ\sigma_{c'}$ with $c'_{i}=\sigma_{c}(x_{i}+x_{d+i})=x_{i}+\sqrt{t}\,x_{d+i}$, that is $c'=g^{t}$. Hence $\lambda\boxplus\mathrm{sc}_{d,t}=\gamma\circ\sigma_{\Gamma_{t}}$.

Clause 2 (Norm bound). By (S), $\sqrt{0}=0$, so $\Gamma_{0}$ has the same maps as the marginal datum $\mathrm{pr}^{1}$ of Affine Data and Affine Substitutions of Noncommutative Polynomials §coordinate, and $\sigma_{\mathrm{pr}^{1}}=\iota^{1}$ by Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §coordinate. By Clause 1, $\lambda\boxplus\mathrm{sc}_{d,0}=\gamma\circ\iota^{1}=\lambda$.

Now let $\lambda\in\Sigma_{d,R}$. Since $\gamma\in\Sigma_{2d}$, it lies in some $\Sigma_{2d,r}$ (Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law), so the multiplication operators $L_{p}$ on $\mathcal{H}_{\gamma}$, $p\in\mathcal{P}_{2d}$, are those 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. Apply Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation to $\gamma$ with $m=2d$, $n=d$ and $a=(x_{1},\dots,x_{d})$, so $\sigma_{a}=\iota^{1}$ and the marginal is $\lambda$, with embedding $\pi:\mathcal{M}_{\lambda}\to\mathcal{M}_{\gamma}$. For $j\in[d]$ the operator $L_{x_{j}}$ on $\mathcal{H}_{\lambda}$ commutes with every $R_{p}$ (Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §commute), so it lies in $\mathcal{M}_{\lambda}$ (The Tracial Algebra of a Noncommutative Law and Its Trace §algebra), and has norm at most $R$ 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 §multiplication. By Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §homomorphism its image $\pi(L_{x_{j}})$ is $L_{\iota^{1}(x_{j})}=L_{x_{j}}$ on $\mathcal{H}_{\gamma}$, and by Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §embedding $\lVert L_{x_{j}}\rVert_{\mathrm{op}}\le R$ on $\mathcal{H}_{\gamma}$. The same argument with $a=(x_{d+1},\dots,x_{2d})$, whose marginal is $\mathrm{sc}_{d}\in\Sigma_{d,2}$ by (SC), gives $\lVert L_{x_{d+j}}\rVert_{\mathrm{op}}\le2$ on $\mathcal{H}_{\gamma}$. 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 §algebra, $L_{g^{t}_{j}}=L_{x_{j}}+\sqrt{t}\,L_{x_{d+j}}$, so by the triangle inequality and homogeneity of the operator norm $\lVert L_{g^{t}_{j}}\rVert_{\mathrm{op}}\le R+2\sqrt{t}$. Since $g^{t}$ is self-adjoint, Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §law with $C=R+2\sqrt{t}>0$ gives $\gamma\circ\sigma_{g^{t}}\in\Sigma_{d,R+2\sqrt{t}}$, which is $\lambda\boxplus\mathrm{sc}_{d,t}$ by Clause 1.

Clause 3 (Moments). First the moments of $\gamma$. For $i,k\in[d]$, by (C) and the marginals of $\gamma$: $\mathrm{m}_{i}(\gamma)=\gamma(\iota^{1}(x_{i}))=\mathrm{m}_{i}(\lambda)$, $\mathrm{m}_{ik}(\gamma)=\gamma(\iota^{1}(x_{i}x_{k}))=\mathrm{m}_{ik}(\lambda)$, $\mathrm{m}_{d+i}(\gamma)=\mathrm{sc}_{d}(x_{i})=0$ and $\mathrm{m}_{d+i,d+k}(\gamma)=\mathrm{sc}_{d}(x_{i}x_{k})=\delta_{ik}$ by (SC). For the mixed moments, $p_{1}=x_{i}-\mathrm{m}_{i}(\lambda)1\in\mathcal{P}_{d}$ and $p_{2}=x_{k}\in\mathcal{P}_{d}$ satisfy $\gamma(\iota^{1}(p_{1}))=\lambda(x_{i})-\mathrm{m}_{i}(\lambda)=0$ and $\gamma(\iota^{2}(p_{2}))=\mathrm{sc}_{d}(x_{k})=0$, so freeness of $\gamma$ (Freeness of Two Groups of Variables under a Noncommutative Law §free with $(e_{1},e_{2})=(1,2)$) gives $\gamma\bigl((x_{i}-\mathrm{m}_{i}(\lambda)1)x_{d+k}\bigr)=0$, hence $\mathrm{m}_{i,d+k}(\gamma)=\mathrm{m}_{i}(\lambda)\,\mathrm{m}_{d+k}(\gamma)=0$; and $\mathrm{m}_{d+k,i}(\gamma)=\mathrm{m}_{i,d+k}(\gamma)=0$ by the symmetry in Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §moments.

By Clause 1, $\lambda\boxplus\mathrm{sc}_{d,t}=\gamma\circ\sigma_{\Gamma_{t}}$, and Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §moments with $T=\Gamma_{t}$ (whose translation part is $0$, and whose row $i$ has the entries $1$ at $i$ and $\sqrt{t}$ at $d+i$) gives

$$\mathrm{m}_{i}(\gamma\circ\sigma_{\Gamma_{t}})=\mathrm{m}_{i}(\gamma)+\sqrt{t}\,\mathrm{m}_{d+i}(\gamma)=\mathrm{m}_{i}(\lambda),$$ $$\mathrm{m}_{ik}(\gamma\circ\sigma_{\Gamma_{t}})=\mathrm{m}_{ik}(\gamma)+\sqrt{t}\,\mathrm{m}_{i,d+k}(\gamma)+\sqrt{t}\,\mathrm{m}_{d+i,k}(\gamma)+t\,\mathrm{m}_{d+i,d+k}(\gamma)=\mathrm{m}_{ik}(\lambda)+t\,\delta_{ik},$$

using $\sqrt{t}\sqrt{t}=t$ (Existence and Uniqueness of the Nonnegative Square Root). Summing over $i=k\in[d]$ gives $M(\lambda\boxplus\mathrm{sc}_{d,t})=M(\lambda)+td$, since $M=\sum_{i}\mathrm{m}_{ii}$ by the statement of Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data.

Clause 4 (Distance to the initial law). Let $e=(g^{t}_{1},\dots,g^{t}_{d},x_{1},\dots,x_{d})$, the tuple of the affine datum from $2d$ to $2d$ variables with translation $0$ whose row $i\in[d]$ has entries $1$ at $i$ and $\sqrt{t}$ at $d+i$, and whose row $d+i$ has entry $1$ at $i$, all other entries $0$. Then $\pi=\gamma\circ\sigma_{e}$ is a tracial state on $\mathcal{P}_{2d}$. By (C), $\pi\circ\iota^{1}=\gamma\circ\sigma_{g^{t}}=\lambda\boxplus\mathrm{sc}_{d,t}$ (Clause 1) and $\pi\circ\iota^{2}=\gamma\circ\sigma_{(x_{1},\dots,x_{d})}=\gamma\circ\iota^{1}=\lambda$, so $\pi\in\Pi(\lambda\boxplus\mathrm{sc}_{d,t},\lambda)$ (Couplings of Two Noncommutative Laws and Their Quadratic Cost §coupling). By (C), $\sigma_{e}(\Delta_{d})=\sum_{j=1}^{d}(e_{j}-e_{d+j})^{2}=\sum_{j=1}^{d}(\sqrt{t}\,x_{d+j})^{2}=t\,\iota^{2}\bigl(\sum_{j=1}^{d}x_{j}x_{j}\bigr)$, so by Couplings of Two Noncommutative Laws and Their Quadratic Cost §cost and (SC)

$$I(\pi)=t\sum_{j=1}^{d}\mathrm{sc}_{d}(x_{j}x_{j})=td.$$

The infimum defining $W_{2}(\lambda\boxplus\mathrm{sc}_{d,t},\lambda)^{2}$ in The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §distance is at most $I(\pi)=td=(\sqrt{td})^{2}$, and (Q) gives $W_{2}(\lambda\boxplus\mathrm{sc}_{d,t},\lambda)\le\sqrt{td}$.

Clause 5 (Contraction). By Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law there are reals $R_{1},R_{2}>0$ with $\mu\in\Sigma_{d,R_{1}}$ and $\nu\in\Sigma_{d,R_{2}}$; with $R=\max(R_{1},R_{2})$, (B) gives $\mu,\nu\in\Sigma_{d,R}$. By The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §attained there is an optimal coupling $\rho\in\Pi(\mu,\nu)$, and $\rho\in\Sigma_{2d,R}\subseteq\Sigma_{2d}$ by Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §bound. Let $\Theta=\rho\star\mathrm{sc}_{d}\in\Sigma_{3d}$ for the split $3d=2d+d$, so $\Theta\circ\iota^{1}_{2d,d}=\rho$ by The Free Product of Two Noncommutative Laws: Existence, Uniqueness, Norm Bound, Substitutions and Weak-Star Continuity §product. Let $f$ be the $2d$-tuple in $\mathcal{P}_{3d}$ with $f_{j}=x_{j}+\sqrt{t}\,x_{2d+j}$ and $f_{d+j}=x_{d+j}+\sqrt{t}\,x_{2d+j}$ ($j\in[d]$), the tuple of an affine datum from $3d$ to $2d$ variables with translation $0$; so $\pi=\Theta\circ\sigma_{f}$ is a tracial state on $\mathcal{P}_{2d}$.

For $k\in\{1,2\}$ apply The Free Product of Two Noncommutative Laws: Existence, Uniqueness, Norm Bound, Substitutions and Weak-Star Continuity §substitution with $m=2d$, $n=d$, $m'=n'=d$, $b=(x_{1},\dots,x_{d})$ in $\mathcal{P}_{d}$, and $a=(x_{1},\dots,x_{d})$ for $k=1$, $a=(x_{d+1},\dots,x_{2d})$ for $k=2$, in $\mathcal{P}_{2d}$; then $\sigma_{a}=\iota^{k}$ (Couplings of Two Noncommutative Laws and Their Quadratic Cost §marginals) and $\sigma_{b}$ is the identity (Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §identity). By (C) the tuple of that clause is $c^{k}=(x_{(k-1)d+1},\dots,x_{kd},x_{2d+1},\dots,x_{3d})$, and

$$\Theta\circ\sigma_{c^{1}}=(\rho\circ\iota^{1})\star\mathrm{sc}_{d}=\mu\star\mathrm{sc}_{d},\qquad\Theta\circ\sigma_{c^{2}}=(\rho\circ\iota^{2})\star\mathrm{sc}_{d}=\nu\star\mathrm{sc}_{d}.$$

By Clause 1 and (C), $\mu\boxplus\mathrm{sc}_{d,t}=(\Theta\circ\sigma_{c^{1}})\circ\sigma_{g^{t}}=\Theta\circ\sigma_{h}$ with $h_{j}=\sigma_{c^{1}}(x_{j}+\sqrt{t}\,x_{d+j})=x_{j}+\sqrt{t}\,x_{2d+j}=f_{j}$, while $\pi\circ\iota^{1}=\Theta\circ\sigma_{(f_{1},\dots,f_{d})}$ by (C); so $\pi\circ\iota^{1}=\mu\boxplus\mathrm{sc}_{d,t}$. Likewise $\sigma_{c^{2}}(x_{j}+\sqrt{t}\,x_{d+j})=x_{d+j}+\sqrt{t}\,x_{2d+j}=f_{d+j}$ gives $\pi\circ\iota^{2}=\nu\boxplus\mathrm{sc}_{d,t}$. Thus $\pi\in\Pi(\mu\boxplus\mathrm{sc}_{d,t},\nu\boxplus\mathrm{sc}_{d,t})$. In the cost the noise cancels: by (C), $\sigma_{f}(\Delta_{d})=\sum_{j=1}^{d}(f_{j}-f_{d+j})^{2}=\sum_{j=1}^{d}(x_{j}-x_{d+j})^{2}=\iota^{1}_{2d,d}(\Delta_{d})$, so

$$I(\pi)=\Theta\bigl(\iota^{1}_{2d,d}(\Delta_{d})\bigr)=\rho(\Delta_{d})=I(\rho)=W_{2}(\mu,\nu)^{2}$$

by Couplings of Two Noncommutative Laws and Their Quadratic Cost §cost and The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §optimal. Hence $W_{2}(\mu\boxplus\mathrm{sc}_{d,t},\nu\boxplus\mathrm{sc}_{d,t})^{2}\le W_{2}(\mu,\nu)^{2}$ by The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §distance, and (Q) gives the claim.

Clause 6 (Semigroup). Use the split $3d=d+2d$ and write $\Theta'=\lambda\star\mathrm{sc}_{2d}\in\Sigma_{3d}$, so $\Theta'\circ\iota^{1}_{d,2d}=\lambda$ by The Free Product of Two Noncommutative Laws: Existence, Uniqueness, Norm Bound, Substitutions and Weak-Star Continuity §product. By Free Semicircular Variables are Characterised by the Schwinger-Dyson Equation Relative to the Other Variables; Semicircular Systems and Rotations §systems, $\mathrm{sc}_{2d}=\mathrm{sc}_{d}\star\mathrm{sc}_{d}$ (split $d+d$), so $\mathrm{sc}_{2d}\circ\iota^{1}=\mathrm{sc}_{d}$ by The Free Product of Two Noncommutative Laws: Existence, Uniqueness, Norm Bound, Substitutions and Weak-Star Continuity §product.

Step 1: an auxiliary law. Let $G$ be the affine datum from $3d$ to $2d$ variables with translation $0$ and matrix $H$ given by $H_{i,i}=1$, $H_{i,d+i}=\sqrt{t}$, $H_{d+i,2d+i}=1$ ($i\in[d]$) and all other entries $0$; its tuple is $(x_{1}+\sqrt{t}\,x_{d+1},\dots,x_{d}+\sqrt{t}\,x_{2d},x_{2d+1},\dots,x_{3d})$. Put $\kappa=\Theta'\circ\sigma_{G}\in\Sigma_{2d}$. Apply The Free Product of Two Noncommutative Laws: Existence, Uniqueness, Norm Bound, Substitutions and Weak-Star Continuity §substitution with $m=d$, $n=2d$, $m'=n'=d$, $a=(x_{1},\dots,x_{d})$ in $\mathcal{P}_{d}$ and $b=(x_{1},\dots,x_{d})$ in $\mathcal{P}_{2d}$, so $\sigma_{a}$ is the identity and $\sigma_{b}=\iota^{1}$: by (C) the tuple of that clause is $c=(x_{1},\dots,x_{2d})$ in $\mathcal{P}_{3d}$ and $\Theta'\circ\sigma_{c}=\lambda\star(\mathrm{sc}_{2d}\circ\iota^{1})=\lambda\star\mathrm{sc}_{d}$. By (C) and Clause 1, $\kappa\circ\iota^{1}=\Theta'\circ\sigma_{(x_{1}+\sqrt{t}x_{d+1},\dots,x_{d}+\sqrt{t}x_{2d})}=(\Theta'\circ\sigma_{c})\circ\sigma_{g^{t}}=(\lambda\star\mathrm{sc}_{d})\circ\sigma_{\Gamma_{t}}=\lambda\boxplus\mathrm{sc}_{d,t}$, since $\sigma_{c}(g^{t}_{j})=g^{t}_{j}$ in $\mathcal{P}_{3d}$.

Step 2: $\kappa=(\lambda\boxplus\mathrm{sc}_{d,t})\star\mathrm{sc}_{d}$. By the forward direction of Free Semicircular Variables are Characterised by the Schwinger-Dyson Equation Relative to the Other Variables; Semicircular Systems and Rotations §characterisation (with $m=d$, $n=2d$, $\alpha=\lambda$, applied to $\Theta'$), $\Theta'$ satisfies the Schwinger-Dyson equation in the last $2d$ variables. Let $j\in[d]$ and $q\in\mathcal{P}_{2d}$. By (C), $\sigma_{G}(x_{d+j}q)=x_{2d+j}\,\sigma_{G}(q)$, so

$$\kappa(x_{d+j}q)=\Theta'\bigl(x_{2d+j}\,\sigma_{G}(q)\bigr)=\partial^{\Theta'}_{2d+j}\bigl(\sigma_{G}(q)\bigr)=\sum_{i=1}^{2d}H_{i,2d+j}\,\partial^{\kappa}_{i}(q)=\partial^{\kappa}_{d+j}(q),$$

the third equality by The Chain Rule for Free Difference Quotients under an Affine Substitution §chain-rule (with $m=3d$, $n=2d$, $T=G$, $k=2d+j$) and the last because $H_{i,2d+j}$ is $1$ for $i=d+j$ and $0$ otherwise. Thus $\kappa$ satisfies the Schwinger-Dyson equation in its last $d$ variables, and since $\kappa\circ\iota^{1}=\lambda\boxplus\mathrm{sc}_{d,t}\in\Sigma_{d}$, the reverse direction of Free Semicircular Variables are Characterised by the Schwinger-Dyson Equation Relative to the Other Variables; Semicircular Systems and Rotations §characterisation (with $m=n=d$) gives $\kappa=(\lambda\boxplus\mathrm{sc}_{d,t})\star\mathrm{sc}_{d}$.

Step 3: the left side. By Clause 1 for the law $\lambda\boxplus\mathrm{sc}_{d,t}$ and the time $s$, Step 2 and (C),

$$(\lambda\boxplus\mathrm{sc}_{d,t})\boxplus\mathrm{sc}_{d,s}=\kappa\circ\sigma_{\Gamma_{s}}=\Theta'\circ\sigma_{u},\qquad u_{i}=\sigma_{G}(x_{i}+\sqrt{s}\,x_{d+i})=x_{i}+\sqrt{t}\,x_{d+i}+\sqrt{s}\,x_{2d+i}\quad(i\in[d]).$$

Step 4: the right side. If $t+s=0$, then $t=s=0$, and both sides equal $\lambda$ by two applications of Clause 2. Otherwise $v=t+s>0$; let $O:[d]\times[2d]\to\mathbb{R}$ have $O_{i,i}=\sqrt{t/v}$, $O_{i,d+i}=\sqrt{s/v}$ and all other entries $0$. For $i,k\in[d]$, $\sum_{l=1}^{2d}O_{il}O_{kl}$ is $0$ if $i\ne k$ and $(\sqrt{t/v})^{2}+(\sqrt{s/v})^{2}=t/v+s/v=1$ if $i=k$, so Free Semicircular Variables are Characterised by the Schwinger-Dyson Equation Relative to the Other Variables; Semicircular Systems and Rotations §rotation (with $k=d$, $n=2d$) gives $\mathrm{sc}_{2d}\circ\sigma_{(O,0)}=\mathrm{sc}_{d}$. Apply The Free Product of Two Noncommutative Laws: Existence, Uniqueness, Norm Bound, Substitutions and Weak-Star Continuity §substitution with $m=d$, $n=2d$, $m'=n'=d$, $a=(x_{1},\dots,x_{d})$ and $b$ the tuple of $(O,0)$, whose entries are $b_{i}=\sqrt{t/v}\,x_{i}+\sqrt{s/v}\,x_{d+i}$: by (C) the tuple of that clause is $c'=(x_{1},\dots,x_{d},b'_{1},\dots,b'_{d})$ with $b'_{i}=\sqrt{t/v}\,x_{d+i}+\sqrt{s/v}\,x_{2d+i}$, and $\Theta'\circ\sigma_{c'}=\lambda\star(\mathrm{sc}_{2d}\circ\sigma_{(O,0)})=\lambda\star\mathrm{sc}_{d}$. By Clause 1 and (C), $\lambda\boxplus\mathrm{sc}_{d,v}=(\Theta'\circ\sigma_{c'})\circ\sigma_{\Gamma_{v}}=\Theta'\circ\sigma_{u'}$ with

$$u'_{i}=\sigma_{c'}(x_{i}+\sqrt{v}\,x_{d+i})=x_{i}+\sqrt{v}\sqrt{t/v}\,x_{d+i}+\sqrt{v}\sqrt{s/v}\,x_{2d+i}=u_{i},$$

because $\sqrt{v}\sqrt{t/v}=\sqrt{t}$ and $\sqrt{v}\sqrt{s/v}=\sqrt{s}$ by (S). Comparing with Step 3 gives the claim.

Clause 7 (Continuity). Time. Fix $p\in\mathcal{P}_{d}$. For a word $w=(w_{1},\dots,w_{k})\in W_{d}$ and $S\subseteq[k]$ let $w^{S}\in W_{2d}$ be the word of length $k$ whose $l$-th letter is $d+w_{l}$ if $l\in S$ and $w_{l}$ otherwise. By Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values, $\sigma_{g^{r}}(x_{w})$ is the product along $w$ of the $g^{r}_{w_{l}}=x_{w_{l}}+\sqrt{r}\,x_{d+w_{l}}$, and an induction on $k$ using distributivity (Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra) and $x_{u}x_{u'}=x_{uu'}$ (Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials) gives

$$\sigma_{g^{r}}(x_{w})=\sum_{S\subseteq[k]}(\sqrt{r})^{|S|}\,x_{w^{S}}\qquad(r\ge0),$$

and $\sigma_{g^{r}}(x_{\varnothing})=1$. Writing $p$ as the finite sum of $p(w)x_{w}$ over the words $w$ in its support, linearity of $\sigma_{g^{r}}$ and of $\gamma$ and Clause 1 give, for $r\ge0$,

$$\operatorname{Re}(\lambda\boxplus\mathrm{sc}_{d,r})(p)=\sum_{w}\sum_{S}(\sqrt{r})^{|S|}\operatorname{Re}\bigl(p(w)\,\gamma(x_{w^{S}})\bigr),$$

and the same with $\operatorname{Im}$, where the complex numbers $p(w)\gamma(x_{w^{S}})$ do not depend on $r$ (for the empty word the inner sum is the single term $p(\varnothing)\gamma(1)$). The map $r\mapsto\sqrt{r}$ is continuous on $[0,\infty)$: for $r\ge r_{0}\ge0$ we have $0\le\sqrt{r}-\sqrt{r_{0}}$ (by (Q), as $r_{0}\le r$) and $(\sqrt{r}-\sqrt{r_{0}})^{2}\le(\sqrt{r}-\sqrt{r_{0}})(\sqrt{r}+\sqrt{r_{0}})=r-r_{0}$, so in all cases $|\sqrt{r}-\sqrt{r_{0}}|^{2}\le|r-r_{0}|$; given $\varepsilon>0$, $\delta=\varepsilon^{2}$ satisfies Continuous Map Between Metric Spaces for the metric of The Absolute Value Metric on the Real Numbers, since if $|r-r_{0}|<\varepsilon^{2}$ and $\varepsilon\le|\sqrt{r}-\sqrt{r_{0}}|$, then $\varepsilon^{2}\le|\sqrt{r}-\sqrt{r_{0}}|^{2}\le|r-r_{0}|<\varepsilon^{2}$, which is absurd. By claims 1 and 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space and induction on $|S|$, each power $r\mapsto(\sqrt{r})^{|S|}$ is continuous on $[0,\infty)$, and by claim 5 of the same theorem so are its real multiples and the finite sums displayed above. This proves the first assertion.

Law. Let $(\mu_{k})$ in $\Sigma_{d,R}$ converge weak-star to $\mu$, and put $R'=\max(R,2)$. By (B) and (SC), every $\mu_{k}$ and $\mathrm{sc}_{d}$ lie in $\Sigma_{d,R'}$, and the constant sequence $(\mathrm{sc}_{d})$ converges weak-star to $\mathrm{sc}_{d}$ by Weak-Star Convergence of Noncommutative Laws §weak-star. By The Free Product of Two Noncommutative Laws: Existence, Uniqueness, Norm Bound, Substitutions and Weak-Star Continuity §continuity, $\mu_{k}\star\mathrm{sc}_{d}\to\mu\star\mathrm{sc}_{d}$ weak-star. For $p\in\mathcal{P}_{d}$, Clause 1 gives $(\mu_{k}\boxplus\mathrm{sc}_{d,t})(p)=(\mu_{k}\star\mathrm{sc}_{d})(\sigma_{\Gamma_{t}}(p))$ and $(\mu\boxplus\mathrm{sc}_{d,t})(p)=(\mu\star\mathrm{sc}_{d})(\sigma_{\Gamma_{t}}(p))$; applying Weak-Star Convergence of Noncommutative Laws §weak-star to the fixed polynomial $\sigma_{\Gamma_{t}}(p)\in\mathcal{P}_{2d}$, the real and imaginary parts of the former converge to those of the latter. Hence $\mu_{k}\boxplus\mathrm{sc}_{d,t}\to\mu\boxplus\mathrm{sc}_{d,t}$ weak-star.