Proof of Free Semicircular Variables are Characterised by the Schwinger-Dyson Equation Relative to the Other Variables; Semicircular Systems and Rotations

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

Notation. Write $N=m+n$. Let $\iota^{2}:\mathcal{P}_{n}\to\mathcal{P}_{N}$ be the substitution of the tuple $(x_{m+1},\dots,x_{N})$, as in Freeness of Two Groups of Variables under a Noncommutative Law §free, and for $j\in[n]$ put $y_{j}=x_{m+j}\in\mathcal{P}_{N}$. For a law $\lambda$ of $k$ variables and $r\in[k]$, $\partial^{\lambda}_{r}$ is the free difference quotient of $\lambda$ in $x_{r}$, and $\delta_{ij}$, $w_{<l}$, $w_{>l}$ are as in the preamble of The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law. Recall that $1=x_{\varnothing}$ by The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §monomials. For a word $v=(v_{1},\dots,v_{h})\in W_{n}$ let $v^{+}=(v_{1}+m,\dots,v_{h}+m)\in W_{N}$, with $\varnothing^{+}=\varnothing$; every word in $W_{m}$ is also regarded as a word in $W_{N}$. Put $\mathcal{A}_{1}=\iota^{1}(\mathcal{P}_{m})$ and $\mathcal{A}_{2}=\iota^{2}(\mathcal{P}_{n})$, subsets of $\mathcal{P}_{N}$. For $\gamma\in\Sigma_{N}$, an element $c\in\mathcal{P}_{N}$ is called $\gamma$-centred if $\gamma(c)=0$.

Step 0: elementary facts. (U) Let $k\in\mathbb{N}$. Two linear maps $\mathcal{P}_{k}\to\mathbb{C}$ that agree on every monomial $x_{w}$, $w\in W_{k}$, are equal, by the uniqueness in claim (a) of Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §linear-extension. (M) For $w\in W_{m}$ and $v\in W_{n}$ we have $\iota^{1}(x_{w})=x_{w}$ and $\iota^{2}(x_{v})=x_{v^{+}}$: by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values these are the products along $w$ and $v$ of the substituted tuples, namely $x_{w_{1}}\cdots x_{w_{h}}$ and $x_{v_{1}+m}\cdots x_{v_{h'}+m}$ ($h,h'$ the lengths), and these equal $x_{w}$ and $x_{v^{+}}$ by $x_{u}x_{u'}=x_{uu'}$ (Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials) and induction on the length, the empty word giving $1$ on both sides. In particular $\iota^{2}(x_{j})=y_{j}$ for $j\in[n]$. (A) $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ contain $1$ and are closed under sums, complex multiples and products, since $\iota^{1}$ and $\iota^{2}$ are linear, map $1$ to $1$ and are multiplicative by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §homomorphism. Hence, for $\gamma\in\Sigma_{N}$, $e\in\{1,2\}$ and $c\in\mathcal{A}_{e}$, the element $c^{\circ}=c-\gamma(c)1$ lies in $\mathcal{A}_{e}$ and is $\gamma$-centred, because $\gamma(1)=1$ by (a) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state; and $c=c^{\circ}+\gamma(c)1$.

Step 1: a product rule. Fix $\lambda\in\Sigma_{N}$ and $r\in[N]$. For $P,Q\in\mathcal{P}_{N}$ let $\ell_{P,Q}:\mathcal{P}_{N}\to\mathbb{C}$ be the linear map with $\ell_{P,Q}(x_{\varnothing})=0$ and

$$\ell_{P,Q}(x_{w})=\sum_{l=1}^{h}\delta_{w_{l}r}\,\lambda(Px_{w_{<l}})\,\lambda(x_{w_{>l}}Q)$$

for every word $w\in W_{N}$ of length $h\in\mathbb{N}$; it exists and is unique by claim (a) of Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §linear-extension. Since $1p=p1=p$ (Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials), $\ell_{1,1}$ agrees on monomials with $\partial^{\lambda}_{r}$ as defined in The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law §difference-quotient, so $\ell_{1,1}=\partial^{\lambda}_{r}$ by (U).

(1.1) For $P,P',Q,Q'\in\mathcal{P}_{N}$ and $c\in\mathbb{C}$, $\ell_{P+cP',Q}=\ell_{P,Q}+c\,\ell_{P',Q}$ and $\ell_{P,Q+cQ'}=\ell_{P,Q}+c\,\ell_{P,Q'}$: both sides of each identity are linear maps that agree on monomials by the distributive laws of Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra and the linearity of $\lambda$, so they are equal by (U).

(1.2) For every integer $s\ge0$ and all $c_{0},\dots,c_{s}\in\mathcal{P}_{N}$, writing $c_{<i}=c_{0}\cdots c_{i-1}$ and $c_{>i}=c_{i+1}\cdots c_{s}$ (empty products being $1$; products are unambiguous by associativity, Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra),

$$\partial^{\lambda}_{r}(c_{0}c_{1}\cdots c_{s})=\sum_{i=0}^{s}\ell_{c_{<i},c_{>i}}(c_{i}).$$

First let every $c_{i}$ be a monomial, $c_{i}=x_{w^{(i)}}$ with $w^{(i)}\in W_{N}$ of length $h_{i}\ge0$, and let $w=w^{(0)}w^{(1)}\cdots w^{(s)}$, of length $h=h_{0}+\dots+h_{s}$. By Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials, $c_{0}\cdots c_{s}=x_{w}$, $c_{<i}=x_{w^{(0)}\cdots w^{(i-1)}}$ and $c_{>i}=x_{w^{(i+1)}\cdots w^{(s)}}$. If $h=0$ both sides vanish, since $\partial^{\lambda}_{r}(1)=0$ and $\ell_{P,Q}(x_{\varnothing})=0$. If $h\ge1$, the assignment $(i,l')\mapsto l=h_{0}+\dots+h_{i-1}+l'$, for $i\in\{0,\dots,s\}$ with $h_{i}\ge1$ and $l'\in[h_{i}]$, is a bijection onto $[h]$, and for corresponding indices $w_{l}=w^{(i)}_{l'}$, $w_{<l}=w^{(0)}\cdots w^{(i-1)}w^{(i)}_{<l'}$ and $w_{>l}=w^{(i)}_{>l'}w^{(i+1)}\cdots w^{(s)}$; hence $x_{w_{<l}}=c_{<i}\,x_{w^{(i)}_{<l'}}$ and $x_{w_{>l}}=x_{w^{(i)}_{>l'}}\,c_{>i}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials. Grouping the $h$ terms of the sum defining $\partial^{\lambda}_{r}(x_{w})$ according to $i$ therefore gives $\sum_{i}\ell_{c_{<i},c_{>i}}(x_{w^{(i)}})$, the indices $i$ with $h_{i}=0$ contributing $\ell_{c_{<i},c_{>i}}(x_{\varnothing})=0$. For general factors, let $E(i_{0})$, for $i_{0}\in\{0,\dots,s+1\}$, be the statement that (1.2) holds for all $(c_{0},\dots,c_{s})$ such that $c_{i}$ is a monomial for every $i\ge i_{0}$. $E(0)$ is the monomial case just proved. Assume $E(i_{0})$ with $i_{0}\le s$, and fix $c_{i}$ for $i\neq i_{0}$, arbitrary for $i<i_{0}$ and monomials for $i>i_{0}$. As functions of $c_{i_{0}}\in\mathcal{P}_{N}$, the left side is linear by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra and the linearity of $\partial^{\lambda}_{r}$, and the right side is linear: the term $i=i_{0}$ because $\ell_{c_{<i_{0}},c_{>i_{0}}}$ is linear, and the terms $i\neq i_{0}$ by (1.1), because $c_{i_{0}}$ enters $c_{<i}$ or $c_{>i}$ linearly by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra. By $E(i_{0})$ the two sides agree when $c_{i_{0}}$ is a monomial, so they agree for all $c_{i_{0}}$ by (U); this is $E(i_{0}+1)$. By induction on $i_{0}$, $E(s+1)$ holds, which is (1.2).

Step 2: reduction to alternating products. Fix $\gamma\in\Sigma_{N}$. For an integer $k\ge0$ let $\mathcal{T}_{k}$ be the set of products $b_{0}a_{1}b_{1}a_{2}b_{2}\cdots a_{k}b_{k}$ with $a_{1},\dots,a_{k}\in\mathcal{A}_{1}$ and $b_{0},\dots,b_{k}\in\mathcal{A}_{2}$ (so $\mathcal{T}_{0}=\mathcal{A}_{2}$), and for $k\in\mathbb{N}$ let $\mathcal{S}_{k}\subseteq\mathcal{T}_{k}$ be the set of those products in which $a_{1},\dots,a_{k}$ and $b_{1},\dots,b_{k-1}$ are $\gamma$-centred ($b_{0}$ and $b_{k}$ arbitrary in $\mathcal{A}_{2}$). (2.1) If $\mathcal{V}\subseteq\mathcal{P}_{N}$ is closed under sums and complex multiples and contains $\mathcal{A}_{2}$ and $\mathcal{S}_{k}$ for every $k\in\mathbb{N}$, then $\mathcal{V}$ contains every monomial of $\mathcal{P}_{N}$.

We show by strong induction on $k\ge0$ that $\mathcal{T}_{k}\subseteq\mathcal{V}$. For $k=0$ this is the hypothesis $\mathcal{A}_{2}\subseteq\mathcal{V}$. Let $k\ge1$, assume $\mathcal{T}_{k'}\subseteq\mathcal{V}$ for every $k'<k$, and let $q=b_{0}a_{1}b_{1}\cdots a_{k}b_{k}\in\mathcal{T}_{k}$. By Step 0 (A) write $a_{i}=a_{i}^{\circ}+\gamma(a_{i})1$ for $i\in[k]$ and $b_{i}=b_{i}^{\circ}+\gamma(b_{i})1$ for $i\in[k-1]$, with $a_{i}^{\circ}\in\mathcal{A}_{1}$ and $b_{i}^{\circ}\in\mathcal{A}_{2}$ $\gamma$-centred. Expanding $q$ by the distributive laws of Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra writes it as a sum of $2^{2k-1}$ terms, one for each choice, for each of these $2k-1$ factors, of its centred part or of its scalar part. The term that chooses every centred part is $b_{0}a_{1}^{\circ}b_{1}^{\circ}\cdots a_{k}^{\circ}b_{k}\in\mathcal{S}_{k}\subseteq\mathcal{V}$. In every other term at least one factor is replaced by a scalar multiple of $1$; pulling out the scalars and multiplying together adjacent factors that lie in the same $\mathcal{A}_{e}$, which stays in $\mathcal{A}_{e}$ by Step 0 (A), writes the term as a complex multiple of an alternating product in $\mathcal{T}_{k'}$, where $k'$ is the number of maximal groups of kept factors $a_{i}^{\circ}$ not separated by a kept factor $b_{i}^{\circ}$ (if no $a_{i}^{\circ}$ is kept, the term lies in $\mathcal{A}_{2}=\mathcal{T}_{0}$). Here $k'<k$: if some $a_{i}$ was replaced, fewer than $k$ factors $a_{i}^{\circ}$ are kept, and otherwise some $b_{i}$ with $i\in[k-1]$ was replaced, which joins $a_{i}^{\circ}$ and $a_{i+1}^{\circ}$ into one group. By the induction hypothesis each such term lies in $\mathcal{V}$, and so does their sum $q$.

Now let $u\in W_{N}$. If no letter of $u$ lies in $[m]$, then $u=v^{+}$ for some $v\in W_{n}$ and $x_{u}=\iota^{2}(x_{v})\in\mathcal{A}_{2}$ by (M). Otherwise let $u_{1},\dots,u_{k}$ ($k\in\mathbb{N}$) be the maximal nonempty segments of consecutive letters of $u$ lying in $[m]$, in order, so that $u=v_{0}^{+}u_{1}v_{1}^{+}u_{2}\cdots u_{k}v_{k}^{+}$ with $v_{0},\dots,v_{k}\in W_{n}$. Then $x_{u}=x_{v_{0}^{+}}x_{u_{1}}x_{v_{1}^{+}}\cdots x_{u_{k}}x_{v_{k}^{+}}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials, with $x_{u_{i}}=\iota^{1}(x_{u_{i}})\in\mathcal{A}_{1}$ and $x_{v_{i}^{+}}=\iota^{2}(x_{v_{i}})\in\mathcal{A}_{2}$ by (M); so $x_{u}\in\mathcal{T}_{k}\subseteq\mathcal{V}$.

Step 3: a vanishing lemma for free laws. Let $\gamma\in\Sigma_{N}$ be free. (3.1) For $k\in\mathbb{N}$, $\gamma$-centred $a_{1},\dots,a_{k}\in\mathcal{A}_{1}$, $\gamma$-centred $b_{1},\dots,b_{k-1}\in\mathcal{A}_{2}$ and any $e\in\mathcal{A}_{2}$,

$$\gamma(e\,a_{1}b_{1}a_{2}\cdots b_{k-1}a_{k})=0.$$

Indeed, $e=e^{\circ}+\gamma(e)1$ by Step 0 (A), so by linearity of $\gamma$ the left side equals $\gamma(e^{\circ}a_{1}b_{1}\cdots b_{k-1}a_{k})+\gamma(e)\,\gamma(a_{1}b_{1}\cdots b_{k-1}a_{k})$. Each element of $\mathcal{A}_{1}$ is $\iota^{1}(p)$ with $p\in\mathcal{P}_{m}$ and each element of $\mathcal{A}_{2}$ is $\iota^{2}(p)$ with $p\in\mathcal{P}_{n}$. The first product has $2k$ $\gamma$-centred factors from $\mathcal{A}_{2},\mathcal{A}_{1},\mathcal{A}_{2},\dots,\mathcal{A}_{1}$ alternately, and the second has $2k-1$ $\gamma$-centred factors from $\mathcal{A}_{1},\mathcal{A}_{2},\dots,\mathcal{A}_{1}$ alternately; so both values are $0$ by the definition of freeness, Freeness of Two Groups of Variables under a Noncommutative Law §free.

Step 4: the direction ($\Rightarrow$) of clause 1. Let $\gamma=\alpha\star\mathrm{sc}_{n}$. By The Free Product of Two Noncommutative Laws: Existence, Uniqueness, Norm Bound, Substitutions and Weak-Star Continuity §product, $\gamma\in\Sigma_{N}$ is free, $\gamma\circ\iota^{1}=\alpha$ and $\gamma\circ\iota^{2}=\mathrm{sc}_{n}$. By The Semicircular Law and Its Scalings §standard, $\mathrm{sc}_{n}$ satisfies the Schwinger-Dyson equation: $\mathrm{sc}_{n}(x_{j}r)=\partial^{\mathrm{sc}_{n}}_{j}(r)$ for $j\in[n]$ and $r\in\mathcal{P}_{n}$. Fix $j\in[n]$ and let $\mathcal{V}$ be the set of $q\in\mathcal{P}_{N}$ with $\gamma(y_{j}q)=\partial^{\gamma}_{m+j}(q)$. Both sides are linear in $q$ (by the linearity of $\gamma$ and of $\partial^{\gamma}_{m+j}$ and by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra), so $\mathcal{V}$ is closed under sums and complex multiples. We show below that $\mathcal{A}_{2}\subseteq\mathcal{V}$ (4a) and that $\mathcal{S}_{k}\subseteq\mathcal{V}$ for every $k\in\mathbb{N}$ (4b), where $\mathcal{S}_{k}$ is formed with this $\gamma$. Then $\mathcal{V}$ contains every monomial by (2.1), so the linear maps $q\mapsto\gamma(y_{j}q)$ and $\partial^{\gamma}_{m+j}$ are equal by (U). As $j\in[n]$ was arbitrary, $\gamma$ satisfies the Schwinger-Dyson equation in the last $n$ variables.

(4a) Let $r\in\mathcal{P}_{n}$. By Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §homomorphism and (M), $y_{j}\,\iota^{2}(r)=\iota^{2}(x_{j}r)$, so $\gamma(y_{j}\,\iota^{2}(r))=\mathrm{sc}_{n}(x_{j}r)=\partial^{\mathrm{sc}_{n}}_{j}(r)$. On the other hand, the linear maps $r\mapsto\partial^{\gamma}_{m+j}(\iota^{2}(r))$ and $\partial^{\mathrm{sc}_{n}}_{j}$ from $\mathcal{P}_{n}$ to $\mathbb{C}$ agree on monomials: at $x_{\varnothing}=1$ both are $0$, as $\iota^{2}(1)=1$; and for $v\in W_{n}$ of length $h\in\mathbb{N}$ we have $\iota^{2}(x_{v})=x_{v^{+}}$ by (M), $(v^{+})_{l}=v_{l}+m$, $(v^{+})_{<l}=(v_{<l})^{+}$ and $(v^{+})_{>l}=(v_{>l})^{+}$, and $\gamma(x_{u^{+}})=\gamma(\iota^{2}(x_{u}))=\mathrm{sc}_{n}(x_{u})$ for $u\in W_{n}$, so by The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law §difference-quotient

$$\partial^{\gamma}_{m+j}(x_{v^{+}})=\sum_{l=1}^{h}\delta_{v_{l}+m,\,m+j}\,\gamma(x_{(v_{<l})^{+}})\,\gamma(x_{(v_{>l})^{+}})=\sum_{l=1}^{h}\delta_{v_{l}j}\,\mathrm{sc}_{n}(x_{v_{<l}})\,\mathrm{sc}_{n}(x_{v_{>l}})=\partial^{\mathrm{sc}_{n}}_{j}(x_{v}).$$

By (U) the two maps are equal, so $\partial^{\gamma}_{m+j}(\iota^{2}(r))=\partial^{\mathrm{sc}_{n}}_{j}(r)=\gamma(y_{j}\,\iota^{2}(r))$, i.e. $\iota^{2}(r)\in\mathcal{V}$.

(4b) Let $k\in\mathbb{N}$ and $q=b_{0}a_{1}b_{1}\cdots a_{k}b_{k}\in\mathcal{S}_{k}$. Left side. By (c) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state, applied to $y_{j}b_{0}a_{1}b_{1}\cdots b_{k-1}a_{k}$ and $b_{k}$, we get $\gamma(y_{j}q)=\gamma(e\,a_{1}b_{1}\cdots b_{k-1}a_{k})$ with $e=b_{k}y_{j}b_{0}$, which lies in $\mathcal{A}_{2}$ by (M) and Step 0 (A); hence $\gamma(y_{j}q)=0$ by (3.1). Right side. Apply (1.2) with $\lambda=\gamma$, $r=m+j$, $s=2k$ and $(c_{0},\dots,c_{2k})=(b_{0},a_{1},b_{1},\dots,a_{k},b_{k})$: $\partial^{\gamma}_{m+j}(q)$ is the sum of the terms $\ell_{P_{i},Q_{i}}(b_{i})$, $i\in\{0,\dots,k\}$, where $P_{i}=b_{0}a_{1}b_{1}\cdots b_{i-1}a_{i}$ (with $P_{0}=1$) and $Q_{i}=a_{i+1}b_{i+1}\cdots a_{k}b_{k}$ (with $Q_{k}=1$), and of terms $\ell_{P,Q}(a_{i})$, $i\in[k]$, for suitable $P,Q\in\mathcal{P}_{N}$. The terms of the second kind vanish: for all $P,Q$ the linear map $\ell_{P,Q}\circ\iota^{1}:\mathcal{P}_{m}\to\mathbb{C}$ vanishes at $x_{\varnothing}$ (as $\iota^{1}(1)=1$) and at $x_{w}$ for every nonempty $w\in W_{m}$, because $\iota^{1}(x_{w})=x_{w}$ by (M) and no letter of $w$ equals $m+j$, so every $\delta_{w_{l},m+j}$ is $0$; hence $\ell_{P,Q}\circ\iota^{1}=0$ by (U). The terms of the first kind vanish as well: fix $i\in\{0,\dots,k\}$; by (U) it suffices to see that $\ell_{P_{i},Q_{i}}\circ\iota^{2}$ vanishes on monomials. It vanishes at $x_{\varnothing}$, and for $v\in W_{n}$ of length $h\in\mathbb{N}$, by (M),

$$\ell_{P_{i},Q_{i}}(\iota^{2}(x_{v}))=\sum_{l=1}^{h}\delta_{v_{l}+m,\,m+j}\,\gamma(P_{i}\,g_{l})\,\gamma(g'_{l}\,Q_{i}),\qquad g_{l}=x_{(v_{<l})^{+}},\quad g'_{l}=x_{(v_{>l})^{+}},$$

where $g_{l},g'_{l}\in\mathcal{A}_{2}$ by (M). If $i\ge1$, then by (c) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state $\gamma(P_{i}g_{l})=\gamma(g_{l}P_{i})=\gamma((g_{l}b_{0})a_{1}b_{1}\cdots b_{i-1}a_{i})$, which is $0$ by (3.1) applied with $i$ in place of $k$ and $e=g_{l}b_{0}\in\mathcal{A}_{2}$. If $i=0$, then by (c) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state $\gamma(g'_{l}Q_{0})=\gamma(g'_{l}a_{1}b_{1}\cdots a_{k}b_{k})=\gamma((b_{k}g'_{l})a_{1}b_{1}\cdots b_{k-1}a_{k})$, which is $0$ by (3.1) with $e=b_{k}g'_{l}\in\mathcal{A}_{2}$. So every summand is $0$, whence $\partial^{\gamma}_{m+j}(q)=0=\gamma(y_{j}q)$ and $q\in\mathcal{V}$.

Step 5: the direction ($\Leftarrow$) of clause 1. Let $\gamma\in\Sigma_{N}$ with $\gamma\circ\iota^{1}=\alpha$ satisfy the Schwinger-Dyson equation in the last $n$ variables, and let $\gamma'=\alpha\star\mathrm{sc}_{n}$. By The Free Product of Two Noncommutative Laws: Existence, Uniqueness, Norm Bound, Substitutions and Weak-Star Continuity §product $\gamma'\circ\iota^{1}=\alpha$, and by Step 4 $\gamma'$ also satisfies the Schwinger-Dyson equation in the last $n$ variables. Let $\mathcal{K}$ be the set of integers $h\ge0$ such that $\gamma(x_{w})=\gamma'(x_{w})$ for every $w\in W_{N}$ of length $h$. We show by strong induction on $h$ that every $h\ge0$ lies in $\mathcal{K}$. For $h=0$, $\gamma(x_{\varnothing})=\gamma(1)=1=\gamma'(1)$ by (a) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state. Let $h\ge1$, assume $h'\in\mathcal{K}$ for every $h'<h$, and let $w\in W_{N}$ have length $h$. If every letter of $w$ lies in $[m]$, then $x_{w}=\iota^{1}(x_{w})$ by (M), so $\gamma(x_{w})=\alpha(x_{w})=\gamma'(x_{w})$. Otherwise there are $l\in[h]$ and $j\in[n]$ with $w_{l}=m+j$; put $u=w_{<l}$ and $v=w_{>l}$, so $x_{w}=x_{u}\,(y_{j}x_{v})$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials. By (c) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state and Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials, and then The Schwinger-Dyson Equation for a Noncommutative Law §relative and The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law §difference-quotient, for $\rho\in\{\gamma,\gamma'\}$

$$\rho(x_{w})=\rho(y_{j}\,x_{vu})=\partial^{\rho}_{m+j}(x_{vu}),$$

which is $0$ if $vu=\varnothing$, and otherwise equals $\sum_{l'=1}^{h-1}\delta_{(vu)_{l'},m+j}\,\rho(x_{(vu)_{<l'}})\,\rho(x_{(vu)_{>l'}})$, where $vu$ has length $h-1$ and the words $(vu)_{<l'}$ and $(vu)_{>l'}$ have lengths $l'-1<h$ and $h-1-l'<h$. By the induction hypothesis the two sums for $\rho=\gamma$ and $\rho=\gamma'$ agree term by term, so $\gamma(x_{w})=\gamma'(x_{w})$ and $h\in\mathcal{K}$. Hence $\gamma$ and $\gamma'$ agree on all monomials, and $\gamma=\gamma'=\alpha\star\mathrm{sc}_{n}$ by (U). Steps 4 and 5 prove clause 1.

Clause 2. Let $\gamma=\mathrm{sc}_{m}\star\mathrm{sc}_{n}\in\Sigma_{N}$. By Step 4 (with $\alpha=\mathrm{sc}_{m}$), $\gamma(x_{m+j}q)=\partial^{\gamma}_{m+j}(q)$ for all $j\in[n]$ and $q\in\mathcal{P}_{N}$. It remains to treat the first $m$ variables. Let $s=(x_{m+1},\dots,x_{N},x_{1},\dots,x_{m})$, an $N$-tuple in $\mathcal{P}_{N}$ with $s_{i}=x_{m+i}$ for $i\in[n]$ and $s_{n+i}=x_{i}$ for $i\in[m]$, and let $\beta=\gamma\circ\sigma_{s}$. By The Free Product of Two Noncommutative Laws: Existence, Uniqueness, Norm Bound, Substitutions and Weak-Star Continuity §symmetry, $\beta=\mathrm{sc}_{n}\star\mathrm{sc}_{m}$, the free product for the split of $n+m$ variables into the first $n$ and the last $m$. Clause 1, already proved and applied with the roles of $m$ and $n$ exchanged and with $\alpha=\mathrm{sc}_{n}$ (the hypothesis $\beta\circ\iota^{1}=\mathrm{sc}_{n}$ holding by The Free Product of Two Noncommutative Laws: Existence, Uniqueness, Norm Bound, Substitutions and Weak-Star Continuity §product), gives

$$\beta(x_{n+i}q')=\partial^{\beta}_{n+i}(q')\qquad(i\in[m],\ q'\in\mathcal{P}_{N}).$$

Let $A:[N]\times[N]\to\mathbb{R}$ be given by $A_{il}=1$ if $s_{i}=x_{l}$ and $A_{il}=0$ otherwise, and let $T=(A,0)$, an affine datum from $N$ to $N$ variables. Each row of $A$ has exactly one entry $1$, so the tuple of $T$ is $s$ and $\sigma_{T}=\sigma_{s}$ by Affine Data and Affine Substitutions of Noncommutative Polynomials §substitution; thus $\beta=\gamma\circ\sigma_{T}$. For $i\in[m]$ the only $i'\in[N]$ with $A_{i'i}\neq0$ is $i'=n+i$, so the chain rule gives $\partial^{\gamma}_{i}(\sigma_{s}(q'))=\partial^{\beta}_{n+i}(q')$ for $q'\in\mathcal{P}_{N}$. Now let $s'=(x_{n+1},\dots,x_{N},x_{1},\dots,x_{n})$. For $j\in[m]$, $\sigma_{s}(s'_{j})=\sigma_{s}(x_{n+j})=s_{n+j}=x_{j}$, and for $j=m+i$ with $i\in[n]$, $\sigma_{s}(s'_{j})=\sigma_{s}(x_{i})=s_{i}=x_{m+i}$, by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values; so Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §composition and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §identity give $\sigma_{s}(\sigma_{s'}(q))=q$ for every $q\in\mathcal{P}_{N}$. Let $i\in[m]$, $q\in\mathcal{P}_{N}$ and $q'=\sigma_{s'}(q)$. Since $x_{i}=s_{n+i}=\sigma_{s}(x_{n+i})$, Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §homomorphism yields

$$\gamma(x_{i}q)=\gamma(\sigma_{s}(x_{n+i}q'))=\beta(x_{n+i}q')=\partial^{\beta}_{n+i}(q')=\partial^{\gamma}_{i}(\sigma_{s}(q'))=\partial^{\gamma}_{i}(q).$$

Hence $\gamma$ satisfies the Schwinger-Dyson equation in all $N$ variables. So does $\mathrm{sc}_{N}$ by The Semicircular Law and Its Scalings §standard, and by The Vacuum Law of the Semicircular Operators is the Unique Noncommutative Law Satisfying the Schwinger-Dyson Equation §unique both equal the vacuum law $\lambda_{S}$ of $N$ semicircular operators; therefore $\mathrm{sc}_{m}\star\mathrm{sc}_{n}=\mathrm{sc}_{m+n}$.

Clause 3. Let $T=(O,0)$, an affine datum from $n$ to $k$ variables, with tuple $a^{T}_{i}=\sum_{l=1}^{n}O_{il}x_{l}$ ($i\in[k]$) by Affine Data and Affine Substitutions of Noncommutative Polynomials §tuple, and let $\mu=\mathrm{sc}_{n}\circ\sigma_{T}$, which lies in $\Sigma_{k}$ by Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §self-adjoint. Let $i\in[k]$ and $p\in\mathcal{P}_{k}$. By Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §homomorphism and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values, $\sigma_{T}(x_{i}p)=a^{T}_{i}\,\sigma_{T}(p)$. Using the linearity of $\mathrm{sc}_{n}$ and Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra, then the Schwinger-Dyson equation of $\mathrm{sc}_{n}$ (The Semicircular Law and Its Scalings §standard, The Schwinger-Dyson Equation for a Noncommutative Law §equation), then the chain rule for $T$ and $\gamma=\mathrm{sc}_{n}$, namely $\partial^{\mathrm{sc}_{n}}_{l}(\sigma_{T}(p))=\sum_{i'=1}^{k}O_{i'l}\,\partial^{\mu}_{i'}(p)$, and finally the hypothesis on $O$, we get

$$\mu(x_{i}p)=\sum_{l=1}^{n}O_{il}\,\mathrm{sc}_{n}(x_{l}\,\sigma_{T}(p))=\sum_{l=1}^{n}O_{il}\,\partial^{\mathrm{sc}_{n}}_{l}(\sigma_{T}(p))=\sum_{i'=1}^{k}\Bigl(\sum_{l=1}^{n}O_{il}O_{i'l}\Bigr)\partial^{\mu}_{i'}(p)=\partial^{\mu}_{i}(p).$$

So $\mu$ satisfies the Schwinger-Dyson equation for $k$ variables. So does $\mathrm{sc}_{k}$ (The Semicircular Law and Its Scalings §standard), and by The Vacuum Law of the Semicircular Operators is the Unique Noncommutative Law Satisfying the Schwinger-Dyson Equation §unique both equal the vacuum law $\lambda_{S}$ of $k$ semicircular operators; hence $\mathrm{sc}_{n}\circ\sigma_{(O,0)}=\mathrm{sc}_{k}$.