Proof of The Wall Energy: Finiteness Forces the Norm Bound, Closed Sublevel Sets, and Monotonicity of the Wall Force along Couplings

Each result cited is applied with the data of its own statement. Throughout, $R>0$ is as in the statement. Linear maps commute with finite sums and the distributive laws of Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra extend to finite sums, by induction along the recursion of Finite Sum Notation in a Vector Space. For a real number $r$ and $z,z'\in\mathbb{C}$: $\operatorname{Re}r=r$, $\operatorname{Re}(z+z')=\operatorname{Re}z+\operatorname{Re}z'$ and $\operatorname{Re}(rz)=r\operatorname{Re}z$. Indeed, write $z=a+bi$ and $z'=a'+b'i$ with $a,b,a',b'$ real (claim 3 of Canonical Form and Arithmetic of Complex Numbers); by claim 1 of that lemma the identities $0,1$ and the additive inverses of real numbers in $\mathbb{C}$ are the real ones, so $r=r+0i$, and by claim 4 of that lemma $z+z'=(a+a')+(b+b')i$ and $rz=(r+0i)(a+bi)=(ra-0b)+(rb+0a)i=ra+(rb)i$, the operations in the parentheses being those of $\mathbb{R}$; the uniqueness in claim 3 of that lemma and Real and Imaginary Parts of a Complex Number give the three identities. Moreover $\overline{-1}=-1$, as $-1$ is real (claim 1 of Canonical Form and Arithmetic of Complex Numbers) and conjugation fixes real numbers (claim 1 of Properties of Complex Conjugation and Modulus); with claim 2 of Elementary Properties of a Complex Inner Product this gives $\langle-u,v\rangle=-\langle u,v\rangle$ in a complex inner product space. For $N\in\mathbb{N}$ read as the real number $\iota(N)$, with $\iota$ the canonical map of $\mathbb{R}$, $0\le N$ by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field. For $k\in\mathbb{N}$, $R^{2k}\neq0$ and $0\le R^{2k}$ by claims 4 and 5 of Properties of Natural Number Powers in a Field, so $R^{-2k}\neq0$ and $0\le R^{-2k}$ by claim 4 of Elementary Arithmetic in an Ordered Field.

Step 0 (Powers of variables). Let $r\in\mathbb{N}$, $j\in[r]$ and $s,t\in\mathbb{N}$, and let $w_{s}\in W_{1}$ be the word of length $s$ all of whose letters are $1$. By Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values, the power $x_{j}^{s}\in\mathcal{P}_{r}$ equals $\sigma(x_{w_{s}})$, where $\sigma:\mathcal{P}_{1}\to\mathcal{P}_{r}$ is the substitution of the $1$-tuple $(x_{j})$.

(P1) $x_{j}^{s}x_{j}^{t}=x_{j}^{s+t}$: by Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal §concatenation, $w_{s}w_{t}$ is the word of length $s+t$ all of whose letters are $1$, i.e. $w_{s+t}$, and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values gives $(x_{j})_{w_{s}w_{t}}=(x_{j})_{w_{s}}(x_{j})_{w_{t}}$.

(P2) $(x_{j}^{s})^{*}=x_{j}^{s}$: $x_{j}^{*}=x_{j}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint; $w_{s}^{\mathrm{rev}}=w_{s}$ by Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal §reversal, so $(x_{w_{s}})^{*}=x_{w_{s}}$ in $\mathcal{P}_{1}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint; and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §adjoint gives $(x_{j}^{s})^{*}=\sigma\bigl((x_{w_{s}})^{*}\bigr)=\sigma(x_{w_{s}})=x_{j}^{s}$.

(P3) For $j\in[d]$, $\iota^{1}(x_{j}^{s})=x_{j}^{s}$ and $\iota^{2}(x_{j}^{s})=x_{d+j}^{s}$, the right sides being powers in $\mathcal{P}_{2d}$. Indeed, by Couplings of Two Noncommutative Laws and Their Quadratic Cost §marginals, $\iota^{1}=\sigma_{b}$ with $b=(x_{1},\dots,x_{d})$ in $\mathcal{P}_{2d}$, and $\sigma_{b}(x_{j})=x_{j}$ 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, applied to the $1$-tuple $(x_{j})$ in $\mathcal{P}_{d}$ and to $b$, gives $\iota^{1}(\sigma(x_{w_{s}}))=\sigma'(x_{w_{s}})$, where $\sigma'$ is the substitution of the $1$-tuple $(x_{j})$ in $\mathcal{P}_{2d}$, and $\sigma'(x_{w_{s}})=x_{j}^{s}$ by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values. The same argument with $b=(x_{d+1},\dots,x_{2d})$ and $\sigma_{b}(x_{j})=x_{d+j}$ gives the claim for $\iota^{2}$.

Step 1 (A squaring inequality). Let $r\in\mathbb{N}$, let $\lambda$ be a tracial state on $\mathcal{P}_{r}$ and let $y\in\mathcal{P}_{r,\mathrm{sa}}$. Then $\lambda(y)$ is real by Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §adjoint, $\lambda(yy)=\lambda(y^{*}y)$ is real and nonnegative by condition (b) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state, and $\lambda(y)^{2}\le\lambda(yy)$. Indeed, $1^{*}=1$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint, $1y=y$ and $11=1$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials, and $\lambda(1)=1$ by condition (a) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state; so Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §cauchy-schwarz with $p=y$ and $q=1$ gives $|\lambda(y)|^{2}\le\lambda(yy)\cdot1$. As $\lambda(y)$ is real, $|\lambda(y)|$ equals $\lambda(y)$ or $-\lambda(y)$ by claim 8 of Properties of Complex Conjugation and Modulus, so $|\lambda(y)|^{2}=\lambda(y)^{2}$.

Step 2 (Part 1: the norm bound). Let $\lambda\in\mathcal{D}_{R}$, and by The Wall Energy of a Given Radius and the Wall Force of a Noncommutative Law §domain let $B$ be real with $E^{R}_{n}(\lambda)\le B$ for every $n\in\mathbb{N}$. Fix $j\in[d]$ and $m\in\mathbb{N}$. Define natural numbers $m_{\ell}$ ($\ell\in\mathbb{N}$) by $m_{1}=m$ and $m_{\ell+1}=m_{\ell}+m_{\ell}$, and put $a_{\ell}=\lambda(x_{j}^{2m_{\ell}})$, real and nonnegative by the preamble of The Wall Energy of a Given Radius and the Wall Force of a Noncommutative Law, and $t_{\ell}=R^{-2m_{\ell}}a_{\ell}$, which satisfies $0\le t_{\ell}$ by claim 5 of Elementary Arithmetic in an Ordered Field.

(i) $t_{\ell}\le B$ for every $\ell$. All summands $R^{-2k}\lambda(x_{i}^{2k})$ of $E^{R}_{m_{\ell}}(\lambda)$ are nonnegative (claim 5 of Elementary Arithmetic in an Ordered Field); $t_{\ell}$ is the summand with indices $i=j$ and $k=m_{\ell}\in[m_{\ell}]$. By claim 6 of Properties of Finite Sums, $t_{\ell}$ is at most the inner sum over $k$ for the index $j$; that inner sum is, by claims 5 and 6 there, a nonnegative summand of the outer sum and at most $E^{R}_{m_{\ell}}(\lambda)$. So $t_{\ell}\le E^{R}_{m_{\ell}}(\lambda)\le B$.

(ii) $t_{\ell}^{2}\le t_{\ell+1}$ for every $\ell$. Let $y=x_{j}^{2m_{\ell}}$, self-adjoint by (P2). Since $2m_{\ell+1}=2m_{\ell}+2m_{\ell}$, (P1) gives $yy=x_{j}^{2m_{\ell+1}}$, so Step 1 (with $r=d$; $\lambda$ is a tracial state by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law) gives $a_{\ell}^{2}\le a_{\ell+1}$. By Addition of Exponents for Natural Number Powers in a Field, $R^{2m_{\ell+1}}=R^{2m_{\ell}}R^{2m_{\ell}}$, so $R^{-2m_{\ell+1}}=R^{-2m_{\ell}}R^{-2m_{\ell}}$, the inverse of a product in a field being the product of the inverses. Here $0\le R^{-2m_{\ell}}R^{-2m_{\ell}}$, by claim 5 of Elementary Arithmetic in an Ordered Field applied to $0\le R^{-2m_{\ell}}$ with the factor $a=R^{-2m_{\ell}}\ge0$ (and $R^{-2m_{\ell}}\cdot0=0$). Multiplying $a_{\ell}^{2}\le a_{\ell+1}$ by this nonnegative element (claim 5 of Elementary Arithmetic in an Ordered Field once more),

$$t_{\ell}^{2}=R^{-2m_{\ell}}R^{-2m_{\ell}}a_{\ell}^{2}\le R^{-2m_{\ell+1}}a_{\ell+1}=t_{\ell+1}.$$

(iii) $t_{1}\le1$. Suppose not. Then $1<t_{1}$, the order of $\mathbb{R}$ being total, and $\delta=t_{1}-1$ satisfies $0\le\delta$ by claim 3 of Elementary Arithmetic in an Ordered Field, and $\delta\ne0$ because $t_{1}\ne1$; so $0<\delta$. We show by induction on $\ell\in\mathbb{N}$ that $1+\ell\delta\le t_{\ell}$ and $1+\delta\le t_{\ell}$, where $\ell$ is read as the real number $\iota(\ell)$. For $\ell=1$ both are equalities, as $\iota(1)=1$ by claim 1 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field. Assume both for $\ell$. Then $1\le1+\delta\le t_{\ell}$ and $0\le t_{\ell}$, so by claim 5 of Elementary Arithmetic in an Ordered Field, applied with the factors $t_{\ell}$ and $\delta$,

$$t_{\ell}(1+\delta)\le t_{\ell}t_{\ell},\qquad\delta\le t_{\ell}\delta .$$

Hence, by (ii) and the compatibility of the order with addition,

$$t_{\ell+1}\ge t_{\ell}^{2}\ge t_{\ell}+t_{\ell}\delta\ge t_{\ell}+\delta\ge1+\ell\delta+\delta=1+(\ell+1)\delta,$$

using $\iota(\ell+1)=\iota(\ell)+1$ (claim 1 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field); and $t_{\ell+1}\ge t_{\ell}+\delta\ge1+\delta+\delta\ge1+\delta$ as $0\le\delta$. By claim 2 of The Archimedean Property of the Real Numbers, with $x=B-1$ and $\varepsilon=\delta$, there is $\ell\in\mathbb{N}$ with $B-1<\ell\delta$, hence $B<1+\ell\delta\le t_{\ell}$, contradicting (i). So $t_{1}\le1$.

Since $t_{1}=R^{-2m}\lambda(x_{j}^{2m})$, multiplying by $R^{2m}\ge0$ (claim 5 of Elementary Arithmetic in an Ordered Field) gives $\lambda(x_{j}^{2m})\le R^{2m}$. As $j\in[d]$ and $m\in\mathbb{N}$ were arbitrary and $\lambda$ is a tracial state on $\mathcal{P}_{d}$, The Norm Bound of a Noncommutative Law: Multiplication by a Variable is Bounded, the Bound is Detected by Even Moments, and Laws Pull Back under Self-Adjoint Substitutions §criterion gives $\lambda\in\Sigma_{d,R}$. Hence $\mathcal{D}_{R}\subseteq\Sigma_{d,R}$.

Moreover $0\le E^{R}_{n}(\lambda)$ for every $n$, by claim 5 of Properties of Finite Sums applied to the inner and then to the outer sum, the summands being nonnegative as noted in (i). By The Wall Energy of a Given Radius and the Wall Force of a Noncommutative Law §energy the sequence $(E^{R}_{n}(\lambda))_{n}$ converges to $\mathcal{W}_{R}(\lambda)$, and the constant sequence $0$ converges to $0$ (Limit of a Sequence of Real Numbers, as $|0-0|=0<\varepsilon$); claim 1 of Order Properties of Limits of Real Sequences gives $0\le\mathcal{W}_{R}(\lambda)$.

Step 3 (Partial energies lie below the wall energy). Let $\lambda\in\mathcal{D}_{R}$ and $n\in\mathbb{N}$. By The Wall Energy of a Given Radius and the Wall Force of a Noncommutative Law §energy the sequence $(E^{R}_{n'}(\lambda))_{n'\in\mathbb{N}}$ is nondecreasing and bounded above, so by A Bounded Monotone Sequence of Real Numbers Converges §nondecreasing it converges to the supremum of its set of terms; by claim 1 of Uniqueness of Limits and Boundedness of Convergent Real Sequences this supremum is $\mathcal{W}_{R}(\lambda)$. Being an upper bound of the terms, it satisfies $E^{R}_{n}(\lambda)\le\mathcal{W}_{R}(\lambda)$.

Step 4 (Part 2: closed sublevel sets). Let $c$, $(\lambda_{m})$ and $\lambda$ be as in claim 2. Fix $n\in\mathbb{N}$. Each $\lambda_{m}$ lies in $\mathcal{D}_{R}\subseteq\Sigma_{d}$ (The Wall Energy of a Given Radius and the Wall Force of a Noncommutative Law §domain). For $j\in[d]$ and $k\in[n]$ the numbers $\lambda_{m}(x_{j}^{2k})$ and $\lambda(x_{j}^{2k})$ are real (preamble of The Wall Energy of a Given Radius and the Wall Force of a Noncommutative Law), hence equal to their real parts, so Weak-Star Convergence of Noncommutative Laws §weak-star with $p=x_{j}^{2k}$ says that the real sequence $(\lambda_{m}(x_{j}^{2k}))_{m}$ converges to $\lambda(x_{j}^{2k})$. By claims 3 and 1 of Arithmetic of Limits of Real Sequences (scalar multiples by $R^{-2k}$, then sums, applied along the recursions of the finite sums over $k$ and over $j$), $(E^{R}_{n}(\lambda_{m}))_{m}$ converges to $E^{R}_{n}(\lambda)$. By Step 3 and the hypothesis, $E^{R}_{n}(\lambda_{m})\le\mathcal{W}_{R}(\lambda_{m})\le c$ for every $m$, so claim 1 of Order Properties of Limits of Real Sequences, compared with the constant sequence $c$, gives $E^{R}_{n}(\lambda)\le c$. As $n$ was arbitrary and $\lambda\in\Sigma_{d}$, $\lambda\in\mathcal{D}_{R}$ by The Wall Energy of a Given Radius and the Wall Force of a Noncommutative Law §domain. By The Wall Energy of a Given Radius and the Wall Force of a Noncommutative Law §energy, $(E^{R}_{n}(\lambda))_{n}$ converges to $\mathcal{W}_{R}(\lambda)$, and claim 1 of Order Properties of Limits of Real Sequences once more gives $\mathcal{W}_{R}(\lambda)\le c$.

Step 5 (Part 3: the marginal images of the force polynomials). Let $\mu$, $\nu$ and $\gamma\in\Pi(\mu,\nu)$ be as in claim 3. By Couplings of Two Noncommutative Laws and Their Quadratic Cost §coupling, $\gamma$ is a tracial state on $\mathcal{P}_{2d}$ with $\gamma\circ\iota^{1}=\mu$ and $\gamma\circ\iota^{2}=\nu$, and $\gamma\in\Sigma_{2d}$ by the preamble of Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws. Hence, by Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §isometries, $V^{1}_{\gamma}\in\mathcal{L}(\mathcal{H}_{\mu},\mathcal{H}_{\gamma})$ and $V^{2}_{\gamma}\in\mathcal{L}(\mathcal{H}_{\nu},\mathcal{H}_{\gamma})$, with $V^{1}_{\gamma}\widehat{p}^{\,\mu}=\widehat{\iota^{1}(p)}^{\,\gamma}$ and $V^{2}_{\gamma}\widehat{p}^{\,\nu}=\widehat{\iota^{2}(p)}^{\,\gamma}$ for $p\in\mathcal{P}_{d}$. For $n\in\mathbb{N}$ and $j\in[d]$ put

$$G^{1}_{n,j}=\sum_{k=1}^{n}2k\,R^{-2k}\,x_{j}^{2k-1},\qquad G^{2}_{n,j}=\sum_{k=1}^{n}2k\,R^{-2k}\,x_{d+j}^{2k-1}\qquad\text{in }\mathcal{P}_{2d}.$$

Since $\iota^{1}$ and $\iota^{2}$ are linear (Substitution of Noncommutative Polynomials into the Variables §substitution), (P3) gives $\iota^{1}(F^{R}_{n,j})=G^{1}_{n,j}$ and $\iota^{2}(F^{R}_{n,j})=G^{2}_{n,j}$; therefore $V^{1}_{\gamma}\widehat{F^{R}_{n,j}}^{\,\mu}=\widehat{G^{1}_{n,j}}^{\,\gamma}$ and $V^{2}_{\gamma}\widehat{F^{R}_{n,j}}^{\,\nu}=\widehat{G^{2}_{n,j}}^{\,\gamma}$.

Step 6 (Continuity of the pairing). Let $H,K$ be complex Hilbert spaces, $T\in\mathcal{L}(H,K)$, $\eta\in K$, and let $(\xi_{n})$ converge to $\xi$ in $H$. Then $(\operatorname{Re}\langle T\xi_{n},\eta\rangle_{K})_{n}$ converges to $\operatorname{Re}\langle T\xi,\eta\rangle_{K}$. Indeed, by the linearity of $T$, claims 1 and 2 of Elementary Properties of a Complex Inner Product with $\overline{-1}=-1$, and the additivity of $\operatorname{Re}$ with $\operatorname{Re}(-z)=-\operatorname{Re}z$ (the case $r=-1$ above), the difference $\operatorname{Re}\langle T\xi_{n},\eta\rangle-\operatorname{Re}\langle T\xi,\eta\rangle$ equals $\operatorname{Re}z_{n}$ with $z_{n}=\langle T(\xi_{n}-\xi),\eta\rangle$. By claim 6 of Properties of Complex Conjugation and Modulus and claim 6 of Properties of the Absolute Value in an Ordered Field, $|\operatorname{Re}z_{n}|\le|z_{n}|$. By Cauchy-Schwarz Inequality in a Complex Inner Product Space and Norm Induced by a Complex Inner Product, $|z_{n}|^{2}\le(\lVert T(\xi_{n}-\xi)\rVert\,\lVert\eta\rVert)^{2}$, so $|z_{n}|\le\lVert T(\xi_{n}-\xi)\rVert\,\lVert\eta\rVert$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field; and $\lVert T(\xi_{n}-\xi)\rVert\le\lVert T\rVert_{\mathrm{op}}\lVert\xi_{n}-\xi\rVert$ because $\lVert T\rVert_{\mathrm{op}}$ is a bound for $T$ (Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §least-bound). Thus, by claim 5 of Elementary Arithmetic in an Ordered Field,

$$\bigl|\operatorname{Re}\langle T\xi_{n},\eta\rangle-\operatorname{Re}\langle T\xi,\eta\rangle\bigr|\le\lVert\eta\rVert\,\lVert T\rVert_{\mathrm{op}}\,\lVert\xi_{n}-\xi\rVert .$$

The real sequence $(\lVert\xi_{n}-\xi\rVert)_{n}$ converges to $0$ by Convergent Sequence in a Metric Space and Limit of a Sequence of Real Numbers, the metric being $(u,u')\mapsto\lVert u-u'\rVert$ (Complex Hilbert Space); so the right side converges to $0$ by claim 3 of Arithmetic of Limits of Real Sequences, and claim 3 of Order Properties of Limits of Real Sequences gives the assertion.

Step 7 (Part 3: passage to the limit). By The Wall Energy of a Given Radius and the Wall Force of a Noncommutative Law §force, for every $j\in[d]$ the sequences $(\widehat{F^{R}_{n,j}}^{\,\mu})_{n}$ and $(\widehat{F^{R}_{n,j}}^{\,\nu})_{n}$ converge to $F^{R}_{j}(\mu)$ in $\mathcal{H}_{\mu}$ and to $F^{R}_{j}(\nu)$ in $\mathcal{H}_{\nu}$. By Step 6 with $T=V^{1}_{\gamma}$, respectively $T=V^{2}_{\gamma}$, and $\eta=\widehat{h_{j}}$, together with Step 5, the real sequences $(\operatorname{Re}\langle\widehat{G^{1}_{n,j}},\widehat{h_{j}}\rangle_{\mathcal{H}_{\gamma}})_{n}$ and $(\operatorname{Re}\langle\widehat{G^{2}_{n,j}},\widehat{h_{j}}\rangle_{\mathcal{H}_{\gamma}})_{n}$ converge to $\operatorname{Re}\langle V^{1}_{\gamma}F^{R}_{j}(\mu),\widehat{h_{j}}\rangle_{\mathcal{H}_{\gamma}}$ and $\operatorname{Re}\langle V^{2}_{\gamma}F^{R}_{j}(\nu),\widehat{h_{j}}\rangle_{\mathcal{H}_{\gamma}}$. By claims 1 and 3 of Arithmetic of Limits of Real Sequences (sums over $j$ along the recursion of the finite sum, then the difference) and Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §coupling-pairing, the real sequence

$$\Delta_{n}=\sum_{j=1}^{d}\operatorname{Re}\langle\widehat{G^{1}_{n,j}},\widehat{h_{j}}\rangle_{\mathcal{H}_{\gamma}}-\sum_{j=1}^{d}\operatorname{Re}\langle\widehat{G^{2}_{n,j}},\widehat{h_{j}}\rangle_{\mathcal{H}_{\gamma}}$$

converges to $\mathcal{J}^{1}_{\gamma}(F^{R}(\mu))-\mathcal{J}^{2}_{\gamma}(F^{R}(\nu))$.

Step 8 (Part 3: each $\Delta_{n}$ is nonnegative). Fix $n\in\mathbb{N}$ and $j\in[d]$, and put $P=G^{1}_{n,j}-G^{2}_{n,j}$. The canonical map is complex-linear by The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §isometry, so, with claims 1 and 2 of Elementary Properties of a Complex Inner Product and $\overline{-1}=-1$,

$$\langle\widehat{G^{1}_{n,j}},\widehat{h_{j}}\rangle_{\mathcal{H}_{\gamma}}-\langle\widehat{G^{2}_{n,j}},\widehat{h_{j}}\rangle_{\mathcal{H}_{\gamma}}=\langle\widehat{P},\widehat{h_{j}}\rangle_{\mathcal{H}_{\gamma}}=\gamma(P^{*}h_{j}),$$

the last equality by The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §isometry, as $\mathcal{H}_{\gamma}$, its inner product and its classes are the completion, pairing and canonical images for the form $(p,q)\mapsto\gamma(p^{*}q)$ (The Complex GNS Space of a Tracial State on Noncommutative Polynomials §gns, The Complex GNS Space of a Tracial State on Noncommutative Polynomials §classes). By the vector space rules of Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §vector-space,

$$P=\sum_{k=1}^{n}2k\,R^{-2k}\bigl(x_{j}^{2k-1}-x_{d+j}^{2k-1}\bigr).$$

Each $x_{j}^{2k-1}$ and $x_{d+j}^{2k-1}$ is self-adjoint by (P2), and the coefficients are real, so $P\in\mathcal{P}_{2d,\mathrm{sa}}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §self-adjoint. Hence, by distributivity and the linearity of $\gamma$,

$$\gamma(P^{*}h_{j})=\gamma(Ph_{j})=\sum_{k=1}^{n}2k\,R^{-2k}\,\gamma\bigl((x_{j}^{2k-1}-x_{d+j}^{2k-1})(x_{j}-x_{d+j})\bigr).$$

Apply Monotonicity of Odd Powers under a Tracial State §monotone with $2d$ in place of $m$, the tracial state $\gamma$ on $\mathcal{P}_{2d}$, $a=x_{j}$ and $b=x_{d+j}$ (self-adjoint by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint), for each $k\in[n]$, with exponent $2k-1$ (the lemma's odd exponent, not the present $n$); its powers are the products along words for the $1$-tuples $(x_{j})$ and $(x_{d+j})$, as here. So each value of $\gamma$ in the last sum is real and nonnegative. Since $0\le2k$ and $0\le R^{-2k}$, claim 5 of Elementary Arithmetic in an Ordered Field and claim 5 of Properties of Finite Sums show that $\gamma(P^{*}h_{j})$ is real and nonnegative. By the additivity of $\operatorname{Re}$ and $\operatorname{Re}r=r$ for real $r$,

$$\operatorname{Re}\langle\widehat{G^{1}_{n,j}},\widehat{h_{j}}\rangle_{\mathcal{H}_{\gamma}}-\operatorname{Re}\langle\widehat{G^{2}_{n,j}},\widehat{h_{j}}\rangle_{\mathcal{H}_{\gamma}}=\gamma(P^{*}h_{j})\ge0 .$$

By claims 2 and 3 of Properties of Finite Sums, $\Delta_{n}$ is the sum over $j\in[d]$ of these differences, so $0\le\Delta_{n}$ by claim 5 there.

Step 9 (Conclusion of Part 3). By Steps 7 and 8 and claim 1 of Order Properties of Limits of Real Sequences, compared with the constant sequence $0$,

$$\mathcal{J}^{1}_{\gamma}\bigl(F^{R}(\mu)\bigr)-\mathcal{J}^{2}_{\gamma}\bigl(F^{R}(\nu)\bigr)\ge0 .$$