Proof of The Free Unitary Heat Generator: Bound, Continuity and Strict Dissipation in the Length-Weighted Gauge

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

Conventions. Throughout, $d\in\mathbb{N}$ is fixed and $\theta=\theta_{d}=\frac{1}{6144\,d}$, so that by The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §weights one has $c_{\varnothing}=1$ and $c_{w}=\gamma_{k}$ for $w\in W^{\circ}_{d,k}$, $k\in\mathbb{N}$, where $\gamma_{k}=\frac{\theta^{k}}{(k+1)^{4}}$; by the same clause $0<\gamma_{k}\le\theta^{k}$. In particular the weight of a nonempty cyclically reduced word depends only on its length. For a word $w$ of length $k$ and $p\in[k]$ we write $\varepsilon_{p}=\varepsilon(w_{p})$, and we use $P(w)$, $\sigma_{p}$, $\tau_{q}$, $y_{pq}=y_{pq}(w)$ and $x_{pq}=x_{pq}(w)$ as in The Free Unitary Heat Generator on Unitary Laws §positions and The Free Unitary Heat Generator on Unitary Laws §pieces. A sum over an empty index set is read as $0$. For finite sums of real numbers the following rules are used without further mention: termwise inequalities pass to the sums by Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §comparison; a sum of nonnegative terms is nonnegative and does not decrease when the index set is enlarged, by Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §nonnegative and Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §monotone; sums are additive and homogeneous, may be reindexed along bijections and agree with sums over a numerical range, by claims 3, 4, 2 and 1 of Properties of a Sum over a Finite Index Set; they split over disjoint unions by Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §disjoint-union; iterated sums are sums over dependent pairs by Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §pairs; and terms equal to $0$ may be dropped by Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §vanishing. For complex numbers we use the multiplicativity of the modulus and the triangle inequality, claims 4 and 7 of Properties of Complex Conjugation and Modulus, and the fact that the modulus of a finite sum is at most the sum of the moduli, Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §modulus. Since every sign is $1$ or $-1$, claim 8 of Properties of Complex Conjugation and Modulus gives $|\varepsilon_{p}|=1$.

Step 0 (elementary facts).

(E1) For every letter $l\in[2d]$ one has $(l^{-1})^{-1}=l$; moreover $\varepsilon(l)=1$ if and only if $l\le d$, in which case $g(l)=l$, and if $\varepsilon(l)=-1$ then $l=d+g(l)$. All of this is read off Words in Unitary Letters: Generators, Signs, Inverse Letters, Lengths and Adjoint Words §letters: if $l\le d$, then $\varepsilon(l)=1$, $g(l)=l$ and $l^{-1}=d+l$, and since $d<d+l$ the inverse letter of $d+l$ is the unique $j$ with $d+l=d+j$, that is $l$; if $d<l$, then $l=d+j$ with $j=g(l)\in[d]$, $\varepsilon(l)=-1$, $l^{-1}=j\le d$, and $j^{-1}=d+j=l$.

(E2) Let $r\in\mathbb{R}$ with $0\le r\le1$ and let $n<n'$ be natural numbers. Then $r^{n'}=r^{n}\,r^{n'-n}$, by induction on $n'-n$ from the recursion, claim 1 of Properties of Natural Number Powers in a Field; and $0\le r^{n}$, $0\le r^{n'-n}\le1^{n'-n}=1$ by claims 5 and 2 of that lemma. Hence $r^{n'}\le r^{n}$ whenever $n\le n'$.

(E3) For every $k\in\mathbb{N}$ one has $k\le2^{k}$ and $k^{4}\le16^{k}$. Indeed $1\le2=2^{1}$, and if $k\le2^{k}$ then, as $1=1^{k}\le2^{k}$ by claims 2 and 5 of Properties of Natural Number Powers in a Field, $k+1\le2^{k}+2^{k}=2^{k+1}$ by claim 1 of that lemma; this is an induction on $k$. By claim 5 of that lemma, $k^{4}\le(2^{k})^{4}$, and $(2^{k})^{4}=2^{k}2^{k}2^{k}2^{k}=4^{k}4^{k}=16^{k}$ by claims 1 and 3 of that lemma. Likewise $1+k\le2^{k}$ for every $k\in\mathbb{N}$: $1+1=2^{1}$, and if $1+k\le2^{k}$ then $1+(k+1)\le2^{k}+1\le2^{k}+2^{k}=2^{k+1}$. Consequently, for $m,r\in\mathbb{N}$, $m+r\le m+mr=m(1+r)\le m\,2^{r}$, where $r\le mr$ because $1\le m$ and $0\le r$.

(E4) Let $\alpha,\beta\in\mathbb{R}$ and let $a,a'\in\mathbb{R}$ be positive with $a\,a'=1$. Then $\alpha\beta\le\frac12a\alpha^{2}+\frac12a'\beta^{2}$, because $a\alpha^{2}-2\alpha\beta+a'\beta^{2}=a'(a\alpha-\beta)^{2}\ge0$, as one checks by expanding and using $aa'=1$.

(E5) Let $s\in\mathbb{R}$ with $0\le s<\frac12$ and let $n\in\mathbb{N}$. Then

$$\sum_{r=1}^{n}s^{r}\le\frac{s}{1-s}.$$

This holds because by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §geometric the left side equals $\frac{s-s^{n+1}}{1-s}$ with $1-s>0$ and $0\le s^{n+1}$.

(E6) For $k\in\mathbb{N}$, $\sum_{i\in[k]}1=\sum_{i=1}^{k}1=k$: the first equality is claim 1 of Properties of a Sum over a Finite Index Set, and the finite sum of $k$ ones is, by definition, the image of $k$ in $\mathbb{R}$ under the canonical map.

(E7) First, $0<\theta\le1$: $\theta$ is positive by The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §weights, and $\theta\le1$ by Order Reversal under Reciprocals, and Summability of the Reciprocals of the Squares §reciprocal applied to $0<1\le6144\,d$. Next, let $n,n'\in\mathbb{N}$ with $n\le n'$. Then $0<n+1\le n'+1$, so $0\le(n+1)^{4}\le(n'+1)^{4}$ by claim 5 of Properties of Natural Number Powers in a Field and $(n+1)^{4}\ne0$ by claim 4 of that lemma; hence Order Reversal under Reciprocals, and Summability of the Reciprocals of the Squares §reciprocal gives

$$0<\frac{1}{(n'+1)^{4}}\le\frac{1}{(n+1)^{4}}.$$

Multiplying inequalities between nonnegative reals, we obtain: (i) if $n\le K$ for some $K\in\mathbb{N}$, then $\frac{\theta^{K}}{(K+1)^{4}}\le\gamma_{n}$, since $0\le\theta^{K}\le\theta^{n}$ by (E2) and claim 5 of Properties of Natural Number Powers in a Field; (ii) if $n<n'$, then $\gamma_{n'}\le\theta^{\,n'-n}\gamma_{n}$, since $\gamma_{n'}=\theta^{\,n'-n}\,\theta^{n}\,\frac{1}{(n'+1)^{4}}$ by the factorisation in (E2), and $\frac{1}{(n'+1)^{4}}\le\frac{1}{(n+1)^{4}}$ with $0\le\theta^{\,n'-n}\theta^{n}$.

Step 1 (the number of pairs, and claim 1). Let $w\in W_{2d}$ have length $k\in\mathbb{N}$. We show

$$\sum_{(p,q)\in P(w)}1\le k^{2}-k.\tag{1.1}$$

Let $D_{k}=\{(p,p):p\in[k]\}$ and $Q_{k}=([k]\times[k])\setminus D_{k}$. Since $p<q$ excludes $p=q$, $P(w)\subseteq Q_{k}$. By The Product of Two Sums over Finite Index Sets is a Sum over the Cartesian Product with $a$ and $b$ constantly $1$, and (E6), $\sum_{[k]\times[k]}1=k\cdot k$; and $p\mapsto(p,p)$ is a bijection from $[k]$ onto $D_{k}$, so $\sum_{D_{k}}1=k$. If $k=1$, then $Q_{k}$ and hence $P(w)$ are empty, and (1.1) reads $0\le0$. If $1<k$, then $Q_{k}$ is nonempty (it contains $(1,2)$), and splitting $[k]\times[k]=Q_{k}\cup D_{k}$ gives $\sum_{Q_{k}}1=k^{2}-k$, whence (1.1) by monotonicity in the index set if $P(w)$ is nonempty; if $P(w)$ is empty, the left side of (1.1) is $0$, and $0\le\sum_{Q_{k}}1=k^{2}-k$ by nonnegativity.

Claim 1. Let $\lambda\in\mathcal{L}_{d}$ and $w\in W_{2d}$. If $w=\varnothing$, then $\Theta\lambda(\varnothing)=0$ by The Free Unitary Heat Generator on Unitary Laws §generator, and $|0|=0=|\varnothing|^{2}$ by claim 3 of Properties of Complex Conjugation and Modulus and Words in Unitary Letters: Generators, Signs, Inverse Letters, Lengths and Adjoint Words §length. If $w$ has length $k\in\mathbb{N}$, then by The Free Unitary Heat Generator on Unitary Laws §generator, the triangle inequality, the modulus of a finite sum, multiplicativity, $|\varepsilon_{p}|=|\varepsilon_{q}|=1$, $|\lambda(v)|\le1$ for all $v\in W_{2d}$ (Unitary Laws: Adjoints, the Bound One, and Reduction to Cyclically Reduced Words §bound) and (1.1),

$$|\Theta\lambda(w)|\le\frac{k}{2}|\lambda(w)|+\sum_{(p,q)\in P(w)}|\lambda(y_{pq})|\,|\lambda(x_{pq})|\le\frac{k}{2}+k^{2}-k\le k^{2}=|w|^{2}.$$

Now let $k\in\mathbb{N}$ and $w\in W^{\circ}_{d,k}$. Since $0\le|\Theta\lambda(w)|\le k^{2}$, Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field (weak form) gives $|\Theta\lambda(w)|^{2}\le k^{4}$, so, as $0<c_{w}\le\theta^{k}$ (Conventions), $0\le c_{w}|\widehat{\Theta}\lambda(w)|^{2}\le\theta^{k}k^{4}$ by The Free Unitary Heat Generator on Unitary Laws §hat. The set $W^{\circ}_{d,k}$ is nonempty by Counting Cyclically Reduced Words: Finiteness and the Bound (2d)^k §finite, so by Counting Cyclically Reduced Words: Finiteness and the Bound (2d)^k §count with $M=\theta^{k}k^{4}$, claim 3 of Properties of Natural Number Powers in a Field and (E3), the $k$-th block sum of $w\mapsto c_{w}|\widehat{\Theta}\lambda(w)|^{2}$ is at most

$$k^{4}\theta^{k}(2d)^{k}=k^{4}(2d\,\theta)^{k}=k^{4}\Bigl(\frac{1}{3072}\Bigr)^{k}\le16^{k}\Bigl(\frac{1}{3072}\Bigr)^{k}=\Bigl(\frac{1}{192}\Bigr)^{k},\tag{1.2}$$

and it is nonnegative. The geometric series of ratio $\frac{1}{192}$ converges by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §geometric, so the series of block sums converges by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §comparison; that is, $\widehat{\Theta}\lambda\in E_{d}$ by The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §sums and The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §space. This proves claim 1.

Step 2 (the pieces of a cyclically reduced word). Let $k\in\mathbb{N}$, $w\in W^{\circ}_{d,k}$ and $(p,q)\in P(w)$. Then $1\le p<q\le k$, so $1<k$, and by The Free Unitary Heat Generator on Unitary Laws §positions

$$\sigma_{p}\le p+1\le q\le\tau_{q}+1,\qquad\tau_{q}\le q\le k,\qquad p\le\sigma_{p},\qquad q-1\le\tau_{q},$$

the last two because $\sigma_{p}\in\{p,p+1\}$ and $\tau_{q}\in\{q-1,q\}$. We show:

(2a) $\sigma_{p}\le\tau_{q}$; hence $y_{pq}=w_{[\sigma_{p},\tau_{q}]}$ has length $|y_{pq}|=\tau_{q}-\sigma_{p}+1\ge1$.

(2b) $|x_{pq}|=k-|y_{pq}|$ and $1\le|x_{pq}|$; hence $1\le|y_{pq}|\le k-1$ and $1\le|x_{pq}|\le k-1$.

(2c) $q-p-1\le|y_{pq}|$ and $k-q+p-1\le|x_{pq}|$, where $q-p-1\ge0$ and $k-q+p-1\ge0$.

(2d) $y_{pq}$ and $x_{pq}$ are reduced.

(2e) $\sigma_{p}$ is determined by $p$ and $\varepsilon_{p}$, $\tau_{q}$ by $q$ and $\varepsilon_{q}$, and conversely $q=\tau_{q}$ if $\varepsilon_{q}=1$ and $q=\tau_{q}+1$ if $\varepsilon_{q}=-1$.

Proof of (2a): otherwise $\sigma_{p}=\tau_{q}+1$ by the displayed chain, and then all its inequalities are equalities: $\sigma_{p}=p+1$, so $\varepsilon_{p}=1$; $q=p+1$; and $\tau_{q}=q-1$, so $\varepsilon_{q}=-1$. By (E1), $w_{p}=g(w_{p})$ and $w_{q}=d+g(w_{q})=d+g(w_{p})=d+w_{p}=(w_{p})^{-1}$, using $g(w_{p})=g(w_{q})$ from the definition of $P(w)$. As $p<k$ and $q=p+1$, this contradicts the reducedness of $w$ (Reduced and Cyclically Reduced Words in Unitary Letters §reduced, Reduced and Cyclically Reduced Words in Unitary Letters §cyclically-reduced). The length formula is the definition of $w_{[a,b]}$ in The Free Unitary Heat Generator on Unitary Laws.

Proof of (2b): by The Free Unitary Heat Generator on Unitary Laws §pieces, $x_{pq}$ is the concatenation of $w_{[\tau_{q}+1,k]}$, of length $k-\tau_{q}$ (read as $0$ if $\tau_{q}=k$), and $w_{[1,\sigma_{p}-1]}$, of length $\sigma_{p}-1$ (read as $0$ if $\sigma_{p}=1$). Hence $|x_{pq}|=k-\tau_{q}+\sigma_{p}-1=k-|y_{pq}|$ by (2a). Suppose $|x_{pq}|=0$. Then $\tau_{q}=k$ and $\sigma_{p}=1$. Since $\tau_{q}\le q\le k$, $q=k=\tau_{q}$, so $\varepsilon_{k}=1$ ($\tau_{q}=q-1$ would give $\tau_{q}<k$); and $\sigma_{p}=1$ excludes $\sigma_{p}=p+1\ge2$, so $\varepsilon_{p}=-1$ and $p=\sigma_{p}=1$. By (E1), $w_{k}=g(w_{k})=g(w_{1})$ and $w_{1}=d+g(w_{1})$, so $(w_{1})^{-1}=g(w_{1})=w_{k}$; since $1<k$ this contradicts Reduced and Cyclically Reduced Words in Unitary Letters §cyclically-reduced. So $1\le|x_{pq}|$, and the bounds follow from $|y_{pq}|+|x_{pq}|=k$.

Proof of (2c): $|y_{pq}|=\tau_{q}-\sigma_{p}+1\ge(q-1)-(p+1)+1=q-p-1$ and $|y_{pq}|\le q-p+1$, so $|x_{pq}|=k-|y_{pq}|\ge k-q+p-1$; the nonnegativity follows from $p<q$, $q\le k$ and $1\le p$.

Proof of (2d): the $i$-th and $(i+1)$-st letters of $y_{pq}$ are $w_{\sigma_{p}+i-1}$ and $w_{\sigma_{p}+i}$, consecutive letters of the reduced word $w$, so $y_{pq}$ is reduced. Two consecutive letters of $x_{pq}$ are consecutive letters of $w$ unless both factors of $x_{pq}$ are nonempty and the two letters are the last letter $w_{k}$ of the first factor and the first letter $w_{1}$ of the second; in that case $w_{1}\ne(w_{k})^{-1}$, since otherwise $(w_{1})^{-1}=((w_{k})^{-1})^{-1}=w_{k}$ by (E1), contradicting Reduced and Cyclically Reduced Words in Unitary Letters §cyclically-reduced (here $1<k$). So $x_{pq}$ is reduced.

(2e) is the definition of $\sigma_{p}$ and $\tau_{q}$ in The Free Unitary Heat Generator on Unitary Laws §positions.

Step 3 (stripping the pieces, and the basic difference estimate). For $k\in\mathbb{N}$, $w\in W^{\circ}_{d,k}$, $(p,q)\in P(w)$ and $\iota\in\{\mathrm{in},\mathrm{out}\}$ put $z^{\mathrm{in}}_{pq}(w)=y_{pq}(w)$ and $z^{\mathrm{out}}_{pq}(w)=x_{pq}(w)$. Each $z^{\iota}_{pq}(w)$ is reduced by (2d), so by Unitary Laws: Adjoints, the Bound One, and Reduction to Cyclically Reduced Words §strip-exists we may choose, and fix once and for all, words $b^{\iota}_{pq}(w)\in W_{2d}$ and $u^{\iota}_{pq}(w)\in W^{\circ}_{d}$ with

$$z^{\iota}_{pq}(w)=b^{\iota}_{pq}(w)\,u^{\iota}_{pq}(w)\,b^{\iota}_{pq}(w)^{*},\qquad|z^{\iota}_{pq}(w)|=2|b^{\iota}_{pq}(w)|+|u^{\iota}_{pq}(w)|.$$

Then:

(3a) $|u^{\iota}_{pq}(w)|\le|z^{\iota}_{pq}(w)|\le k-1$ by (2b); consequently, using $|y_{pq}|+|x_{pq}|=k$, the natural numbers $k-|u^{\mathrm{in}}_{pq}(w)|$ and $k-|u^{\mathrm{out}}_{pq}(w)|$ satisfy $k-|u^{\mathrm{in}}_{pq}(w)|\ge|x_{pq}|\ge1$ and $k-|u^{\mathrm{out}}_{pq}(w)|\ge|y_{pq}|\ge1$.

(3b) $\lambda(z^{\iota}_{pq}(w))=\lambda(u^{\iota}_{pq}(w))$ for every $\lambda\in\mathcal{L}_{d}$, by Unitary Laws: Adjoints, the Bound One, and Reduction to Cyclically Reduced Words §strip-invariance.

(3c) Let $\mu,\nu\in\mathcal{L}_{d}$ and put $\Delta_{v}=\mu(v)-\nu(v)$ for $v\in W_{2d}$; then $\Delta_{\varnothing}=1-1=0$ by Laws of d-Tuples of Unitaries §normalised. For $k\in\mathbb{N}$ and $w\in W^{\circ}_{d,k}$ put

$$R(w)=\sum_{(p,q)\in P(w)}\Bigl(\bigl|\Delta_{u^{\mathrm{in}}_{pq}(w)}\bigr|+\bigl|\Delta_{u^{\mathrm{out}}_{pq}(w)}\bigr|\Bigr).$$

Then

$$\Bigl|\Theta\mu(w)-\Theta\nu(w)+\frac{k}{2}\Delta_{w}\Bigr|\le R(w).\tag{3.1}$$

Indeed, by The Free Unitary Heat Generator on Unitary Laws §generator and additivity of finite sums, $\Theta\mu(w)-\Theta\nu(w)+\frac{k}{2}\Delta_{w}=-\sum_{(p,q)\in P(w)}\varepsilon_{p}\varepsilon_{q}\bigl(\mu(y_{pq})\mu(x_{pq})-\nu(y_{pq})\nu(x_{pq})\bigr)$ (both sides $0$ if $P(w)$ is empty). By field arithmetic and (3b),

$$\mu(y_{pq})\mu(x_{pq})-\nu(y_{pq})\nu(x_{pq})=\Delta_{y_{pq}}\,\mu(x_{pq})+\nu(y_{pq})\,\Delta_{x_{pq}}=\Delta_{u^{\mathrm{in}}_{pq}(w)}\,\mu(x_{pq})+\nu(y_{pq})\,\Delta_{u^{\mathrm{out}}_{pq}(w)},$$

whose modulus is at most $|\Delta_{u^{\mathrm{in}}_{pq}(w)}|+|\Delta_{u^{\mathrm{out}}_{pq}(w)}|$ by the triangle inequality, multiplicativity and Unitary Laws: Adjoints, the Bound One, and Reduction to Cyclically Reduced Words §bound. The modulus of the sum, with $|\varepsilon_{p}\varepsilon_{q}|=1$, gives (3.1).

Step 4 (claim 2). By The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §metric, $(\mathcal{L}_{d},d_{\mathcal{L}})$ is a metric space; by The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §inner-product, $E_{d}$ is a real inner product space with norm $\lVert\cdot\rVert_{d}$, so $(x,y)\mapsto\lVert x-y\rVert_{d}$ is a metric on $E_{d}$ by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §metric. We verify continuity at an arbitrary $\lambda\in\mathcal{L}_{d}$. Fix $\lambda\in\mathcal{L}_{d}$ and a real $\epsilon>0$. We choose, in this order, first $K\in\mathbb{N}$ depending only on $\epsilon$, and then $\eta>0$ depending only on $\epsilon$ and $K$.

Choice of $K$. By Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §geometric the sequence $((\frac{1}{192})^{n})_{n\in\mathbb{N}}$ converges to $0$, so there is $K\in\mathbb{N}$ with $(\frac{1}{192})^{n}<\frac{191\,\epsilon^{2}}{1536}$ for all $n\ge K$; in particular

$$\frac{768}{191}\Bigl(\frac{1}{192}\Bigr)^{K+1}<\frac{\epsilon^{2}}{2}.\tag{4.1}$$

Choice of $\eta$. Put $\Lambda=(6144\,d)^{K}(K+1)^{4}$, which is positive by claims 5 and 4 of Properties of Natural Number Powers in a Field, $C=4K^{5}\Lambda\,(2d)^{K}$, which is positive, and $\eta=\min\{1,\frac{\epsilon^{2}}{2C}\}>0$.

Now let $\mu\in\mathcal{L}_{d}$ with $D:=d_{\mathcal{L}}(\lambda,\mu)<\eta$. Apply (3c) to the pair $(\lambda,\mu)$ in place of $(\mu,\nu)$, so that $\Delta_{v}=\lambda(v)-\mu(v)$; by The Embedding of Unitary Laws into the Word Gauge Space and the Gauge Distance §distance and The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §sobolev, $D=\lVert\iota_{d}(\lambda)-\iota_{d}(\mu)\rVert_{d}$ where $\iota_{d}(\lambda)-\iota_{d}(\mu)$ is the map $v\mapsto\Delta_{v}$ on $W^{\circ}_{d}$. Let $g=\widehat{\Theta}\lambda-\widehat{\Theta}\mu\in E_{d}$ (claim 1 and The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §space), the map $v\mapsto\Theta\lambda(v)-\Theta\mu(v)$ on $W^{\circ}_{d}$; $g(\varnothing)=0$ by The Free Unitary Heat Generator on Unitary Laws §generator. For $k\in\mathbb{N}$ put $H_{k}=\sum_{w\in W^{\circ}_{d,k}}c_{w}|g(w)|^{2}$ and $X_{k}=\sum_{w\in W^{\circ}_{d,k}}c_{w}|\Delta_{w}|^{2}$, both nonnegative. By The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §norm and The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §sums, the square of a nonnegative square root being the number itself, and $g(\varnothing)=0=\Delta_{\varnothing}$,

$$\lVert g\rVert_{d}^{2}=\sum_{k=1}^{\infty}H_{k},\qquad D^{2}=\sum_{k=1}^{\infty}X_{k}.$$

Tail. Let $k\in\mathbb{N}$ and $w\in W^{\circ}_{d,k}$. By the triangle inequality and claim 1, $|g(w)|\le|\Theta\lambda(w)|+|\Theta\mu(w)|\le2k^{2}$, so $|g(w)|^{2}\le4k^{4}$ by Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field; by Counting Cyclically Reduced Words: Finiteness and the Bound (2d)^k §count and the computation (1.2), $0\le H_{k}\le4(\frac{1}{192})^{k}$. Apply Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §tail-bound with $\mu_{k}=(\frac{1}{192})^{k}$, whose series converges by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §geometric, $M=4$ and $w_{k}=192^{k}H_{k}\in[0,4]$ (note $(\frac{1}{192})^{k}192^{k}=1^{k}=1$ by claims 3 and 2 of Properties of Natural Number Powers in a Field); with the tail formula of Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §geometric and (4.1),

$$\sum_{k=1}^{\infty}H_{k}-\sum_{k=1}^{K}H_{k}\le4\cdot\frac{(\frac{1}{192})^{K+1}}{1-\frac{1}{192}}=\frac{768}{191}\Bigl(\frac{1}{192}\Bigr)^{K+1}<\frac{\epsilon^{2}}{2}.$$

Head. Let $v\in W^{\circ}_{d}$ with $1\le|v|=n\le K$. Then $c_{v}|\Delta_{v}|^{2}\le X_{n}\le\sum_{j=1}^{n}X_{j}\le\sum_{j=1}^{\infty}X_{j}=D^{2}$, by monotonicity in the index set, nonnegativity of the $X_{j}$ and Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates. By (E7)(i), $\frac{\theta^{K}}{(K+1)^{4}}\le\gamma_{n}=c_{v}$, so $\frac{\theta^{K}}{(K+1)^{4}}|\Delta_{v}|^{2}\le D^{2}$; multiplying by $\Lambda>0$ and using $\frac{\theta^{K}}{(K+1)^{4}}\Lambda=\theta^{K}(6144\,d)^{K}=(6144\,d\,\theta)^{K}=1^{K}=1$ (claims 3 and 2 of Properties of Natural Number Powers in a Field) gives $|\Delta_{v}|^{2}\le\Lambda D^{2}$, which also holds for $v=\varnothing$ as $\Delta_{\varnothing}=0$. Let $e\ge0$ be the nonnegative square root of $\Lambda D^{2}$ (Existence and Uniqueness of the Nonnegative Square Root); by Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field (weak form), $|\Delta_{v}|\le e$ for every $v\in W^{\circ}_{d}$ with $|v|\le K$.

Let $k\in[K]$ and $w\in W^{\circ}_{d,k}$. Every $u^{\iota}_{pq}(w)$ lies in $W^{\circ}_{d}$ and has length at most $k-1<K$ by (3a), so $R(w)\le\sum_{(p,q)\in P(w)}2e\le2(k^{2}-k)e$ by (1.1). By (3.1) and the triangle inequality, $|g(w)|\le\frac{k}{2}|\Delta_{w}|+R(w)\le(\frac{k}{2}+2k^{2}-2k)e\le2k^{2}e$, hence $|g(w)|^{2}\le4k^{4}e^{2}=4k^{4}\Lambda D^{2}\le4K^{4}\Lambda D^{2}$ by Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field and claim 5 of Properties of Natural Number Powers in a Field. Since $0\le\theta\le1$ by (E7), claims 5 and 2 of Properties of Natural Number Powers in a Field and the Conventions give $0<c_{w}\le\theta^{k}\le1^{k}=1$, so $0\le c_{w}|g(w)|^{2}\le|g(w)|^{2}\le4K^{4}\Lambda D^{2}$. By Counting Cyclically Reduced Words: Finiteness and the Bound (2d)^k §count, $H_{k}\le4K^{4}\Lambda D^{2}(2d)^{k}\le4K^{4}\Lambda(2d)^{K}D^{2}$, where $(2d)^{k}\le(2d)^{K}$ because $(2d)^{K}=(2d)^{k}(2d)^{K-k}$ for $k<K$ and $(2d)^{K-k}\ge1^{K-k}=1$ (as in (E2), with claim 5 of Properties of Natural Number Powers in a Field). Summing over $k\in[K]$ with (E6),

$$\sum_{k=1}^{K}H_{k}\le K\cdot4K^{4}\Lambda(2d)^{K}D^{2}=C\,D^{2}.$$

Since $0\le D<\eta$ and $0<\eta\le1$, Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field (strict form) gives $D^{2}<\eta^{2}\le\eta\le\frac{\epsilon^{2}}{2C}$, so $\sum_{k=1}^{K}H_{k}\le\frac{\epsilon^{2}}{2}$.

Conclusion. Adding head and tail, $\lVert g\rVert_{d}^{2}<\frac{\epsilon^{2}}{2}+\frac{\epsilon^{2}}{2}=\epsilon^{2}$, so $\lVert g\rVert_{d}<\epsilon$ by Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field (strict form), both numbers being nonnegative. By symmetry of the metric of $E_{d}$, $\lVert\widehat{\Theta}\mu-\widehat{\Theta}\lambda\rVert_{d}=\lVert g\rVert_{d}<\epsilon$. As $\mu$ with $d_{\mathcal{L}}(\lambda,\mu)<\eta$ was arbitrary, $\lambda\mapsto\widehat{\Theta}\lambda$ is continuous at $\lambda$ relative to $\mathcal{L}_{d}$, and, $\lambda$ being arbitrary, continuous on $\mathcal{L}_{d}$ (Continuous Map Between Metric Spaces). This proves claim 2.

Step 5 (row sums). Put $\rho=\frac{1}{24}$, and for $j\in\{0\}\cup\mathbb{N}$ let $\rho_{j}=\rho$ if $j=0$ and $\rho_{j}=\rho^{j}$ if $j\in\mathbb{N}$. For $k\in\mathbb{N}$ and $w\in W^{\circ}_{d,k}$ put, with the exponents natural numbers by (3a),

$$S(w)=\sum_{(p,q)\in P(w)}\Bigl(\rho^{\,k-|u^{\mathrm{in}}_{pq}(w)|}+\rho^{\,k-|u^{\mathrm{out}}_{pq}(w)|}\Bigr).$$

We show

$$S(w)\le\frac{47\,k}{276}.\tag{5.1}$$

If $P(w)$ is empty, $S(w)=0$. Otherwise let $(p,q)\in P(w)$. By (3a) and (E2), $\rho^{\,k-|u^{\mathrm{in}}_{pq}(w)|}\le\rho^{|x_{pq}|}$ and $\rho^{\,k-|u^{\mathrm{out}}_{pq}(w)|}\le\rho^{|y_{pq}|}$. By (2b) and (2c), $|x_{pq}|\ge1$ and $|x_{pq}|\ge k-q+p-1$, so (E2) gives $\rho^{|x_{pq}|}\le\rho_{k-q+p-1}$ (compare with $\rho^{k-q+p-1}$ if $k-q+p-1\ge1$, and with $\rho^{1}$ if $k-q+p-1=0$); likewise $|y_{pq}|\ge1$ and $|y_{pq}|\ge q-p-1$ give $\rho^{|y_{pq}|}\le\rho_{q-p-1}$. The set $P(w)$ is contained in the set of pairs $(p,q)$ with $p\in[k-1]$ and $p<q\le k$, and all terms are nonnegative, so

$$S(w)\le\sum_{p=1}^{k-1}\ \sum_{q=p+1}^{k}\bigl(\rho_{k-q+p-1}+\rho_{q-p-1}\bigr).$$

For fixed $p$, as $q$ runs over $\{p+1,\dots,k\}$, $j=q-p-1$ runs bijectively over $\{0,\dots,k-p-1\}$ and $j'=k-q+p-1$ runs bijectively over $\{p-1,\dots,k-2\}$, both subsets of $\{0,\dots,k-2\}$. Hence each of the two inner sums is at most $\sum_{j=0}^{k-2}\rho_{j}=\rho+\sum_{j=1}^{k-2}\rho^{j}\le\rho+\frac{\rho}{1-\rho}=\frac{1}{24}+\frac{1}{23}=\frac{47}{552}$, by (E5) with $s=\rho$ (the last sum being empty if $k=2$). Therefore $S(w)\le(k-1)\cdot2\cdot\frac{47}{552}\le\frac{47k}{276}$, using (E6).

Step 6 (counting the words whose pieces strip to a given word). Let $m,k\in\mathbb{N}$ with $m<k$ and let $u\in W^{\circ}_{d,m}$. Let $I(u,k)$ be the set of triples $\xi=(w,(p,q),\iota)$ with $w\in W^{\circ}_{d,k}$, $(p,q)\in P(w)$, $\iota\in\{\mathrm{in},\mathrm{out}\}$ and $u^{\iota}_{pq}(w)=u$; it is a subset of the finite set $W^{\circ}_{d,k}\times([k]\times[k])\times\{\mathrm{in},\mathrm{out}\}$ (Counting Cyclically Reduced Words: Finiteness and the Bound (2d)^k §finite, claim 1 of Finiteness of Cartesian Products, Tuple Sets, and Permutation Sets), hence finite by claim 3 of Basic Properties of Finite Sets. Put $N(u,k)=\sum_{\xi\in I(u,k)}1$. We show

$$N(u,k)\le16\,k\,(2d)^{k-m}.\tag{6.1}$$

Let $T=\{t\in\{0\}\cup\mathbb{N}:m+2t\le k-1\}$, which contains $0$, and $\Pi=\{\mathrm{in},\mathrm{out}\}\times[k]\times\{1,-1\}\times\{1,-1\}\times T$. To $\xi=(w,(p,q),\iota)\in I(u,k)$ assign $\pi(\xi)=(\iota,p,\varepsilon_{p},\varepsilon_{q},t)$ with $t=|b^{\iota}_{pq}(w)|$; since $|z^{\iota}_{pq}(w)|=m+2t\le k-1$ by Step 3 and (3a), $\pi(\xi)\in\Pi$. For $\pi\in\Pi$ let $I_{\pi}=\{\xi\in I(u,k):\pi(\xi)=\pi\}$; these sets are pairwise disjoint with union $I(u,k)$.

Fix $\pi=(\iota,p,e,e',t)\in\Pi$ with $I_{\pi}$ nonempty, and put $L=m+2t$ and $n_{\pi}=k-m-t$. Let $\xi=(w,(p,q),\iota)\in I_{\pi}$ and $z=z^{\iota}_{pq}(w)$, so $|z|=L$. By (2e), $\sigma:=\sigma_{p}$ depends only on $(p,e)$; by (2a) and (2b), $\tau:=\tau_{q}$ equals $L+\sigma-1$ if $\iota=\mathrm{in}$ and $k-L+\sigma-1$ if $\iota=\mathrm{out}$; and by (2e), $q$ is determined by $\tau$ and $e'$. So $\sigma$, $\tau$ and $q$ depend only on $\pi$. Let $r(\xi)=w_{[1,\sigma-1]}\,w_{[\tau+1,k]}$ (empty factors omitted) if $\iota=\mathrm{in}$, and $r(\xi)=w_{[\sigma,\tau]}=y_{pq}$ if $\iota=\mathrm{out}$; in both cases $r(\xi)$ is the other piece rearranged, of length $k-L\ge1$ by (2b). Put $\varphi_{\pi}(\xi)=b^{\iota}_{pq}(w)\,r(\xi)$, a word of length $t+(k-L)=n_{\pi}\ge1$, that is an element of $[2d]^{n_{\pi}}$. The map $\varphi_{\pi}$ is injective on $I_{\pi}$: from $v=\varphi_{\pi}(\xi)$ one reads off $b=b^{\iota}_{pq}(w)$ as the first $t$ letters and $r=r(\xi)$ as the last $k-L$ letters, then $z=b\,u\,b^{*}$ (Step 3), and then $w$ letter by letter: if $\iota=\mathrm{in}$, $w_{i}=r_{i}$ for $i<\sigma$, $w_{i}=z_{i-\sigma+1}$ for $\sigma\le i\le\tau$ and $w_{i}=r_{i-L}$ for $\tau<i\le k$; if $\iota=\mathrm{out}$, $w_{i}=z_{k-\tau+i}$ for $i<\sigma$, $w_{i}=r_{i-\sigma+1}$ for $\sigma\le i\le\tau$ and $w_{i}=z_{i-\tau}$ for $\tau<i\le k$ (these are the definitions of $w_{[a,b]}$ and of the pieces in The Free Unitary Heat Generator on Unitary Laws §pieces, read backwards); finally $\xi=(w,(p,q),\iota)$ with $(p,q,\iota)$ determined by $\pi$. So $\varphi_{\pi}$ is a bijection from $I_{\pi}$ onto a nonempty set $B_{\pi}\subseteq[2d]^{n_{\pi}}$, and by reindexing and Counting Cyclically Reduced Words: Finiteness and the Bound (2d)^k §count with $a$ constantly $1$ and $M=1$,

$$\sum_{\xi\in I_{\pi}}1=\sum_{v\in B_{\pi}}1\le(2d)^{n_{\pi}}\le(2d)^{k-m}\Bigl(\frac12\Bigr)^{t},$$

the last inequality being trivial if $t=0$ and following for $t\ge1$ from $(2d)^{k-m}=(2d)^{n_{\pi}}(2d)^{t}$ (claim 1 of Properties of Natural Number Powers in a Field, as in (E2)), $(2d)^{t}\ge2^{t}$ (claim 5 of that lemma) and $2^{t}(\frac12)^{t}=1$ (claims 3 and 2). Splitting $I(u,k)$ over the nonempty $I_{\pi}$ and then enlarging to all $\pi\in\Pi$ (nonnegative terms), and writing the sum over $\Pi$ as an iterated sum whose summand depends only on $t$, with (E6),

$$N(u,k)\le2\cdot k\cdot2\cdot2\cdot(2d)^{k-m}\sum_{t\in T}\Bigl(\frac12\Bigr)^{t}\le8k(2d)^{k-m}\cdot2,$$

since $\sum_{t\in T}(\frac12)^{t}\le1+\sum_{t=1}^{k}(\frac12)^{t}\le1+1$ by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §geometric (the sum of the halving series is $1$) and Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates. This is (6.1).

Step 7 (claim 3). Let $\mu,\nu\in\mathcal{L}_{d}$, and let $\Delta_{v}=\mu(v)-\nu(v)$ and $R(w)$ be as in (3c). By The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §sobolev, $x:=\iota_{d}(\mu)-\iota_{d}(\nu)\in E_{d}$ is the map $v\mapsto\Delta_{v}$ on $W^{\circ}_{d}$, the map $v\mapsto c_{v}|v||\Delta_{v}|^{2}$ is summable, and $\lVert x\rVert_{d,1}$ is defined. By claim 1 and The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §space, $h:=\widehat{\Theta}\mu-\widehat{\Theta}\nu\in E_{d}$, and it is the map $v\mapsto\Theta\mu(v)-\Theta\nu(v)$ on $W^{\circ}_{d}$. For $k\in\mathbb{N}$ put

$$X_{k}=\sum_{w\in W^{\circ}_{d,k}}c_{w}|\Delta_{w}|^{2}\ge0,\qquad P_{k}=\sum_{w\in W^{\circ}_{d,k}}c_{w}\operatorname{Re}\bigl(\overline{\Delta_{w}}\,h(w)\bigr).$$

(7a) The two gauges and the inner product. The $k$-th block sum of $v\mapsto c_{v}|v||\Delta_{v}|^{2}$ is $kX_{k}$, since $|w|=k$ on $W^{\circ}_{d,k}$ (homogeneity). As $\Delta_{\varnothing}=0$, The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §norm, The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §sobolev and The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §sums give, both series converging,

$$\lVert x\rVert_{d}^{2}=\sum_{k=1}^{\infty}X_{k},\qquad\lVert x\rVert_{d,1}^{2}=\sum_{k=1}^{\infty}kX_{k}.$$

By The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §expansion, the series $\sum_{k=1}^{\infty}P_{k}$ converges and $\langle x,h\rangle_{d}=\operatorname{Re}(\overline{\Delta_{\varnothing}}\,h(\varnothing))+\sum_{k=1}^{\infty}P_{k}=\sum_{k=1}^{\infty}P_{k}$, since $\overline{0}\,h(\varnothing)=0$ and $\operatorname{Re}0=0$.

(7b) Pointwise bound. Let $k\in\mathbb{N}$ and $w\in W^{\circ}_{d,k}$, and put $\beta(w)=h(w)+\frac{k}{2}\Delta_{w}$, so that $|\beta(w)|\le R(w)$ by (3.1). Then $\overline{\Delta_{w}}\,h(w)=-\frac{k}{2}|\Delta_{w}|^{2}+\overline{\Delta_{w}}\,\beta(w)$ by claim 3 of Properties of Complex Conjugation and Modulus. The real part of a real number is that number, and real parts add, by the uniqueness of the canonical form in Real and Imaginary Parts of a Complex Number; with claims 6, 4 and 3 of Properties of Complex Conjugation and Modulus,

$$\operatorname{Re}\bigl(\overline{\Delta_{w}}\,h(w)\bigr)=-\frac{k}{2}|\Delta_{w}|^{2}+\operatorname{Re}\bigl(\overline{\Delta_{w}}\,\beta(w)\bigr)\le-\frac{k}{2}|\Delta_{w}|^{2}+|\Delta_{w}|\,R(w).$$

(7c) Splitting the cross terms. For $(p,q)\in P(w)$ and $\iota\in\{\mathrm{in},\mathrm{out}\}$ let $n=k-|u^{\iota}_{pq}(w)|\in\mathbb{N}$ (by (3a)) and apply (E4) with $\alpha=|\Delta_{w}|$, $\beta=|\Delta_{u^{\iota}_{pq}(w)}|$, $a=\rho^{n}$ and $a'=24^{n}$, which are positive with $aa'=(\rho\cdot24)^{n}=1^{n}=1$ by claims 3 and 2 of Properties of Natural Number Powers in a Field. Summing over $(p,q)\in P(w)$ and $\iota$, and using the definition of $S(w)$ in Step 5 and (5.1),

$$|\Delta_{w}|\,R(w)\le\frac12S(w)|\Delta_{w}|^{2}+\frac12\sum_{(p,q)\in P(w)}\ \sum_{\iota\in\{\mathrm{in},\mathrm{out}\}}24^{\,k-|u^{\iota}_{pq}(w)|}\,\bigl|\Delta_{u^{\iota}_{pq}(w)}\bigr|^{2}\le\frac{47k}{552}|\Delta_{w}|^{2}+\frac12\,G(w),$$

where $G(w)$ denotes the double sum. Multiplying (7b) and this bound by $c_{w}=\gamma_{k}>0$ and summing over $w\in W^{\circ}_{d,k}$ gives, for every $k\in\mathbb{N}$,

$$P_{k}\le\Bigl(-\frac12+\frac{47}{552}\Bigr)kX_{k}+\frac12G_{k},\qquad G_{k}:=\sum_{w\in W^{\circ}_{d,k}}\gamma_{k}G(w).\tag{7.1}$$

(7d) Resumming the second part by the stripped word. Let $J\in\mathbb{N}$ (a new letter, unrelated to the $K$ of Step 4). We show

$$\sum_{k=1}^{J}G_{k}\le\frac{16}{63}\sum_{m=1}^{J}m\,X_{m}.\tag{7.2}$$

Let $T_{J}$ be the set of triples $\xi=(w,(p,q),\iota)$ with $w\in W^{\circ}_{d}$, $1\le|w|\le J$, $(p,q)\in P(w)$ and $\iota\in\{\mathrm{in},\mathrm{out}\}$, and for such $\xi$ with $|w|=k$ put $f(\xi)=\gamma_{k}24^{\,k-|u^{\iota}_{pq}(w)|}|\Delta_{u^{\iota}_{pq}(w)}|^{2}\ge0$. Writing iterated sums as sums over dependent pairs, and noting that words with $P(w)$ empty contribute $0$, $\sum_{k=1}^{J}G_{k}=\sum_{\xi\in T_{J}}f(\xi)$ (both sides $0$ if $T_{J}$ is empty). Terms with $u^{\iota}_{pq}(w)=\varnothing$ vanish, as $\Delta_{\varnothing}=0$. For every other $\xi\in T_{J}$ the word $u=u^{\iota}_{pq}(w)$ lies in $W^{\circ}_{d,m}$ for some $m\in\mathbb{N}$ with $m\le|w|-1\le J-1$ by (3a); so these $\xi$ form the disjoint union of the sets $I(u,k)$ of Step 6 over $m\in[J-1]$, $u\in W^{\circ}_{d,m}$ and $k\in\mathbb{N}$ with $m<k\le J$, and on $I(u,k)$ the function $f$ has the constant value $\gamma_{k}24^{k-m}|\Delta_{u}|^{2}$. Hence (the right side being $0$ if $J=1$)

$$\sum_{\xi\in T_{J}}f(\xi)=\sum_{m=1}^{J-1}\ \sum_{u\in W^{\circ}_{d,m}}|\Delta_{u}|^{2}\sum_{k=m+1}^{J}N(u,k)\,\gamma_{k}24^{k-m}.$$

By (6.1), by (E7)(ii) in the form $\gamma_{k}\le\theta^{k-m}\gamma_{m}=\theta^{k-m}c_{u}$ (here $m<k$ and $u\in W^{\circ}_{d,m}$), and by claim 3 of Properties of Natural Number Powers in a Field,

$$N(u,k)\,\gamma_{k}24^{k-m}\le16k\,c_{u}\,(2d\cdot24\cdot\theta)^{k-m}=16k\,c_{u}\Bigl(\frac{1}{128}\Bigr)^{k-m},$$

because $2d\cdot24\cdot\theta=\frac{48d}{6144d}=\frac{1}{128}$. Reindex by $r=k-m$, which runs over $[J-m]$ with $J-m\in\mathbb{N}$. By (E3), $k=m+r\le m\,2^{r}$, and $2^{r}(\frac{1}{128})^{r}=(\frac{1}{64})^{r}$ by claim 3 of Properties of Natural Number Powers in a Field; applying (E5) with $s=\frac{1}{64}$, for which $\frac{s}{1-s}=\frac{1}{63}$,

$$\sum_{k=m+1}^{J}k\Bigl(\frac{1}{128}\Bigr)^{k-m}=\sum_{r=1}^{J-m}(m+r)\Bigl(\frac{1}{128}\Bigr)^{r}\le m\sum_{r=1}^{J-m}\Bigl(\frac{1}{64}\Bigr)^{r}\le\frac{m}{63}.$$

Therefore $\sum_{\xi\in T_{J}}f(\xi)\le16\sum_{m=1}^{J-1}\frac{m}{63}\sum_{u\in W^{\circ}_{d,m}}c_{u}|\Delta_{u}|^{2}=\frac{16}{63}\sum_{m=1}^{J-1}m\,X_{m}$, and adding the nonnegative term with $m=J$ gives (7.2). (No term independent of $m$ arises here: this is where $m\ge1$, that is $\Delta_{\varnothing}=0$, is used to absorb the factor $k=m+r$ into $m\,2^{r}$.)

(7e) Truncated inequality. Let $J\in\mathbb{N}$. Summing (7.1) over $k\in[J]$ and using (7.2),

$$\sum_{k=1}^{J}P_{k}\le\sum_{k=1}^{J}\Bigl(-\frac12+\frac{47}{552}+\frac{8}{63}\Bigr)kX_{k}\le\sum_{k=1}^{J}\Bigl(-\frac{1}{4}\Bigr)kX_{k},$$

the last step because $kX_{k}\ge0$ and, with the common denominator $552\cdot63=34776$,

$$-\frac12+\frac{47}{552}+\frac{8}{63}=\frac{-17388+2961+4416}{34776}=-\frac{10011}{34776}\le-\frac{8694}{34776}=-\frac14.$$

(7f) Passage to the limit. By (7a), $\sum_{k=1}^{\infty}kX_{k}$ converges with sum $\lVert x\rVert_{d,1}^{2}$, so by Elementary Properties of Series of Real Numbers §linearity the series with terms $-\frac14kX_{k}$ converges with sum $-\frac14\lVert x\rVert_{d,1}^{2}$. By Series of Real Numbers §convergent, the two sides of the inequality in (7e) are the $J$-th partial sums of convergent series, which converge to $\langle x,h\rangle_{d}$ (by (7a)) and to $-\frac14\lVert x\rVert_{d,1}^{2}$ respectively. Claim 1 of Order Properties of Limits of Real Sequences gives

$$\bigl\langle\iota_{d}(\mu)-\iota_{d}(\nu),\,\widehat{\Theta}\mu-\widehat{\Theta}\nu\bigr\rangle_{d}\le-\frac14\lVert\iota_{d}(\mu)-\iota_{d}(\nu)\rVert_{d,1}^{2}.$$

This proves claim 3.