Proof of Feedback Drifts: the Energy Bound in the Word Gauge, the Momentum Pairing and Two-Sided Tangency

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

Word polynomials, the $e_{w}$, products, adjoints and evaluations are those of Word Polynomials in Unitary Letters: the Normed *-Algebra of Finitely Supported Word Functions and Evaluation of Word Polynomials by a Unitary Law. Finite sums are those of Sum over a Finite Index Set; reindexing, additivity and homogeneity refer to claims 2, 3 and 4 of Properties of a Sum over a Finite Index Set. By The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §weights, $\theta_{d}=\frac{1}{6144\,d}$ and $0<c_{w}\le\theta_{d}^{\,k}$ for $w\in W^{\circ}_{d,k}$, while $c_{\varnothing}=1$. Every set $W^{\circ}_{d,k}$ is nonempty and finite (Counting Cyclically Reduced Words: Finiteness and the Bound (2d)^k §finite), and $\varnothing\in W^{\circ}_{d}$ (Reduced and Cyclically Reduced Words in Unitary Letters §cyclically-reduced). For a word $w$ of length $k$ and $m\in[k]$ we write $j_{m}=g(w_{m})$, $\varepsilon_{m}=\varepsilon(w_{m})$ and $r_{m}=r_{m}(w)$ (Words in Unitary Letters: Generators, Signs, Inverse Letters, Lengths and Adjoint Words §letters, Rotations and Cyclic Derivatives of Words in Unitary Letters §rotations).

Step 0 (preliminaries). Let $\lambda\in\mathcal{L}_{d}$.

(Z) For the zero word polynomial $0$ and every word polynomial $X$, $X0=0$ and $\lambda(0)=0$. Every value of $X0$ is $0$ or a sum of terms $X(u)\cdot0=0$, and $\lambda(0)$ is a sum of terms $0\cdot\lambda(w)$; both vanish by Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §vanishing.

(S) Let $Q$ be a nonempty finite set, $X$ and $Y_{q}$ ($q\in Q$) word polynomials and $z_{q}\in\mathbb{C}$. Then $\lambda\bigl(X\sum_{q\in Q}z_{q}Y_{q}\bigr)=\sum_{q\in Q}z_{q}\lambda(XY_{q})$, the inner sum being pointwise (Rotations and Cyclic Derivatives of Words in Unitary Letters). This follows from Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §algebra and Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §linear by induction on the number of elements of $Q$.

(R) For every self-adjoint $X$ and every $r\in W_{2d}$, $|\lambda(Xe_{r})|^{2}\le\lambda(XX)$. Since $X=X^{*}$, Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §cauchy-schwarz gives $|\lambda(Xe_{r})|^{2}=|\lambda(X^{*}e_{r})|^{2}\le\lambda(X^{*}X)\,\lambda(e_{r}^{*}e_{r})$. By Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §algebra, $e_{r}^{*}e_{r}=e_{r^{*}}e_{r}=e_{r^{*}r}$, and $\lambda(e_{r^{*}r})=\lambda(r^{*}r)$ by Evaluation of Word Polynomials by a Unitary Law §evaluation. Put $y=r^{*}$. Then $y^{*}=r$, as recorded in Word Polynomials in Unitary Letters: the Normed *-Algebra of Finitely Supported Word Functions §adjoint, and $r^{*}r=y\,\varnothing\,y^{*}$ by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §monoid. So $\lambda(r^{*}r)=\lambda(\varnothing)=1$ by Unitary Laws: Adjoints, the Bound One, and Reduction to Cyclically Reduced Words §strip-invariance and Laws of d-Tuples of Unitaries §normalised. Finally, $\lambda(X^{*}X)=\lambda(XX)$, and this number is real and nonnegative by Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §positive.

(B1) Let $a\in\mathcal{P}_{d}$. Then $b_{a}(\lambda)(\varnothing)=0$. For every $w\in W^{\circ}_{d}$ of length $k\in\mathbb{N}$,

$$b_{a}(\lambda)(w)=\sum_{m\in[k]}\mathrm{i}\,\varepsilon_{m}\,\lambda\bigl(a^{j_{m}}e_{r_{m}}\bigr).$$

First, $D^{i}_{\varnothing}=0$ (Rotations and Cyclic Derivatives of Words in Unitary Letters §derivative), so every term of $b_{a}(\lambda)(\varnothing)$ is $0$ by (Z), and the sum is $0$ by Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §vanishing. For $w$ of length $k$ and $i\in[d]$, put $M_{i}=\{m\in[k]:j_{m}=i\}$. If $M_{i}=\emptyset$, then $D^{i}_{w}=0$ and $\lambda(a^{i}D^{i}_{w})=0$ by (Z). Otherwise (S) gives $\lambda(a^{i}D^{i}_{w})=\sum_{m\in M_{i}}\mathrm{i}\varepsilon_{m}\lambda(a^{i}e_{r_{m}})$. The set $I=\{i\in[d]:M_{i}\neq\emptyset\}$ contains $j_{1}$. By Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §vanishing, $b_{a}(\lambda)(w)=\sum_{i\in I}\lambda(a^{i}D^{i}_{w})$ (Feedback Drifts of Polynomial Controls on Unitary Laws §drift). By Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §pairs, this is the sum of $\mathrm{i}\varepsilon_{m}\lambda(a^{j_{m}}e_{r_{m}})$ over the pairs $(i,m)$ with $i\in I$ and $m\in M_{i}$. The map $(i,m)\mapsto m$ is a bijection of the set of these pairs onto $[k]$, with inverse $m\mapsto(j_{m},m)$, and reindexing gives (B1).

(K) For $k\in\mathbb{N}$, $k\le2^{k}$ and $k^{2}\le4^{k}$ (natural powers in $\mathbb{R}$). By induction, $1\le2=2^{1}$, and if $k\le2^{k}$ then $k+1\le2^{k}+2^{k}=2^{k+1}$, since $1=1^{k}\le2^{k}$ (claims 1, 2 and 5 of Properties of Natural Number Powers in a Field). Then $k^{2}\le2^{k}2^{k}=4^{k}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field and claim 3 of Properties of Natural Number Powers in a Field.

Step 1 (clause 1, the constant). Let $k\in\mathbb{N}$ and $w\in W^{\circ}_{d,k}$. Then $|w|$ is the image of $k$ (Words in Unitary Letters: Generators, Signs, Inverse Letters, Lengths and Adjoint Words §length), so by (K) and claim 5 of Elementary Arithmetic in an Ordered Field, $0\le c_{w}|w|^{2}\le\theta_{d}^{\,k}4^{k}$. By Counting Cyclically Reduced Words: Finiteness and the Bound (2d)^k §count and claim 3 of Properties of Natural Number Powers in a Field, the $k$-th block sum $C_{k}$ of $w\mapsto c_{w}|w|^{2}$ satisfies

$$0\le C_{k}\le\theta_{d}^{\,k}4^{k}(2d)^{k}=(8d\,\theta_{d})^{k}=\bigl(\tfrac{1}{768}\bigr)^{k};$$

here $0\le C_{k}$ holds by Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §nonnegative. The geometric series of ratio $\frac1{768}$ converges by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §geometric. By Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §comparison, $\sum_{k=1}^{\infty}C_{k}$ converges. The map $w\mapsto c_{w}|w|^{2}$ takes nonnegative values, its value at $\varnothing$ being $1\cdot0=0$. It is therefore summable (The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §sums), with sum $\kappa_{d}^{2}=0+\sum_{k=1}^{\infty}C_{k}$. This sum is nonnegative by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates, as in The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §norm, and $\kappa_{d}$ is its nonnegative square root (Existence and Uniqueness of the Nonnegative Square Root).

Step 2 (clause 2, the energy bound). Let $a\in\mathcal{P}_{d}$, $\lambda\in\mathcal{L}_{d}$, and write $b=b_{a}(\lambda)$. Each $a^{j}$ is self-adjoint (Polynomial Controls on Unitary Laws and Their Energy §controls), so $\alpha_{j}=\lambda(a^{j}a^{j})$ is a nonnegative real for every $j\in[d]$ (Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §positive). By Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §monotone with $E=\{j\}$ and by Polynomial Controls on Unitary Laws and Their Energy §energy, $\alpha_{j}\le\sum_{i\in[d]}\alpha_{i}=\lVert a\rVert_{\lambda}^{2}$. Hence (R) gives $|\lambda(a^{j}e_{r})|^{2}\le\lVert a\rVert_{\lambda}^{2}$, that is, $|\lambda(a^{j}e_{r})|\le\lVert a\rVert_{\lambda}$ for all $j\in[d]$ and $r\in W_{2d}$, by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field.

For $w=\varnothing$ we have $|b(\varnothing)|=0=|\varnothing|\,\lVert a\rVert_{\lambda}$ by (B1). Now let $w\in W^{\circ}_{d}$ have length $k\in\mathbb{N}$. Each $\varepsilon_{m}\in\{1,-1\}$, so $|\mathrm{i}\varepsilon_{m}|=1$ (claims 3, 4 and 8 of Properties of Complex Conjugation and Modulus). Then (B1), Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §modulus, claim 4 of Properties of Complex Conjugation and Modulus and Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §comparison give

$$|b(w)|\le\sum_{m\in[k]}\bigl|\lambda\bigl(a^{j_{m}}e_{r_{m}}\bigr)\bigr|\le\sum_{m\in[k]}\lVert a\rVert_{\lambda}=k\,\lVert a\rVert_{\lambda}=|w|\,\lVert a\rVert_{\lambda}.$$

Here the sum of a constant over $[k]$ is $k$ times the constant, by claim 1 of Properties of a Sum over a Finite Index Set and induction on $k$.

Consequently, for every $w\in W^{\circ}_{d}$ we have $c_{w}|b(w)|^{2}\le\lVert a\rVert_{\lambda}^{2}\,c_{w}|w|^{2}$, by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field and claim 5 of Elementary Arithmetic in an Ordered Field. So the $k$-th block sum $B_{k}$ of $w\mapsto c_{w}|b(w)|^{2}$ satisfies $0\le B_{k}\le\lVert a\rVert_{\lambda}^{2}C_{k}$, by Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §nonnegative, Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §comparison and homogeneity. The series $\sum_{k}\lVert a\rVert_{\lambda}^{2}C_{k}$ converges to $\lVert a\rVert_{\lambda}^{2}\kappa_{d}^{2}$ by Step 1 and Elementary Properties of Series of Real Numbers §linearity. By Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §comparison, $\sum_{k}B_{k}$ converges, with sum at most $\lVert a\rVert_{\lambda}^{2}\kappa_{d}^{2}$. Therefore $b\in E_{d}$ (The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §space), and $\lVert b\rVert_{d}^{2}=c_{\varnothing}|b(\varnothing)|^{2}+\sum_{k=1}^{\infty}B_{k}\le(\kappa_{d}\lVert a\rVert_{\lambda})^{2}$ (The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §norm). Both $\lVert b\rVert_{d}$ and $\kappa_{d}\lVert a\rVert_{\lambda}$ are nonnegative, so $\lVert b\rVert_{d}\le\kappa_{d}\lVert a\rVert_{\lambda}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field.

Step 3 (clause 3, the momentum pairing). Let $a\in\mathcal{P}_{d}$, $\lambda\in\mathcal{L}_{d}$ and $p\in E_{d}$, and write $b=b_{a}(\lambda)$, which lies in $E_{d}$ by Step 2.

(a) Reality. Let $n\in\mathbb{N}$ and $i\in[d]$, and write $A=a^{i}$ and $Z=Z^{i}_{p,n}$. By Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §algebra and Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §linear, $\lambda(A\,\Xi^{i}_{p,n})=\frac12\lambda(AZ)+\frac12\lambda(AZ^{*})$ (The Truncated Cyclic Gradient of a Gauge Vector §gradient). We have $\lambda(AZ^{*})=\lambda(Z^{*}A)$ by Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §tracial. Since $A^{*}=A$, Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §algebra gives $Z^{*}A=Z^{*}A^{*}=(AZ)^{*}$, so $\lambda(Z^{*}A)=\overline{\lambda(AZ)}$ by Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §adjoint. By claim 2 of Properties of Complex Conjugation and Modulus,

$$\lambda\bigl(a^{i}\,\Xi^{i}_{p,n}\bigr)=\tfrac12\bigl(\lambda(AZ)+\overline{\lambda(AZ)}\bigr)=\operatorname{Re}\lambda\bigl(a^{i}Z^{i}_{p,n}\bigr),$$

which is a real number.

(b) Partial sums. By The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §expansion with $x=p$ and $y=b$, the series $\sum_{k}P_{k}$ with $P_{k}=\sum_{w\in W^{\circ}_{d,k}}c_{w}\operatorname{Re}\bigl(\overline{p(w)}\,b(w)\bigr)$ converges, and $\langle p,b\rangle_{d}=\operatorname{Re}\bigl(\overline{p(\varnothing)}\,b(\varnothing)\bigr)+\sum_{k=1}^{\infty}P_{k}=\sum_{k=1}^{\infty}P_{k}$, because $b(\varnothing)=0$ by (B1). Let $n\in\mathbb{N}$. Each word has only one length (Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal §words), so $(k,w)\mapsto w$ is a bijection from the pairs with $k\in[n]$ and $w\in W^{\circ}_{d,k}$ onto $W^{\circ}_{\le n}$. Claim 1 of Properties of a Sum over a Finite Index Set, Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §pairs and reindexing therefore give $s_{n}:=\sum_{k=1}^{n}P_{k}=\sum_{w\in W^{\circ}_{\le n}}c_{w}\operatorname{Re}\bigl(\overline{p(w)}\,b(w)\bigr)$.

The real part commutes with finite sums and with real factors. Indeed, $2\operatorname{Re}\zeta=\zeta+\overline{\zeta}$ (claim 2 of Properties of Complex Conjugation and Modulus); combine this with Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §conjugate, additivity, homogeneity and claim 1 of the same lemma. Hence $s_{n}=\operatorname{Re}\sum_{w\in W^{\circ}_{\le n}}c_{w}\overline{p(w)}\,b(w)$. By Feedback Drifts of Polynomial Controls on Unitary Laws §drift and homogeneity, the summand equals $\sum_{i\in[d]}c_{w}\overline{p(w)}\lambda(a^{i}D^{i}_{w})$. We exchange the two finite sums: Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §pairs applies twice, with reindexing along $(w,i)\mapsto(i,w)$. By (S) with $Q=W^{\circ}_{\le n}$ and The Truncated Cyclic Gradient of a Gauge Vector §gradient, we then obtain

$$s_{n}=\operatorname{Re}\sum_{i\in[d]}\ \sum_{w\in W^{\circ}_{\le n}}c_{w}\overline{p(w)}\,\lambda\bigl(a^{i}D^{i}_{w}\bigr)=\operatorname{Re}\sum_{i\in[d]}\lambda\bigl(a^{i}Z^{i}_{p,n}\bigr)=\sum_{i\in[d]}\lambda\bigl(a^{i}\,\Xi^{i}_{p,n}\bigr),$$

the last equality by (a) and the commutation of $\operatorname{Re}$ with finite sums. The partial sums $s_{n}$ of the convergent series $\sum_{k}P_{k}$ converge to its sum $\langle p,b\rangle_{d}$ in $(\mathbb{R},d_{\mathbb{R}})$ (Series of Real Numbers §convergent), which proves clause 3.

Step 4 (clause 4: operator preliminaries). Fix $a\in\mathcal{P}_{d}$ and $\lambda\in\mathcal{L}_{d}$, and write $b=b_{a}(\lambda)$. Operators are handled in the notation of Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation. For a tracial W*-probability space and a $d$-tuple $V$ of unitaries in it, we use the notation $V^{(l)}$, $V^{(w)}$ and $\lambda_{V}$ fixed in the statement of Unitary Laws Are the Laws of Unitary Tuples, and Are Realised Together with a Free Semicircular Family. By Unitary Laws Are the Laws of Unitary Tuples, and Are Realised Together with a Free Semicircular Family §realisation there are a tracial W*-probability space $(H,M,\Omega)$ with trace $\tau=\tau_{M}$ and a $d$-tuple $U$ of unitary operators on $H$ belonging to $M$ with $\lambda_{U}=\lambda$. The self-adjoint operators $S_{j}$ provided there are not used. The triple $(H,M,\Omega)$ is a cyclic tracial operator algebra (Tracial W*-Probability Spaces §space).

(O1) For $T\in M$, $|\tau(T)|=|\langle\Omega,T\Omega\rangle|\le\lVert\Omega\rVert\,\lVert T\Omega\rVert\le\lVert T\rVert_{\mathrm{op}}$. The equality is Tracial W*-Probability Spaces §trace with Cyclic Tracial Operator Algebras and Their Traces §trace. For the first inequality, Cauchy-Schwarz Inequality in a Complex Inner Product Space gives $|\langle\Omega,T\Omega\rangle|^{2}\le\lVert\Omega\rVert^{2}\lVert T\Omega\rVert^{2}$; both sides of the claimed inequality are nonnegative, so claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field applies. For the second, $\lVert T\rVert_{\mathrm{op}}$ is a bound for $T$ by 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, so $\lVert T\Omega\rVert\le\lVert T\rVert_{\mathrm{op}}\lVert\Omega\rVert$ by Bounded Linear Maps between Complex Normed Spaces and the Operator Norm §bounded, and $\lVert\Omega\rVert=1$ by Cyclic Tracial Operator Algebras and Their Traces §cyclic.

(O2) $\tau$ is linear, $\tau(I)=1$, $\tau(ST)=\tau(TS)$ and $\tau(S^{*})=\overline{\tau(S)}$ for $S,T\in M$ (The Trace, the Conjugation and the Right Action of a Cyclic Tracial Operator Algebra §trace). $M$ contains $I$ and is closed under sums, complex multiples, products and adjoints (Cyclic Tracial Operator Algebras and Their Traces §star-algebra).

(O3) For every letter $l$, $\lVert U^{(l)}\rVert_{\mathrm{op}}\le1$. Indeed, by claim 3 of Properties of Unitary Operators each $U_{j}$ is a bounded linear operator on $H$ with operator norm at most $1$; by 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, $U_{j}\in\mathcal{L}(H)$ and that operator norm is $\lVert U_{j}\rVert_{\mathrm{op}}$, so $\lVert U_{j}\rVert_{\mathrm{op}}\le1$. Moreover $\lVert U_{j}^{*}\rVert_{\mathrm{op}}=\lVert U_{j}\rVert_{\mathrm{op}}$ (Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint). Hence every product of such operators, including the empty product $I$, has operator norm at most $1$. This follows by submultiplicativity and $\lVert I\rVert_{\mathrm{op}}\le1$ (Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §operations) and induction on the number of factors.

(O4) $U^{(u)}U^{(v)}=U^{(uv)}$ for $u,v\in W_{2d}$, since $U^{(\varnothing)}=I$ and the letters of $uv$ are those of $u$ followed by those of $v$ (Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal §concatenation). By induction, a product of consecutive factors $U^{(w_{m})}$ of a word $w$ is $U^{(y)}$ for the corresponding subword $y$.

(O5) A map $T\in\mathcal{L}(H)$ with $T^{*}T=TT^{*}=I$ is unitary. Every $v$ equals $T(T^{*}v)$. Also $\langle Tu,Tv\rangle=\langle u,T^{*}Tv\rangle=\langle u,v\rangle$, because $T$ is the adjoint of $T^{*}$ (Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint-calculus, Adjoint of a Linear Map between Complex Inner Product Spaces §adjoint). Conversely, each $U_{j}$ is a bounded linear operator on $H$ by claim 3 of Properties of Unitary Operators, so $U_{j}\in\mathcal{L}(H)$ by 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. It has an adjoint $U_{j}^{*}\in\mathcal{L}(H)$ by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint, and $U_{j}^{*}U_{j}=U_{j}U_{j}^{*}=I$. Indeed, $\langle u,U_{j}^{*}U_{j}v\rangle=\langle U_{j}u,U_{j}v\rangle=\langle u,v\rangle$ with $u=U_{j}^{*}U_{j}v-v$ gives $\lVert u\rVert^{2}=\langle u,u\rangle=\langle u,U_{j}^{*}U_{j}v\rangle-\langle u,v\rangle=0$, so $u=0$ by definiteness (claim 4 of Elementary Properties of a Complex Inner Product), that is, $U_{j}^{*}U_{j}v=v$. Writing $v=U_{j}u$ by surjectivity, $U_{j}U_{j}^{*}v=U_{j}(U_{j}^{*}U_{j}u)=v$.

(O6) $(U^{(w)})^{*}=U^{(w^{*})}$ for every $w\in W_{2d}$. First let $l$ be a letter. If $\varepsilon(l)=1$, then $l^{-1}=d+g(l)$ has generator $g(l)$ and sign $-1$ (Words in Unitary Letters: Generators, Signs, Inverse Letters, Lengths and Adjoint Words §letters), so $U^{(l^{-1})}=U_{g(l)}^{*}=(U^{(l)})^{*}$. If $\varepsilon(l)=-1$, then $l^{-1}=g(l)$ has sign $1$, and $(U^{(l)})^{*}=(U_{g(l)}^{*})^{*}=U_{g(l)}=U^{(l^{-1})}$, because $T$ is the adjoint of $T^{*}$ (Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint-calculus). For $w=\varnothing$ we have $I^{*}=I$. For $w$ of length $k$, the same clause and induction on $k$ give $(U^{(w)})^{*}=(U^{(w_{k})})^{*}\cdots(U^{(w_{1})})^{*}=U^{(w_{k}^{-1})}\cdots U^{(w_{1}^{-1})}$. This is $U^{(w^{*})}$, since the $i$-th letter of $w^{*}$ is $(w_{j})^{-1}$ with $i+j=k+1$ (Words in Unitary Letters: Generators, Signs, Inverse Letters, Lengths and Adjoint Words §adjoint, Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal §reversal).

Step 5 (the control operators). Let $i\in[d]$. Let $F_{i}$ be a support set of $a^{i}$ with $n_{i}$ elements, and $\varphi_{i}:[n_{i}]\to F_{i}$ a bijection. Put

$$A'_{i}=\sum_{q=1}^{n_{i}}a^{i}\bigl(\varphi_{i}(q)\bigr)\,U^{(\varphi_{i}(q))},\qquad A^{i}=\tfrac12\bigl(A'_{i}+A'^{*}_{i}\bigr),$$

finite sums in the complex vector space $\mathcal{L}(H)$ (Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §operations). Both operators lie in $M$ by (O2). By Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint-calculus, $(A^{i})^{*}=\frac12(A'^{*}_{i}+A'_{i})=A^{i}$, so $A^{i}$ is self-adjoint. Put $N=1+\sum_{i\in[d]}\lVert A^{i}\rVert_{\mathrm{op}}$. Then $1\le N$, and $\lVert A^{i}\rVert_{\mathrm{op}}\le N$ for every $i$ (Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §monotone).

(T) $\tau(A^{i}U^{(r)})=\lambda(a^{i}e_{r})$ for every $r\in W_{2d}$. By (O2), (O4) and distributivity of composition over sums (Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §operations), $\tau(A'_{i}U^{(r)})=\sum_{q=1}^{n_{i}}a^{i}(\varphi_{i}(q))\,\tau(U^{(\varphi_{i}(q)r)})$. Since $\lambda_{U}=\lambda$ and by Sum over a Finite Index Set, this equals $\sum_{v\in F_{i}}a^{i}(v)\lambda(vr)$. On the other hand, $a^{i}=\sum_{v\in F_{i}}a^{i}(v)e_{v}$ pointwise: at $w\in F_{i}$ only the term $v=w$ is nonzero (Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §vanishing), and outside $F_{i}$ both sides vanish. So Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §algebra and Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §linear (by induction on the number of summands), together with $\lambda(e_{vr})=\lambda(vr)$, give $\lambda(a^{i}e_{r})=\sum_{v\in F_{i}}a^{i}(v)\lambda(e_{v}e_{r})=\sum_{v\in F_{i}}a^{i}(v)\lambda(vr)=\tau(A'_{i}U^{(r)})$. Next, by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint-calculus, $A'^{*}_{i}U^{(r)}$ is the adjoint of $(U^{(r)})^{*}A'_{i}=U^{(r^{*})}A'_{i}$, using (O6). So (O2) and the case just proved give $\tau(A'^{*}_{i}U^{(r)})=\overline{\tau(U^{(r^{*})}A'_{i})}=\overline{\tau(A'_{i}U^{(r^{*})})}=\overline{\lambda(a^{i}e_{r^{*}})}$. By Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §adjoint, Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §algebra (with $(e_{r^{*}})^{*}=e_{r}$, since $(r^{*})^{*}=r$), self-adjointness of $a^{i}$, and Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §tracial:

$$\overline{\lambda(a^{i}e_{r^{*}})}=\lambda\bigl((a^{i}e_{r^{*}})^{*}\bigr)=\lambda\bigl(e_{r}\,a^{i}\bigr)=\lambda\bigl(a^{i}e_{r}\bigr).$$

Hence $\tau(A^{i}U^{(r)})=\frac12\lambda(a^{i}e_{r})+\frac12\lambda(a^{i}e_{r})=\lambda(a^{i}e_{r})$.

Step 6 (Cayley transforms). Put $s_{0}=\frac{1}{4N}>0$. Let $s$ be real with $|s|\le s_{0}$, so that $|s|N\le\frac14$, and let $i\in[d]$. Put $X_{i}=\frac{\mathrm{i}s}{2}A^{i}\in M$. Then $\lVert X_{i}\rVert_{\mathrm{op}}=\frac{|s|}{2}\lVert A^{i}\rVert_{\mathrm{op}}\le\frac18<1$ by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §operations. By The Neumann Series, Inverses in Double Commutants, and Invertibility of A - iyI for Self-Adjoint A §neumann, $T_{i}=I-X_{i}$ is a bijection of $H$ onto $H$ with $T_{i}^{-1}\in\mathcal{L}(H)$ and $\lVert T_{i}^{-1}\rVert_{\mathrm{op}}\le\frac{1}{1-1/8}=\frac87$. As $T_{i}\in M$ and $M=M''$ (Tracial W*-Probability Spaces §space), $T_{i}^{-1}\in M$ by The Neumann Series, Inverses in Double Commutants, and Invertibility of A - iyI for Self-Adjoint A §commutant. Put $T'_{i}=I+X_{i}$ and $C_{i}=T'_{i}T_{i}^{-1}\in M$.

(a) $C_{i}^{*}C_{i}=C_{i}C_{i}^{*}=I$. As $s$ is real and $A^{i}$ is self-adjoint, $X_{i}^{*}=\overline{\tfrac{\mathrm{i}s}{2}}A^{i}=-X_{i}$ (Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint-calculus, Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint-unique); so $T_{i}^{*}=T'_{i}$ and $T'^{*}_{i}=T_{i}$. Also $T_{i}T'_{i}=I-X_{i}^{2}=T'_{i}T_{i}$, so $T_{i}^{-1}$ commutes with $T'_{i}$ by The Neumann Series, Inverses in Double Commutants, and Invertibility of A - iyI for Self-Adjoint A §commutant. Since $T_{i}^{-1}\in\mathcal{L}(H)$ and $H$ is a complex Hilbert space, $T_{i}^{-1}$ has an adjoint $(T_{i}^{-1})^{*}\in\mathcal{L}(H)$ by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint. From $T_{i}T_{i}^{-1}=T_{i}^{-1}T_{i}=I$ and Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint-calculus we get $(T_{i}^{-1})^{*}T'_{i}=I$ and $T'_{i}(T_{i}^{-1})^{*}=I$. So $T'_{i}$ is a bijection with inverse $T'^{-1}_{i}=(T_{i}^{-1})^{*}$, and $C_{i}^{*}=T'^{-1}_{i}T_{i}$. Then $C_{i}^{*}C_{i}=T'^{-1}_{i}T_{i}T'_{i}T_{i}^{-1}=T'^{-1}_{i}T'_{i}T_{i}T_{i}^{-1}=I$. Multiplying $T_{i}^{-1}T'_{i}=T'_{i}T_{i}^{-1}$ by $T'^{-1}_{i}$ on both sides gives $T'^{-1}_{i}T_{i}^{-1}=T_{i}^{-1}T'^{-1}_{i}$, and hence $C_{i}C_{i}^{*}=T'_{i}T_{i}^{-1}T'^{-1}_{i}T_{i}=T'_{i}T'^{-1}_{i}T_{i}^{-1}T_{i}=I$.

(b) The path. Put $V_{i}=C_{i}U_{i}\in M$ and $V(s)=(V_{1},\dots,V_{d})$. By (O5) and (a), $V_{i}^{*}V_{i}=U_{i}^{*}C_{i}^{*}C_{i}U_{i}=I$ and $V_{i}V_{i}^{*}=C_{i}U_{i}U_{i}^{*}C_{i}^{*}=I$, so $V_{i}$ is unitary by (O5). Define $\gamma(s)=\lambda_{V(s)}$; it belongs to $\mathcal{L}_{d}$ by Unitary Laws Are the Laws of Unitary Tuples, and Are Realised Together with a Free Semicircular Family §law. This defines $\gamma$ on $\{s\in\mathbb{R}:-s_{0}\le s\le s_{0}\}$. For $s=0$ we have $X_{i}=0$, $C_{i}=I$ and $V(0)=U$, so $\gamma(0)=\lambda_{U}=\lambda$.

(c) Expansion. Since $T'_{i}=T_{i}+2X_{i}$, $C_{i}=I+2X_{i}T_{i}^{-1}$. Also $T_{i}^{-1}=I+X_{i}T_{i}^{-1}$, because $T_{i}^{-1}-X_{i}T_{i}^{-1}=T_{i}T_{i}^{-1}=I$. Hence, with $2X_{i}=\mathrm{i}s\,A^{i}$,

$$D_{i}:=C_{i}-I=\mathrm{i}s\,A^{i}+R_{i},\qquad R_{i}=2X_{i}^{2}T_{i}^{-1},\qquad\lVert R_{i}\rVert_{\mathrm{op}}\le2\Bigl(\frac{|s|N}{2}\Bigr)^{2}\frac87\le s^{2}N^{2}.$$

By Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint-calculus, $D_{i}^{*}=C_{i}^{*}-I=-\mathrm{i}s\,A^{i}+R_{i}^{*}$, and $\lVert R_{i}^{*}\rVert_{\mathrm{op}}=\lVert R_{i}\rVert_{\mathrm{op}}$ by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint. By Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §operations and $|s|N\le\frac14$, both $D_{i}$ and $D_{i}^{*}$ have operator norm at most $|s|N+s^{2}N^{2}\le2|s|N\le\frac12$.

Step 7 (expansion along a word). Let $0<|s|\le s_{0}$ and let $w\in W^{\circ}_{d}$ have length $k\in\mathbb{N}$. For $m\in[k]$ put $L_{m}=U^{(w_{m})}$, and

$$B_{m}=D_{j_{m}}L_{m}\ \text{ if }\varepsilon_{m}=1,\qquad B_{m}=L_{m}D_{j_{m}}^{*}\ \text{ if }\varepsilon_{m}=-1 .$$

Then $V^{(w_{m})}=L_{m}+B_{m}$. If $\varepsilon_{m}=1$, $V^{(w_{m})}=C_{j_{m}}U_{j_{m}}=(I+D_{j_{m}})L_{m}$. If $\varepsilon_{m}=-1$, $V^{(w_{m})}=(C_{j_{m}}U_{j_{m}})^{*}=U_{j_{m}}^{*}C_{j_{m}}^{*}=L_{m}(I+D_{j_{m}}^{*})$ (Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint-calculus). By (O3) and Step 6(c), $\lVert B_{m}\rVert_{\mathrm{op}}\le2|s|N$. For $J\subseteq[k]$ let $\Pi^{J}=F^{J}_{1}\cdots F^{J}_{k}$, where $F^{J}_{m}=B_{m}$ for $m\in J$ and $F^{J}_{m}=L_{m}$ for $m\notin J$. Multiplying out $V^{(w)}=(L_{1}+B_{1})\cdots(L_{k}+B_{k})$ uses distributivity of composition over sums (Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §operations) and induction on $k$. It gives $V^{(w)}=\sum_{J\subseteq[k]}\Pi^{J}$, a sum over the $2^{k}$ subsets of $[k]$ (their number, by induction on $k$). By (O3) and submultiplicativity, $\lVert\Pi^{J}\rVert_{\mathrm{op}}\le\prod_{m\in J}\lVert B_{m}\rVert_{\mathrm{op}}$.

(i) $J=\varnothing$: $\Pi^{\varnothing}=U^{(w)}$ by (O4), so $\tau(\Pi^{\varnothing})=\lambda_{U}(w)=\lambda(w)$.

(ii) $J=\{m\}$. If $\varepsilon_{m}=1$, let $P=L_{1}\cdots L_{m-1}$ and $Q=L_{m}\cdots L_{k}$. By (O4), $P=U^{(w_{[1,m-1]})}$ ($P=I$ if $m=1$), $Q=U^{(w_{[m,k]})}$, and $\Pi^{\{m\}}=PD_{j_{m}}Q$. By (O4) and Rotations and Cyclic Derivatives of Words in Unitary Letters §rotations, $QP=U^{(w_{[m,k]}w_{[1,m-1]})}=U^{(r_{m})}$ if $1<m$, and $QP=U^{(w)}=U^{(r_{1})}$ if $m=1$. If $\varepsilon_{m}=-1$, let $P=L_{1}\cdots L_{m}=U^{(w_{[1,m]})}$ and $Q=L_{m+1}\cdots L_{k}=U^{(w_{[m+1,k]})}$ ($Q=I$ if $m=k$). Then $\Pi^{\{m\}}=PD_{j_{m}}^{*}Q$, and again $QP=U^{(r_{m})}$ by Rotations and Cyclic Derivatives of Words in Unitary Letters §rotations. In both cases Step 6(c) gives $\Pi^{\{m\}}=P\bigl(\mathrm{i}\varepsilon_{m}s\,A^{j_{m}}+R\bigr)Q$ with $R=R_{j_{m}}$ or $R=R_{j_{m}}^{*}$. By (O2) and (T), $\tau(PA^{j_{m}}Q)=\tau(A^{j_{m}}QP)=\tau(A^{j_{m}}U^{(r_{m})})=\lambda(a^{j_{m}}e_{r_{m}})$. By (O1), (O3) and submultiplicativity, $|\tau(PRQ)|\le\lVert R\rVert_{\mathrm{op}}\le s^{2}N^{2}$. Hence, by the linearity of $\tau$, $\tau(\Pi^{\{m\}})=s\,\mathrm{i}\varepsilon_{m}\lambda(a^{j_{m}}e_{r_{m}})+\rho_{\{m\}}$ with $\rho_{\{m\}}:=\tau(PRQ)$, and $|\rho_{\{m\}}|\le s^{2}N^{2}\le4s^{2}N^{2}$.

(iii) $J$ with at least two elements. As $0\le2|s|N\le\frac12\le1$, $\lVert\Pi^{J}\rVert_{\mathrm{op}}\le(2|s|N)^{2}=4s^{2}N^{2}$. So $\rho_{J}:=\tau(\Pi^{J})$ satisfies $|\rho_{J}|\le4s^{2}N^{2}$ by (O1).

By the linearity of $\tau$, $\gamma(s)(w)=\tau(V^{(w)})=\sum_{J\subseteq[k]}\tau(\Pi^{J})$. Let $\mathcal{J}$ be the set of nonempty subsets of $[k]$; it is nonempty since $[k]\in\mathcal{J}$ ($k\ge1$), and it has $2^{k}-1<2^{k}$ elements. For $J\in\mathcal{J}$ put $\varphi(J)=s\,\mathrm{i}\varepsilon_{m}\lambda(a^{j_{m}}e_{r_{m}})$ if $J=\{m\}$ and $\varphi(J)=0$ if $J$ has at least two elements; by (ii) and (iii), $\tau(\Pi^{J})=\varphi(J)+\rho_{J}$ with $|\rho_{J}|\le4s^{2}N^{2}$ for every $J\in\mathcal{J}$. Splitting off $J=\varnothing$ by Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §disjoint-union (with $F=\{\varnothing\}$ and $G=\mathcal{J}$), and using (i) and additivity,

$$\gamma(s)(w)=\lambda(w)+\sum_{J\in\mathcal{J}}\varphi(J)+\sum_{J\in\mathcal{J}}\rho_{J}.$$

The set $\mathcal{S}$ of one-element subsets of $[k]$ is a nonempty subset of $\mathcal{J}$ outside which $\varphi$ vanishes, and $m\mapsto\{m\}$ is a bijection from $[k]$ onto $\mathcal{S}$. So Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §vanishing, reindexing, homogeneity and (B1) give $\sum_{J\in\mathcal{J}}\varphi(J)=\sum_{m\in[k]}s\,\mathrm{i}\varepsilon_{m}\lambda(a^{j_{m}}e_{r_{m}})=s\,b(w)$. Finally, Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §modulus, Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §comparison and the count of $\mathcal{J}$ (a sum of a nonnegative constant over a set with $n$ elements is $n$ times the constant, as in Step 2) give

$$\bigl|\gamma(s)(w)-\lambda(w)-s\,b(w)\bigr|=\Bigl|\sum_{J\in\mathcal{J}}\rho_{J}\Bigr|\le(2^{k}-1)\cdot4s^{2}N^{2}\le4\cdot2^{k}s^{2}N^{2}\qquad(0<|s|\le s_{0},\ w\in W^{\circ}_{d}\text{ of length }k).\tag{$\ast$}$$

Step 8 (the two-sided tangent vector). For $0<|s|\le s_{0}$ let $v_{s}=\frac1s\bigl(\iota_{d}(\gamma(s))-\iota_{d}(\lambda)\bigr)-b$. It lies 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 §space, The Embedding of Unitary Laws into the Word Gauge Space and the Gauge Distance §embedding and Step 2. By The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §sobolev and the pointwise operations, $v_{s}(w)=\frac1s\bigl(\gamma(s)(w)-\lambda(w)\bigr)-b(w)$ for $w\in W^{\circ}_{d}$. At $w=\varnothing$, $v_{s}(\varnothing)=0$: indeed $\gamma(s)(\varnothing)=\tau(I)=1=\lambda(\varnothing)$ by (O2) and Laws of d-Tuples of Unitaries §normalised, and $b(\varnothing)=0$ by (B1). For $w\in W^{\circ}_{d,k}$, ($\ast$) and claims 4 and 8 of Properties of Complex Conjugation and Modulus give $|v_{s}(w)|=|s|^{-1}|\gamma(s)(w)-\lambda(w)-s\,b(w)|\le4\cdot2^{k}|s|N^{2}$. Hence $c_{w}|v_{s}(w)|^{2}\le16\,s^{2}N^{4}\,(4\theta_{d})^{k}$, by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field and claim 3 of Properties of Natural Number Powers in a Field. By Counting Cyclically Reduced Words: Finiteness and the Bound (2d)^k §count and claim 3 of Properties of Natural Number Powers in a Field, the $k$-th block sum satisfies

$$0\le\sum_{w\in W^{\circ}_{d,k}}c_{w}|v_{s}(w)|^{2}\le16\,s^{2}N^{4}(8d\,\theta_{d})^{k}=16\,s^{2}N^{4}\bigl(\tfrac1{768}\bigr)^{k}.$$

By Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §geometric and Elementary Properties of Series of Real Numbers §linearity, $\sum_{k=1}^{\infty}16\,s^{2}N^{4}(\frac1{768})^{k}=\frac{16}{767}s^{2}N^{4}\le s^{2}N^{4}$. So by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §comparison and The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §norm, $\lVert v_{s}\rVert_{d}^{2}=0+\sum_{k=1}^{\infty}\sum_{w\in W^{\circ}_{d,k}}c_{w}|v_{s}(w)|^{2}\le(|s|N^{2})^{2}$. Hence $\lVert v_{s}\rVert_{d}\le|s|N^{2}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field.

Now let $\varepsilon>0$ be given and choose $\delta=\varepsilon N^{-2}>0$. For every real $s$ with $0<|s|\le s_{0}$ and $|s|<\delta$, $\lVert v_{s}\rVert_{d}\le|s|N^{2}<\varepsilon$. By Step 6(b), $\gamma$ maps $\{s:-s_{0}\le s\le s_{0}\}$ into $\mathcal{L}_{d}$ with $\gamma(0)=\lambda$. So by Inward and Two-Sided Tangent Vectors to the Space of Unitary Laws §two-sided, $b_{a}(\lambda)\in T^{\pm}_{\lambda}$, which is the set of Unitary Laws with Free Unitary Noise: Standing Data §noise. $\blacksquare$