Proof of The Free Unitary Heat Generator Points into the Space of Unitary Laws

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

Operators are handled in the notation of Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation, and for a tracial W*-probability space and a $d$-tuple $V$ of unitaries in it we use the notation $V^{(l)}$, $V^{(w)}$, $\mathcal{A}_{V}$ 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. For letters we use Words in Unitary Letters: Generators, Signs, Inverse Letters, Lengths and Adjoint Words §letters: a letter $l$ with $\varepsilon(l)=1$ is the letter $g(l)$, one with $\varepsilon(l)=-1$ is the letter $d+g(l)$.

Fix $\lambda\in\mathcal{L}_{d}$. 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 $\tau=\tau_{M}$, a $d$-tuple $U$ of unitary operators on $H$ belonging to $M$ with $\lambda_{U}=\lambda$, and self-adjoint $S_{1},\dots,S_{d}\in M$ with $\lVert S_{j}\rVert_{\mathrm{op}}\le2$, satisfying Unitary Laws Are the Laws of Unitary Tuples, and Are Realised Together with a Free Semicircular Family §centred and Unitary Laws Are the Laws of Unitary Tuples, and Are Realised Together with a Free Semicircular Family §covariance. We fix these data.

Step 0 (elementary facts). (O1) For $T\in M$, $|\tau(T)|=|\langle\Omega,T\Omega\rangle|\le\lVert\Omega\rVert\,\lVert T\Omega\rVert\le\lVert T\rVert_{\mathrm{op}}$. Here 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 $|\langle\Omega,T\Omega\rangle|^{2}\le\langle\Omega,\Omega\rangle\langle T\Omega,T\Omega\rangle=\lVert\Omega\rVert^{2}\lVert T\Omega\rVert^{2}=(\lVert\Omega\rVert\,\lVert T\Omega\rVert)^{2}$, and since both $|\langle\Omega,T\Omega\rangle|$ and $\lVert\Omega\rVert\,\lVert T\Omega\rVert$ are nonnegative, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field yields $|\langle\Omega,T\Omega\rangle|\le\lVert\Omega\rVert\,\lVert T\Omega\rVert$. 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 $\lVert\Omega\rVert=1$ (Cyclic Tracial Operator Algebras and Their Traces §cyclic). (O2) $\tau(AB)=\tau(BA)$ for $A,B\in M$ (Cyclic Tracial Operator Algebras and Their Traces §tracial), $\tau$ is linear and $\tau(I)=1$ (The Trace, the Conjugation and the Right Action of a Cyclic Tracial Operator Algebra §trace). (O3) For every letter $l$, $\lVert U^{(l)}\rVert_{\mathrm{op}}\le1$, by claim 3 of Properties of Unitary Operators and $\lVert U_{j}^{*}\rVert_{\mathrm{op}}=\lVert U_{j}\rVert_{\mathrm{op}}$ (Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint). (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); in particular every $U^{(u)}$ lies in $\mathcal{A}_{U}$ and every product of operators $U^{(u)}$ lies in $\mathcal{A}_{U}$. (O5) A map $T\in\mathcal{L}(H)$ with $T^{*}T=TT^{*}=I$ is unitary: every $v$ equals $T(T^{*}v)$, and $\langle Tu,Tv\rangle=\langle u,T^{*}Tv\rangle=\langle u,v\rangle$, 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, and hence $U_{j}$ 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; it satisfies $U_{j}^{*}U_{j}=U_{j}U_{j}^{*}=I$: from $\langle u,U_{j}^{*}U_{j}v\rangle=\langle U_{j}u,U_{j}v\rangle=\langle u,v\rangle$ with $u=U_{j}^{*}U_{j}v-v$ we get $U_{j}^{*}U_{j}v=v$, and writing $v=U_{j}u$ by surjectivity, $U_{j}U_{j}^{*}v=U_{j}(U_{j}^{*}U_{j}u)=v$. (O6) For $0\le t\le\frac1{16}$ let $\sqrt t$ be the nonnegative square root; then $\sqrt t\le\frac14$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, hence $t=\sqrt t\,\sqrt t\le\frac14\sqrt t$ and $t\sqrt t\le\frac14 t\le\frac1{16}\sqrt t$.

Step 1 (Cayley transforms). Let $0\le t\le\frac1{16}$ and $j\in[d]$. Put $X_{j}=\frac{i\sqrt t}{2}S_{j}\in M$; then $\lVert X_{j}\rVert_{\mathrm{op}}\le\sqrt t\le\frac14<1$ 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, $T_{j}=I-X_{j}$ is a bijection of $H$ onto $H$ with $T_{j}^{-1}\in\mathcal{L}(H)$ and $\lVert T_{j}^{-1}\rVert_{\mathrm{op}}\le\frac{1}{1-1/4}=\frac43$; as $T_{j}\in M$ and $M=M''$ (Tracial W*-Probability Spaces §space), $T_{j}^{-1}\in M$ by The Neumann Series, Inverses in Double Commutants, and Invertibility of A - iyI for Self-Adjoint A §commutant. Put $T'_{j}=I+X_{j}$ and $C_{j}=T'_{j}T_{j}^{-1}\in M$ (Cyclic Tracial Operator Algebras and Their Traces §star-algebra); we write $C_{j}(t)$ when $t$ matters.

(a) $C_{j}^{*}C_{j}=C_{j}C_{j}^{*}=I$. Since $S_{j}$ is self-adjoint, $S_{j}^{*}=S_{j}$ and $X_{j}^{*}=\overline{\tfrac{i\sqrt t}{2}}S_{j}=-X_{j}$ (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 $T_{j}^{*}=T'_{j}$ and $T'^{*}_{j}=T_{j}$. Also $T_{j}T'_{j}=I-X_{j}^{2}=T'_{j}T_{j}$, so $T_{j}^{-1}$ commutes with $T'_{j}$ by The Neumann Series, Inverses in Double Commutants, and Invertibility of A - iyI for Self-Adjoint A §commutant. From $T_{j}T_{j}^{-1}=T_{j}^{-1}T_{j}=I$ 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, $(T_{j}^{-1})^{*}T'_{j}=(T_{j}T_{j}^{-1})^{*}=I$ and $T'_{j}(T_{j}^{-1})^{*}=I$; so $T'_{j}$ is a bijection with inverse $T'^{-1}_{j}=(T_{j}^{-1})^{*}$, and $C_{j}^{*}=T'^{-1}_{j}T_{j}$. Then $C_{j}^{*}C_{j}=T'^{-1}_{j}T_{j}T'_{j}T_{j}^{-1}=T'^{-1}_{j}T'_{j}T_{j}T_{j}^{-1}=I$. Multiplying $T_{j}^{-1}T'_{j}=T'_{j}T_{j}^{-1}$ by $T'^{-1}_{j}$ on both sides gives $T'^{-1}_{j}T_{j}^{-1}=T_{j}^{-1}T'^{-1}_{j}$, and hence $C_{j}C_{j}^{*}=T'_{j}T_{j}^{-1}T'^{-1}_{j}T_{j}=T'_{j}T'^{-1}_{j}T_{j}^{-1}T_{j}=I$.

(b) The path. Put $V_{j}=U_{j}C_{j}\in M$ and $V=V(t)=(V_{1},\dots,V_{d})$. By (O5) and (a), $V_{j}^{*}V_{j}=C_{j}^{*}U_{j}^{*}U_{j}C_{j}=I$ and $V_{j}V_{j}^{*}=U_{j}C_{j}C_{j}^{*}U_{j}^{*}=I$, so $V_{j}$ is unitary by (O5). Define $\gamma(t)=\lambda_{V(t)}$; 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. For $t=0$ we have $X_{j}=0$, $T_{j}=T'_{j}=T_{j}^{-1}=I$, $C_{j}=I$, $V=U$, so $\gamma(0)=\lambda_{U}=\lambda$.

(c) Expansion. Since $T'_{j}=T_{j}+2X_{j}$, $C_{j}=I+2X_{j}T_{j}^{-1}$; and $T_{j}^{-1}=I+X_{j}T_{j}^{-1}$, because $T_{j}^{-1}-X_{j}T_{j}^{-1}=T_{j}T_{j}^{-1}=I$. Substituting twice, $X_{j}T_{j}^{-1}=X_{j}+X_{j}^{2}+X_{j}^{3}T_{j}^{-1}$, so, with $2X_{j}=i\sqrt t\,S_{j}$ and $2X_{j}^{2}=-\frac t2S_{j}^{2}$,

$$D_{j}:=C_{j}-I=i\sqrt t\,S_{j}-\tfrac t2S_{j}^{2}+R_{j},\qquad R_{j}=2X_{j}^{3}T_{j}^{-1},\qquad\lVert R_{j}\rVert_{\mathrm{op}}\le2(\sqrt t)^{3}\tfrac43=\tfrac83t\sqrt t,$$

and, 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, $D_{j}^{*}=C_{j}^{*}-I=-i\sqrt t\,S_{j}-\frac t2S_{j}^{2}+R_{j}^{*}$ with $\lVert R_{j}^{*}\rVert_{\mathrm{op}}=\lVert R_{j}\rVert_{\mathrm{op}}$ (Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint). 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 (O6), for $e\in\{1,-1\}$ and $D^{(1)}_{j}=D_{j}$, $D^{(-1)}_{j}=D_{j}^{*}$,

$$\lVert D^{(e)}_{j}\rVert_{\mathrm{op}}\le2\sqrt t+2t+\tfrac83t\sqrt t\le\bigl(2+\tfrac12+\tfrac16\bigr)\sqrt t\le3\sqrt t,\qquad\lVert D^{(e)}_{j}-ie\sqrt t\,S_{j}\rVert_{\mathrm{op}}\le2t+\tfrac83t\sqrt t\le3t.\tag{B}$$

Step 2 (expansion along a word). Let $0<t\le\frac1{16}$ and let $w\in W_{2d}$ have length $k\in\mathbb{N}$. For $m\in[k]$ put $j_{m}=g(w_{m})$, $\varepsilon_{m}=\varepsilon(w_{m})$, $A_{m}=U^{(w_{m})}$, $Y_{m}=i\varepsilon_{m}\sqrt t\,S_{j_{m}}$, and

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

Then $V^{(w_{m})}=A_{m}+B_{m}$: if $\varepsilon_{m}=1$, $V^{(w_{m})}=U_{j_{m}}C_{j_{m}}=A_{m}(I+D_{j_{m}})$; if $\varepsilon_{m}=-1$, $V^{(w_{m})}=(U_{j_{m}}C_{j_{m}})^{*}=C_{j_{m}}^{*}U_{j_{m}}^{*}=(I+D_{j_{m}}^{*})A_{m}$ (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). Write $B_{m}=G_{m}+E_{m}$, where $G_{m}=A_{m}Y_{m}$ if $\varepsilon_{m}=1$ and $G_{m}=Y_{m}A_{m}$ if $\varepsilon_{m}=-1$. By (O3), (B) and Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §operations,

$$\lVert B_{m}\rVert_{\mathrm{op}}\le3\sqrt t,\qquad\lVert G_{m}\rVert_{\mathrm{op}}\le2\sqrt t,\qquad\lVert E_{m}\rVert_{\mathrm{op}}\le3t .$$

For $J\subseteq[k]$ let $\Pi^{J}=F^{J}_{1}F^{J}_{2}\cdots F^{J}_{k}$ with $F^{J}_{m}=B_{m}$ for $m\in J$ and $F^{J}_{m}=A_{m}$ for $m\notin J$. Multiplying out the product $V^{(w)}=(A_{1}+B_{1})\cdots(A_{k}+B_{k})$ by 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; induction on $k$) gives $V^{(w)}=\sum_{J\subseteq[k]}\Pi^{J}$, a sum over the $2^{k}$ subsets of $[k]$ (their number, by induction on $k$). Since all $A_{m}$ have norm at most $1$, $\lVert\Pi^{J}\rVert_{\mathrm{op}}\le\prod_{m\in J}\lVert B_{m}\rVert_{\mathrm{op}}$.

Step 3 (the terms). (i) $J=\varnothing$: $\Pi^{\varnothing}=A_{1}\cdots A_{k}=U^{(w)}$ by (O4) and induction on the number of factors, and $\tau(\Pi^{\varnothing})=\lambda_{U}(w)=\lambda(w)$.

(ii) $J=\{m\}$. Let $P=A_{1}\cdots A_{m}$ and $Q=A_{m+1}\cdots A_{k}$ if $\varepsilon_{m}=1$, and $P=A_{1}\cdots A_{m-1}$ and $Q=A_{m}\cdots A_{k}$ if $\varepsilon_{m}=-1$ (empty products being $I$). By (O4) and induction on the number of factors, $P=U^{(u)}$ and $Q=U^{(u')}$ for words with $uu'=w$, $PQ=U^{(w)}$, and $QP=U^{(u'u)}\in\mathcal{A}_{U}$. By (c), $\Pi^{\{m\}}=P\bigl(Y_{m}-\frac t2S_{j_{m}}^{2}+R\bigr)Q$ with $R=R_{j_{m}}$ or $R=R_{j_{m}}^{*}$. By (O2), Unitary Laws Are the Laws of Unitary Tuples, and Are Realised Together with a Free Semicircular Family §centred with $Z=QP$, and Unitary Laws Are the Laws of Unitary Tuples, and Are Realised Together with a Free Semicircular Family §covariance with $i=l=j_{m}$, $Z=I=U^{(\varnothing)}$ and $Z'=QP$:

$$\tau(PY_{m}Q)=i\varepsilon_{m}\sqrt t\,\tau(S_{j_{m}}QP)=0,\qquad\tau(PS_{j_{m}}^{2}Q)=\tau(S_{j_{m}}IS_{j_{m}}QP)=\tau(I)\tau(QP)=\tau(PQ)=\lambda(w),$$

and $|\tau(PRQ)|\le\lVert PRQ\rVert_{\mathrm{op}}\le\lVert P\rVert_{\mathrm{op}}\lVert R\rVert_{\mathrm{op}}\lVert Q\rVert_{\mathrm{op}}\le\lVert R\rVert_{\mathrm{op}}\le\frac83t\sqrt t$ by (O1) and submultiplicativity (Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §operations), since $P$ and $Q$ are products of the $A_{n}$, each of operator norm at most $1$ by (O3) (an empty product $I$ having norm at most $1$ by the same clause), so that $\lVert P\rVert_{\mathrm{op}}\le1$ and $\lVert Q\rVert_{\mathrm{op}}\le1$ by induction on the number of factors. Hence $\tau(\Pi^{\{m\}})=-\frac t2\lambda(w)+e_{\{m\}}$ with $|e_{\{m\}}|\le3t\sqrt t$.

(iii) $J=\{p,q\}$ with $p<q$. Let $\Pi^{J}_{G}$ be $\Pi^{J}$ with $B_{p},B_{q}$ replaced by $G_{p},G_{q}$. By distributivity, $\Pi^{J}-\Pi^{J}_{G}$ is the sum of the product with $B_{p}$ and $E_{q}$ in the places $p,q$ and of the product with $E_{p}$ and $G_{q}$ there, so

$$\lVert\Pi^{J}-\Pi^{J}_{G}\rVert_{\mathrm{op}}\le3\sqrt t\cdot3t+3t\cdot2\sqrt t=15\,t\sqrt t .$$

Let $\sigma_{p},\tau_{q}$ be defined by the same formulas as in The Free Unitary Heat Generator on Unitary Laws §positions, which involve only $p$, $q$ and the signs $\varepsilon_{p},\varepsilon_{q}$ and not the generators $g(w_{p}),g(w_{q})$; thus $\sigma_{p}\in\{p,p+1\}$ and $\tau_{q}\in\{q-1,q\}$. Since $1\le p<q\le k$, we get $1\le p\le\sigma_{p}\le p+1\le q\le\tau_{q}+1$ and $1\le q-1\le\tau_{q}\le q\le k$, so $\sigma_{p},\tau_{q}\in[k]$ and $\sigma_{p}\le q\le\tau_{q}+1$. In $\Pi^{J}_{G}$ the factor $Y_{p}$ stands right after $A_{p}$ if $\varepsilon_{p}=1$ and right before $A_{p}$ if $\varepsilon_{p}=-1$; in both cases the factors $A_{n}$ to its left are those with $n<\sigma_{p}$ and those to its right begin with $n=\sigma_{p}$. Likewise $Y_{q}$ stands right after $A_{q}$ if $\varepsilon_{q}=1$ ($\tau_{q}=q$) and right before $A_{q}$ if $\varepsilon_{q}=-1$ ($\tau_{q}=q-1$), so the factors $A_{n}$ to its left end with $n=\tau_{q}$ and those to its right are those with $n>\tau_{q}$. By (O4) and induction on the number of factors, and the definition of $w_{[a,b]}$ in The Free Unitary Heat Generator on Unitary Laws,

$$\Pi^{J}_{G}=U^{(a)}\,Y_{p}\,U^{(y)}\,Y_{q}\,U^{(b)},\qquad a=w_{[1,\sigma_{p}-1]},\quad y=w_{[\sigma_{p},\tau_{q}]},\quad b=w_{[\tau_{q}+1,k]},$$

with $a=\varnothing$ if $\sigma_{p}=1$, $b=\varnothing$ if $\tau_{q}=k$, and $y=\varnothing$ (so $U^{(y)}=I$) if $\tau_{q}<\sigma_{p}$; when $(p,q)\in P(w)$, $y=y_{pq}(w)$ and $ba=x_{pq}(w)$ by The Free Unitary Heat Generator on Unitary Laws §pieces. By (O2), (O4) and Unitary Laws Are the Laws of Unitary Tuples, and Are Realised Together with a Free Semicircular Family §covariance with $Z=U^{(y)}$ and $Z'=U^{(ba)}$,

$$\tau(\Pi^{J}_{G})=(i\varepsilon_{p}\sqrt t)(i\varepsilon_{q}\sqrt t)\,\tau\bigl(S_{j_{p}}U^{(y)}S_{j_{q}}U^{(ba)}\bigr)=-\varepsilon_{p}\varepsilon_{q}\,t\,\delta_{j_{p}j_{q}}\,\lambda(y)\,\lambda(ba).$$

So $\tau(\Pi^{J})=-t\,\varepsilon_{p}\varepsilon_{q}\lambda(y_{pq}(w))\lambda(x_{pq}(w))+e_{J}$ if $(p,q)\in P(w)$, and $\tau(\Pi^{J})=e_{J}$ if $g(w_{p})\neq g(w_{q})$, where $|e_{J}|\le15t\sqrt t$ by (O1).

(iv) $J$ with at least three elements: as $3\sqrt t\le\frac34\le1$, $\lVert\Pi^{J}\rVert_{\mathrm{op}}\le(3\sqrt t)^{3}=27t\sqrt t$, so $\tau(\Pi^{J})=e_{J}$ with $|e_{J}|\le27t\sqrt t$ by (O1).

Since $\gamma(t)(w)=\tau(V^{(w)})=\sum_{J\subseteq[k]}\tau(\Pi^{J})$ by the linearity of $\tau$, and the pairs $J=\{p,q\}$ with $g(w_{p})=g(w_{q})$ correspond to the elements $(p,q)$ of $P(w)$, the main terms of (i)-(iii) add up to $\lambda(w)-\frac{k}{2}t\lambda(w)-t\sum_{(p,q)\in P(w)}\varepsilon(w_{p})\varepsilon(w_{q})\lambda(y_{pq}(w))\lambda(x_{pq}(w))=\lambda(w)+t\,\Theta\lambda(w)$ by The Free Unitary Heat Generator on Unitary Laws §generator. Every nonempty $J$ has $|e_{J}|\le27t\sqrt t$ and there are fewer than $2^{k}$ of them, so by Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §modulus

$$\bigl|\gamma(t)(w)-\lambda(w)-t\,\Theta\lambda(w)\bigr|\le27\cdot2^{k}\,t\sqrt t\qquad(0<t\le\tfrac1{16},\ w\text{ of length }k).\tag{$\ast$}$$

Step 4 (the inward tangent vector). Put $s_{0}=\frac1{16}$; by Step 1(b), $\gamma$ maps $\{s:0\le s\le s_{0}\}$ into $\mathcal{L}_{d}$ with $\gamma(0)=\lambda$. For $0<s\le s_{0}$ let $v_{s}=\frac1s\bigl(\iota_{d}(\gamma(s))-\iota_{d}(\lambda)\bigr)-\widehat{\Theta}\lambda\in E_{d}$ (The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §space, The Embedding of Unitary Laws into the Word Gauge Space and the Gauge Distance §embedding, The Free Unitary Heat Generator: Bound, Continuity and Strict Dissipation in the Length-Weighted Gauge §bound); its value at $w\in W^{\circ}_{d}$ is $v_{s}(w)=\frac1s(\gamma(s)(w)-\lambda(w))-\Theta\lambda(w)$, the operations being pointwise. At $w=\varnothing$, $v_{s}(\varnothing)=0$, since $\gamma(s)(\varnothing)=\lambda(\varnothing)=1$ (Laws of d-Tuples of Unitaries §normalised) and $\Theta\lambda(\varnothing)=0$. For $w\in W^{\circ}_{d,k}$, ($\ast$) gives $|v_{s}(w)|\le27\cdot2^{k}\sqrt s$, so $c_{w}|v_{s}(w)|^{2}\le729\,s\,(4\theta_{d})^{k}$ by 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

$$0\le\sum_{w\in W^{\circ}_{d,k}}c_{w}|v_{s}(w)|^{2}\le729\,s\,(8d\,\theta_{d})^{k}=729\,s\,\bigl(\tfrac1{768}\bigr)^{k},$$

as $\theta_{d}=\frac1{6144d}$ (The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §weights). By Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §geometric and Elementary Properties of Series of Real Numbers §linearity, $\sum_{k=1}^{\infty}729\,s\,(\frac1{768})^{k}=729\,s\cdot\frac{1/768}{1-1/768}=\frac{729}{767}s\le s$, 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, $\lVert v_{s}\rVert_{d}^{2}=0+\sum_{k=1}^{\infty}\sum_{w\in W^{\circ}_{d,k}}c_{w}|v_{s}(w)|^{2}\le s$.

Now let $\varepsilon>0$ be given, and choose $\delta=\varepsilon^{2}>0$. For every real $s$ with $0<s\le s_{0}$ and $s<\delta$ we get $\lVert v_{s}\rVert_{d}^{2}\le s<\varepsilon^{2}$, hence $\lVert v_{s}\rVert_{d}\le\varepsilon$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. By Inward and Two-Sided Tangent Vectors to the Space of Unitary Laws §inward, $\widehat{\Theta}\lambda\in T^{+}_{\lambda}$. $\blacksquare$