Proof of The Quadratic Control Hamiltonian: Legendre Formula, Bounds, Differences in the Law and Admissibility

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

Notation and conventions. $W^{\circ}_{d}$, the sets $W^{\circ}_{d,k}$ ($k\in\mathbb{N}$), the weights $c_{w}$ and the lengths $|w|$ are as in The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge; every $c_{w}$ 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. For $n\in\mathbb{N}$, $W^{\circ}_{\le n}$ is the finite set of The Truncated Cyclic Gradient of a Gauge Vector §gradient, the union of the $W^{\circ}_{d,k}$ with $k\in[n]$. For $m,n\in\mathbb{N}$ with $m<n$ let $V_{m,n}$ be the set of $w\in W^{\circ}_{d}$ whose length $k$ satisfies $m<k\le n$. Words of different lengths are different, so $W^{\circ}_{\le n}$ is the disjoint union of $W^{\circ}_{\le m}$ and $V_{m,n}$, and $V_{m,n}$ (respectively $W^{\circ}_{\le n}$) is the disjoint union of the sets $W^{\circ}_{d,k}$ with $m<k\le n$ (respectively $k\in[n]$), each nonempty and finite by Counting Cyclically Reduced Words: Finiteness and the Bound (2d)^k §finite.

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 $\lVert\cdot\rVert_{d}$ whose zero vector is the zero map, so The Cauchy-Schwarz Inequality in a Real Inner Product Space, Elementary Identities in a Real Inner Product Space and The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity apply to it. Absolute values of real numbers are handled by Properties of the Absolute Value in an Ordered Field (claim 2: $|-x|=|x|$; claim 3: $-|x|\le x\le|x|$; claim 5: the triangle inequality; claim 6: $|x|\le c$ if and only if $-c\le x\le c$); by its claim 8 the absolute value of a real number is its modulus as a complex number. Order manipulations are those of Elementary Order Arithmetic in an Ordered Field and Elementary Arithmetic in an Ordered Field. Sums over nonempty finite index sets are those of Sum over a Finite Index Set; they are additive and homogeneous by claims 3 and 4 of Properties of a Sum over a Finite Index Set, split over disjoint unions by Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §disjoint-union, compare termwise by Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §comparison, and satisfy $\bigl|\sum_{x\in F}f(x)\bigr|\le\sum_{x\in F}|f(x)|$ by Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §modulus. A sum over $[d]$ of real terms each at most a real number $c$ is at most $d\,c$: by the comparison just cited, claim 1 of Properties of a Sum over a Finite Index Set and the recursion in claim 1 of Properties of Finite Sums, it is at most $c$ added $d$ times, which is $d\,c$ with $d$ read in $\mathbb{R}$.

Sums and complex multiples of word polynomials are pointwise, and so are finite sums of word polynomials (Rotations and Cyclic Derivatives of Words in Unitary Letters); we write $-X=(-1)X$ and $X-Y=X+(-1)Y$. From the triangle inequality in Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §l1, by induction on the number of terms along the recursive definition of finite sums,

$$\Bigl\lVert\sum_{j\in J}X_{j}\Bigr\rVert_{1}\le\sum_{j\in J}\lVert X_{j}\rVert_{1}\tag{N}$$

for word polynomials $X_{j}$ indexed by a nonempty finite set $J$. Since $Y-X=(-1)(X-Y)$ pointwise and $\lVert(-1)Z\rVert_{1}=\lVert Z\rVert_{1}$ by Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §l1, $\lVert Y-X\rVert_{1}=\lVert X-Y\rVert_{1}$; and $X-X$ is the zero word polynomial, whose $\ell^{1}$ norm is $0$ by Word Polynomials in Unitary Letters: the Normed *-Algebra of Finitely Supported Word Functions §norm.

Convergence of real sequences is that of Limit of a Sequence of Real Numbers, which agrees with convergence in $(\mathbb{R},d_{\mathbb{R}})$ by The Real Numbers: Standing Notation and Background §sequences. We use the limit laws Arithmetic of Limits of Real Sequences, the order properties Order Properties of Limits of Real Sequences (claim 1: comparison; claim 3: domination by a null sequence; claim 4: absolute values), uniqueness of limits (claim 1 of Uniqueness of Limits and Boundedness of Convergent Real Sequences), the fact that a constant sequence converges to its value, and the following consequence of Limit of a Sequence of Real Numbers: (T) if $(a_{n})$ converges to $L$ and $m\in\mathbb{N}$, then $n\mapsto a_{n+m}$ converges to $L$, since $n+m\ge n$; hence claim 1 of Order Properties of Limits of Real Sequences applies to inequalities $a_{n}\le b_{n}$ known only for $n\ge m$ (apply it to the shifted sequences).

For $\lambda\in\mathcal{L}_{d}$, $p\in E_{d}$ and $a\in\mathcal{P}_{d}$ write $v_{a}(\lambda,p)=-\langle p,b_{a}(\lambda)\rangle_{d}-\frac12\lVert a\rVert_{\lambda}^{2}$, so that $H_{Q}(\lambda,p)$ is the supremum of $\{v_{a}(\lambda,p):a\in\mathcal{P}_{d}\}$ by The Quadratic Control Hamiltonian on Unitary Laws §hamiltonian. By Upper Bound and Least Upper Bound: (S1) $v_{a}(\lambda,p)\le H_{Q}(\lambda,p)$ for every $a\in\mathcal{P}_{d}$; (S2) $H_{Q}(\lambda,p)\le c$ for every real $c$ with $v_{a}(\lambda,p)\le c$ for all $a\in\mathcal{P}_{d}$.

Step 1 (length vectors). Let $p\in E_{d}$. For $k\in\mathbb{N}$ put

$$\beta_{k}(p)=\sum_{w\in W^{\circ}_{d,k}}c_{w}\,|p(w)|\,k,\qquad K_{k}=\sum_{w\in W^{\circ}_{d,k}}c_{w}\,k^{2},\qquad s_{n}=\sum_{k=1}^{n}K_{k}\ (n\in\mathbb{N}),$$

nonnegative reals by Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §nonnegative. By Feedback Drifts: the Energy Bound in the Word Gauge, the Momentum Pairing and Two-Sided Tangency §constant, $w\mapsto c_{w}|w|^{2}$ is summable; its block sums are the $K_{k}$, since $|w|=k$ on $W^{\circ}_{d,k}$, and since $|\varnothing|=0$ its sum is $\sum_{k=1}^{\infty}K_{k}=\kappa_{d}^{2}$ (The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §sums). Thus $s_{n}\to\kappa_{d}^{2}$ by Series of Real Numbers §convergent.

Define maps $W^{\circ}_{d}\to\mathbb{C}$ with nonnegative real values by $\widehat{p}(w)=|p(w)|$, $\ell(w)=|w|$ and, for $m\in\mathbb{N}$, $\ell_{m}(w)=|w|$ if $|w|>m$ and $\ell_{m}(w)=0$ otherwise; so $\ell(\varnothing)=\ell_{m}(\varnothing)=0$. By claim 8 of Properties of Complex Conjugation and Modulus the modulus of a nonnegative real is the number itself, so $|\widehat{p}(w)|^{2}=|p(w)|^{2}$, $|\ell(w)|^{2}=|w|^{2}$ and $|\ell_{m}(w)|^{2}$ is $|w|^{2}$ or $0$. Hence $w\mapsto c_{w}|\widehat{p}(w)|^{2}$ is the summable map $w\mapsto c_{w}|p(w)|^{2}$, so $\widehat{p}\in E_{d}$ and $\lVert\widehat{p}\rVert_{d}=\lVert p\rVert_{d}$ (The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §space, The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §norm); $w\mapsto c_{w}|\ell(w)|^{2}$ is $w\mapsto c_{w}|w|^{2}$, so $\ell\in E_{d}$ with $\lVert\ell\rVert_{d}^{2}=\kappa_{d}^{2}$, whence $\lVert\ell\rVert_{d}=\kappa_{d}$ by the uniqueness in Existence and Uniqueness of the Nonnegative Square Root; and the block sums of $w\mapsto c_{w}|\ell_{m}(w)|^{2}$ are $Q^{(m)}_{k}=0$ for $k\le m$ and $Q^{(m)}_{k}=K_{k}$ for $k>m$, so $0\le Q^{(m)}_{k}\le K_{k}$ and $\ell_{m}\in E_{d}$ by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §comparison. Put $\tau_{m}=\lVert\ell_{m}\rVert_{d}\ge0$; it does not depend on $p$.

The numbers $\tau_{m}$ tend to $0$. Let $m\in\mathbb{N}$. For $n>m$, splitting the finite sum at $m$ (claim 1 of Properties of Finite Sums, by induction on $n$) gives $\sum_{k=1}^{n}Q^{(m)}_{k}=s_{n}-s_{m}$, which converges to $\kappa_{d}^{2}-s_{m}$ as $n\to\infty$ by claim 3 of Arithmetic of Limits of Real Sequences and (T). As $c_{\varnothing}|\ell_{m}(\varnothing)|^{2}=0$, uniqueness of limits gives $\tau_{m}^{2}=\kappa_{d}^{2}-s_{m}$, and so $\tau_{m}^{2}\to0$ as $m\to\infty$ by the limit laws. Given a real $\varepsilon>0$, $\varepsilon^{2}>0$ by claim 5 of Elementary Order Arithmetic in an Ordered Field; choose $N$ with $\tau_{m}^{2}=|\tau_{m}^{2}-0|<\varepsilon^{2}$ for $m\ge N$; then $\tau_{m}<\varepsilon$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. Hence

$$\tau_{m}\to0\quad(m\to\infty).\tag{1}$$

Pairings. Apply The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §expansion to $\widehat{p}$ and $\ell$. The values being nonnegative reals, $\overline{\widehat{p}(w)}=\widehat{p}(w)$ by claim 1 of Properties of Complex Conjugation and Modulus, and the real part of the real number $|p(w)|\,|w|$ is itself by Real and Imaginary Parts of a Complex Number; so the terms $P_{k}$ there are the $\beta_{k}(p)$ and the term at $\varnothing$ is $0$. Thus $\sum_{k=1}^{\infty}\beta_{k}(p)$ converges with sum $\langle\widehat{p},\ell\rangle_{d}$. In the same way, for $m\in\mathbb{N}$, $\langle\widehat{p},\ell_{m}\rangle_{d}$ is the sum of the convergent series with nonnegative terms $0$ for $k\le m$ and $\beta_{k}(p)$ for $k>m$, whose $n$-th partial sum is $\sum_{k=m+1}^{n}\beta_{k}(p)$ for $n>m$. By Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates, $x\le|x|$ and The Cauchy-Schwarz Inequality in a Real Inner Product Space, for all $m,n\in\mathbb{N}$ with $m<n$,

$$\sum_{k=1}^{n}\beta_{k}(p)\le\langle\widehat{p},\ell\rangle_{d}\le\lVert p\rVert_{d}\,\kappa_{d},\qquad \sum_{k=m+1}^{n}\beta_{k}(p)\le\langle\widehat{p},\ell_{m}\rangle_{d}\le\lVert p\rVert_{d}\,\tau_{m}.\tag{2}$$

Step 2 ($\ell^{1}$ bounds for the truncated gradients). First, $\lVert D^{i}_{w}\rVert_{1}\le|w|$ for $i\in[d]$ and every word $w$ of length $k\in\mathbb{N}$. Let $M$ be the set of $m\in[k]$ with $g(w_{m})=i$ (Rotations and Cyclic Derivatives of Words in Unitary Letters §derivative). If $M$ is empty, $D^{i}_{w}$ is the zero word polynomial, of $\ell^{1}$ norm $0$. Otherwise, by (N) and Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §l1,

$$\lVert D^{i}_{w}\rVert_{1}\le\sum_{m\in M}\bigl|\mathrm{i}\,\varepsilon(w_{m})\bigr|\,\lVert e_{r_{m}(w)}\rVert_{1}=\sum_{m\in M}1\le\sum_{m\in[k]}1=k,$$

because $|\mathrm{i}\,\varepsilon(w_{m})|=|\mathrm{i}|\,|\varepsilon(w_{m})|=1$ (claim 4 of Properties of Complex Conjugation and Modulus, $\varepsilon(w_{m})$ being $1$ or $-1$), $\lVert e_{u}\rVert_{1}=|1|=1$ computed with the support set $\{u\}$ (Word Polynomials in Unitary Letters: the Normed *-Algebra of Finitely Supported Word Functions §norm), and by Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §monotone and claim 1 of Properties of a Sum over a Finite Index Set.

Let $p\in E_{d}$, $i\in[d]$ and $m,n\in\mathbb{N}$ with $m<n$. Evaluating at each word and splitting $W^{\circ}_{\le n}$ into $W^{\circ}_{\le m}$ and $V_{m,n}$, $Z^{i}_{p,n}=Z^{i}_{p,m}+\sum_{w\in V_{m,n}}c_{w}\overline{p(w)}D^{i}_{w}$ (The Truncated Cyclic Gradient of a Gauge Vector §gradient), so $Z^{i}_{p,n}-Z^{i}_{p,m}$ is this last sum. Since $|c_{w}\overline{p(w)}|=c_{w}|p(w)|$ (claims 3, 4 and 8 of Properties of Complex Conjugation and Modulus, $c_{w}$ being positive), (N), Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §l1 and the bound on $\lVert D^{i}_{w}\rVert_{1}$ give

$$\lVert Z^{i}_{p,n}-Z^{i}_{p,m}\rVert_{1}\le\sum_{w\in V_{m,n}}c_{w}|p(w)|\,|w|=\sum_{k=m+1}^{n}\beta_{k}(p),$$

the equality by splitting $V_{m,n}$ into the sets $W^{\circ}_{d,k}$ and claim 1 of Properties of a Sum over a Finite Index Set; in the same way $\lVert Z^{i}_{p,n}\rVert_{1}\le\sum_{k=1}^{n}\beta_{k}(p)$ for every $n\in\mathbb{N}$. By Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §algebra (adjoints are additive and $(zX)^{*}=\overline{z}X^{*}$), $\Xi^{i}_{p,n}-\Xi^{i}_{p,m}=\frac12(Z^{i}_{p,n}-Z^{i}_{p,m})+\frac12(Z^{i}_{p,n}-Z^{i}_{p,m})^{*}$, and by Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §l1 (triangle inequality, multiples, $\lVert X^{*}\rVert_{1}=\lVert X\rVert_{1}$) $\lVert\Xi^{i}_{p,n}-\Xi^{i}_{p,m}\rVert_{1}\le\lVert Z^{i}_{p,n}-Z^{i}_{p,m}\rVert_{1}$; likewise $\lVert\Xi^{i}_{p,n}\rVert_{1}\le\lVert Z^{i}_{p,n}\rVert_{1}$. With (2), and with $\Xi^{i}_{p,n}-\Xi^{i}_{p,n}=0$ for the case $m=n$,

$$\lVert\Xi^{i}_{p,n}\rVert_{1}\le\kappa_{d}\lVert p\rVert_{d}\quad(n\in\mathbb{N}),\qquad \lVert\Xi^{i}_{p,n}-\Xi^{i}_{p,m}\rVert_{1}\le\lVert p\rVert_{d}\,\tau_{m}\quad(m\le n).\tag{3}$$

Now let $p,q\in E_{d}$ and $n\in\mathbb{N}$. Pointwise $(p-q)(w)=p(w)-q(w)$, and $\overline{p(w)-q(w)}=\overline{p(w)}-\overline{q(w)}$ by claim 1 of Properties of Complex Conjugation and Modulus; so, evaluating at each word and using additivity and homogeneity of finite sums, $Z^{i}_{p-q,n}=Z^{i}_{p,n}-Z^{i}_{q,n}$, and then $\Xi^{i}_{p-q,n}=\Xi^{i}_{p,n}-\Xi^{i}_{q,n}$ by Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §algebra. By (3) applied to $p-q$,

$$\lVert\Xi^{i}_{p,n}-\Xi^{i}_{q,n}\rVert_{1}\le\kappa_{d}\lVert p-q\rVert_{d}.\tag{4}$$

Step 3 (evaluation estimate). Each $\Xi^{i}_{p,n}$ is self-adjoint, as recalled in the statement. For word polynomials $X,X',Y,Y'$ and $\lambda\in\mathcal{L}_{d}$, Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §algebra gives $(X-X')Y=XY-X'Y$ and $X'(Y-Y')=X'Y-X'Y'$, so by Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §linear $\lambda(XY)-\lambda(X'Y')=\lambda\bigl((X-X')Y\bigr)+\lambda\bigl(X'(Y-Y')\bigr)$; by the triangle inequality for the modulus (claim 7 of Properties of Complex Conjugation and Modulus) and Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §l1 ($|\lambda(Z)|\le\lVert Z\rVert_{1}$ and $\lVert ZZ'\rVert_{1}\le\lVert Z\rVert_{1}\lVert Z'\rVert_{1}$),

$$|\lambda(XY)-\lambda(X'Y')|\le\lVert X-X'\rVert_{1}\lVert Y\rVert_{1}+\lVert X'\rVert_{1}\lVert Y-Y'\rVert_{1}.\tag{5}$$

Step 4 (truncated Hamiltonians). For $\lambda\in\mathcal{L}_{d}$, $p\in E_{d}$ and $n\in\mathbb{N}$ put $h_{n}(\lambda,p)=\frac12\sum_{i\in[d]}\lambda\bigl(\Xi^{i}_{p,n}\Xi^{i}_{p,n}\bigr)$, a real number, and $C_{p}=d\,\kappa_{d}\lVert p\rVert_{d}^{2}\ge0$. For $m\le n$, (5) with $X=Y=\Xi^{i}_{p,n}$, $X'=Y'=\Xi^{i}_{p,m}$ and (3) give $|\lambda(\Xi^{i}_{p,n}\Xi^{i}_{p,n})-\lambda(\Xi^{i}_{p,m}\Xi^{i}_{p,m})|\le2\kappa_{d}\lVert p\rVert_{d}^{2}\tau_{m}$; summing over $[d]$ (modulus of a sum at most the sum of moduli) and halving,

$$|h_{n}(\lambda,p)-h_{m}(\lambda,p)|\le C_{p}\,\tau_{m}\qquad(m\le n).\tag{6}$$

For $p,q\in E_{d}$ and $n\in\mathbb{N}$, (5) with $X=Y=\Xi^{i}_{p,n}$, $X'=Y'=\Xi^{i}_{q,n}$, together with (3) and (4), gives $|\lambda(\Xi^{i}_{p,n}\Xi^{i}_{p,n})-\lambda(\Xi^{i}_{q,n}\Xi^{i}_{q,n})|\le\kappa_{d}^{2}\bigl(\lVert p\rVert_{d}+\lVert q\rVert_{d}\bigr)\lVert p-q\rVert_{d}$, hence

$$|h_{n}(\lambda,p)-h_{n}(\lambda,q)|\le\tfrac{d}{2}\,\kappa_{d}^{2}\bigl(\lVert p\rVert_{d}+\lVert q\rVert_{d}\bigr)\lVert p-q\rVert_{d}.\tag{7}$$

Clause 1 (Legendre formula). Let $\lambda\in\mathcal{L}_{d}$ and $p\in E_{d}$, and write $h_{n}=h_{n}(\lambda,p)$.

Convergence. By (1) and claim 3 of Arithmetic of Limits of Real Sequences, $C_{p}\tau_{m}\to0$. Given a real $\varepsilon>0$, choose $N$ with $C_{p}\tau_{m}=|C_{p}\tau_{m}-0|<\varepsilon$ for $m\ge N$. For $m,n\ge N$, (6) applied with the smaller of the two indices in the role of $m$ (and $|x-y|=|y-x|$) gives $|h_{n}-h_{m}|<\varepsilon$. So $(h_{n})$ is a Cauchy sequence and converges, by Every Cauchy Sequence of Real Numbers Converges, to a real number $h$. Fix $m\in\mathbb{N}$; as $n\to\infty$, $|h_{n}-h_{m}|\to|h-h_{m}|$ by claim 3 of Arithmetic of Limits of Real Sequences and claim 4 of Order Properties of Limits of Real Sequences, so (6), (T) and claim 1 of Order Properties of Limits of Real Sequences give

$$|h-h_{m}|\le C_{p}\,\tau_{m}\qquad(m\in\mathbb{N}).\tag{8}$$

Upper bound. Let $a\in\mathcal{P}_{d}$ and $n\in\mathbb{N}$, and put $\pi_{n}=\sum_{i\in[d]}\lambda(a^{i}\Xi^{i}_{p,n})$, real by Feedback Drifts: the Energy Bound in the Word Gauge, the Momentum Pairing and Two-Sided Tangency §pairing. For $i\in[d]$, $A_{i}=a^{i}+\Xi^{i}_{p,n}$ is self-adjoint by Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §algebra, so $0\le\lambda(A_{i}A_{i})$ by Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §positive. 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, $\lambda(A_{i}A_{i})=\lambda(a^{i}a^{i})+\lambda(a^{i}\Xi^{i}_{p,n})+\lambda(\Xi^{i}_{p,n}a^{i})+\lambda(\Xi^{i}_{p,n}\Xi^{i}_{p,n})$, and $\lambda(\Xi^{i}_{p,n}a^{i})=\lambda(a^{i}\Xi^{i}_{p,n})$ by Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §tracial. Summing over $[d]$ and using Polynomial Controls on Unitary Laws and Their Energy §energy and Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §nonnegative,

$$0\le\sum_{i\in[d]}\lambda(A_{i}A_{i})=\lVert a\rVert_{\lambda}^{2}+2\pi_{n}+2h_{n},\qquad\text{so}\qquad-\pi_{n}-\tfrac12\lVert a\rVert_{\lambda}^{2}\le h_{n}.$$

By Feedback Drifts: the Energy Bound in the Word Gauge, the Momentum Pairing and Two-Sided Tangency §pairing, $\pi_{n}\to\langle p,b_{a}(\lambda)\rangle_{d}$, so the left side converges to $v_{a}(\lambda,p)$ by the limit laws, and claim 1 of Order Properties of Limits of Real Sequences gives $v_{a}(\lambda,p)\le h$. As $a$ was arbitrary, (S2) gives $H_{Q}(\lambda,p)\le h$.

Lower bound. For $m,n\in\mathbb{N}$ put $g_{m,n}=\sum_{i\in[d]}\lambda(\Xi^{i}_{p,m}\Xi^{i}_{p,n})$, real by Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §positive. Fix $m\in\mathbb{N}$ and let $a_{m}=(-\Xi^{1}_{p,m},\dots,-\Xi^{d}_{p,m})$. By Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §algebra, $(-X)^{*}=\overline{(-1)}X^{*}=-X$ for self-adjoint $X$, so $a_{m}\in\mathcal{P}_{d}$ (Polynomial Controls on Unitary Laws and Their Energy §controls); $(-X)(-Y)=(-1)(-1)(XY)=XY$ and $(-X)Y=(-1)(XY)$, so with Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §linear, $\lVert a_{m}\rVert_{\lambda}^{2}=2h_{m}$ and $\sum_{i\in[d]}\lambda\bigl((-\Xi^{i}_{p,m})\Xi^{i}_{p,n}\bigr)=-g_{m,n}$. By Feedback Drifts: the Energy Bound in the Word Gauge, the Momentum Pairing and Two-Sided Tangency §pairing, $-g_{m,n}\to\langle p,b_{a_{m}}(\lambda)\rangle_{d}$ as $n\to\infty$, so by claim 3 of Arithmetic of Limits of Real Sequences $g_{m,n}\to g_{m}$, where $g_{m}=-\langle p,b_{a_{m}}(\lambda)\rangle_{d}$, and $v_{a_{m}}(\lambda,p)=g_{m}-h_{m}$. For $n\ge m$, (5) with $X=\Xi^{i}_{p,m}$, $X'=Y=Y'=\Xi^{i}_{p,n}$ and (3) give $|\lambda(\Xi^{i}_{p,m}\Xi^{i}_{p,n})-\lambda(\Xi^{i}_{p,n}\Xi^{i}_{p,n})|\le\lVert p\rVert_{d}\tau_{m}\cdot\kappa_{d}\lVert p\rVert_{d}$, since $\lVert\Xi^{i}_{p,m}-\Xi^{i}_{p,n}\rVert_{1}=\lVert\Xi^{i}_{p,n}-\Xi^{i}_{p,m}\rVert_{1}$ and the second term of (5) vanishes, $Y-Y'$ being the zero word polynomial; summing over $[d]$, $|g_{m,n}-2h_{n}|\le C_{p}\tau_{m}$ for $n\ge m$. As $n\to\infty$, $|g_{m,n}-2h_{n}|\to|g_{m}-2h|$ (limit laws, claim 4 of Order Properties of Limits of Real Sequences), so $|g_{m}-2h|\le C_{p}\tau_{m}$ by (T) and claim 1 of Order Properties of Limits of Real Sequences. With (8) and $-|x|\le x$,

$$v_{a_{m}}(\lambda,p)=h+(g_{m}-2h)+(h-h_{m})\ge h-2C_{p}\tau_{m},$$

so $h-2C_{p}\tau_{m}\le H_{Q}(\lambda,p)$ by (S1), for every $m\in\mathbb{N}$. By (1) and the limit laws $h-2C_{p}\tau_{m}\to h$, and comparison with the constant sequence $H_{Q}(\lambda,p)$ (claim 1 of Order Properties of Limits of Real Sequences) gives $h\le H_{Q}(\lambda,p)$.

Hence $h=H_{Q}(\lambda,p)$, which is clause 1, and (8) becomes

$$|H_{Q}(\lambda,p)-h_{m}(\lambda,p)|\le C_{p}\,\tau_{m}\qquad(\lambda\in\mathcal{L}_{d},\ p\in E_{d},\ m\in\mathbb{N}).\tag{9}$$

Clause 2 (bounds). Let $\lambda\in\mathcal{L}_{d}$ and $p\in E_{d}$. The upper bound holds because $\frac12\kappa_{d}^{2}\lVert p\rVert_{d}^{2}$ is an upper bound of $\{v_{a}(\lambda,p):a\in\mathcal{P}_{d}\}$ by The Quadratic Control Hamiltonian on Unitary Laws §hamiltonian, and (S2). For the lower bound let $0$ also denote the tuple of zero word polynomials, which lies in $\mathcal{P}_{d}$ by Polynomial Controls on Unitary Laws and Their Energy §controls. The zero word polynomial is $0X$ for any word polynomial $X$, so $0\,Y=(0X)Y=0(XY)$ is the zero word polynomial by Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §algebra, and its evaluation is $0$ by Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §linear. Hence every term of $b_{0}(\lambda)(w)=\sum_{i\in[d]}\lambda(0\,D^{i}_{w})$ (Feedback Drifts of Polynomial Controls on Unitary Laws §drift) is $0$, and $b_{0}(\lambda)$ is the zero map by Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §vanishing, the zero vector of $E_{d}$; so $\langle p,b_{0}(\lambda)\rangle_{d}=0$ by Elementary Identities in a Real Inner Product Space §zero. Likewise $\lVert 0\rVert_{\lambda}^{2}=\sum_{i\in[d]}\lambda(0\cdot0)=0$. Thus $v_{0}(\lambda,p)=0$, and $0\le H_{Q}(\lambda,p)$ by (S1).

Clause 3 (differences in the law). Let $\mu,\nu\in\mathcal{L}_{d}$ and $p\in E_{d}$; the word polynomials $\Xi^{i}_{p,n}$ do not depend on the law. By claims 3 and 4 of Properties of a Sum over a Finite Index Set, $-\frac12\sum_{i\in[d]}\bigl(\mu(\Xi^{i}_{p,n}\Xi^{i}_{p,n})-\nu(\Xi^{i}_{p,n}\Xi^{i}_{p,n})\bigr)=h_{n}(\nu,p)-h_{n}(\mu,p)$. By clause 1 and claim 3 of Arithmetic of Limits of Real Sequences, $h_{n}(\nu,p)-h_{n}(\mu,p)\to H_{Q}(\nu,p)-H_{Q}(\mu,p)$, and $\sum_{i\in[d]}(\mu(\cdot)-\nu(\cdot))=2h_{n}(\mu,p)-2h_{n}(\nu,p)$ converges to $2H_{Q}(\mu,p)-2H_{Q}(\nu,p)$; multiplying this limit by $-\frac12$ gives the displayed identity, the limit existing.

Clause 4 (admissibility). We check the two properties of Admissible Hamiltonians on Unitary Laws.

Dependence on the momentum. Let $\lambda\in\mathcal{L}_{d}$ and $p,q\in E_{d}$. By clause 1 and the limit laws, $h_{n}(\lambda,p)-h_{n}(\lambda,q)\to H_{Q}(\lambda,p)-H_{Q}(\lambda,q)$, so, by claim 4 and claim 1 of Order Properties of Limits of Real Sequences applied to (7) against a constant sequence,

$$|H_{Q}(\lambda,p)-H_{Q}(\lambda,q)|\le\tfrac{d}{2}\,\kappa_{d}^{2}\bigl(\lVert p\rVert_{d}+\lVert q\rVert_{d}\bigr)\lVert p-q\rVert_{d}.\tag{10}$$

Dependence on the law. Let $\mu,\nu\in\mathcal{L}_{d}$, $p\in E_{d}$ and $m\in\mathbb{N}$. By the triangle inequality and (9) for $\mu$ and for $\nu$,

$$|H_{Q}(\mu,p)-H_{Q}(\nu,p)|\le2C_{p}\tau_{m}+|h_{m}(\mu,p)-h_{m}(\nu,p)|.\tag{11}$$

Moreover, if $(\lambda_{j})_{j\in\mathbb{N}}$ converges to $\lambda_{0}$ in $(\mathcal{L}_{d},d_{\mathcal{L}})$, then $h_{m}(\lambda_{j},p)\to h_{m}(\lambda_{0},p)$. Indeed, for $i\in[d]$ let $Y^{i}=\Xi^{i}_{p,m}\Xi^{i}_{p,m}$, with a support set $F_{i}$. By The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §convergence, for each $w\in W_{2d}$, $\lambda_{j}(w)\to\lambda_{0}(w)$ in $(\mathbb{C},d_{\mathbb{C}})$, that is (Convergent Sequence in a Metric Space), the real sequence $|\lambda_{j}(w)-\lambda_{0}(w)|$ converges to $0$. By Evaluation of Word Polynomials by a Unitary Law §evaluation, additivity and homogeneity of finite sums, the modulus bound for finite sums and claim 4 of Properties of Complex Conjugation and Modulus,

$$|\lambda_{j}(Y^{i})-\lambda_{0}(Y^{i})|=\Bigl|\sum_{w\in F_{i}}Y^{i}(w)\bigl(\lambda_{j}(w)-\lambda_{0}(w)\bigr)\Bigr|\le\sum_{w\in F_{i}}|Y^{i}(w)|\,|\lambda_{j}(w)-\lambda_{0}(w)|,$$

and the right side converges to $0$ as $j\to\infty$ by claims 1 and 3 of Arithmetic of Limits of Real Sequences and induction on the number of terms. By claim 3 of Order Properties of Limits of Real Sequences, $\lambda_{j}(Y^{i})\to\lambda_{0}(Y^{i})$ (these are real numbers), and the limit laws give $h_{m}(\lambda_{j},p)\to h_{m}(\lambda_{0},p)$.

Continuity. Write $d_{\times}$ for the product metric on $\mathcal{L}_{d}\times E_{d}$ of Admissible Hamiltonians on Unitary Laws §continuous, built from the metric spaces $(\mathcal{L}_{d},d_{\mathcal{L}})$ (The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §metric) and $(E_{d},(x,y)\mapsto\lVert x-y\rVert_{d})$ (The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §metric); it is a metric by claim 1 of The Product Metric is a Metric. Let $(\lambda_{0},p_{0})\in\mathcal{L}_{d}\times E_{d}$ and let $\bigl((\lambda_{j},p_{j})\bigr)_{j\in\mathbb{N}}$ converge to it in $(\mathcal{L}_{d}\times E_{d},d_{\times})$. By claim 1 of Coordinatewise Convergence, Sequential Compactness and Density in a Product Metric Space, $\lambda_{j}\to\lambda_{0}$ in $(\mathcal{L}_{d},d_{\mathcal{L}})$ and $p_{j}\to p_{0}$ in $E_{d}$, that is, $\lVert p_{j}-p_{0}\rVert_{d}\to0$; and $\lVert p_{j}\rVert_{d}\to\lVert p_{0}\rVert_{d}$ by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §continuity. So by the limit laws $e_{j}=\frac{d}{2}\kappa_{d}^{2}\bigl(\lVert p_{j}\rVert_{d}+\lVert p_{0}\rVert_{d}\bigr)\lVert p_{j}-p_{0}\rVert_{d}\to0$. Let a real $\varepsilon>0$ be given. By (1) and the limit laws choose $m\in\mathbb{N}$ with $2C_{p_{0}}\tau_{m}<\varepsilon/3$; then, by the preceding paragraph with $p=p_{0}$, choose $J\in\mathbb{N}$ such that for $j\ge J$ both $e_{j}<\varepsilon/3$ and $|h_{m}(\lambda_{j},p_{0})-h_{m}(\lambda_{0},p_{0})|<\varepsilon/3$ (take the larger of the two indices provided). For $j\ge J$, by the triangle inequality, (10) for $\lambda_{j}$, $p_{j}$, $p_{0}$, and (11) for $\lambda_{j}$, $\lambda_{0}$, $p_{0}$,

$$|H_{Q}(\lambda_{j},p_{j})-H_{Q}(\lambda_{0},p_{0})|\le|H_{Q}(\lambda_{j},p_{j})-H_{Q}(\lambda_{j},p_{0})|+|H_{Q}(\lambda_{j},p_{0})-H_{Q}(\lambda_{0},p_{0})|<\tfrac{\varepsilon}{3}+\tfrac{\varepsilon}{3}+\tfrac{\varepsilon}{3}=\varepsilon .$$

Thus $H_{Q}(\lambda_{j},p_{j})\to H_{Q}(\lambda_{0},p_{0})$ in $(\mathbb{R},d_{\mathbb{R}})$. By Continuity Between Metric Spaces is Equivalent to Sequential Continuity §on-subset, applied with $A$ the whole space $\mathcal{L}_{d}\times E_{d}$ (on which the restricted metric is $d_{\times}$ itself), $H_{Q}$ is continuous on $\mathcal{L}_{d}\times E_{d}$; this is property Admissible Hamiltonians on Unitary Laws §continuous.

Tangential invariance. Let $\lambda\in\mathcal{L}_{d}$ and $p,p'\in E_{d}$ with $\langle p-p',v\rangle_{d}=0$ for every $v\in T^{\pm}_{\lambda}$. Let $a\in\mathcal{P}_{d}$. Then $b_{a}(\lambda)\in T^{\pm}_{\lambda}$ by Feedback Drifts: the Energy Bound in the Word Gauge, the Momentum Pairing and Two-Sided Tangency §tangent, so by Elementary Identities in a Real Inner Product Space §bilinear, $\langle p,b_{a}(\lambda)\rangle_{d}-\langle p',b_{a}(\lambda)\rangle_{d}=\langle p-p',b_{a}(\lambda)\rangle_{d}=0$, and hence $v_{a}(\lambda,p)=v_{a}(\lambda,p')$. As $a$ was arbitrary, the two sets whose suprema define $H_{Q}(\lambda,p)$ and $H_{Q}(\lambda,p')$ in The Quadratic Control Hamiltonian on Unitary Laws §hamiltonian coincide, and so do their suprema: $H_{Q}(\lambda,p)=H_{Q}(\lambda,p')$. This is property Admissible Hamiltonians on Unitary Laws §tangential, and $H_{Q}$ is an admissible Hamiltonian.