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

In the setting of Unitary Laws with Free Unitary Noise: Standing Data, with the conventions of Unitary Laws with Free Unitary Noise: Standing Data §spaces and the sets $T^{\pm}_{\lambda}$ of two-sided tangent vectors, let $[d]$ be the initial segment determined by $d$, let $W^{\circ}_{d}$ be the set of cyclically reduced words with lengths $|w|$, let $c_{w}$ be the weights, let $\mathcal{P}_{d}$ be the set of polynomial controls with energies $\lVert a\rVert_{\lambda}^{2}$, let $b_{a}(\lambda)$ be the feedback drift, let $\Xi^{i}_{p,n}$ be the truncated cyclic gradients, and let $\lambda(X)$ be the evaluation of a word polynomial. Let $\mathbb{C}$ be the complex numbers with modulus $|z|$. Then the following hold.

1. (Constant)¶ The map $w\mapsto c_{w}|w|^{2}$ on $W^{\circ}_{d}$ is summable; its sum is nonnegative 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 $\kappa_{d}$ denotes its nonnegative square root.

2. (Energy bound)¶ For every $a\in\mathcal{P}_{d}$, $\lambda\in\mathcal{L}_{d}$ and $w\in W^{\circ}_{d}$, $|b_{a}(\lambda)(w)|\le|w|\,\lVert a\rVert_{\lambda}$. Consequently $b_{a}(\lambda)\in E_{d}$ and $\lVert b_{a}(\lambda)\rVert_{d}\le\kappa_{d}\lVert a\rVert_{\lambda}$.

3. (Momentum pairing)¶ For every $a\in\mathcal{P}_{d}$, $\lambda\in\mathcal{L}_{d}$ and $p\in E_{d}$, the number $\lambda\bigl(a^{i}\,\Xi^{i}_{p,n}\bigr)$ is real for every $n\in\mathbb{N}$ and every $i\in[d]$, and

$$\bigl\langle p,b_{a}(\lambda)\bigr\rangle_{d}=\lim_{n\to\infty}\sum_{i\in[d]}\lambda\bigl(a^{i}\,\Xi^{i}_{p,n}\bigr),$$

the limit existing in $(\mathbb{R},d_{\mathbb{R}})$.

4. (Tangency)¶ For every $a\in\mathcal{P}_{d}$ and $\lambda\in\mathcal{L}_{d}$, $b_{a}(\lambda)\in T^{\pm}_{\lambda}$.