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

In the setting of The Real Numbers: Standing Notation and Background, let $\mathbb{N}$ be the natural numbers, let $d\in\mathbb{N}$, let $\mathcal{L}_{d}$ be the set of unitary laws of $d$-tuples, let $\Theta$ be the free unitary heat generator, and for $\lambda\in\mathcal{L}_{d}$ let $\widehat{\Theta}\lambda$ be the restriction of $\Theta\lambda$ to the set of cyclically reduced words, which lies in the gauge space $E_{d}$ by The Free Unitary Heat Generator: Bound, Continuity and Strict Dissipation in the Length-Weighted Gauge §bound. The embedding $\iota_{d}$ and the norm $\lVert\cdot\rVert_{d}$ entering the definition of inward tangent vectors are those of The Embedding of Unitary Laws into the Word Gauge Space and the Gauge Distance and The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge. Let $T^{+}_{\lambda}$ be the set of inward tangent vectors to $\mathcal{L}_{d}$ at $\lambda$.

(Inward direction)¶ For every $\lambda\in\mathcal{L}_{d}$, $\widehat{\Theta}\lambda\in T^{+}_{\lambda}$.