The iterated dual-sigma word action: applying $\sigma_{s_1}^* \sigma_{s_2}^* \cdots$ in left-to-right order. This is the "co-action" version of the reflection action.
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The empty word acts as the identity.
Cons step: prepending $s$ to the word means first applying the dual sigma at $s$.
Concatenation: the word action is the composition $\sigma^*_{ws_2} \circ \sigma^*_{ws_1}$.
Singleton word action: $\sigma^*_s$ applied to $y$.
Applying the reverse word cancels: $\sigma^*_{\overline{ws}} \circ \sigma^*_{ws} = \mathrm{id}$.
Symmetric cancellation: $\sigma^*_{ws} \circ \sigma^*_{\overline{ws}} = \mathrm{id}$.
The iterated word action is $\mathbb{R}$-linear: composition of linear maps.
The Tits cone is closed under the dual word action: $\sigma^*_w$ sends $\mathcal U$ to itself.
The simple reflection $\sigma_s^*$ fixes any vector lying on the wall $\nu_s = 0$.
Transpose identity: $\langle \sigma_s^* y, v\rangle = \langle y, \sigma_s v\rangle$, expressing that $\sigma$ and $\sigma^*$ are mutually adjoint.
Coordinate formula via transpose: $(\sigma^*_w y)(s) = \sum_t y_t \cdot (\sigma_w e_s)_t$.
The word action depends only on the group element $\mathrm{wordProd}(ws)$, not on the word: two words representing the same Coxeter group element induce the same dual word action.
Descent detection: if the dual word action sends a nonnegative vector to a vector with a negative $s$-coordinate, then $s$ is a right descent of the group element $\mathrm{wordProd}(ws)$.
Geometric exchange condition: when $(\sigma^*_w y)(s) < 0$ for $y \ge 0$, the vector $\sigma_s^*(\sigma^*_w y)$ can be re-expressed via a shorter word $ws'$, shrinking word length via the strong exchange property.
The wall parameter $t_0 = \min_{s \in \mathrm{neg\_set}} \frac{x_s}{x_s - \mu_s}$: the smallest $t \in [0,1]$ for which the convex combination $(1-t)x + t\mu$ reaches the wall $\{z : z_s = 0\}$ for some $s$.
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The wall parameter is the minimum over the negative set, so it bounds each individual ratio.
The wall parameter is attained: some $s_0 \in \mathrm{neg\_set}$ realises the minimum ratio $x_{s_0} / (x_{s_0} - \mu_{s_0})$.
The wall parameter is nonnegative, since each individual ratio is.
The wall parameter lies in $[0, 1]$: for each $s$ in the negative set $x_s / (x_s - \mu_s) \le 1$ since $\mu_s < 0$.
The wall point $(1-t_0)\,x + t_0\,\mu$ at the wall parameter has nonnegative coordinates at every $s'$: at indices in the negative set the choice of $t_0$ keeps the coordinate $\ge 0$; at others both summands are nonnegative.
The wall point hits zero at the minimising index: some $s_0$ has wall-point coordinate equal to $0$.
Strong induction on word length: any convex combination of $x \in \mathcal D_0$ and $\sigma^*_w y$ (with $y$ nonneg) lies in the Tits cone. This is the key step of convexity; the wall-parameter / exchange-condition machinery shrinks $|w|$.
Bounded version of convex_closure_titsCone: not only does the convex combination lie in $\mathcal U$, but it is realised as $\sigma^*_{ws'} y'$ with word length bounded by the input bound $n$.
The Tits cone $\mathcal U \subset (B \to \mathbb R)$ is convex.
Scalar compatibility of $\sigma^*_s$: $\sigma^*_s(c \cdot x) = c \cdot \sigma^*_s(x)$.
Scalar compatibility of the iterated word action: $\sigma^*_w(c \cdot x) = c \cdot \sigma^*_w(x)$.
The Tits cone $\mathcal U$ is closed under nonnegative scaling: if $x \in \mathcal U$ and $c \ge 0$ then $c \cdot x \in \mathcal U$.
Existence in the fundamental domain: every $x \in \mathcal U$ is $\sigma^*_w$-image of some $y$ in the closed fundamental chamber $\overline{C}$.
The face $w \cdot F_I = \sigma^*_w(F_I)$: image of the standard face indexed by $I \subseteq S$ under the word action of $w$.
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Uniform bound: along the segment $[x, \mu] \subset \mathcal U$, every point can be written as $\sigma^*_w(y)$ for $y \ge 0$ with $|w|$ bounded by a fixed $N$ depending only on $x, \mu$.
The segment $[x, \mu]$ is covered by finitely many faces $w \cdot F_I$ with $|w|$ bounded by a constant $N$ depending only on the endpoints.
Face intersection rigidity: if $w \cdot F_I \cap F_J \ne \emptyset$ then $I = J$ and $w$ fixes $F_I$ pointwise. This is the key combinatorial property of the face decomposition of $\mathcal U$.
Every point of the closed fundamental chamber lies on the face $F_I$ where $I$ is its zero-coordinate set $\{s : y_s = 0\}$.
Uniqueness in the fundamental domain: if $y_1, y_2 \in \overline{C}$ and $\sigma^*_w(y_2) = y_1$ for some $w$, then $y_1 = y_2$. Combined with existence, this shows $\overline{C}$ is a strict fundamental domain for the $W$-action on $\mathcal U$.
Local finiteness of the face stratification along a segment: a segment $[x,\mu] \subset \mathcal U$ meets only finitely many faces $w \cdot F_I$.
Main theorem on the Tits cone $\mathcal U$. Bundles convexity, cone closure under nonnegative scaling, existence and uniqueness of the closed fundamental chamber $\overline{C}$, the face intersection rigidity property, and uniform finiteness of the face covering on any segment.