D4 chamber labels and complete adjacency graph

For each row $(x,i)$, apply the reflection rule in [F1] from right to left to ${\alpha }_i$. The result is $x{\alpha }_i={\alpha }_2$ in every row. Therefore $x{s}_i{x}^{-1}=s_2$, so $W_{J}x=x{W}_{\{i\}}$. Also $x{\alpha }_i$ is positive; by [F4], $\mathrm{\ell }(x{s}_i)>\mathrm{\ell }(x)$. Since $W_{\{i\}}=\{e,s_i\}$, $x$ is shortest in its right coset. Thus each row is a valid label in [F2].

For an independently reproducible chamber certificate, let $f_i({\alpha }_i)=0$ and $f_i({\alpha }_j)=1$ for $j\mathrm{\ne }i$. For each row calculate the signs of $(x{f}_i)(r)$ in the order of the seven roots in [F3], applying the letters of $x$ from left to right to each root before evaluating with $f_i$. The resulting sign strings for vertices 1 through 32, in order, are $--+-+++,-++-+++,-++--++,-++++++,-++--+-,-+++-++,+++++++,--+-++-,-+-+-++,+-+-+++,--+--+-,-+---+-,-+-+-+-,++-+-++,++-++++,+-+++++,--+----,-+-----,-+-+---,++-+--+,++-++-+,+-+++-+,--+-+--,++-+---,+-+-+-+,-------,+--+---,+--++-+,+------,+--++--,+---+--,+-+-+--$. No entry is zero and all strings are distinct. Each $f_i$ lies in $C_{\{i\}}$, so [F2] places $x{f}_i$ in the chamber represented by its row. The 32 distinct chambers exhaust the 32 chambers of [F3].

Two regions of a finite real hyperplane arrangement are adjacent exactly when their sign strings differ in one position. Indeed, adjacent regions differ only across their common wall. Conversely, for points in two regions differing only at one hyperplane, their joining segment stays strictly on the common side of every other hyperplane and crosses the remaining hyperplane at a point in no other hyperplane; a small neighbourhood there supplies a common facet. Comparing the strings in step 2.1 gives exactly the 48 pairs in the Example. In particular no adjacency is omitted.

For any listed adjacency from $A=(x,I)$ to $B=(y,K)$, [F5] gives $y=x{v}_{a,I}$, so [F6] identifies its arrow label as $x^{-1}y=v_{a,I}$. The reverse arrow has label $y^{-1}x$ and is its inverse in this groupoid. The $D_4$ system has four simple generators, so [F7] applies. Since step 3.1 lists every adjacency, these are precisely the adjacent-crossing arrows and they generate $\widetilde{\mathcal{B}\mathcal{H}}(J)$.

Vertex 7 is the fundamental label $(e,2)$. The listed graph is connected, as can be checked by following its edges from vertex 7 to every listed vertex. Following such a path recovers each displayed label by the iterative simple wall-crossing formula in [F5]; the bijection in [F2] ensures that the recovered label is the one in its row. Together with the exhaustive chamber and adjacency certificates and the generating-arrow identification, this proves the example.