The characterization of pure-tone simplicial arrangements

Let $c\mathcal{A}$ be the cone over a real line arrangement $\mathcal{A}$, and suppose that $c\mathcal{A}$ is a simplicial arrangement. Let $F_{\mathcal{A}}$ denote the Milnor fiber, let $H^1(F_{\mathcal{A}})_{\neq 1}$ be the part of its first cohomology on which the monodromy eigenvalue is not $1$, and let a multinet structure mean the corresponding combinatorial structure on $c\mathcal{A}$. A real line arrangement is pure-tone when its relevant chamber-boundary map has a nontrivial linear relation, as in the preceding discussion. The characterization of pure-tone simplicial arrangements. The following conditions are equivalent:

$$c\mathcal{A}=\mathcal{A}(6m,1)\quad\text{for some }m>0;$$ $$H^1(F_{\mathcal{A}})_{\neq 1}\neq 0;$$

$\mathcal{A}$ is pure-tone; $c\mathcal{A}$ has a $k$-multinet structure for some $k\geq 3$; and $c\mathcal{A}$ has a $3$-multinet structure. These five conditions are equivalent.