Gain-tightness characterization for symmetric sphere frameworks with horizontal or improper rotational symmetry

Let $\mathcal{S}$ denote the sphere. For a $S$-symmetric framework, let $S$ be either the group $\mathcal{C}_{mh}$ with $m$ even or the group $\mathcal{S}_{2m}$ with $m$ odd. Let $(G,p)$ be an $S$-generic realisation on $\mathcal{S}$, and let $(G_0,\psi)$ be its quotient $S$-gain graph. A gain graph is $(2,3,1)^i$-gain-tight when it is $(2,3,1)^i$-gain-sparse and has $|F|=2|V(F)|-1$ for its full edge set, where the sparsity bounds are $|F|\leq 2|V(F)|-3$ when a nonempty edge set $F$ has vertex-generated subgroup $\mathcal{C}_1$ or $\mathcal{C}_i$, and $|F|\leq 2|V(F)|-1$ otherwise. The sphere gain-tightness conjecture. The framework $(G,p)$ is $S$-isostatic if and only if $(G_0,\psi)$ is $(2,3,1)^i$-gain-tight. This would complete the sphere characterizations for the remaining groups $\mathcal{C}_{mh}$ with even $m$ and $\mathcal{S}_{2m}$ with odd $m$; the source presents these cases as conjectural, while the analogous cases listed in the surrounding summary are established by the cited theorems.