The geometric subsystem quantisation programme extracts finite‑dimensional symplectic manifolds from the moduli spaces of distinguished classical solutions of the sine–Gordon equation. Using the inverse scattering description and the additive nature of the symplectic form, we prove a complete classification theorem for these manifolds as Hamiltonian torus spaces, within the standard IST additive reduction for bound‑state clusters. The only invariant is the number of breathers in each bound‑state cluster; all moduli spaces with the same breather count are equivariantly symplectomorphic. The classification reveals that the internal geometry depends only on the number of breathers and is independent of the number of attached kinks, and it exhausts all possible finite‑dimensional internal phase spaces arising from the soliton spectrum within the IST‑based bound‑cluster framework. As a consequence, the analytically continued scattering slice, with an unbounded internal moment map, is rigorously excluded from the genuine bound‑state sector. The classification turns the obstruction theorem of the companion paper into a structural statement and completes the geometric picture of the quantum particle multiplets in the sine–Gordon model.
Timmermans et al. (Wed,) studied this question.