01. Geometric Subsystem Quantization of the Sine--Gordon Two‑Soliton Scattering Sector and the Analytic Continuation to a Bound‑State Slice (Obstruction Theorem)

The geometric subsystem quantization programme extracts a finite‑dimensional symplectic manifold from the moduli space of distinguished solutions of a classical field theory. We complete the quantization of the sine–Gordon two‑soliton (kink–antikink) scattering sector using the Hamiltonian theory of the inverse scattering transform (IST). The two‑soliton moduli space with zero total topological charge is proved to be symplectomorphic to \ (R⁴\) with the canonical product symplectic form \ (₂ = da₁ dp₁ + da₂ dp₂\) ; the explicit global Darboux chart is constructed. Deformation quantization via the Moyal product is therefore immediate. We further perform an analytic continuation of the classical two‑soliton phase space, obtaining a real symplectic slice whose internal sector is \ (S^1 R+\), while the previously quantized breather has an internal sector \ (S^1 (0, 8) \). The two manifolds are not symplectomorphic as Hamiltonian \ (S^1\) -spaces (their natural internal circle actions have moment maps with different images) ; hence the bound‑state slice obtained by this simple continuation is physically distinct from the breather. The paper, together with the existing single‑kink and breather quantizations, establishes the geometric subsystem programme for the scattering sector and clarifies the nature of the analytic continuation.