Geometric Subsystem Quantization of Multi‑Breather and Multi‑Wobble Bound States in the Sine-Gordon Model: Extending the IST reduction to arbitrary bound clusters

We extend the geometric subsystem quantisation programme to arbitrary bound‑state clusters of the sine–Gordon equation. A general cluster consists of m kinks and n breathers sharing the same velocity and centre position. Using the additive property of the inverse scattering transform (IST) symplectic form and the bound‑state reduction, we derive the canonical symplectic form \[ ωC = dPC daC + ∑ⱼ₌₁ⁿ dθ_j dI_j, I_j = 16φ_j ∈ (0,8π), \] with total momentum PC = 8Θ\,(m + 2∑ⱼ₌₁ⁿsinφⱼ). The internal phase space is the product (S¹)ⁿ×(0,8π)ⁿ with the standard Hamiltonian Tⁿ-action. Deformation quantisation via the adapted Moyal product follows immediately. Multi‑cluster configurations, where each cluster moves with a different rapidity, factorise into a product of single‑cluster spaces. The classification of all such internal spaces by breather count alone, proved in the companion paper~{Classification}, is reviewed and placed in the context of the explicit constructions. The present work provides the detailed, sector‑by‑sector verification that the general structural claims of the classification are realised by explicit IST reductions.