The Geometric Subsystem Quantisation Programme - From the TSO classification to deformation quantisation: a comprehensive review

We present the complete logical and historical development of the geometric subsystem quantisation programme. The narrative begins with the classification of ODE reductions of nonlinear wave equations (the Time‑Shared Object, TSO), continues through the embedding of finite‑energy sectors into the field phase space (the PTSO), the pullback of the canonical symplectic form, and the application of deformation quantisation to the resulting finite‑dimensional moduli spaces. We explain how the universal kink theorem emerged, how the integrable sine–Gordon model was fully solved using the inverse scattering transform, and how the programme extended to on‑integrable theories including shape modes, coupled dynamics, and meson truncations. Every result is stated with its precise assumptions; we distinguish between rigorous non‑perturbative theorems, perturbative approximations, and open problems. The article is intended as a roadmap and synthesis, clearly distinguishing between rigorous non‑perturbative theorems, perturbative results, and open problems, and explicitly stating the programme’s scope boundaries.