02. Geometric Subsystem Quantization of Double Sine-Gordon Multi‑Kinks

We extend the geometric subsystem quantisation programme to the multi‑kink sector of the double sine–Gordon (DSG) equation. For each topological charge n, the DSG model admits a static, finite‑energy 2 n‑kink configuration — a bound state of n elementary sub‑kinks with fixed separations. Boosting the multi‑kink yields a two‑parameter family of travelling waves parametrised by a centre position a and a velocity v. We embed the corresponding moduli space into the classical phase space (the PTSO) of the DSG field, pull back the canonical field‑theoretic symplectic form, and show that the result is the canonical Darboux form dP da with the relativistic momentum P = Mₙ v/1-v^2, where Mₙ is the static mass of the n‑kink. Deformation quantisation via the Moyal product yields the commutation relation a, P=i and a free semi‑relativistic quantum particle of mass Mₙ. The derivation uses only Lorentz invariance and the exponential decay of the static profile; no integrability or inverse scattering is required. The result provides a rigorous geometric foundation for the quantum mechanics of multi‑kinks and demonstrates that the geometric subsystem programme is not limited to elementary solitons.