We present a complete, step‑by‑step derivation of the one‑loop meson (radiative) correction to the rest mass of the sine–Gordon kink within the geometric subsystem quantisation programme. The derivation begins with the exact symplectic embedding of the kink plus meson fluctuations into the field‑theoretic phase space (the PTSO). We prove that, at quadratic order in the fluctuations, the pullback of the canonical symplectic form decouples the translational mode from all meson modes exactly and without any perturbative approximation. The resulting quadratic Hamiltonian is identical to that used in the conventional collective‑coordinate method. Quantisation leads to a sum over zero‑point energies that, after standard renormalisation, yields the well‑known Dashen–Hasslacher–Neveu (DHN) result M = -1/ (in units m=1, =1). The calculation explicitly verifies that the geometric subsystem programme is fully consistent with the established semiclassical quantisation of solitons and provides a rigorous geometric foundation for the separation of the zero mode.
Kalmykov et al. (Wed,) studied this question.