Universal Quantization of the Translational Mode of Relativistic Kinks - A geometric pullback theorem for scalar field theories with vacuum degeneracy

We prove a general theorem for the geometric subsystem quantization of the centre‑of‑mass degree of freedom of any relativistic kink in a scalar field theory with degenerate vacua. The only input is a Lorentz‑invariant Lagrangian density with a smooth potential V () that has at least two degenerate global minima. The translational moduli space of a boosted kink is immersed into the classical phase space of the field theory, and the pullback of the canonical symplectic form is computed exactly. The result is the universal Darboux form dP da with relativistic momentum P = M v, where the static kink mass M is expressed by the Bogomolny integral M = _-^+2V () \, d over the kink's field range. Deformation quantization via the Moyal product yields the canonical commutation relation a, P=i and the reduced centre‑of‑mass quantum Hamiltonian H = -^2₀^{2+M^2} for the translational collective coordinate. The derivation uses only the first integral of the static kink equation and Lorentz invariance; no explicit profile, integrability, or inverse scattering is required. The theorem unifies and simplifies the geometric subsystem quantization of all well‑known kinks (^4, ^6, double sine–Gordon, sine–Gordon, etc. ) and shows that the quantum translational mode is insensitive to the details of the potential beyond the single mass parameter M.