The two‑loop energy shift induced by the quartic kinetic vertex that arises in the rigorous local symplectic reduction of the sine–Gordon phase space around the static kink is derived. The symmetric vertex is shown to be proportional to (p²-k²) ² csch (k+p) 2 and vanishes for equal momenta, as required by Bose symmetry. The proper four‑meson interaction Hamiltonian is constructed with full combinatorial rigour, leading to the two‑loop energy shift. The resulting integral is ultraviolet finite; no counterterm subtraction is needed. The complete O (⁴) contribution to the kink mass from this vertex is presented as a single, absolutely convergent three‑dimensional integral, including all interference terms.
Kalmykov et al. (Mon,) studied this question.