09.02. Exact Symplectic Form for the Kink-Antikink Superposition Ansatz - a rigorous geometric foundation for collective‑coordinate approximations.

We compute the exact pullback of the canonical field‑theoretic symplectic form to the four‑dimensional parameter space of a kink‑antikink superposition ansatz in a general relativistic scalar field theory with degenerate vacua. The diagonal blocks of the symplectic matrix reproduce the free‑particle forms $dP_i\wedge da_i$ with the relativistic momenta $P_i = M\gamma_i v_i$, where $M$ is the kink mass. The off‑diagonal (interaction) blocks are explicitly expressed in closed form as a finite set of overlap integrals of the static kink profile and its derivatives; these integrals can be further reduced using the first integral of the static field equation. No asymptotic approximation is made; the result is exact for arbitrary separation of the centres. The pulled‑back form is closed; its non‑degeneracy holds at least for large separations, where the off‑diagonal components are exponentially small, and extends by continuity to an open neighbourhood of the asymptotic region. The result provides a rigorous geometric starting point for the collective‑coordinate method: the exact field‑theoretic symplectic structure on the physically motivated ansatz is obtained directly from the canonical phase space.