Asymptotic Factorisation of the Symplectic Form for N Well-Separated Kinks - Multi--kink scattering in non--integrable scalar field theories

We extend the asymptotic factorisation theorem for the canonical field--theoretic symplectic form from a single kink--antikink pair to an arbitrary number N of well--separated kinks in a general relativistic scalar field theory with degenerate vacua. The configuration is modelled by a superposition of N single--kink Cauchy data with independent centres and velocities. We prove that the pullback of the canonical symplectic form to the $2N$--dimensional parameter space factorises into the sum of N free--kink symplectic forms plus off--diagonal corrections that are exponentially small in the minimum separation between any two solitons. The error is bounded by O\!(e^-μ Dₘᵢₙ/2), where μ is the decay rate of the static kink and Dₘᵢₙ the minimal separation. The result is universal and does not rely on integrability or closed--form solutions. The leading--order symplectic structure is a product of free--particle phase spaces, providing a rigorous geometric foundation for the quantum mechanics of asymptotically free multi--kink states in non--integrable models.