04. No-Go Theorem for Finite-Energy Heteroclinic Amsler Reductions under Nondegenerate-Vacuum Potentials

We investigate the pure bilinear (Amsler) reduction  $\omega(u,v)=f(\gamma uv+\delta)$ of the nonlinear wave equation $\omega_{uv}=H(\omega)$ for a smooth potential $V$ ($H=-V'$) with finitely many nondegenerate local minima. The reduction yields a second‑order ODE with a singular point at $\zeta=0$.  Under the assumptions that the solution is globally smooth, that it connects two local minima at spatial and temporal infinities, and that its derivative decays appropriately, we prove that finite total energy at all times forces the two asymptotic vacua to have equal depth and ultimately forces the solution to be the constant vacuum configuration.  Consequently, there is no continuous moduli space of finite‑energy particle‑like states arising from the pure Amsler reduction, and the geometric subsystem quantisation programme cannot be applied to this sector.  This result explains why the kink (travelling wave) sector remains the primary source of finite‑energy particle‑like objects in the sine–Gordon model and its generalisations.