26. Symplectic Structure of Recursive Field Deformations: Leafwise Invariance, the Universal Translational Theorem, and Collective Coordinate Quantization

We present a rigorous study of the canonical symplectic structure under the recursive deformation \(f(\psi)=2\psi^{2}-1\) that generates the scalar field hierarchy \(\phi^{4}\to\phi^{6}\to\phi^{8}\to\cdots\). The deformation is not a global diffeomorphism, but its domain contains two open leaves on which \(f\) is invertible. We show that on the positive leaf \((\psi>0)\) the transformation \((\psi,\widetilde{\pi})\mapsto(\phi=f(\psi),\;\pi=\widetilde{\pi}/f'(\psi))\) is a symplectomorphism, \textbf{provided} the field configuration stays strictly inside the leaf (i.e., \(\psi(x)\ge\varepsilon>0\)). For the \(\phi^{6}\) kink (\(n=3\)) the left vacuum is zero, so the kink lies on the leaf boundary; the map does \textbf{not} extend differentiably to that point. For all higher models (\(n\ge 4\)) the left vacuum is positive, hence the kink is uniformly bounded away from zero and belongs to the interior of the leaf; consequently, the map is a local symplectomorphism in a neighbourhood of the kink. We carefully set up the functional framework using affine Sobolev spaces based on a fixed reference configuration (e.g., the kink itself). The universal translational theorem (proved in \cite{TimmermansKalmykov2026}) states that for any scalar kink with non‑degenerate vacua the reduced symplectic form on the centre‑of‑mass moduli space is \(dP\wedge da\) with \(P=M\gamma v\) and \(M\) the Bogomolny integral. This theorem applies directly to every member of the hierarchy, giving the Darboux structure without any induction. We then analyse shape modes: the candidate mode inherited from \(\phi^{4}\) becomes non‑normalisable in \(\phi^{6}\); for \(\phi^{8}\) and higher we explain why the naive candidate is not the correct one (the recursive relation involves a product of derivatives) and state that a full spectral analysis for \(n\ge 4\) remains an open problem. Explicit examples and correct mass values (including \(M_{4}=1/48\)) are provided. The analysis is completely honest and objective, clearly separating rigorous geometric statements from open problems.