Contact Renormalization of One-Mode Electron Correlation: Validation in Four- and Ten-Electron Systems

A stationary many-electron eigenstate has one dynamical phase although its internal response can be strongly multimode. We ask whether the stationary Hartree–Fock information carrier contains enough information to determine the scalar correlation energy through one effective return. In regularized Hartree–Fock, Gaussian primitive products fix a minimum-information relative-coordinate mode before any one-mode reduction; virial completion defines the correlation source, while exact contact measures the additional return restriction created by neighboring occupied pairs. The Be-like $1s^{2}2s^2$ sequence establishes the one-valence-pair boundary: with no neighboring occupied valence pair, the source returns through the universal floor $D_0=\sqrt{1+{\delta }^2}$. Ten-electron molecules and Ne-like ions then test how finite-pair contact dresses that return. The same parameter-free source–return construction reproduces the established molecular and ionic correlation trends with quantitative accuracy. The numerical agreement therefore serves as validation of a physical separation between source strength and return restriction, rather than as the objective by itself.