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

Abstract

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 $1s^22s^2$ sequence establishes the one-valence-pair boundary: with no neighboring occupied valence pair, the source returns through the universal floor $D_0=\sqrt{1+δ^2}$. 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.