Stationarity as a One-Mode Constraint on Quantum Correlation

Abstract

A stationary quantum state has one dynamical phase factor and one scalar energy even when its internal response spans a large Hilbert space. Our argument concerns not the value or observability of an absolute phase, but the one-ness of stationary time evolution. In regularized Hartree--Fock (reg-HF), the connection to the stationary mean field is established before any one-mode reduction: the primitive-product relative-coordinate Gaussian measure induced by the Gaussian HF representation fixes a covariance and, at that covariance, is simultaneously the minimum-Fisher-information and maximum-Shannon-differential-entropy distribution and the equality case of the Stam bound. The Coulomb operator admits a compatible Gaussian scale resolution, and virial completion converts the resulting regularized pair displacement into a stationary-compatible response. Stationarity motivates a one-mode constraint on the terminal scalar correlation response. The decisive reg-HF hypothesis is a one-mode exclusion axiom: covariance carried by response components orthogonal to the retained pair mode is excluded from the terminal scalar energy property, without asserting that the unrestricted response is one dimensional. Helium provides the minimal nontrivial test. The parameter-free retained Coulomb carrier $-δ/e$ and finite-$Z$ insertion $8δ/9$ closely reproduce independent He-like coefficients and energies, while an explicit Bethe--Goldstone calculation displays a broad non-single-denominator return spectrum even though its scalar energy is exactly moment-compressible. The result separates microscopic spectral complexity from one-mode energetic sufficiency.