In the conventional theory of the ground state of heliu m ato m, any solution of the two-electron Schrodinger equation is wrong! That is caused by a strong imbalance of proportions of the attraction versus repulsion operators' acting on a wave function, in the initial differential Schrodinger equation and in its integral algebraic form. In the differential equation, the attraction operators act on the radial part of wave function independently fro m one another obeying the true proportion: one attraction versus one repulsion per electron. But the resulting wave function turns out to satisfy the wrong integral proportion: two attraction energies versus one repulsion energy per electron. The underlying contradiction is un-coincidence of the additivity rules of energies and energy operators: whereas energies of the electrons may be summed the corresponding energy operators referred to different electrons may not. Consequences of this principal contradiction are d iscussed.
We prove that when two identical localized sources attempt to occupy the same normal mode of a nonlinear elastic medium, the doubling of the displacement field couples with the density-dependent stiffness to produce an excess elastic energy. This excess energy is derived analytically as a function of the separation distance, yielding an effective finite-range repulsive potential that decays exponentially in the far field and is regularized by nonlinear hardening in the near field. The result provides a mechanical formulation of Pauli repulsion without invoking algebraic antisymmetrization, offering an alternative mathematical pathway for many-body ground-state calculations that avoids the fermion sign problem at the foundational level.