Fermion Sign Problem: Mechanism, Guarantee, and Solution

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.