The sign-structure of a quantum many-body ground state is a central quantity in fixed-node approaches to mitigating the Fermion sign problem. For continuum electronic-structure Hamiltonians, the correct sign-structure is sufficient to compute exact gr
The sign-structure of a quantum many-body ground state is a central quantity in fixed-node approaches to mitigating the Fermion sign problem. For continuum electronic-structure Hamiltonians, the correct sign-structure is sufficient to compute exact ground state properties; on the other hand, lattice fixed-node approaches retain an additional dependence on trial wave-function amplitudes which are improved upon non-optimally while preserving their sign structure. Here we consider an iterative map obtained by repeatedly replacing the fixed-node trial state with the ground state of its associated lattice fixed-node Hamiltonian (independent of whether and how a practical algorithm could implement this iteration). We show the fixed points of this map are eigenstates of the Hamiltonian and reduced Hamiltonian resulting in at most one fixed point in the interior of each sign chamber. We prove that the ground state fixed point is stable and that the boundary of the ground state sign chamber is repulsive. This shows the amplitude dependence beyond having the correct ground state sign-structure disappears under self-consistent iteration. In other sign chambers, energy descent can be obstructed by the imposed sign structure; this tension leads to the iteration driving select amplitudes to zero producing `support collapse' onto chamber boundaries and corresponding to reduced Hamiltonians. Excited-state fixed points are therefore provably unstable and flow toward lower-energy boundary fixed points. We show sign chamber boundaries have a directional stability; boundary points attractive in one direction are repulsive under a sign flip. By revealing the geometry and stability structure of iterative fixed-node dynamics, our results clarify the foundations of a central approach to the Fermion sign problem and motivate potential new algorithmic strategies.