This paper completes the geometric foundation of the coupled Dirac–Lambda framework by resolving the remaining structural questions of fixed-point existence and topology selection. Paper 4 established that, on a fixed compact geometry with dominance margin greater than one, the KKT system uniquely determines all nine charged fermion masses and mixing moduli with zero free continuous parameters. Two issues remained: whether the geometric self-consistency map admits a stable fixed point, and whether the round three-sphere geometry is uniquely selected within a natural geometric class without invoking additional postulates. We resolve both questions within a precise analytic framework. First, for compact positively curved Einstein three-manifolds satisfying admissibility conditions (dominance margin greater than one and a Lipschitz bound on the self-consistency operator), we prove via the Banach implicit function theorem that the self-consistency map admits a unique stable local fixed-point metric. Nondegeneracy of the linearized operator follows from a Neumann-series criterion applied to the DeTurck-gauged Einstein operator with matter backreaction. Second, restricting to spherical space forms at fixed curvature normalization, we prove topology selection by volume rigidity. The round three-sphere uniquely maximizes volume and, consequently, uniquely maximizes the dominance margin. Nontrivial quotients have strictly smaller volume and reduced Dirac multiplicities, which lowers the spectral budget and strictly decreases the dominance margin. Combining fixed-point stability with spectral rigidity yields unconditional structural closure on the admissible spherical class: the round three-sphere is uniquely selected, a stable geometric fixed point exists, and the mass and mixing closure results of Paper 4 hold at that fixed point without additional structural assumptions. The framework therefore achieves complete analytic closure within the specified geometric class. The only remaining structural input is the choice of admissible spectral filter exponent, selected minimally within the convergence constraints.
Rodgers Jeremy (Thu,) studied this question.