Geometric Fixed-Point Existence and Spectral Rigidity in the Coupled Dirac–Lambda System

This paper completes the geometric foundation of the coupled Dirac-Lambda framework by resolving the remaining structural questions of geometric 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 questions remained: whether the geometric self-consistency map admits a stable fixed point, and whether the round three-sphere is uniquely selected within a natural admissible geometric class without additional postulates.

The paper resolves both questions within a precise analytic setting.

First, for compact positively curved Einstein three-manifolds satisfying the admissibility conditions, the geometric self-consistency map is shown to admit a unique stable local fixed-point metric. Stability follows from nondegeneracy of the linearized, DeTurck-gauged Einstein operator with matter backreaction and a corresponding local inversion criterion.

Second, restricting to spherical space forms at fixed curvature normalization, the paper proves topology selection by volume rigidity. The round three-sphere uniquely maximizes volume and therefore uniquely maximizes the dominance margin. Nontrivial quotients have strictly smaller volume and reduced Dirac multiplicities, which lower the spectral budget and strictly decrease the dominance margin.

Combining fixed-point stability with spectral rigidity yields 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.