Elliptic de Sitter Space as the Dominant Non-Orientable No-Boundary Geometry

In the elliptic interpretation of de Sitter space, antipodal points are identified. The interpretation has been studied as a realisation of observer complementarity and, recently, as the setting of a no-boundary density matrix, but why the universe should be elliptic rather than global de Sitter has not been addressed. Within the no-boundary proposal the question is sharp: the Euclidean four-sphere outweighs its antipodal quotient RP⁴ by a factor exp(3π/2GΛ). We show that a single principle, adopted rather than derived, suffices to select the elliptic interpretation: the no-boundary ground state carries no orientation. Gauging CRT requires non-orientable geometries to be included in the gravitational path integral but does not exclude orientable ones, so the principle is a genuine additional input. Granted it, the selection of RP⁴ is a theorem. For complete non-orientable Riemannian four-manifolds with R_ab = Λg_ab and Λ > 0, the volume, and hence the magnitude of the on-shell action, is at most half that of the round four-sphere, with equality only for RP⁴. The bound is attained in every even dimension and in no odd one. RP⁴ is therefore the unique dominant saddle within the non-orientable class, and its Lorentzian continuation is elliptic de Sitter space. The identification acts as CPT, the spacetime is not time-orientable, spinors require a Pin⁺ structure, which RP⁴ admits, and the vacuum has w = −1. Every input is labelled as derived, cited or adopted. Without the orientation principle, global de Sitter space remains the dominant geometry.