Topological quantization of the vacuum energy from pre-geometric gravity

We show that pre-geometric gravity à la Mac Dowell–Mansouri predicts the topological quantization of the cosmological constant. In this framework, spacetime emerges from the spontaneous breaking of a de Sitter gauge symmetry, and a gravitational θ θ -term relates the vacuum energy to a periodic topological angle. The discrete periodicity of this angle forces the cosmological constant to take only quantized values, with the observed dark energy density selecting a vacuum labeled by a topological index k ~ 10¹²⁰ k ∼ 10 120 . We derive Golden Equations linking the cosmological constant, Planck mass, Gauss–Bonnet coupling, Immirzi parameter, and θ θ -angle, showing that the Immirzi parameter emerges as the ratio of two topological couplings. The pre-geometric Higgs potential is periodic, with minima separated by Planck-scale barriers that exponentially suppress vacuum decay. The de Sitter entropy is itself quantized and equals the topological index, providing a holographic interpretation of the vacuum selection.