Topological Sector of Generalized Pre-geometric Gravity

We investigate the topological sector of pre-geometric gravity formulated as a constrained BF gauge theory. Starting from the Generalized Wilczek Gravity – a theory where spacetime emerges from spontaneous symmetry breaking of a (anti)de Sitter gauge group – we demonstrate that the Gauss-Bonnet, Pontryagin, and Nieh-Yan topological couplings are not continuous parameters but are subject to discrete quantization. This quantization induces a periodic potential for the gravitational Higgs field, with an infinite tower of degenerate vacua labeled by topological quantum numbers [Formula: see text]. We derive the complete set of quantization rules for all emergent gravitational parameters: the cosmological constant, the Planck mass, the Immirzi parameter, and the topological couplings. We show that the MacDowell-Mansouri sector is the essential ingredient for this topological structure: when it is switched off, all quantization rules disappear and the theory becomes purely continuous. Conversely, when the MM sector is present, the Planck mass and Immirzi parameter acquire discrete spectra, with the observed cosmological constant selecting the topological sector [Formula: see text]. This provides a topological interpretation of the cosmological hierarchy: the large number [Formula: see text] is not a fine-tuning but a topological quantum number labeling the vacuum sector of the pre-geometric theory.