Emergent Fermion Condensation and Immirzi parameter from Pre-metric Gravity

We generalize Wilczek’s pre-geometric gravity by incorporating a kinetic mixing term which, upon spontaneous symmetry breaking (SSB), gives rise to the Holst action, thereby providing a dynamical origin for the Barbero-Immirzi parameter. Torsion-mediated four-fermion interactions arise in the Higgsed phase, with their coupling strength controlled by pre-geometric parameters. An attractive channel permits fermion condensation via a gravitational Nambu-Jona-Lasinio (NJL) mechanism akin to BCS superconductivity. The pre-geometric gauge theory inherently supplies a UV completion that may cure unitarity problems plaguing effective four-fermion interactions. The Einstein Equivalence Principle is not automatically recovered, permitting species-dependent cutoffs with phenomenological consequences for neutrino physics, dark matter and dark energy. Our construction furnishes a unified account of the Immirzi parameter, fermion condensation, and early universe cosmology grounded in fundamental pre-geometric principles.