A Unified Nieh-Gauss-Bonnet Topological Invariant in f ( R , Σ, T ) Gravity: Unifying Inflation and Dark Energy through Curvature and Torsion

This paper introduces a novel, unified topological invariant within the framework of Parameterized Absolute Parallelism (PAP) geometry. This invariant elegantly combines a generalized Gauss–Bonnet scalar, which incorporates curvature, with the Nieh–Yan topological density, which arises from torsion. We construct a modified theory of gravity based on this invariant, derive the complete set of field equations, and apply them to a homogeneous and isotropic universe. The resulting cosmological model exhibits rich and novel phenomenology, including scenarios for torsion-driven inflation, non-singular cosmological solutions, and topological phase transitions. This work establishes a comprehensive foundation for PAP–Nieh–Gauss–Bonnet gravity, presenting it as a natural and powerful topological extension of existing theories that modify gravity through both curvature and torsion.