The discovery of the universe’s accelerated expansion has posed significant challenges to the standard cosmological model based solely on General Relativity and a cosmological constant. Modified gravity theories offer an intriguing geometric alternative to dark energy models, in which the acceleration arises from modifications to gravity itself. Among these, the class of teleparallel gravity theories—in which gravity is described by torsion instead of curvature—has emerged as a theoretical framework. In particular, \(f\left(T\right)\) gravity generalizes the Teleparallel Equivalent of General Relativity by promoting the torsion scalar \(T\) to an arbitrary function \(f\left(T\right)\) , formulated on the Weitzenböck spacetime. In this work, a novel and viable \(f\left(T\right)\) gravity model of the form: \(f\left(T\right)=T+\frac{\alpha {(-T)}^\frac{1}{n}}{1+\beta {(-T)}^\frac{1}{n}}\) is proposed, where α, β are model parameters and \(n\) governs the onset and scaling behavior of the torsion-induced corrections. This construction ensures a smooth interpolation between early time General Relativity and late-time accelerated expansion sourced by an effective torsional dark energy density \({\rho }_{DE}\sim {a}^{-\frac{2}{n}}\) , where \(a(t)\) is the scale factor. The novelty of the formulation lies in its sigmoid-type rational structure that smoothly activates the torsional dark energy sector, preserves early time consistency with standard cosmology, and predicts a late-time accelerating universe driven by a dynamically induced energy density. We have also proposed a novel extension involving the inclusion of parity-violating Chern–Simons-like terms and the Nieh–Yan topological invariant.
