The Karahan Framework: A Consistent Minimal Nonlinear Torsion Extension of Cosmology

We present the Karahan Framework, a consistent minimal nonlinear extension of teleparallel gravity based on the torsion function �. The framework introduces a single effective cosmological parameter while preserving the standard �CDM limit.At the cosmological level, the model leads to a modified Friedmann equation of the form�,which admits an exact analytical solution and remains well defined under explicit physical admissibility conditions. The background evolution is constructed to satisfy the normalization condition �, ensuring internal consistency and compatibility with observational cosmology.We impose the conditions�,�,and�,which guarantee real, stable, and non-singular cosmological solutions.Using late-time observational probes, including Hubble expansion measurements �, baryon acoustic oscillations (BAO), and Type Ia supernovae, we obtain first constraints indicating that the torsion parameter � is consistent with zero within current uncertainties. This demonstrates that the framework reproduces standard cosmology at late times without introducing observable tension.Beyond the homogeneous background, the framework predicts measurable deviations in strong-gravity regimes. In particular, torsion induces angular-dependent deformations in black hole shadow geometries of the form�,providing a direct observational pathway through horizon-scale imaging.The Karahan Framework therefore represents a mathematically consistent, observationally viable, and testable extension of cosmology, with potential signatures emerging in extreme gravitational environments.