The Karahan Framework v54.0: A Minimal Nonlinear Teleparallel Cosmology with Exact Analytical Solution

We present the Karahan Framework v54.0 as a minimal nonlinear extension of teleparallel gravity based on the model F(T) = T + ξT². The construction follows the principle of simplicity, introducing a single additional parameter while preserving the General Relativity limit. At the level of homogeneous and isotropic cosmology, the model yields a closed modified Friedmann equation with an exact analytical solution for the normalized expansion function. A multi-component formulation including matter and radiation ensures correct normalization and consistency with late-time cosmology. Fits to H(z) data constrain the effective torsion parameter α to small values, indicating that the model reproduces ΛCDM behavior at low redshift while allowing controlled deviations at high energy densities. The framework is explicitly constructed to be empirically testable using standard cosmological probes and admits a well-defined physical parameter domain. In the strong-gravity regime, a phenomenological regular-core metric is introduced to illustrate how torsion effects may resolve central singularities while preserving asymptotic Schwarzschild behavior. The Karahan Framework is not presented as a complete theory of gravity, but as a minimal, mathematically consistent, and observationally testable model. Its primary result is that nonlinear torsion corrections can be introduced in a controlled and analytically tractable way without violating precision cosmological constraints.