The Karahan Framework: Complete Theoretical Formulation of a Minimal Nonlinear Torsion Cosmology

We present the complete theoretical formulation of the Karahan Framework, a minimal nonlinear extension of teleparallel gravity defined by the quadratic torsion function �. This work consolidates the full conceptual development, mathematical derivations, cosmological modeling, and physical interpretation of the framework into a single unified document.Starting from the modified torsion action, we derive the cosmological dynamics and obtain a nonlinear Friedmann equation of the form�,which admits an exact analytical solution and reduces consistently to the standard �CDM model in the limit �. The background evolution is constructed to satisfy the normalization condition �, ensuring internal consistency and compatibility with observational cosmology.We establish explicit physical admissibility conditions that guarantee real, stable, and non-singular cosmological solutions. The framework is further connected to observational data through a combined likelihood approach using late-time probes, including Hubble expansion measurements, baryon acoustic oscillations (BAO), and Type Ia supernovae. First constraints indicate that the torsion parameter remains consistent with zero within current uncertainties, demonstrating agreement with standard cosmology at late times.Beyond homogeneous cosmology, the framework provides a natural extension to strong-gravity regimes. In particular, torsion induces angular-dependent deformations in black hole shadow geometries of the form�,offering a direct observational pathway via horizon-scale imaging.This document represents the complete theoretical structure of the Karahan Framework, including derivation logic, consistency conditions, parameter interpretation, and physical implications. The model provides a mathematically consistent, observationally viable, and testable extension of cosmology, with potential signatures emerging in extreme gravitational environments.