The Karahan Framework v76.0: A Minimal Teleparallel Cosmology with Analytic Expansion and Torsion-Induced Black Hole Shadow Signatures

We present the Karahan Framework v76.0, a minimal and analytically tractable extension of teleparallel gravity defined by the nonlinear function �. The model preserves the General Relativity limit while introducing a single additional parameter that controls torsion-induced corrections.At the cosmological level, the framework yields a closed-form solution for the normalized expansion function �, ensuring exact analytical control and a well-defined physical domain. Observational constraints from late-time expansion data restrict the torsion parameter to small values, demonstrating consistency with the standard �CDM model while allowing for controlled deviations at high energy densities.In the strong-gravity regime, the framework incorporates a regularized compact-object model that replaces central singularities with finite cores. An axisymmetric torsion-vortex extension introduces a minimal deformation mechanism for black hole shadows, parameterized through a direction-dependent impact function. Current observational precision constrains these deformations to be small, maintaining agreement with nearly circular shadow geometries.A unified statistical structure combines cosmological and compact-object observables within a joint likelihood framework. The parameter space is explored using Markov Chain Monte Carlo methods under strict physical admissibility conditions.The Karahan Framework is not intended 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 manner, remain compatible with current data, and provide a clear pathway toward detecting new gravitational physics with next-generation observations.