Testing Exact Anisotropic Bianchi Type-V Cosmology in $f(T)$ Gravity with Cosmic Chronometers

Abstract Anisotropic cosmologies in modified gravity are useful only when exact solvability is accompanied by isotropization, viable matter properties, and an acceptable mean expansion history. We examine these requirements for Bianchi type-V cosmology in $f(T)$ gravity with a comoving perfect-fluid source. Within the pure-tetrad formulation, the field equations select a linear torsion sector yielding three exact power-law solution classes. We also derive the covariant tetrad-spin-connection equations, including the off-diagonal constraints. The $n>0$, $n\neq3$ anisotropic family satisfies the covariant compatibility conditions in the linear, GR-equivalent sector, while the $n=0$ ansatz admits an isotropically expanding solution with $K=0$ and unrestricted $l>0$. In the nonlinear isotropic sector, we reconstruct $f(T)$ parametrically for general $n$ and obtain an explicit nonlinear function for $n=1$, without assuming its functional form. For the three pure-tetrad classes, we analyze the directional Hubble rates, mean expansion, shear, normalized anisotropy, deceleration and jerk parameters, reconstructed density, pressure, equation of state, and energy conditions. Interestingly, only the first class approaches an isotropic matter-like regime, with decaying shear and normalized anisotropy and a mean expansion approaching matter-era behavior. Comparison with 33 cosmic-chronometer measurements shows that this class reproduces the qualitative increase of $H(z)$ with redshift, but its law $H(z)=H_0(1+z)^{3/2}$ is too rigid for the late-time expansion history; the fixed-$\Omega_m$ flat $\Lambda$CDM model provides a better fit. Moreover, the reconstructed matter sector admits positive-density regions satisfying the null and dominant energy conditions, while the strong energy condition is violated for the representative positive-coupling choice. The remaining classes expand too slowly and retain finite normalized anisotropy. In the isotropic Bianchi type-V limit, we further obtain an exact parametric reconstruction of nonlinear $f(T)$ models, including an explicit functional form in the $n=1$ branch.