Chaplygin and Polytropic Kantowski--Sachs Solutions in Teleparallel $F(T)$ Gravity
A covariant reconstruction framework for Kantowski--Sachs (KS) geometries sourced by Chaplygin-type and polytropic fluids in teleparallel $F(T)$ gravity is developed using the coframe--spin-connection formalism and the teleparallel invariant approach. The matter sector is modelled by nonlinear equations of state, including the generalized Chaplygin gas $p=-A/\rho^{\alpha}$ and a polytropic law $p=K\rho^{\Gamma}$. The corresponding conservation laws determine the dependence of the fluid density on the anisotropic KS volume $V=A_2A_3^2$. These source scalings are then inserted into the symmetric part of the covariant teleparallel field equations and used to reconstruct the functional form of $F(T)$ directly from the KS dynamics. Power-law and exponential ans\"atze generate distinct invariant reconstruction branches. In the power-law sector, the Chaplygin fluid produces mixed constant-plus-power source terms, while the polytropic sector generates density powers controlled by the polytropic index. In the exponential sector, the natural reconstruction variable is the shifted invariant $X=T_0-T$, leading to shifted teleparallel de Sitter branches. The reconstructed models are interpreted as local anisotropic cosmological sectors and, for contracting angular KS scale factors, as local Kantowski--Sachs black-hole-interior reconstruction branches. The analysis is local and branch-dependent; leading-order viability is assessed through \(F_T>0\) and \(F_{TT}>0\), while a complete perturbative stability analysis is left for future work. The reconstruction is entirely driven by nonlinear matter conservation laws, thereby reversing the standard reconstruction strategy in which the gravitational Lagrangian is prescribed a priori.