Anisotropic Thin Shell Around a 3 Solar Mass Core: Static Equilibrium, Radial Stability, Vacuum Limit, Thickness Constraints, Equation of State, Constant-Sound-Speed Closure and Finite-Thickness Test

This monograph investigates whether collapsing matter with a mass of approximately three solar masses can avoid gravitational singularity formation by developing a rigid thin shell with anisotropic stresses surrounding a regular fluid core. Combining Parts I through IV into a single volume, the work explores the theoretical feasibility of non-singular compact objects supported by pressure anisotropy and high-density surface layers. ​Key aspects covered in the monograph include: ​Static Equilibrium and Stability: Analysis of the Darmois-Israel junction conditions and linear radial stability, establishing exact closed-form expressions for the critical stiffness required to maintain a stable configuration. ​Equation of State Constraints: Proof demonstrating that standard linear closures are unstable across all compactness parameters, and the derivation of an allowable stability and causality window for polytropic and constant-sound-speed surface equations of state. ​Core and Thickness Constraints: Evaluation of self-consistency requirements regarding shell thickness, revealing that ultra-high nuclear densities are necessary for the thin-shell approximation, along with the incorporation of an anisotropic Tolman-Oppenheimer-Volkoff interior core. ​Finite-Thickness and Energy Conditions: Testing the robustness of the thin-shell limit by replacing it with a finite-thickness layer (modeled as an Einstein-cluster limit) and examining local energy conditions, alongside preliminary estimates for external gravitational-wave echo delays.