Zero-Complexity Hyperbolically Symmetric Static Fluids with an Exact Radial Stability Boundaries

Abstract The structure-scalar formulation for hyperbolically symmetric fluids \cite{herrera2021hyperbolically} is generalized to the closed radial stability problem within general relativity. We keep the inner boundary condition on the mass function and find the linear metric constraints, density variation and oscillation operators for a particular Lagrangian pressure dependence and fixed anisotropy. The exact zero-complexity configuration, trapped between rigid boundaries, becomes an inverse-square spectral problem. Applying the logarithmic variable change results in a necessary and sufficient condition on the sign of the principal eigenvalue, a precise estimate of the number of unstable radial modes, and an analytic expression for marginal eigenfunctions. The stability diagram depends on the response coefficient and the annulus thickness, independent shooting and grid refinement verify the numerical spectrum. We also calculate the first-order complexity response and apply the variational method to a separate spherical constant-density reference star scaled to published NICER mass and radius values for PSR J0030+0451. A fixed-density mass--radius family also permits a high-mass comparison. This stellar calculation is illustrative rather than an observational fit and the hyperbolic result applies to the confined fluid sector. The principal contribution is an exact and reproducible constitutive stability diagram that adds dynamical information to Herrera's hyperbolic equilibrium framework.