An Exact Finite-Density Regular Black-Hole Geometry with Universal Tidal Zones

Whether a black-hole interior can be modeled by matter at finite density is posed by exact geometries whose relation to the singularity theorems is stated. We solve the Einstein equations in closed form for the quartic-Lorentzian source $\rho ={\rho }_0{[1+{(r/a)}^4]}^{-1}$ with $p_r=-\rho$: the mass function is elementary, and the transverse pressure obeys the exact anisotropic equation of state $p_t=\rho -2{\rho }^2/{\rho }_0$. The geometry is the magnetically charged solution of a rational nonlinear electrodynamics with Maxwell weak-field limit, obtained previously by Ma and Kruglov and re-derived here from a selection principle. All static curvature invariants are bounded, with a conformally flat de Sitter core. Radially freely falling observers of every energy measure identical tidal eigenvalues (a cancellation equivalent to $\rho +p_r=0$), whose certified sign changes partition the spacetime into four zones at radii universal in units of a. The horizon structure is a trichotomy in $\kappa =M/a$, with ${\kappa }_{\mathrm{ext}}=1.762157$. Extensions develop an exact generalized-Vaidya interpolation with prescribed positive flux, closed-form thermodynamics, two-method test-scalar quasinormal frequencies, conditional shadow translations, and a candidate rotating geometrical extension. The principal limitation is the inner horizon: under standard Ori-type growth assumptions, mass inflation drives interior curvature to the Planck scale in finite advanced time; no stability of the two-horizon interior is claimed, while the exact static construction stands unconditionally.