Single-horizon regular black holes and the fate of the inner-horizon instability
Regular black holes (RBHs) are often constructed with a regular point-like center and two horizons, including an inner Cauchy horizon susceptible to mass inflation. We show that a two-horizon structure is not a necessary consequence of regularity. For static, spherically symmetric geometries with an asymptotically static exterior, we establish a horizon-parity theorem: a regular point-like center requires an even number of sign-changing zeros of $f$. We then construct two classes of single-horizon RBHs with a finite-area core, distinguished by whether the horizon lies outside the core or coincides with it, and analyze their matter content, energy conditions, and thermodynamic properties. The absence of an inner Cauchy horizon eliminates the classical mass-inflation mechanism on the single-horizon patch. We further establish mode stability against scalar perturbations for these finite-area-core configurations by showing that the effective potential is non-negative. Our results establish how the central geometry and asymptotic structure constrain horizon multiplicity and identify single-horizon configurations that avoid the classical inner-horizon instability.