Fold-Point Scaling and Phenomenological Cascade Closures in Regular Black Hole Spacetimes

Abstract

Semiclassical analyses of regular black holes suggest that inner-horizon instabilities can drive the trapped region to evaporate faster than the Hawking time and seed a transient anti-trapped region. Whether this repeats into a cascade terminating in a horizon-free configuration requires a self-consistent solution of the semiclassical Einstein equations, which we do not attempt. Instead, within the static Bardeen family, we show that the renormalized-stress-tensor (RSET) result of Arrechea, Liberati and Spadafora (arXiv:2608.03538) does not by itself force any damping of a hypothetical cascade near extremality, since its near-extremal limit at fixed cycle duration is finite and nonzero. We then prove, under explicit nondegeneracy hypotheses on any static metric family whose horizons merge at a fold point, that the inner-horizon surface gravity scales as |kappa_-| ~ (M-M_crit)^{1/2}, and verify this in two regular black hole families, Bardeen and Hayward, with a quantified systematic uncertainty. A phenomenological closure built on this scaling gives a mass gap decaying as x_n ~ n^{-2}, generalizing to x_n ~ n^{-1/(pq)} for a fold exponent p and an uncalibrated closure exponent q; the required cycle count is a property of the closure, not a physical prediction. As a separate exercise, we build an effective adiabatic model for the outer-horizon mass from the exact Einstein tensor of a generalized Vaidya-Bardeen ansatz matched to the cited RSET flux; the resulting law includes an O(1) correction absent from the naive Schwarzschild-Vaidya relation, relaxes to M_crit only asymptotically, and gives a timescale shorter than the inner-horizon amplification timescale by up to four orders of magnitude away from extremality but longer very close to extremality. We distinguish what is derived under stated assumptions from what the closure simply postulates.