Recasting the Quantum--Black-Hole Boundary with a Line-Density Diagnostic: From Schwarzschild to Kerr--Newman

We introduce a line-density diagnostic, defined as mass divided by a natural length scale, in order to compare quantum localization and black-hole horizon formation within the same physical dimension. On the quantum side, we use the quantum line density ρq = m/λc = m²c/ℏ, based on the reduced Compton wavelength λc = ℏ/(mc). On the black-hole side, we define the horizon density ρbh = M/r+ as a horizon-radius-based line-density diagnostic, where r+ is the outer event-horizon radius. In the Schwarzschild limit, this gives ρbh = c²/(2G), and the matching condition with the quantum line density yields Mcrit = mP/√2. The main result of this paper is that the single critical mass obtained in the Schwarzschild limit is promoted, for Kerr–Newman black holes, to the critical surface Mcrit(χ, ζ)/mP = (1 + √(1 − χ² − ζ²))^(-1/2), with χ² + ζ² ≤ 1. Thus the critical mass varies continuously over the range mP/√2 ≤ Mcrit ≤ mP. This work does not propose a complete theory of quantum gravity. Rather, it is a short semiclassical note that reorganizes the boundary between quantum localization, set by the reduced Compton wavelength, and horizon formation as a comparison between quantum line density and horizon density.