The Quantum Boundary of Black Hole Interiors: Termination of the Sum over Geometries at Planck Curvature

Abstract

Classical general relativity predicts a singularity at the center of every black hole. We argue that this singularity is never reached. We propose that the gravitational functional integral loses support at the Planck curvature threshold ($\mathcal{K} \sim \ell_P^{-4}$), forming a quantum boundary, $\mathcal{B}_Q$, that truncates the spacetime manifold at a finite, positive radius. The mechanism relies on an ambiguity--definedness--support chain: at Planck scales, no unique metric is physically selected by the semiclassical action or Compton localization. This intrinsic ambiguity implies no definite spacetime geometry exists, assigning such configurations vanishing wavefunctional support ($\Psi = 0$). Geometrically, this acts as topological excision: regions lacking a definite metric behave as microscopic holes. As the rising curvature drives them to merge, no continuous configuration remains in the admissible domain of the Feynman-DeWitt measure. The location of $\mathcal{B}_Q$ is set by accretion history. For a spinning black hole, mass inflation carries curvature to the threshold at the inner horizon $r_{-}$, placing $\mathcal{B}_Q$ at a macroscopic radius and leaving the Cauchy horizon, ring, and deeper extensions outside the physical manifold. $\mathcal{B}_Q$ thereby acts as a quantum-geometric cutoff for the mass-inflation instability, capping the internal mass parameter at a finite amplification $n_{\rm qb} \approx (r_{-}/r_g)^3 (r_g/\ell_P)^2/\sqrt{48}$. Evaluating the Gibbons-Hawking-York boundary term over this terminal slice yields a finite interior action, $S_{GHY}^{qb} \approx \frac{3}{2} n_{\rm qb} Mc^2\,\Delta t$. Without invoking trans-Planckian degrees of freedom, these results suggest the classical singularity is not a physical event but the terminal boundary of the geometry's domain of definition.