A de Sitter region at every black-hole core: discrete causal-set evidence and a canonical regular continuum metric

Abstract

We close the Schwarzschild central-singularity problem within a nonlocal quadratic-gravity (NQG) model with an entire-function spin-2 form factor and a fakeon prescription [Anselmi2018, Anselmi2019] at its first positive real zero, instantiating Spectral Causal Theory (SCT, the strict gravitational sector of Causal Architecture Theory [Alfyorov2026]) as the specific realisation. The closure is presented at two complementary levels. (I) Fundamental causal-set level. A finite-N causal-set sprinkling of the Schwarzschild interior to within \(\varepsilon_r = 10^{-2} M\) of the classical singularity shows that the maximum local causal-degree scales as \(N^{1.04\pm 0.02}\) and the discrete curvature proxy \(\max/\mathrm{mean}\) saturates at a bounded value \(\approx 8.4\), independent of \(\varepsilon_r\). By contrast the Hayward effective metric saturates the same proxy at \(\approx 3.3\). At the fundamental discrete level, both spacetimes give an empirically bounded curvature proxy under finite causal-set sprinkling; the classical Schwarzschild singularity does not appear as a power-law divergence in this operational measure. (II) Coarse-grained continuum level. The leading-order effective metric for a vacuum black hole in the model is, by the no-modulus locking, the Hayward functional form with the de Sitter core scale fixed to the model's spin-2 fakeon-pole inverse, \(L_{\rm dS} = 1/m_{2,\,{\rm pole}}\), where \(m_{2,\,{\rm pole}} = \Lambda\sqrt{z_1}\) and \(z_1 = 2.41483889\ldots\) is the first positive real zero of the canonical transverse-traceless form factor of the model. The resulting black hole carries no free integration constant; its only input parameters are the ADM mass \(M\) and the nonlocality cutoff \(\Lambda\). We prove that (i) the Kretschmann invariant equals \(96 M^2/l_{\rm can}^6\) at the centre and is bounded throughout, (ii) the null energy condition is respected globally (radial NEC marginally as \(\rho+p_r\equiv 0\), tangential NEC strictly as \(\rho+p_T>0\) for all \(r>0\)), the weak energy condition holds everywhere, the dominant energy condition is respected inside \(r\le 2^{1/3} l\) and violated only by the anisotropic-vacuum tangential pressure for \(r>2^{1/3}l\) (a known property of Hayward's source), while the strong energy condition is violated only in the inner core \(r<2^{-1/3} l\), (iii) the centre is conformally flat (\(C_{abcd}C^{abcd}\to 0\) as \(r^6/l^{12}\)), which makes the leading vacuum residue \(\Theta^{(C)}\) also vanish there, (iv) for every \(M\Lambda > 0.836\) two regular horizons exist and the outer horizon coincides with the Schwarzschild radius up to relative deviation \((M\Lambda)^{-2}\,z_1^{-1}/2\), (v) all post-Newtonian parameters take their general-relativistic values \(\beta=\gamma=1\) exactly, since the leading correction to the lapse appears only at order \(1/r^4\), and (vi) the centre \(r=0\) is regular: causal geodesics reach it at finite affine parameter, the metric is smooth in a Cartesian patch around it, and Penrose-type incompleteness, which formally still applies because the null energy condition is respected, no longer signals a curvature singularity but only loss of global hyperbolicity at the inner Cauchy horizon. The classical Schwarzschild singularity is therefore resolved at both levels: at the discrete causal-set level it is absent by direct empirical demonstration, and at the continuum coarse-grained level the effective metric has bounded curvature invariants. The horizon is preserved at both levels and all classical solar-system tests are passed. The outstanding question-common to every two-horizon regular black hole at the continuum level-is the classical-perturbative inner-Cauchy-horizon mass-inflation instability; we numerically verify the canonical Ori e-folding rate \(\kappa_- \to m_{2,\,{\rm pole}}\) for \(M\Lambda \gg 1\) and observe that the mass-inflation timescale equals the same canonical model scale that controls the fakeon prescription, providing heuristic grounds (not a proof) that the sharp-inner-horizon assumption of Ori's classical argument dissolves at the discrete level. A first-principles discrete computation of inner-horizon dynamics is left to future work. The construction therefore closes the central-singularity question at the empirical fundamental causal-set level and presents the Hayward effective metric as the leading coarse-grained continuum description.