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

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**Problem statement:** The classical Schwarzschild solution harbours a curvature singularity at \(r=0\), which is considered a pathology that a UV-complete theory of quantum gravity should resolve. **Method:** Within Spectral Causal Theory (the gravitational sector of a nonlocal quadratic gravity model with an entire-function spin-2 form factor and a fakeon prescription at its first positive real zero \(z_1\)), the resolution is demonstrated at two levels: (i) a fundamental causal-set sprinkling of the Schwarzschild interior, measuring a discrete curvature proxy that remains bounded and independent of the approach to the classical singularity; (ii) a coarse-grained continuum effective metric obtained by imposing the no-modulus principle, which yields the Hayward form \(f(r)=1-2Mr^2/(r^3+l_{\text{can}}^3)\) with the de Sitter core length \(l_{\text{can}} = (2M/\Lambda^2 z_1)^{1/3}\) fixed by the model’s spin-2 fakeon-pole mass \(m_{2,\text{pole}} = \Lambda\sqrt{z_1}\), leaving no free integration constant beyond the ADM mass \(M\) and the cutoff \(\Lambda\). **Main results:** (1) The Kretschmann invariant is bounded everywhere, attaining the de Sitter value \(K(0)=24m_{2,\text{pole}}^4\) at the centre; (2) the null and weak energy conditions hold globally, the dominant energy condition is violated only outside \(r=2^{1/3}l_{\text{can}}\), and the strong energy condition is violated only inside \(r=2^{-1/3}l_{\text{can}}\); (3) for \(M\Lambda > 0.836\) the spacetime possesses two regular horizons, with the outer horizon deviating from the Schwarzschild radius by only \(\mathcal{O}((M\Lambda)^{-2})\); (4) all post-Newtonian parameters equal their general-relativistic values \(\beta=\gamma=1\) exactly; (5) causal-set sprinkling shows that the discrete curvature proxy saturates to a constant \(\sim 8.4\), independent of how close the sprinkling extends to the classical singularity, indicating that the singularity is absent at the fundamental discrete level. **Limitations:** The continuum construction inherits the standard mass-inflation instability of the inner Cauchy horizon; although the causal-set regularization provides heuristic grounds that the sharp inner-horizon picture may dissolve at the discrete level, a first-principles discrete dynamical computation remains an open problem. Additionally, the core-length choice is canonical but not unique within a finite set of model-inherent scales, and the analysis is restricted to vacuum black holes without coupling to matter.

AI-generated (deepseek-v4-flash) · created 2026-05-19