The Lemaître–Tolman–Bondi Metric with a Central Pointlike Mass

We present a comprehensive general relativistic analysis of the Lemaître–Tolman–Bondi (LTB) metric, incorporating a cosmological constant Λ and a central pointlike mass Md at the geometric origin. Within this framework, Md is identified as the material source of dark matter in cosmology, yielding a scale-dependent total matter–density parameter Ωm(L) characterized by an L−3 decay of its dark component Ωd(L). We demonstrate that the Hubble and S8 tensions are not independent anomalies but interconnected consequences of spacetime inhomogeneity. These discrepancies arise from a combination of physical and methodological factors: the probing of radial gradients at different characteristic scales and the subsequent interpretation of these data through a global FLRW template. This approach, compounded by the practice of isotropic sky averaging, masks the underlying LTB geometry and converts the physical variation of the manifold into the observed cosmological tensions. Our framework provides a self-consistent geometric explanation for current anomalies while preserving the Copernican principle, identifying the crisis in cosmology as arising from the application of homogeneous models to a manifold characterized by radial gradients and scale-dependent dynamics, where the observer and probes reside within the same inhomogeneous regime.