This paper introduces a novel geometric modification of General Relativity that systematically eliminates mathematical singularities by modeling the three-dimensional spatial manifold as an elastic membrane possessing a finite physical thickness. We propose a structural decoupling of time into a uniform, monotonic cosmic global time () exerting a constant background pressure, and a local proper time () representing the intrinsic micro-thickness of the 3D spatial membrane. Within a gravitational potential well, the localized mass causes the proper time axis to tilt, generating a non-linear elastic restoring force (vacuum field resistance) that counterbalances gravitational collapse. At the center of a black hole, this surface tension drives the net spatial curvature to zero, completely neutralizing the unphysical divergence of curvature invariants. Utilizing an empirical mass-density scaling analysis, we derive an approximate analytical formulation for the physical core radius () as a function of mass () and a threshold mass parameter (). Crucially, we model the internal dynamics of this non-singular core as a physical, ultra-compact state of matter structurally equivalent to a hyper-dense neutron star. We demonstrate that while independent photons are trapped at the event horizon due to the proper time axis tilting to (rendering the outbound speed of light ), continuous static fields—such as gravitational and electromagnetic fields—propagate unimpeded through the horizon via continuous spatial field coupling. This framework explains why black holes possess intense, core-driven magnetic fields and inherently rules out the existence of primordial mini black holes.