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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 ($t$) exerting a constant background pressure, and a local proper time ($\tau$) 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 ($r$) as a function of mass ($M$) and a threshold mass parameter ($M_{th} \approx 5 \times 10^{30}\text{ kg}$). 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 $90^\circ$ (rendering the outbound speed of light $c_{\text{out}} = 0$), 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.