The Wafold: Curvature-Driven Termination and Dimensional Compression in Black Holes

This work explores a geometric description of black holes in which spacetime terminates on a curvature-triggered hypersurface rather than extending to an interior singularity. We study the implications of a scenario in which, upon reaching a critical curvature threshold, the three-dimensional spatial geometry compresses into a thin, closed boundary identified here as the wafold. Beyond this, the manifold would no longer continue, and all mass-energy and information would be confined to the hypersurface itself. This framework combines two well-explored paths: (1) curvature-driven geometric compression, in which extreme curvature forces the bulk degrees of freedom to become supported on a thin hypersurface (without altering the underlying dimensionality of spacetime), and (2) the motivation underlying the holographic principle, namely that black-hole entropy scales with surface area rather than volume, suggesting that information is governed by a boundary geometry rather than a bulk volume. We elaborate a dimensional conversion law that would be required to describe the collapse of spatial volume into surface area as a conserved flux of geometric capacity across the wafold, and we analyze the resulting consequences of treating this hypersurface as the terminal boundary of the manifold.