Singularity Resolution in the Davis-Wilson Framework: A Topological Approach to Black Hole Physics

This work presents the Davis-Wilson framework's approach to gravitational singularities, addressing the foundational results established by Nobel Laureate Sir Roger Penrose in his landmark 1965 paper "Gravitational Collapse and Space-time Singularities." Penrose's singularity theorems—for which he shared the 2020 Nobel Prize in Physics—demonstrated that singularities are inevitable consequences of Einstein's general relativity under physically reasonable conditions. Building on the mathematical foundations laid by Raychaudhuri (1955) and extended by Hawking and Penrose (1970), these theorems represent one of the most profound results in gravitational physics. What This Paper Does We introduce a decomposition of physical states into continuous geometry (Φ) and discrete winding sectors (r): $$C = (\Phi, r)$$ This structure provides a natural regularization mechanism at high curvatures while preserving the classical predictions of general relativity at low energies. The discrete topological sector prevents true infinities while maintaining unitarity and information conservation. Scope of Results The following results are established with full mathematical rigor: Spherically symmetric spacetimes — where the Misner-Sharp quasi-local mass is well-defined and curvature invariants admit explicit control (Theorems 7.7, 7.11) Mini-superspace models — where the Hamiltonian constraint can be solved exactly and bounce dynamics are proven (Theorem 7.19) Regge calculus discretizations — where thousands of geometric degrees of freedom are active and convergence under lattice refinement is observed What remains open: Extension to generic (non-spherically symmetric) trapped surfaces, rotating (Kerr) black holes, and cosmological singularities with arbitrary anisotropy. The lattice results provide evidence that the mechanism is not an artifact of symmetry reduction, but a general analytic proof remains an important open problem. Numerical Validation This work includes results from GPU-accelerated Monte Carlo simulations: 40,000+ thermalization steps 128 vertices with thousands of geometric degrees of freedom Statistical validation with p < 10⁻⁶ for key signatures Correlation analysis confirming extended time continuation through high-curvature regions Relation to Prior Work This paper is part of the Davis-Wilson geometric program, which has addressed: Yang-Mills Mass Gap — DOI: 10.5281/zenodo.17942784 Navier-Stokes Regularity — DOI: 10.5281/zenodo.18216597 P ≠ NP Geometric Separation — DOI: 10.5281/zenodo.18210109 Poincaré Conjecture (Independent Proof) — DOI: 10.5281/zenodo.18251382 The Field Equations of Semantic Coherence —DOI:10.5281/zenodo.17771796 Intellectual Property Notice Portions of this work are covered by the following U.S. Provisional Patent Applications: 63/933,299 — System and Method for Modulation of Quantum Vacuum Topology via Non-Abelian Gauge Field Configurations 63/933,103 — System and Method for Geometric Verification and Optimization of Gauge Field Configurations via Topological Cache Mapping 63/927,445 — Systems and Methods for Fixed-Size Reasoning State Representation via Topological Residue in Neural Networks Academic use and research are not restricted by these filings. Key Features Rigorous mathematical proofs including derivation of winding-curvature bounds from equations of motion Unitarity established via explicit Hamiltonian construction and Stone's theorem UV finiteness at all loop orders Honest delineation of proven results vs. conjectures (Section 11: Domain of Proven Results) Extensive scholarly citations including Einstein (1915), Raychaudhuri (1955), Penrose (1965), Hawking-Penrose (1970), Wilson (1974), Bekenstein (1973), Schoen-Yau (1979), and Engelhardt-Wall (2015) Files Included singularity_resolution.pdf — Main paper (69 pages) singularity_resolution.tex — LaTeX source Supporting figures, code and numerical data Keywords black holes, singularity resolution, quantum gravity, Penrose singularity theorem, topological quantum field theory, lattice gauge theory, information paradox, Hawking radiation, winding numbers, holonomy, Regge calculus, Monte Carlo simulation Related Identifiers Penrose (1965): DOI 10.1103/PhysRevLett.14.57 Hawking (1975): DOI 10.1007/BF02345020 Bekenstein (1973): DOI 10.1103/PhysRevD.7.2333 License Creative Commons Attribution 4.0 International (CC BY 4.0) Author Bee Rosa Davisbee_davis@alumni.brown.edu Sixty-one years after Penrose demonstrated the inevitability of singularities in general relativity (January 18, 1965), this work presents one approach to their resolution—not by abandoning Einstein's theory, but by recognizing that physical configuration space has discrete topological structure alongside continuous geometry.