Resolution of Black Hole Singularities via Planck-Scale Modified Gravity

Abstract:S(r) Theory introduces a novel approach to quantum gravity by proposing that spacetime at its fundamental quantum limit behaves as a 2D surface rather than a traditional 4D volume. By establishing a scale-dependent modification factor S(r), where S(r) = 0 at the minimal radius (r_min), the theory successfully removes the non-physical gravitational singularity found at the center of classical black holes. The framework utilizes a core modified metric function to explain gravity, dark energy, and the quantum origin of mass without relying on arbitrary free parameters, bounding the physical criteria strictly via fundamental Planck-scale variables. Version 2.0 Update Notes (18 June 2026):- Complete Mathematical Derivation: Added the rigorous, step-by-step derivation of the Kretschmann curvature scalar (K).- Singularity Resolution Proof: Mathematically proved that the central singularity is fully resolved, showing that K remains strictly finite as r approaches 0 (K(r→0) < ∞).- Formatting: Updated the manuscript layout to a standard, clean two-column journal formatting for enhanced readability.Note: The foundational core equation S(r) remains completely unchanged from the initial v1.0 version published on 17 June 2026. "Version 9 is the final verified version of the manuscript"।