Geometric Resolution of the Black Hole Singularity in Rotational‑Order Geometry

We demonstrate that the rotational-order geometric unification framework naturally resolves the black hole singularity problem. 

By applying the rigorous variational foundation established in Paper 38, we solve the modified Einstein field equations in a spherically symmetric, static background. We show that the internal S³ orientation field develops a non-trivial topological gradient in the deep gravitational well. The resulting geometric stress-energy tensor provides a strong repulsive effective pressure. 

As the radial coordinate r → 0, this topological pressure counteracts the gravitational collapse, replacing the infinite curvature singularity with a finite-sized topological core. The Kretschmann scalar is proven to be bounded. For a stellar-mass black hole with benchmark parameters, K_max ∼ 10⁶² m⁻⁴, which is far below the Planck scale (10¹⁴⁰ m⁻⁴), rendering the spacetime geodesically complete. 

This geometric core also predicts observable gravitational wave echoes (reserved as P51 in our catalog) testable by future detectors like LISA and the Einstein Telescope.