Bounded Oscillatory Near-Core Dynamics in a Saturating-Scalar Black-Hole Interior: Numerical Evidence in Spherical Symmetry

Abstract: Previous characteristic simulations of scalar-field collapse within the Frame-Transformed Asymptotic Regularization (FTAR) framework suggested mass inflation at the origin even with saturating potentials. We demonstrate that this result was a numerical artifact arising from discrete commutator failure ([Dv,Dr]=0) in our Eddington-Finkelstein implementation near r=0, an issue plausibly generic to mixed-characteristic discretizations. We reformulate the problem in a constraint-consistent 3+1 Cauchy framework using polar-areal coordinates, eliminating the artifact. With recalibrated initial data that genuinely accesses the saturation regime (∣ϕ∣>ϕ⋆), the minimal FTAR model exhibits a long-lived oscillatory near-core transient (Regime III)—a third dynamical outcome distinct from both de Sitter plateau formation (Regime I) and monotonic kinetic runaway (Regime II). Regime III is characterized by repeated crossings of the saturation threshold without convergence to a static state, and is robust under resolution refinement. Key Highlights & Methodological Advances in Version 2.3: Resolution of the Mass Inflation Artifact: Provides a rigorous autopsy of the discrete commutator failure in characteristic (Eddington-Finkelstein) coordinates, offering a crucial methodological lesson for numerical relativity. Constraint-Consistent 3+1 Architecture: Implements a polar-areal Cauchy framework where the Hamiltonian constraint is solved as a radial ODE on each time slice, guaranteeing regularity at r=0 and eliminating spurious central energy injection. Discovery of "Regime III" Dynamics: Uncovers a novel, robust oscillatory metastable state in the near-core region, proving that scalar field collapse in saturating theories does not reduce to a simple binary (static core vs. runaway) outcome. High-Resolution Verification: Results are rigorously cross-validated across multiple grid resolutions (Nr∈{600,900,1200}) to ensure the oscillatory regime is a physical feature, not grid ringing. Precise Parameter Mapping: Establishes the exact amplitude boundary of the near-threshold saturation window at 0.0485