A Saturating Scalar Effective Model for Black-Hole Interiors: Regularity Criteria and Dynamical Regimes

We develop a minimal effective Einstein–scalar model for studying high-curvature black-hole interiors using a bounded saturating scalar potential. The construction is intended as an effective closure rather than a fundamental theory or a claim of a complete regular black-hole solution. The paper formulates explicit regularity criteria, identifies the transition to the saturation regime, and classifies the qualitative dynamical regimes admitted by the effective equations. Particular care is taken to distinguish properties of the effective model from statements about their dynamical realization. The potential is written as> V(φ)=Vₚₗₐₜₑₐᵤ^2(φ/φ_), >with the relation to the notation used in earlier versions stated explicitly. Numerical realization of the model is addressed separately in the companion Paper III, which evolves constraint-consistent Cauchy data in a horizon-penetrating BSSN formulation and tests trapped-region formation. This revised companion edition updates the scope and terminology of Paper I to reflect the results of the subsequent numerical validation campaign. It does not reinterpret the numerical results as predictions of the effective model.