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

ABSTRACT We present the Fundamental Theory of Attractive Repulsion (FTAR), a classical field-theoretic extension of General Relativity that resolves the black hole singularity and the cosmological Big Bang singularity through a single mechanism: a stability field coupled to the trace of the energy-momentum tensor. The theory is derived from a minimal variational action principle. In the black hole sector, FTAR yields a regular metric with a de Sitter core, a cold stable remnant at the Planck scale (a Dark Matter candidate), and quasi-normal mode (QNM) spectra consistent with current LIGO/Virgo observations. In cosmology, the modified Friedmann equation produces a non-singular bounce at the saturation density, preventing the initial singularity. KEY RESULTS IN THIS VERSION (v4): Variational Principle: Full derivation of field equations from the FTAR Action, ensuring mathematical consistency with standard field theory and reducing to General Relativity in the weak-field limit. Regular Black Holes & Remnants: The solution naturally leads to a "Planck star" remnant structure. We demonstrate that the effective potential for gravitational perturbations remains indistinguishable from Schwarzschild gravity at astrophysical scales, deviating only near the Planck-scale core. Observational Consistency: We show that deviations in the ringdown phase (QNM) are of the order 10^-76, rendering the theory consistent with current gravitational wave observations while predicting detectable "echoes" for future high-sensitivity detectors. Bounce Cosmology: The theory replaces the Big Bang with a non-singular bounce, consistent with Big Bang Nucleosynthesis (BBN) constraints. TESTABILITY The paper identifies three independent observational channels to test the theory: Gravitational-wave echo delays (distinct from standard GR noise). Existence of Planck-mass remnant dark matter. A suppressed tensor-to-scalar ratio in the CMB, testable by future missions like LiteBIRD. STATUS Draft v4 (February 2026). Prepared for submission to Physical Review D. "Version 5.3: Updated numeric solver and metric derivations" "Version 5.5: This upload supersedes earlier drafts previously deposited at Zenodo. The following substantive changes were introduced relative to the last archived version: *Formal completeness.* The explicit scalar stress-energy tensor Tμν(Φ)Tμν(Φ) is now derived and stated (Eq. 3), closing the gap between the action and the modified Einstein equations. The derivation chain Action → Field equations → Effective stress-energy → Modified Friedmann equation → Bounce is now complete for the cosmological sector. Honest scope of the black hole sector. The regular metric is consistently labelled as phenomenological throughout the manuscript, including the abstract. Preliminary numerical integration of the self-consistent Einstein–Φ system is reported: the interior geometry differs structurally from the Hayward ansatz, with the effective mass function acquiring negative values inside the horizon — interpreted as a manifestation of the negative-energy NEC-violation character of the scalar field. Full interior analysis is deferred to Paper II. The exterior solution family (Phase Diagram scan) is validated and reported in Appendix G. *Radiation sector.* The structural limitation of the ΦT coupling for conformal matter is addressed with a quantitative order-of-magnitude estimate of the QCD trace anomaly at Planck density (O(αsρsat)O(αsρsat)), supplementing the qualitative argument present in earlier drafts. Two-sector no-go theorem. The structural necessity of distinct saturation functions for the BH and cosmological sectors is now supported by a formal analogy with Loop Quantum Gravity, where the same situation arises from a single underlying discrete geometry. EFT framing. The framework is presented throughout as an effective field theory valid at the Planck scale, with explicit open problems enumerated (UV completion, universal coupling, self-consistent BH interior, cosmological perturbations, remnant abundance, QNM spectrum, Poisson–Israel instability). Format. First release in PDF format compatible with Physical Review D submission requirements."