Relational Stability Gravity (FTAR): Regular Black Holes, Cold Remnants, and Bounce Cosmology from a Variational Action Principle

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.