Gravitational Wave Echoes from Saturated Compact Cores: A Variable Compactness Model with Quantized Formation

The black hole singularity predicted by the General Relativity signals a breakdown at high densities. In this paper, We propose a phenomenological model where the Schwarzschild radius is a physical surface of a saturated core formed via quantized mass energy accretion. The photon sphere emerges as a pre-horizon reflective barrier during formation. The core’s radius offset ϵ ≡ (Rcore − Rs)/Rs is not universal but varies with the equation of state (EOS). This yields a spectrum of gravitational wave echo signatures multiple chirps, single chirps, or none, offering a unified explanation for conflicting LIGO echo searches. We derive the echo time delay, present a classification scheme, and compare predictions to reported echo candidates.