We present a comprehensive, mathematically rigorous analysis of the repulsive Euclidean core (rcore) mechanism within Dynamic Scalar Field Theory (DSFT). Moving beyond standard general relativity, where cosmological and black hole singularities are inevitable, DSFT intro- duces a diffeomorphism-invariant framework modulated by two non-linear coupling functions: α(ϕ) = 1 − ϕ 2/Λ 2 α and β(ϕ) = 1 + ϕ 2/Λ 2 β . Inside black holes and during the pre-Big Bang epoch, as α(ϕ) → 0, the metric smoothly undergoes a spontaneous dynamical signature change, replacing physical singularities with a metastable embryonic Euclidean state. We derive the exact inhomogeneous, non-linear perturbation equations governing tensor modes (hk) within this regime, establishing the existence of a structural high-pass filter that triggers ultra-violet gravitational wave (GW) saturation. This filter enforces a strict physical lock on stochastic pri- mordial power at ΩGWh 2 ≈ 4.1×10−9 and modulates quasinormal modes (QNMs). We confront these exact numerical results with the newly released LIGO-Virgo-KAGRA Fifth Gravitational Wave Transient Catalog (GWTC-5.0). DSFT precisely accounts for the reported multi-mode ringdown anomalies, overtone damping deviations, and sub-millisecond periodic features in high- significance events like GW250114, while simultaneously resolving the global Hubble tension (H0 = 73.042 ± 0.011 km/s/Mpc) through dynamic cosmic amplitude corrections