Dynamical 4-D Gauss-Bonnet action from matter-graviton interactions in a curved background

The Glavan-Lin proposal for 4D Einstein-Gauss-Bonnet (EGB) gravity introduces a singular dimensional scaling to bypass Lovelock's theorem, though its fundamental origin remains debated. In this work, we demonstrate that this specific dimension-dependent scaling naturally emerges from the one-loop self-energy corrections of gravitons. By employing real-space techniques to evaluate graviton interactions with minimally coupled scalar and electromagnetic fields in a de Sitter background, we show that the $1/(D-4)$ pole universally generates a dynamical Gauss-Bonnet term. This confirms that the scaling is not an ad-hoc classical limit but a necessary consequence of quantum field-theoretic renormalization. Furthermore, canceling the remaining divergences strictly requires the inclusion of quadratic curvature counterterms, specifically Weyl-squared and $R^2$ invariants. We discuss the implications of this in the early-Universe and consequences in strong gravity regime.