The EGUP-Induced Critical Radius: A New Holographic Scale for Quantum Gravity

We present a unified framework incorporating both the Generalized and Extended Uncertainty Principles (GUP/EUP) in Anti-de Sitter space. This reveals a fundamental quantum gravity scale, the \textit{critical radius} $r_{\rm crit}=(\beta/\gamma)^{1/4}\sqrt{\ell_{P}L}$, which marks a phase transition where quantum gravitational ($\beta$) and AdS curvature ($\gamma$) effects equilibrate. At this scale, we demonstrate three interconnected phenomena: (i) a breakdown of the standard holographic duality, signaled by the exact vanishing of the boundary stress tensor $\langle T_{\mu\nu}\rangle=0$ under the self-duality condition $\partial_z g_{\mu\nu}|_{z=r_{\rm crit}}=0$; (ii) a topological transition manifested by the complexification of the central charge, $c_{\rm eff}=c\left(1+\frac{i}{2}\sqrt{\kappa}\ell_{P}^{2}\right)$ with $\kappa=(\beta/\gamma)^{1/4}\sqrt{\beta\gamma\ell_P/L}$; and (iii) a mechanism as a scenario for information paradox resolution, where information is recovered via topological storage in Chern-Simons states, modifying the Page curve with $\Delta S_{\rm recovery}>0$. These effects establish a consistency condition $L>\sqrt{\beta}\ell_{P}$ for a valid AdS/CFT correspondence and identify $r_{\rm crit}$ as the thermodynamic critical point where black holes transition to stringy remnants and information is topologically scrambled.