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 $\mathit{\text{critical}}\mathit{\text{radius}}$ $r_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}=(\beta \mathrm{/}\gamma {)}^{1\mathrm{/}4}\sqrt{{\mathrm{\ell }}_{P}L}$, 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 $⟨T_{\mu \nu }⟩=0$ under the self-duality condition ${\mathrm{\partial }}_z{g}_{\mu \nu }{\mathrm{\mid }}_{z=r_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}}=0$; (ii) a topological transition manifested by the complexification of the central charge, $c_{\mathrm{e}\mathrm{f}\mathrm{f}}=c(1+\frac{i}{2}\sqrt{\kappa }{\mathrm{\ell }}_P^2)$ with $\kappa =(\beta \mathrm{/}\gamma {)}^{1\mathrm{/}4}\sqrt{\beta \gamma {\mathrm{\ell }}_P\mathrm{/}L}$; 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 $\mathrm{\Delta }{S}_{\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{y}}>0$. These effects establish a consistency condition $L>\sqrt{\beta }{\mathrm{\ell }}_P$ for a valid AdS/CFT correspondence and identify $r_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$ as the thermodynamic critical point where black holes transition to stringy remnants and information is topologically scrambled.