Extended uncertainty principle inspired black hole in a G\"odel Universe

We explore analytically the implications of a curvature-modified extended uncertainty principle (EUP) derived in a rotating G\"odel spacetime and apply it to the construction of a semiclassical black hole model. Adapting techniques from corpuscular black hole frameworks, we reinterpret the G\"odel-type uncertainty relation as an effective energy bound, leading to a modified lapse function with explicit dependence on the global rotation parameter $a$ and the radial coordinate $r_0$. Analytic expressions are derived for key gravitational features, including the event horizon, photon spherehere, shadow radius, and deflection angle, with curvature corrections scaling as $a^{-2}$ and $r_0^2 / a^4$. Series expansion in the limit $a \to \infty$ shows that global rotation consistently increases all observables relative to the Schwarzschild case. Applying these results to astrophysical data, we use Event Horizon Telescope (EHT) measurements of Sgr A* and M87* to infer lower bounds of $a/M \sim 10^5$, while solar system light-bending observations in the parametrized post-Newtonian (PPN) framework yield $a / M_\odot \sim 5 \times 10^4$. These large but finite values validate the asymptotic expansion and confirm that G\"odel-type rotation remains observationally suppressed, yet theoretically coherent. Our results demonstrate that global rotation, when treated semiclassically via curvature-modified uncertainty, introduces detectable signatures in principle, though well below current observational sensitivity. The framework offers a consistent path toward exploring the quantum-gravitational interplay between global geometry and local black hole structure.