  Antoniadis, Mazur and Mottola (AMM) two years ago computed the intrinsic
Hausdorff dimension of spacetime at the infrared fixed point of the quantum
conformal factor in 4D Gravity. The fractal dimension was determined by the
coefficient of the Gauss-Bonnet topological term associated with the conformal
gravitational anomaly and was found to be greater than 4. It is explicitly
shown how one can relate the value of the Hausdorff dimension computed by AMM
to the universal dimensional fluctuation of spacetime $\epsilon $ given by
$\phi^3/2$, where $\phi$ is the Golden Mean $0.618..$. Based on the infrared
scaling limit of the theory and using recent Renormalization Group arguments by
El Naschie, we conjecture that the unknown coefficient $Q^2$, associated with
the four dimensional gravitational conformal anomaly, could be precisely equal
to the inverse fine structure constant values ranging between $137.036 $ and
137.081. Our results generate decimal digits up to any $arbitrary$ number.
