Since quantum field theory itself has no mathematically rigorous basis, and the AdS/CFT correspondence links quantum field theory to something as exotic as quantum gravity, I don't think there is any hope for precise mathematical statements coming out of that correspondence in the foreseeable future.

If I interpret the question as "explicit computational consequences of the AdS/CFT correspondence", then perhaps the Ryu–Takayanagi formula stands out. This formula for the entanglement entropy in a quantum field theory can be derived from the AdS/CFT correspondence, and checked with explicit calculations in some cases.

From the physics perspective, the strongest observational consequence of the AdS/CFT correspondence is the Kovtun–Son–Starinets bound on the viscosity/entropy ratio, which can be measured in a quark-gluon plasma.

Since the AdS/CFT correspondence links quantum field theory to something as exotic as quantum gravity, I don't think there is any hope for precise mathematical statements coming out of that correspondence in the foreseeable future.

If I interpret the question as "explicit computational consequences of the AdS/CFT correspondence", then perhaps the Ryu–Takayanagi formula stands out. This formula for the entanglement entropy in a quantum field theory can be derived from the AdS/CFT correspondence, and checked with explicit calculations in some cases.

From the physics perspective, the strongest observational consequence of the AdS/CFT correspondence is the Kovtun–Son–Starinets bound on the viscosity/entropy ratio, which can be measured in a quark-gluon plasma.