Mathematical predictions of AdS/CFT

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.

Although it might seem futile, given how far most of the activity on AdS/CFT is from rigorous mathematics, I think this a good question, provided one is happy, for now, with (very) baby versions of this correspondence.

An example of nontrivial mathematical prediction, using a baby version of AdS/CFT (the Caffarelli-Silvestre extension) is the conformal invariance of the scaling limit of critical long-range Ising (or $\phi^4$) models in 3D. You can find this discussed in the article "Conformal Invariance in the Long-Range Ising Model" by Paulos, Rychkov, van Rees and Zan. For a precise formulation of the prediction as a mathematical conjecture see, e.g., my article "Towards Three-Dimensional Conformal Probability", journal version, preprint version.

Also note that a lot of work has been done as far as making CFT in 2D rigorous, most notably for the Ising model. Carlo mentioned the Ryu-Takayanagi formula. This is perhaps something one could prove (maybe someone did that already, I don't know).

Good to know also: A mathematical result which has the flavor of a very baby AdS/CFT correspondence is the representation of a temperate distribution in $\mathscr{S}'(\mathbb{R}^d)$, using the wavelet transform, as an integral over the Euclidean AdS, i.e., the hyperbolic space $\mathbb{H}^{d+1}$, with the right invariant metric/measure. See for example the book "Wavelets, an Analysis Tool" by Holschneider. If one introduces a cut-off hypersuface approaching the conformal boundary in order to restrict the domain of integration in $\mathbb{H}^{d+1}$, one can use this operation as a way of regulating probability measures on $\mathscr{S}'(\mathbb{R}^d)$ (e.g. the Euclidean functional integrals of a CFT on the boundary). One can show these cut-off measures converge weakly to the given probability measure, as the cut-off surface approaches the boundary.

I am not an expert in this area but I know that Costello, Gaiotto, Paquette, and others have been studying topological and holomorphic twists of ADS/CFT. Unlike the full correspondence this one seems amenable to mathematical analysis.