We establish how the Breitenlohner-Freedman (BF) bound is realized on tilings
of two-dimensional Euclidean Anti-de Sitter space. For the continuum, the BF
bound states that on Anti-de Sitter spaces, fluctuation modes remain stable for
small negative mass-squared $m^2$. This follows from a real and positive total
energy of the gravitational system. For finite cutoff $\varepsilon$, we solve
the Klein-Gordon equation numerically on regular hyperbolic tilings. When
$\varepsilon\to0$, we find that the continuum BF bound is approached in a
manner independent of the tiling. We confirm these results via simulations of a
hyperbolic electric circuit. Moreover, we propose a novel circuit including
active elements that allows to further scan values of $m^2$ above the BF bound.
