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.

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