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The lower-critical dimension for the existence of the Ising spin-glass phaseis calculated, numerically exactly, as $d_L = 2.520$ for a family ofhierarchical lattices, from an essentially exact (correlation coefficent $R^2 =0.$) near-linear fit to 23 different diminishing fractional dimensions.To obtain this result, the phase transition temperature between the disorderedand spin-glass phases, the corresponding critical exponent $y_T$, and therunaway exponent $y_R$ of the spin-glass phase are calculated for consecutivehierarchical lattices as dimension is lowered.

Lower-Critical Spin-Glass Dimension from 23 Sequenced Hierarchical Models