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In the spirit of the qualitative description of the toy model we disregard this complica- tion. 52 Within the toy model, this choice of labeling the states of higher energy as “bonding” is just a convention, since we do not really have a continuum from the valence band. However we will see that this convention is consistent with the results of the numerical calculations for the real system, where a valence band quasi-continuum does exist. 53 Non-additive Coulomb correction means that the As neigh- boring the two Mn, receives V$_{off}$ rather than 2 V$_{off}$ in the V$_{corr}$ term in Eq. (1). 54 Gaussians with a magnitude proportional to the calculated lattice-model LDOS are placed at each atomic position in the (110) plane. The Gaussians are given a full width at half maximum equal to half the nearest neighbor distance in order to emulate the finite spatial resolution in the STM experiments, and a logarithmic color scale is applied to facilitate comparison with the constant-current-mode STM images. 55 One exception is again the LSDA calculation of Ref. [9], which finds J = 15 meV, in good agreement with our re- sult, despite their small supercell corresponding to x = 0 . 009.