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We construct a generalized gradient approximation (GGA) for the density n xc (r, r+u) at position r+u of the exchange-correlation hole surrounding an electron at r, or more precisely for its system and spherical average< n xc (u)>=(4π)− 1∫ d Ω u N− 1∫ d 3 r n (r) n xc (r, r+u). Starting from the second-order density gradient expansion, which involves the local spin densities n↑(r), n↓(r) and their gradients∇ n↑(r),∇ n↓(r), we cut off the spurious large-u contributions to restore those exact conditions on the hole that the local spin density (LSD) approximation respects. Our GGA hole recovers the Perdew-Wang 1991 and Perdew-Burke-Ernzerhof GGA’s for the exchange-correlation energy, which therefore respect the same powerful hole constraints as LSD. When applied to real systems, our hole model provides a more detailed test of these energy functionals, and also predicts the observable electron-electron …