The infinite-layer structure nickelate Ba 2 NiO 2 (AgSe) 2 (BNOAS) with d 8 Ni ions and a peculiar susceptibility χ (T), synthesized at high pressure, is studied with correlated density functional methods. The overriding feature of the calculations is a violation of Hund's rule coupled with complete but unconventional spin-orbital polarization, leading to an unexpected low-spin 1 B 1 , "off-diagonal singlet" textured by an internal orbital structure of compensating d ↑ x 2 −y 2 and d ↓ z 2 spins. This unconventional configuration has a lower energy than conventional high-spin or low-spin alternatives. An electronic transition is obtained at a critical Ni-O separation d Ni-O c = 2.03 Å, which corresponds closely to the observed critical value of 2.00-2.05 Å, above which Ni becomes magnetic in square planar NiO 2 compounds. We propose scenarios for the signature of magnetic reconstruction in χ (T) at T m = 130 K without any Curie-Weiss background (no moment) that invokes ordering of Ni d 8 moieties that are largely this generalized Kondo singlet. Because hole states are primarily Se 4p rather than O 2p, the usual issue of Mott insulator versus charge transfer insulator is supplanted by a character in which electrons and holes are separated in real space. The underlying physics of this system is modeled by a Kondo sieve spin model (two-dimensional Kondo necklace) of a "Kondo" d z 2 spin on each site, coupled to a d x 2 −y 2 spin that is itself strongly coupled to neighboring like-spins within the layer. The observed magnetic order places BNOAS below the quantum critical point of the Kondo sieve model, providing a realization of the long-range ordered near-singlet weak antiferromagnetic phase. We propose electron doping experiments that would drive the system toward a d 9−δ configuration and possible superconductivity with a similarity to the recently reported hole-doped infinite-layer cuprate Ba 2 CuO 3.2 that superconducts at 73 K.