NaFeP: first pairing prediction for an unmade 111 iron phosphide, and the pnictogen-height rule inside one compound

Abstract

Companion to arXiv:2609.10614, which built a physics judge that classifies the gap symmetry of a superconductor candidate from its structure and showed that generation in the wire families re-finds the known canon. Here the judge is applied end to end to the single compound that survived its own screen. From 58 iron-pnictide compositions in the electronic window of BaFe$_2$As$_2$ absent from the Materials Project, a zero-cost literature-and-stability screen leaves one with no bulk synthesis and no band-structure, magnetic or pairing calculation: NaFeP (111-type, P4/nmm, Fe$^{2+}$ d$^6$, arsenic-free). The judge, refined with an explicit Stoner gate and a frame-independent symmetry reader, returns a paramagnetic DFT ground state and an s$\pm$ gap: a clean sign reversal between the hole cylinders at Gamma and the electron sheets at the zone corner, at nesting vector (1/2,1/2,1/2), in every basis and cell tried. Whether the gap is nodeless is where the instrument is tested. The phosphorus height (1.16 Å) lies below the empirical threshold of about 1.33 Åseparating nodal from nodeless pnictides, so the structural rule calls NaFeP nodal. The sealed verdict on a metal-only basis said nodeless; a d+p basis that passes the same gates puts the inner hole cylinder at the node threshold (min|g|/max|g| = 0.07-0.10), and a scan of the phosphorus height reproduces, inside one compound, the ingredients of the height rule: a third hole pocket, a switch of the nesting vector to (1/2,1/2,0), a monotonic approach to the Stoner instability, and near-degeneracy of s$\pm$ and d channels beyond 1.33 Å. The prediction we stand behind is the d+p reading: an s$\pm$ superconductor with a near-nodal inner hole sheet, on the LiFeP side of the rule, with the disagreement between bases reported, not resolved by choice. No Tc is asserted; the discriminating measurement, nodal or nodeless, is stated in advance.