Sixteen-State Energy Mapping for First-Principles Four-Spin Ring Exchange: Validation on $La_2CuO_4$ and $SrFeO_2$

Four-spin ring (cyclic) exchange $J_{ring}$ is an essential ingredient of the Heisenberg spin Hamiltonian of cuprates and other square-lattice magnets, yet it has lacked the kind of direct, local first-principles extraction that the four-state method provides for bilinear exchange, $J$. We supply it by generalizing that method to a sixteen-state ($2^4$) scheme. Symmetry reduces the sixteen configurations to six or eight inequivalent energies, so the cost is modest. The derivation also shows that the conventional four-state magnetic coupling, $J$, is itself ring-renormalized, by $\pm 2 J_{ring} S^2$ with the sign set by the reference state. T-La$_2$CuO$_4$ confirms this quantitatively: three independent routes agree on $J_{ring}$ to $0.2\%$, giving $J_{ring}/J_1 = 0.25$, and a four-state $J_1$ quoted without naming its reference is wrong by $12\%$ in this material. The direct sixteen-state extraction itself proves reference-dependent, the N\'eel and ferromagnetic baths bracketing the mapping value: a fourth-order fingerprint of interactions beyond the pair-plus-ring model, which additional reference baths resolve into a bare $J_{ring}$ and a converging tower of six- and eight-spin loop couplings. SrFeO$_2$ ($S = 2$), with the same plaquette yet $J_{ring}/J = 0.006$, provides the negative control: a plaquette is necessary for ring exchange, far from sufficient. The complete workflow, including the band-gap and local-moment diagnostics that certify any such extraction, is implemented in the openly available Mag4 package, so that $J_{ring}$ costs no more effort to obtain than $J$.