LZ-Motivated Pseudo-Dirac Higgsinos in the Supersymmetric 331 Model from the Supersymmetric \texorpdfstring{$SU(6)$}{SU(6)} GUT Model

The high-recoil event reported by LUX–ZEPLIN motivates a neutral electroweak state with an inelastic splitting in the sub-MeV range. We propose supersymmetric 331 model from the supersymmetric $SU(6)$ GUT model, and we study the electroweakino sector relevant to this recoil. Above the $331$-breaking scale hypercharge is not an independent gauge factor, and the leading bino–wino cancellation is replaced by a relation between the physical $U(1{)}_X$ and $SU(3{)}_L$ gaugino invariants, $I_X=-I_L$. A singlet–adjoint gauge-kinetic source also correlates the colored invariant. We diagonalize the neutral gauge–Higgs fermion sector of the renormalizable model and determine the tree-level Higgsino splitting without introducing an effective bino mass by hand. The required solutions form a continuous locus in the gaugino and $331$-breaking parameters. At the reference point, the solution lies close to the leading sum rule, while variation of the $331$-breaking coupling moves the locus continuously. The same calculation tests the ordering of the additional neutral states, the gauge/gaugino contribution to the $331$-breaking scalar mass, and the gravitino ordering. We compare this construction with recent $SU(5)$, singlino-assisted, electroweak-multiplet, nuclear-response and solar-capture analyses of the LZ event.