Modular $S_4$ Inverse Seesaw at the LZ High-Recoil Frontier

The extension of the LUX–ZEPLIN nuclear-recoil analysis to approximately $270\mathrm{k}\mathrm{e}\mathrm{V}$ probes a qualitatively new dark-matter regime in which endothermic scattering can suppress conventional low-energy recoils and concentrate the surviving spectrum near the Galactic high-velocity tail. We construct a modular-$S_4$ realization in which a minimal $2N_R+2S$ inverse seesaw accounts for lepton flavor, while a separate pair of Standard-Model-singlet fermions forms the pseudo-Dirac state relevant for high-recoil xenon scattering. The sectors share a common complex modulus $\tau$, so the modular forms selected by charged-lepton and neutrino data also enter the operators generating the dark Dirac mass and Majorana splitting. Our scan yields $181447$ lepton-sector solutions satisfying ${\chi }_{\mathrm{\ell }+\nu }^2<30$, distributed among several disconnected modular regions, and predicts one effectively massless light neutrino at leading order. Restoring the physical scales allows TeV-scale heavy quasi-Dirac neutral fermions with small non-unitarity. We subsequently scan $2\times {10}^6$ dark-sector points only over flavor-accepted values of $\tau$ and identify common-modulus solutions in the LZ-oriented region $300\mathrm{G}\mathrm{e}\mathrm{V}\le m_{\chi }\le 2\mathrm{T}\mathrm{e}\mathrm{V}$ and $300\mathrm{k}\mathrm{e}\mathrm{V}\le \delta \le 400\mathrm{k}\mathrm{e}\mathrm{V}$. An exploratory two-bin Poisson likelihood, including endothermic xenon kinematics, the Helm form factor, Standard-Halo-Model velocities, Gaussian energy smearing, and profiling over the signal normalization, gives a best grid point at $(m_{\chi },\delta )\simeq (1.847\mathrm{T}\mathrm{e}\mathrm{V},371.5\mathrm{k}\mathrm{e}\mathrm{V})$, close to the terrestrial inelastic kinematic ceiling. The construction provides a direct link between modular lepton flavor, TeV-scale inverse-seesaw dynamics, and high-recoil inelastic dark matter.