Modular $S_4$ Inverse Seesaw at the LZ High-Recoil Frontier
The extension of the LUX--ZEPLIN nuclear-recoil analysis to approximately $270\,\mathrm{keV}$ 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 $τ$, 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 $χ^2_{\ell+ν}<30$, 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 $2\times10^6$ dark-sector points only over flavor-accepted values of $τ$ and identify common-modulus solutions in the LZ-oriented region $300\,\mathrm{GeV}\le m_χ\le2\,\mathrm{TeV}$ and $300\,\mathrm{keV}\leδ\le400\,\mathrm{keV}$. 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 $(m_χ,δ)\simeq(1.847\,\mathrm{TeV},371.5\,\mathrm{keV})$, 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.