Asymmetric dark matter from leptogenesis in type-III seesaw framework with modular $S_4$ symmetry

We present a unified framework for neutrino masses, baryogenesis, and dark matter based on a modular $S_4$ symmetry combined with a type-III seesaw mechanism. All Yukawa couplings, CP phases, and flavor textures originate from a single complex modulus $\tau$, whose vacuum expectation value controls both visible and dark sector dynamics. The same modular parameter fixes the neutrino mass matrix, determines the CP asymmetries driving resonant leptogenesis, and correlates the resulting baryon and dark matter abundances. A detailed numerical analysis shows that the model reproduces all neutrino oscillation data within the $3\sigma$ NuFIT~5.2 (2024) ranges for normal ordering, predicting $\delta_{\rm CP} \simeq \pm (150^\circ-180^\circ)$, $\sum m_\nu\simeq(0.06-0.08)~\mathrm{eV}$, and an effective Majorana mass $m_{\beta\beta} \simeq (8 - 18)\times 10^{-3}~\mathrm{eV}$, testable in next-generation neutrinoless double-beta decay experiments. The same modular Yukawas yield resonantly enhanced CP asymmetries $|\epsilon_{L,\chi}| \sim 10^{-9}-10^{-6}$ at $M_\Sigma \sim 10^{7}~\mathrm{GeV}$, successfully generating the observed baryon asymmetry $\eta_B\simeq6\times10^{-10}$ and dark relic density $\Omega_\chi h^2\simeq0.12$ without additional free parameters. The predicted correlation $\Omega_\chi/\Omega_B\simeq5.4$ fixes the dark matter mass to $m_\chi\simeq0.1-2~\mathrm{GeV}$, consistent with all current constraints. This framework therefore realizes a fully predictive baryon$-$dark matter co-genesis, where the geometry of the modular symmetry links the origin of flavor, CP violation, and the cosmic matter asymmetry.