This study presents the first comprehensive Bayesian inference of neutron star matter, incorporating $\mathrm{\Delta }$-resonances alongside hyperons and nucleons within a density-dependent relativistic hadron framework. Using constraints from nuclear saturation properties, chiral effective field theory ($\chi \mathrm{EFT}$), NICER radius measurements, and tidal deformability data from GW170817, we systematically explore the impact of $\mathrm{\Delta }$-resonances on the equation of state (EOS) of dense matter and neutron star observables. Our results demonstrate that the inclusion of $\mathrm{\Delta }$ baryons softens the EOS at low densities while maintaining sufficient stiffness at high densities to support $2M_{\odot }$ neutron stars. This naturally reconciles neutron star radius constraints with the recent observation of the low-mass compact object in HESS J1731-347 while simultaneously exhibiting excellent agreement with GW170817 tidal deformability constraints, reinforcing the astrophysical viability of $\mathrm{\Delta }$-admixed neutron stars. Additionally, $\mathrm{\Delta }$-resonances are found to populate the outer layers of the neutron star core, which may have implications for neutron star mergers and their cooling. Furthermore, we show that the presence of $\mathrm{\Delta }$ baryons might significantly influence neutron star cooling via the direct Urca process. We also investigate quasinormal $f$-mode oscillations within a fully general relativistic framework, revealing strong correlations between the $f$-mode frequency, neutron star compactness, and tidal deformability. With the inclusion of $\mathrm{\Delta }$-resonances and adherence to astrophysical constraints, we obtain $f_{1.4}={1.97}_{-0.22}^{+0.17}\mathrm{kHz}$ and the damping time ${\tau }_{f_{1.4}}=0.19_{-0.03}^{+0.05}\mathrm{s}$ at the $1\sigma$ confidence level.

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• Nuclear astrophysics