We investigate core composition $g_1$ modes of cold, nonrotating neutron stars containing antikaon condensates, hyperons, and $\mathrm{\Delta }(1232)$ baryons. To our knowledge, this is the first fully general-relativistic calculation of the continuous-composition core $g_1$-mode frequency and gravitational-wave damping time for stars with a $K^{-}$ condensate. Using the BigApple equation of state (EOS), we compute frequencies and damping times in full general relativity (GR) and evaluate frozen-composition tidal overlaps from full-GR eigenfunctions within a leading-order hybrid resonance prescription. A species-resolved Ledoux decomposition, cross-checked through mode-frequency sensitivities, identifies the driving channels. We compare frozen composition with a fast-$K$ limit for $n\leftrightarrow p+K^{-}$ and a strong-equilibrium limit for the $\mathrm{\Delta }$ quartet. Fast-$K$ equilibration retains 36%–44% of the peak local value of $c_s^2-c_e^2$. At the physical maxima or EOS-validity end points, the frequencies retain 65.7%–73.4% of their frozen values and the gravitational-wave damping times increase by factors of 14.3–31.9, while the kaon-associated mode remains above the nucleonic band. Strong $\mathrm{\Delta }$ equilibration removes most of the direct $\mathrm{\Delta }$-induced enhancement, returning the $N\mathrm{\Delta }$ mode toward the nucleonic band, whereas the high-frequency $NY\mathrm{\Delta }$ branch survives because of the frozen $\mathrm{\Lambda }$ gradient. Eigenfunction tracking confirms a continuous $g_1$ branch, and DD-ME2 reproduces this hierarchy. For circular, equal-mass binary neutron-star inspirals, the frozen-composition phase-shift estimates satisfy $|\mathrm{\Delta }{\mathrm{\Phi }}_{g_1}|\le 1.410\times {10}^{-3}\mathrm{rad}$, a factor of 21 below the 0.03-rad favorable-event benchmark reported for the Einstein Telescope. An exotic species therefore produces a distinct composition mode only if its composition gradient, or a coupled slowly equilibrating gradient, survives over the oscillation period.

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• Nuclear astrophysics
• Nuclear matter in neutron stars