Thiago Ferreira, Earl P. Bellinger, Ebraheem Farag, Christopher J. Lindsay

Low-mass stars from the first epoch of star formation may still persist in the Milky Way and its satellite dwarf galaxies today; however, their detection is confounded by surface pollution from interstellar accretion and internal mixing, which obscure their primordial composition and blur their distinction from second-generation stars. Asteroseismology offers a probe of the internal structure and evolutionary state of stars, and hence may aid in the search for primordial stars. In this second paper of the series, we present the first non-radial adiabatic pulsation analysis of low-mass, metal-free stellar models. We use a $0.85\,M_\odot$ red giant as a case study and compare its seismic signatures with those of higher-metallicity models. At the same central hydrogen fractions, Pop III main-sequence models display systematically higher $r_{02}\equivδν_{02}/Δν$ ratio and lower $Δν$ than metal-enriched analogues, a direct consequence of their larger internal sound speeds and steeper core-envelope stratification. To interpret the structural dependence during giant evolution, we introduce a composite asteroseismic diagnostic, $ψ\equivΔν/ΔΠ_1$, which traces how metallicity influences the balance between acoustic and buoyancy cavities through its imprint on opacity, core contraction, and mean molecular-weight gradients. Pop III models occupy a distinct locus in the $ψ-ΔΠ_1$ plane due to their radiative interiors with lower mean densities and delayed development of core mean molecular weight gradients. We find that asteroseismology is a powerful diagnostic for identifying relic Pop III stars despite potentially polluted surfaces, providing a clear pathway for future searches of the Galaxy's oldest surviving stars with upcoming surveys.