Detailed theoretical analysis of core Helium-burning stars: Mixed mode patterns I. Impact of the He-flash discontinuity and of induced semi-convection

Context. Recent space missions such as CoRoT, Kepler, and TESS have made asteroseismology a powerful tool for studying the internal structure of stars. Red giants, in particular, are central in these studies due to their rich oscillation spectra, which provide details of both their core and envelope through their mixed oscillation modes. Despite these advances, models of core helium-burning red giants struggle to reproduce the observed oscillation spectra, particularly because of uncertainties in the treatment of mixing processes such as overshooting and semi-convection. These discrepancies highlight the need for further asteroseismic constraints to improve stellar models. Aims. We aim to identify the key structural features influencing asteroseismic observations and how seismic signatures relate to internal chemical composition by investigating the asteroseismic properties of core helium-burning stars with a consistent treatment of induced semi-convection and overshooting. Methods. We use a new version of the Liège stellar evolution code and the Liège adiabatic oscillation code to compute and analyse the mixed-mode oscillation patterns of various models of core helium-burning stars. Results. We find that sharp transitions in the chemical composition and overshooting in the central part of the models significantly affect the mixed-mode oscillation spectra of core helium-burning stars. Overshooting variations alter the size of the semi-convective zone, which locally modifies the Brunt─Väisälä frequency and, thereby, the observed period spacing. Notably, our models indicate that modifications in overshooting are balanced by adjustments in the semi-convective layers, maintaining a consistent total mixed-core size across models. Thus, stellar evolution is minimally affected by these internal adjustments, unlike the seismic signatures, as seen in the Brunt─Väisälä frequency profile. The semi-convective zone also introduces additional seismic trapping, although for advanced models, modes confined within this region are unlikely to be detectable due to their high energy density and minimal impact on surrounding modes. Conclusions. We highlight the importance of detailed seismic studies to characterise mixing processes near the convective core in core helium-burning stars. This provides a first step towards constraining the chemical composition gradient. In addition, we notice a balance between the extension of overshooting and semi-convective zones, affecting the oscillation spectra.