Abstract Asteroseismic studies of red giants have primarily relied on two global parameters: the large frequency separation (Δν) and the frequency of maximum power (νmax). Meanwhile, the p-mode phase shift (ϵ) and small frequency separations (δν01, δν02), which offer additional constraints on stellar interiors, remain underexplored due to measurement challenges. Here we develop an automated pipeline based on collapsed échelle diagrams and apply it to ~16, 000 Kepler red giants, jointly measuring Δν, ϵ, δν01, and δν02 and assembling the largest homogeneous catalogue of these quantities to date, together with updated Δν values and formal internal uncertainties. Using this catalogue, we quantify evolutionary trends across the red-giant branch and core-helium-burning phase. We find that δν02/Δν stays nearly constant for RGB stars and, for core-helium-burning stars, organises into two sequences that are systematically offset but partially overlap, broadly separating stars in the red-clump and secondary-clump regimes. We also trace the mass- and metallicity-dependent helium-flash transition. Meanwhile, ϵ follows a single Δν–ε relation common to both evolutionary phases. Comparisons with stellar-evolution models reveal systematic offsets in ϵ and δν01, which we interpret as signatures of near-surface and outer-envelope modelling deficiencies. These comparisons further suggest that dipole-mode small separations are sensitive to mode-dependent surface terms in evolved stars. Overall, our results demonstrate that ϵ and the small separations provide important diagnostics of core structure, convective-boundary mixing, and helium ignition that are complementary to those provided by Δν and νmax alone. The resulting catalogue offers a reference for testing and calibrating future stellar-evolution models.
Wang et al. (Thu,) studied this question.