Asteroseismic Diagnostics for Red Giants with Kepler: Measuring epsilon and Small Frequency Separations in 16,000 Stars

Asteroseismic studies of red giants have primarily relied on two global parameters: the large frequency separation (Dnu) and the frequency of maximum power (numax). Meanwhile, the p-mode phase shift (epsilon) and small frequency separations (dnu01, dnu02), which offer additional constraints on stellar interiors, remain underexplored due to measurement challenges. Here we develop an automated pipeline based on collapsed echelle diagrams and apply it to about 16,000 Kepler red giants, jointly measuring Dnu, epsilon, dnu01, and dnu02 and assembling the largest homogeneous catalogue of these quantities to date, together with updated Dnu 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 dnu02/Dnu 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, epsilon follows a single Dnu-epsilon relation common to both evolutionary phases. Comparisons with stellar-evolution models reveal systematic offsets in epsilon and dnu01, 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 epsilon and the small separations provide important diagnostics of core structure, convective-boundary mixing, and helium ignition that are complementary to those provided by Dnu and numax alone. The resulting catalogue offers a reference for testing and calibrating future stellar-evolution models.