Four ages of rotating stars in the rotation--activity relationship and gyrochronology

Gyrochronology and the rotation--activity relationship are standard techniques used to determine the evolution phase.The mismatch in the definition of the evolutionary phases has so far raised many issues in physics and mathematics and hampered the understanding of how the internal dynamo processes affect the observable properties.To address this problem, we seek a unified scheme that shows a one-to-one mapping from gyrochronology to the rotation--activity relationship.We obtained the chromospheric activity $R'_{\rm HK}$ of 6846 stars and their rotation periods to investigate the rotation--activity relationship.We apply a three-interval model to fit the relationship and find that it is best fitted by three intervals in the range of Ro$<0.7$.We associate those intervals to the convective, gap and interface phases of gyrochronology.Furthermore,we suggest an additional epoch at late times of the I phase.We further use the three-interval models to fit the period--activity relationship in temperature bins and determine the duration of the transition phase as a function of effective temperature.By comparing the critical temperature and period of the g-to-I transition with the slowly rotating sequence of 10 young open clusters whose ages range from 1 Myr to 2.5 Gyr, we conclude that our new model finds the pure I sequence without fast rotating outliers, which defines the zero-age I sequence (ZAIS).We propose that there is an ambiguous consensus on when the I sequence starts to work.This ambiguity is from the visually convergent sequence of the color--period diagrams in open clusters.This visually convergent sequence is younger than the ZAIS and is actually the pre-I sequence that can be associated with the stall of the spin-down. Our results unify the rotation--activity relationship and gyrochonology for the stellar evolution of low-mass stars, for which we coined the ``CgIW" scenario.