On the common index jump theorem and further developments

In [LZ02], Long and Zhu established originally the common index jump theorem (CIJT) for symplectic paths in 2002, which was later generalized to its enhanced version (ECIJT) by Duan, Long and Wang in [DLW16] in 2016. Started from [GGM18] of 2018 and [GG20] of 2020, and finally in [CGG24] of 2024, a similar index theorem was obtained, i.e., Theorem 3.3 of [CGG24], which was called the index recurrence theorem there. In this short note, we give detailed proofs to show that the first 4 assertions dealing with index iterations in the total of 5 assertions in Theorem 3.3 of [CGG24] actually coincide completely with results in (ECIJT) of [DLW16]