Nearest hamming cycle period in MD5
Jelly, 37 bytes

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Try it online!

Try it online! - using full implementation of MD5

I'll comment it when I have time. Jelly doesn't have an MD5 built-in, so the header for the first TIO implements MD5 for a fixed length (32 bytes) input. I've also now extended this to a general purpose MD5 routine (second TIO link). For the purposes of the challenge, there is a monadic link just above the challenge code that takes a 32 character string and returns the MD5 hash. The remaining code (as displayed above here) calculates the hamming distance between the original input and sequential hashes until it matches the first one.