Exercise (mostly I've seen more intense than walking, but cardio like running, cycling, rowing, etc. - I haven't come across much for resistance training like weightlifting) has been very well studied to improve cognitive ability, as well as mental health (depression/anxiety). Depending on what you mean by proven, it's not gravity, but it's pretty good evidence compared to just about anything in medicine (just look at the numerous citations testing various aspects/study designs/different populations).
Just a few studies (there are countless): https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3951958/ https://pubmed.ncbi.nlm.nih.gov/20065132/ https://pubmed.ncbi.nlm.nih.gov/35153701/ https://ju.se/download/18.4662178a174aa5f82061c573/160103080...