From one author's home page (Jean-Pierre Eckmann -
https://fiteoweb.unige.ch/~eckmannj/publications.html), the latest version of full paper:
Walks in Rotation Spaces Return Home when Doubled and Scaled (with Tsvi Tlusty) Physical Review Letters 135, 147201 (2025)
https://fiteoweb.unige.ch/~eckmannj/ps_files/ETPRL.pdf