Periodic ringsets in the Lipschitz quaternions: the complete six-letter classification

Abstract

Let L be the ring of Lipschitz quaternions and let U be its six square roots of -1. For a periodic word w: Z -> U, this manuscript gives a complete necessary-and-sufficient criterion for S_w={a+w(a):a in Z} to be a ringset for every prescribed period. Ramified and nonsplit highest prime-power position fibres must contain at least two letters. At a split prime, the exact additional obstruction is a square-root-separated pair of singleton fibres carrying antipodal letters. Explicit null-polynomial obstructions and a local-global argument prove the classification, which strictly generalizes the preceding squarefree-period and all-odd-period binary results. Status: Public Beta v0.1-beta; internally verified candidate proof; external review pending.