Periodic Two-Letter Ringsets in the Lipschitz Quaternions: The General Odd-Period Classification

Let L be the ring of Lipschitz quaternions and let w: Z to {i,j} be a periodic binary word of arbitrary odd period m. This manuscript proves that the set S_w={a+w(a):a in Z} is a ringset if and only if w is nonconstant and, for every prime p dividing m with p congruent to 3 modulo 4, every position class modulo p^{v_p(m)} contains both letters. The result strictly generalizes the preceding odd-squarefree-period classification (doi:10.5281/zenodo.22069614). The new ingredient is an explicit higher-prime-power fixed-divisor obstruction for monochromatic position classes. Status: Public Beta; internally verified candidate proof; external mathematical review and journal peer review pending.