Periodic Infinite Ringsets in the Lipschitz Quaternions

Let L be the ring of Lipschitz quaternions and let w: Z → {i, j} have prescribed odd squarefree period m. For the infinite set Sw = {a + w(a) : a ∈ Z}, we prove an exact local-global ringset criterion. The set is a ringset if and only if w is nonconstant and, for every prime p dividing m with p ≡ 3 (mod 4), every residue class modulo p contains both letters. The proof combines split matrix-ring idempotents, Frobenius obstruction polynomials, periodic congruence orbits, and the Chinese remainder theorem. Status: Public Beta v0.1; internally verified candidate proof; external mathematical review pending.