Published August 23, 2026
| Version v0.1-beta
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Periodic Infinite Ringsets in the Lipschitz Quaternions
Authors/Creators
Description
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.
Notes
OpenAI Codex assisted with literature search, symbolic exploration, verification, and manuscript preparation. Carptopus is the sole author and assumes responsibility for the mathematical claims and final text. The residue-null-ideal criterion, finite quaternion ringset theory, finite core-set classifications, and the previously known sufficient pairing mechanism remain prior work. Absolute priority is qualified pending broader public and expert review.