A Sparse-Defect Counterexample to Abelian-Border Periodicity

Charlier, Harju, Puzynina and Zamboni asked whether an infinite word whose sufficiently long factors are all Abelian bordered must be Abelian periodic. We give a negative answer over the binary alphabet. The construction combines a four-height periodic nearest-neighbor path with sparse finite replacement walks. The resulting binary word is not Abelian ultimately periodic, although every factor of length at least 141 has an Abelian border. Exact finite certificates verify the periodic palette, all 28 replacement blocks, the five-witness contamination bound and a finite sparse-prefix positive control. Status: Public Beta; internally verified candidate proof; external mathematical review pending.