Sharp Initial Thresholds for Abelian-Bordered Binary Infinite Words

Let mu_ab(x) denote the maximum length of an Abelian-unbordered factor of a binary infinite word x. We determine the first sharp threshold layers: mu_ab(x) at most 13 forces ordinary ultimate periodicity with sharp eventual-period bound 13; the least finite value attained by a non-ordinarily-periodic word is 14; and mu_ab(x) at most 14 forces Abelian ultimate periodicity with sharp Abelian-period bound 14. The upper bounds follow from exact finite overlap-graph classifications with independently reproducible Python and C++ verification. We also prove that if the frequency of 1 is the reduced fraction p/q and mu_ab(x) is less than 2q, then the tail splits into length-q blocks of weight p. The full bounded-Abelian-unbordered-factor question remains open outside these regimes. Status: Public Beta v0.1; internally verified candidate proof; external mathematical review pending.