A Ternary Word Attaining the Abelian Maximal Pattern Complexity Bound at Pattern Size Three

Define the length-three substitution 0->001, 1->020, 2->000 and let alpha be its one-sided fixed point beginning with 0. This preprint proves that alpha is recurrent and aperiodic by projection and that its Abelian maximal pattern complexity at pattern size three is exactly 7. Thus the general lower bound (r-1)k+1 is attained at the first parameter pair r=3, k=3 not covered by the previously known equality cases. The global upper bound is certified by an exact 938-state base-three patch automaton, and the pattern {0,2,9} supplies sharpness. Status: Public Beta; internally verified candidate proof; external mathematical review pending.