For every repetition threshold α with (8/3)^+≤α<3, the lengths of α-free uniform binary morphisms are exactly the positive integers other than 3 and 6; each admitted length has a complementary witness.
For every repetition threshold α with (8/3)^+≤α<3, the lengths of α-free uniform binary morphisms are exactly the positive integers other than 3 and 6; each admitted length has a complementary witness. At the strict threshold 8/3, length 5 is impossible.
For every repetition threshold α with (8/3)^+≤α<3, the lengths of α-free uniform binary morphisms are exactly the positive integers other than 3 and 6; each admitted length has a complementary witness. At the strict threshold 8/3, length 5 is impossible.
If correct, settles the stated stability question in the interval from (8/3)^+ to 3 and identifies a genuine change at the strict 8/3 endpoint.
Shallit, Shur and Zorcic ask for the phase transitions in admissible lengths of uniform binary morphisms preserving repetition avoidance. In Section 4 of Power-free complementary binary morphisms, they conjecture that the set of admissible lengths stays constant whenever $(8/3)^+\leq\alpha<3$, with another transition at the strict threshold $8/3$. Here a morphism is $\alpha$-free if it sends every $\alpha$-free binary input word to an $\alpha$-free output; the plus convention permits exponent exactly $8/3$. Does this length-set stability hold, and is the strict endpoint different?
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