We revisit the topic of power-free morphisms, focusing on the properties of the class of complementary binary morphisms. Such morphisms map binary letters 0 and 1 to complementary words. We prove that every prefix of the famous Thue-Morse word $\mathbf{t}$ gives a complementary morphism that is $3^+$-free and then $\alpha$-free for any real $\alpha>3$. We also describe the lengths of all prefixes of $\mathbf{t}$ that give cubefree complementary morphisms by a 4-state binary finite automaton. Next we show that cubefree complementary morphisms of length $k$ exist for all $k\ne\{3,6\}$. Moreover, if $k$ is not representable as $3\cdot2^n$, then the images of letters can be chosen to be factors of $\mathbf{t}$. In addition to more traditional techniques of combinatorics on words, we also rely on the Walnut theorem-prover. Its use and limitations are discussed.

翻译：我们重新探讨了无幂词态射这一主题，重点关注互补二元态射类的性质。此类态射将二元字母0和1映射为互补词。我们证明了著名的Thue-Morse词$\mathbf{t}$的每个前缀都能给出一个$3^+$-自由态射，进而对任意实数$\alpha>3$均为$\alpha$-自由。我们还通过一个四状态二元有限自动机描述了$\mathbf{t}$中所有能给出无立方词互补态射的前缀长度。接下来我们证明，对任意$k\ne\{3,6\}$，长度为$k$的无立方词互补态射均存在。此外，若$k$不能表示为$3\cdot2^n$，则可选择字母的像为$\mathbf{t}$的因子。除使用更传统的词组合学技巧外，我们还借助了Walnut定理证明器，并讨论了其使用方式与局限性。