On Examples of Rank-Two Symbolic Shifts

The uploaded paper proves: (i) the Thue–Morse word M is not rank-one (via Lemmas 2.9–2.10 and Theorem 2.11), and also gives an explicit two-word inductive construction showing M is at most rank two; hence M is rank two . It further proves the symbolic system (XM,σ) is not rank-one (Theorem 5.8) , and that all (binary) substitution fixed points are at most rank two (Theorem 6.2) . For Sturmian sequences, it establishes non–rank-one for both types and that quadratic (substitutive) Sturmian systems are rank-two (see Example and Proposition 6.5) , in line with the abstract’s claims . The candidate solution states the same high-level conclusions, but several key proof steps are incorrect or incomplete. Notably: (a) its non–rank-one argument for M relies on a gap-subsequence contradiction that is not justified (a nonconstant sequence can have constant modular subsequences), (b) its proposed rank-2 construction for M based on pairing zeros (blocks 00 and 0110) is not shown to give the required inductive rank-2 scheme and misstates how 010 arises across block boundaries, and (c) its claim that every binary substitution fixed point follows from a two-return-words S-adic argument is asserted but not proven. By contrast, the paper’s proofs are explicit and complete.