(Don’t) Mind the Gap Complexity of Gapped Digit Substitutions

The paper’s Theorem 6 states p_S(n)=2,4, 6n+14 (n=3,4,5), and 8n+12 (n≥6). That implies p_S(3)=32 and p_S(5)=44, which is impossible for a binary sequence (where p(3)≤8 and p(5)≤32). Moreover, the paper’s own earlier computational table (Figure 5, row S(a)=b,a,b; S(b)=a,b,a) lists p(n) for n=2..10 as (4,8,14,20,28,36,44,52,60), matching the model’s formula p(3)=8, p(4)=14, p(5)=20, and p(n)=8n−20 for n≥6. The paper’s proof sketch via higher-block recoding and a right-special tree is conceptually sound, but the final constants (+14 and +12) in Theorem 6 are wrong—likely from misapplying the higher-block shift or (binary-only) Proposition 2 after recoding to an 8-letter alphabet. The model’s derivation (3-adic parent recursion, finite right-special base, and stabilization with 8 right-special factors for n≥5) is consistent with the paper’s own computational evidence and yields the correct piecewise complexity.