A PROFINITE APPROACH TO COMPLETE BIFIX DECODINGS OF RECURRENT LANGUAGES

The paper proves the claim cleanly: Theorem 4.5 (“Every charged complete bifix decoding of a uniformly recurrent language is uniformly recurrent”) follows by combining Proposition 8.2 (which, using the F-charged hypothesis and a Green-relations argument in the syntactic monoid, identifies the decoding as Fin_X(e)) with Proposition 7.1 (which shows Fin_X(e) is uniformly recurrent for any finite code X inside a uniformly recurrent F) . By contrast, the model’s proof outline misattributes the equality J_X(F∩X*) = JA(F) ∩ X̂* to the charged condition (it actually holds for any finite F-complete bifix code; see Proposition 6.2(i)) and then treats that equality as sufficient to deduce recurrence, which is not justified in the outline . It also conflates JA(·) over Â* with J_X(·) over X̂*. Hence the model’s argument is incomplete/incorrect in its core recurrence step, even though its final conclusion matches the paper.