A NON-BOREL SPECIAL ALPHA-LIMIT SET IN THE SQUARE

The paper proves (i) every special α-limit set sα(x) is analytic by working in the natural extension X̂⊂XN and expressing sα(x) as the projection of a Borel relation R, see Theorem 1 and its proof (sα(x)=π(R∩(A×X))) , and (ii) constructs a subshift X with sα(ω0) Σ1^1-complete (Theorem 4) and embeds it as a totally invariant subsystem of a surjective triangular map F:I^2→I^2 (Proposition 6), yielding Theorem 5: a surjective map on the square with Σ1^1-complete sα(x0) . The candidate’s Part (1) mirrors the paper’s analytic argument and is correct in substance. However, Part (2) is flawed: the proposed zero-dimensional coding map f0 is not continuous at points with H=0 (images along gates converge to (t,u*,0) while f0 is constant to x0 there), the Tietze-based extension to I^2 does not ensure surjectivity, and no condition is imposed to prevent incoming arrows into the embedded subsystem, so the reduction to Σ1^1-hardness can be invalidated. Hence the paper’s arguments are correct while the model’s construction fails on key topological/dynamical details.