Borel complexity of the family of attractors for weak IFSs

The paper proves Gδσ-hardness of wIFSd by building a continuous coding ϕ whose preimage ϕ^{-1}[wIFS1] is the complement of an Fσδ-complete set {A ⊆ N^2 : ∃∞ n ∀∞ m (n,m) ∈ A} (Lemma 3.2 and Lemma 3.4), and then lifts to higher dimensions via the zero-embedded product; see the abstract, Lemma 3.2, Lemma 3.3 (the two-consecutive-members lemma), Lemma 3.4 (the construction of ϕ), and the statement of Theorem 3.5 . By contrast, the candidate solution reduces from a Σ^0_3-complete set C defined by “for all but finitely many rows the vertical section is infinite,” and then incorrectly equates Σ^0_3 with Gδσ to claim Gδσ-hardness for wIFSd. This conflates Σ^0_3 (= Fσδ) with Π^0_3 (= Gδσ) and breaks the hardness and the non-Fσδ conclusion. The paper’s geometric scaffold and counting obstruction match the candidate’s sketch, but the model’s complexity-theoretic reduction is incorrect; the paper’s is correct.