Existence of Σ¹₃ instances in RFN^{-1}(ATR)

Determine whether there exists any Σ¹₃ sentence σ such that, over ACA, the ω-model reflection RFN(σ) is equivalent to ATR; equivalently, determine whether the class RFN^{-1}(ATR) contains any theories finitely axiomatizable by a Σ¹₃ sentence.

Background

The paper introduces RFN^{-1}(ATR) as the class of theories T for which RFN(T) is equivalent to ATR over ACA, and shows that this class approximates hyperarithmetic analysis (HA) in several respects, including closure properties and inclusion of many known HA theories. In HA, there are no theories finitely axiomatized by Σ¹₃ sentences, and the author investigates whether a similar restriction holds for RFN^{-1}(ATR).

Partial progress is given: the paper proves that no theory of the form ∀X∃!Yθ(X,Y) (with arithmetic θ) belongs to RFN^{-1}(ATR), ruling out a broad subclass of Π¹₂-form sentences. However, the general question for Σ¹₃ sentences remains unresolved.