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Characterizing Σ1_2-altitude via transitive models of ATR0^set

Show that for every Σ1_2-singleton real R, the Σ1_2-altitude Alt_{Σ1_2}(R) equals the least height of a transitive model M of ATR_0^{set} in which R is a Σ_1-definable class in the language of set theory.

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Background

The paper introduces the Σ1_2-altitude Alt_{Σ1_2}(R) of a Σ1_2-singleton real R via genedendrons and their climaxes, motivated by connecting Π1_2 dilators and intermediate pointclasses. In the final remarks, the author seeks a recursion-theoretic or set-theoretic characterization of this new rank.

The conjectured characterization relates Alt_{Σ1_2}(R) to the minimal height of a transitive model of ATR_0{set} where R is Σ_1-definable as a class, providing a model-theoretic interpretation of the altitude.

References

We may ask if the $\Sigma1_2$-altitude has recursion-theoretic or set-theoretic characterization, and the following conjecture suggests one possibility: \begin{conjecture} Let $R$ be a $\Sigma1_2$-singleton real. $Alt_{\Sigma1_2}(R)$ is equal to the least height of a transitive model $M$ of $ATR_0\mathsf{set}$ on which $R$ is a $\Sigma_1$-definable class in the language of set theory. \end{conjecture}

— Proof-theoretic dilator and intermediate pointclasses (2501.11220 - Jeon, 20 Jan 2025) in Section: Final remarks