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Relating Π1_3 dilator sections at 0^♯ to Π1_1[0^♯]-proof theory

Prove that, working over ZFC with the existence of 0^♯, for every Π1_3-sound extension T of ACA_0, the equality |T|_{Π1_3}(0^♯) = |T + ∃0^♯|_{Π1_1[0^♯]} holds.

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Background

A central theme of the paper is connecting Π1_2-proof-theoretic dilators with intermediate pointclasses such as Π1_1[R]. In the final remarks, the author proposes a higher-level analogue involving 0♯ and Π1_3 proof theory.

Establishing this equality would extend the main correspondence of the paper to a stronger setting (Π1_3), relating the section of the Π1_3 proof-theoretic dilator at 0♯ to the Π1_1[0♯]-proof-theoretic ordinal of T augmented by the existence of 0♯.

References

As an example, the author conjectures the following: Working over $ZFC$ with the existence of $0\sharp$, let $T$ be a $\Pi1_3$-sound extension of $ACA_0$. If we view $0\sharp$ as a dilator (cf. ), we have \begin{equation*} |T|{\Pi1_3}(0\sharp) = |T + \exists 0\sharp|{\Pi1_1[0\sharp]}. \end{equation*}

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