Pseudo-Hyperjump Inversion Fails for Turing Degrees

Abstract

Jananthan and Simpson asked in Conjecture~4.5 whether every pseudo-hyperjump ${\mathrm{HJ}}_e(X)=X\oplus V_e^X$, with $V_e^X$ uniformly ${\Pi }_1^1(X)$, admits Turing-degree inversion above Kleene's $\mathcal{O}$. We settle their conjecture in the negative. We construct one index $e_{\ast }$ such that $X{<}_{\mathrm{T}}{\mathrm{HJ}}_{e_{\ast }}(X)$ for every real $X$, while the range of ${\mathrm{HJ}}_{e_{\ast }}$ omits every Turing degree $\mathbf{d}$ such that ${°}_{\mathrm{T}}(\mathcal{O})\le \mathbf{d}<{°}_{\mathrm{T}}({\mathcal{O}}^{\mathcal{O}})$. Thus Conjecture~4.5 fails even for an operator which strictly raises the Turing degree of every input. Each of Properties~4.6--4.8 implies Conjecture~4.5; consequently, the accompanying class-characterisation problems collapse: no class of reals, even without a definability assumption, satisfies any one of these properties.

arXiv comment5 pages