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Pseudo-Hyperjump Inversion Fails for Turing Degrees
Abstract
Jananthan and Simpson asked in Conjecture~4.5 whether every pseudo-hyperjump HJe(X)=X⊕ VeX, with VeX uniformly Π11(X), admits Turing-degree inversion above Kleene's 𝒪. We settle their conjecture in the negative. We construct one index e* such that X<THJe*(X) for every real X, while the range of HJe* omits every Turing degree d such that °T(𝒪)≤d<°T(𝒪𝒪). 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.
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BibTeX
@misc{cintioli2026pseudohyperjump,
title = {{Pseudo-Hyperjump Inversion Fails for Turing Degrees}},
author = {Patrizio Cintioli},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37478},
eprint = {2609.37478},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37478},
}
RIS
TY - UNPB
TI - Pseudo-Hyperjump Inversion Fails for Turing Degrees
AU - Patrizio Cintioli
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.37478
UR - https://arxiv.org/abs/2609.37478
AB - Jananthan and Simpson asked in Conjecture~4.5 whether every pseudo-hyperjump HJ_e(X)=X⊕ V_e^X, with V_e^X uniformly Π^1_1(X), admits Turing-degree inversion above Kleene's 𝒪. We settle their conjecture in the negative. We construct one index e_* such that X<_THJ_e_*(X) for every real X, while the range of HJ_e_* omits every Turing degree d such that °_T(𝒪)≤d<°_T(𝒪^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.
ER -