Bistationary Traces, Wide Levels, and Branch-Cover Rigidity for an Unrestricted Typed Variant of the Hayut-Magidor Forcing
Abstract
For every uncountable regular cardinal α, 𝕊∗(α) is an explicitly typed four-coordinate forcing motivated by the ladder-system construction of Hayut and Magidor. The forcing is σ-closed and, after adjoining a formal maximum, α-strategically closed. For α≥ω2, every nonempty countable family of designated generic branches has a stationary and costationary common trace on the generic ladder-coordinate set Lα, while no countable family of cofinal branches generates Lα. These conclusions persist under a Kurepa-style level-size bound. In the unrestricted forcing, for every infinite cardinal μ<α in the ground model, some level of the generic tree contains a copy of (μ2)V. Consequently, the endpoint-corrected restriction family indexed by 𝒫ω2α is too wide, whereas the scaled restriction system indexed by 𝒫αα has all levels of size less than α exactly when α is strongly inaccessible in the ground model. When these equivalent conditions hold, the branch-covering number of Lα relative to the scaled system is at least ω1. The low-cofinality empty-value convention also ensures that the set of domains of Lα contains no club in 𝒫αα. The unrestricted tree clause of the motivating presentation is retained, without asserting forcing equivalence.
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BibTeX
@misc{hu2026bistationary,
title = {{Bistationary Traces, Wide Levels, and Branch-Cover Rigidity for an Unrestricted Typed Variant of the Hayut-Magidor Forcing}},
author = {Xing-Yu Hu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.25033},
eprint = {2609.25033},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25033},
}
RIS
TY - UNPB
TI - Bistationary Traces, Wide Levels, and Branch-Cover Rigidity for an Unrestricted Typed Variant of the Hayut-Magidor Forcing
AU - Xing-Yu Hu
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2609.25033
UR - https://arxiv.org/abs/2609.25033
AB - For every uncountable regular cardinal α, 𝕊^∗(α) is an explicitly typed four-coordinate forcing motivated by the ladder-system construction of Hayut and Magidor. The forcing is σ-closed and, after adjoining a formal maximum, α-strategically closed. For α≥ω_2, every nonempty countable family of designated generic branches has a stationary and costationary common trace on the generic ladder-coordinate set L_α, while no countable family of cofinal branches generates L_α. These conclusions persist under a Kurepa-style level-size bound. In the unrestricted forcing, for every infinite cardinal μ<α in the ground model, some level of the generic tree contains a copy of (^μ2)^V. Consequently, the endpoint-corrected restriction family indexed by 𝒫_ω_2α is too wide, whereas the scaled restriction system indexed by 𝒫_αα has all levels of size less than α exactly when α is strongly inaccessible in the ground model. When these equivalent conditions hold, the branch-covering number of L_α relative to the scaled system is at least ω_1. The low-cofinality empty-value convention also ensures that the set of domains of L_α contains no club in 𝒫_αα. The unrestricted tree clause of the motivating presentation is retained, without asserting forcing equivalence.
ER -