Tootfinder

Cleared off the driveway from the second snow. It was nice and powdery, so no big deal.
Then I had to clear in front of the garage doors from the first snowstorm. Snow drifts with frozen rain on top had turned them into glaciers.
After some trial and error, I found that using the maul would break the ice into boulders. Then it was a function of breaking the boulders to the point where I could lift them and toss them clear.
They were not light and I am very tired.

Dense Chains, Antichains, and Universal Partial Orders Inside a Bounded Finite-One Degree
Patrizio Cintioli
https://arxiv.org/abs/2603.27901 https://arxiv.org/pdf/2603.27901 https://arxiv.org/html/2603.27901
arXiv:2603.27901v1 Announce Type: new
Abstract: We construct a nonrecursive set \(A\le_T\emptyset'\) and a uniformly computable family of sets \(C_0,C_1,\dots\), all bounded finite-one equivalent to \(A\), such that the corresponding \(1\)-degrees form a copy of the dense linear order \((\mathbb Q,\le)\). Motivated by a recent preprint of Richter, Stephan, and Zhang, which shows that bounded finite-one degrees can be as rigid as a discrete \(\omega\)-chain and asks whether there are bounded finite-one degrees consisting exactly of a dense linearly ordered set of \(1\)-degrees, we introduce a block-density profile method for controlling one-one reducibility inside a single bounded finite-one degree.
As further applications, in the same bounded finite-one degree we obtain an infinite antichain of \(1\)-degrees and, more generally, an embedded copy of every countable partial order. A single bounded finite-one degree can already exhibit dense, incomparable, and universal order-theoretic behaviour.
Our main technical tool is a profile theorem based on computable block-density codings. The witness set constructed here is not \(m\)-rigid, so the phenomena obtained in this paper arise from a mechanism different from earlier \(m\)-rigidity-based constructions. Although our results do not settle the exact realization problem posed by Richter, Stephan, and Zhang, we show that density itself is not the obstruction: a single bounded finite-one degree may already contain a copy of \((\mathbb Q,\le)\), an infinite antichain, and embeddings of all countable partial orders.
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