Dense Chains, Antichains, and Universal Partial Orders Inside a Bounded Finite-One Degree

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.