\section{Consequences and further questions}\label{sec:consequences} The bound on the number of total orders consistent with the answers (Theorem~\ref{thm:fixed-stage}) also yields comparison lower bounds when the algorithm starts or finishes with a partial order. The two settings have different information bounds: initial relations reduce what remains to be learned, whereas a target partial order specifies what the algorithm must establish. \subsection{Producing a prescribed partial order} Let $R$ be a partial order on $n$ slots. In \emph{partial-order production}, the input consists of $n$ keys with no known relations. The algorithm must assign one key to each slot so that the key in slot $u$ is smaller than the key in slot $v$ whenever $u<_R v$. Sorting corresponds to a chain. Finding the minimum corresponds to one slot preceding all the others, with no other required relations. A fixed assignment is valid for exactly $e(R)$ total orders of the keys. The information-theoretic lower bound is therefore \[ I(R)=\log(n!)-\log e(R). \] Cardinal et al.~\cite{cardinal-production} give a deterministic polynomial-time algorithm using $I(R)+o(I(R))+O(n)$ comparisons. \begin{corollary}[Partial-order production]\label{cor:production} The adversary of Theorem~\ref{thm:main} forces every correct comparison algorithm producing an $n$-element partial order $R$ to make at least \[ (1-o(1))I(R) \] comparisons. The $o(1)$ term depends only on $n$. \end{corollary} \begin{proof} See Appendix~\ref{app:consequences}. \end{proof} Its leading term matches Cardinal et al.'s upper bound when $I(R)=\omega(n)$. The information bound can be smaller than the task's comparison complexity. For example, producing a minimum has $I(R)=\log n$ and requires $n-1$ comparisons. \subsection{Sorting from supplied partial information} Now let $P$ describe relations already known between the input keys, supplied as a DAG. The sorter must identify one of its $e(P)$ linear extensions, so the information bound is $\log e(P)$. \begin{corollary}[Supplied partial information]\label{cor:initial-poset} There is a deterministic adversary that can be initialized from any $n$-element partial order $P$ in polynomial time, gives legal answers in polynomial time per comparison, and forces every correct sorter to make at least \[ (1-o(1))\log e(P)-O(n\log\log n) \] additional comparisons. Both error terms are uniform over $P$. \end{corollary} \begin{proof} See Appendix~\ref{app:consequences}. \end{proof} The construction uses Cardinal et al.'s polynomial-time procedure~\cite{cardinal-production} to extend $P$ to a \emph{weak order}: a sequence of antichain layers, with every key in an earlier layer preceding every key in a later one. The additional relations reduce $\log e(P)$ by at most $O(n\log\log n)$. Run an independent copy of our adversary within each layer and answer between layers from their fixed order. All copies use the same parameters, chosen from the total size $n$, so the guarantee also covers small layers. The comparison bound approaches $\log e(P)$ when $\log e(P)=\omega(n\log\log n)$. Kahn and Kim's adversary already forces $\Omega(\log e(P))$ for every $P$. The improvement here is the coefficient approaching one in this regime. \subsection{Sampling linear extensions} Huber~\cite{huber-sampling} gives a near-linear expected-time exact sampler for height-two posets of bounded degree. Our contraction estimate also gives a sampler for any fixed degree and directed-height bounds. \begin{corollary}[Exact sampling]\label{cor:exact-sampling} Fix integers $d,h\ge1$. Given adjacency lists of an $n$-vertex DAG of degree at most $d$ and with directed paths of at most $h$ vertices, one can sample a linear extension exactly uniformly using $O(n\log n)$ arithmetic operations and fair random bits in expectation. \end{corollary} \begin{proof} See Appendix~\ref{app:exact-sampling}. \end{proof} \subsection{Open questions} Can a deterministic polynomial-time adversary force $(1-o(1))\log e(P)$ further comparisons for every supplied partial order $P$? The additive loss in Corollary~\ref{cor:initial-poset} can overwhelm the information that remains. For two disjoint chains of length $n/2$, $\log e(P)=\Theta(n)$. For a chain on $n-1$ items and one unrelated item, $\log e(P)=\log n$. Reducing the additive loss to $O(n)$ would already extend the useful range. Cardinal et al.~\cite[Section~4]{cardinal-production} exhibit partial orders $P$ for which every weak-order extension $W$ satisfies $\log e(W)\le\log e(P)-\Omega(n\log\log n)$. A sharper construction must therefore preserve more of the possible interleavings between layers. Initial partial information also arises when sorting points whose locations are known only to lie in specified intervals \cite{imprecise-points,interval-orders}. Extending our adversary to those models would require answers compatible with realizable coordinates and with the permitted geometric operations. \paragraph{Volume approximation for order-restriction DAGs.} The closures of the regions $K(D)$ are \emph{marked order polytopes} \cite{marked-order-polytopes}: fixed marked elements encode the interval endpoints. This connects the local calculations of this paper to a global approximation problem. For fixed integers $d,h$, can a deterministic algorithm, given an order-restriction DAG of degree at most $d$, directed paths of at most $h$ vertices, rational interval endpoints, and $0<\varepsilon<1$, compute \[ (1-\varepsilon)V(D)\le\widehat V\le(1+\varepsilon)V(D) \] in time polynomial in the input bit length and $1/\varepsilon$, with the polynomial allowed to depend on $d,h$? Assume $V(D)>0$. This asks for a deterministic fully polynomial-time approximation scheme (FPTAS). Theorem~\ref{thm:local} supplies local enclosing intervals, but its enumeration bound is not polynomial in the inverse accuracy. A concrete first case is $h=2$, $d=3$, with every interval equal to $(0,1)$. Orient the edges of a bipartite graph from one side to the other. Reflecting the coordinates on the second side changes $X_a