Attractor Stability in Finite Asynchronous Biological System Models

The paper explicitly states as Fact 4 that for any vertex v the set Acyc_v(G) is a complete set of κ-class representatives and that picking one linear extension per O ∈ Acyc_v(G) exhausts all possible cycle structures for F_π when varying the update order, citing prior work for the proof. It operationalizes this via Algorithm 1, which constructs Acyc_v(G) by enumerating acyclic orientations of G \ v and reattaching v as a unique source, thereby yielding one representative per κ-class. These claims and their intended use are consistent and correct in the paper’s framework . The candidate’s solution provides a different proof sketch: a constructive surjection from κ-classes to Acyc_v(G) via maximal sequences of clicks avoiding v, followed by a counting argument (|Acyc_v(G)| = κ(G)) to conclude bijectivity, and then the SDS cycle-equivalence step. Substantively, it agrees with the paper’s claims and is largely correct. One minor gap is the use of “maximal length” click sequences without ruling out repeats; this is easily repaired by restricting to repetition-free sequences (or giving a monotone potential function). The candidate also appropriately adds the connectedness hypothesis for the nonemptiness of Acyc_v(G), which the paper assumes implicitly. Overall, both are correct; the paper presents the result as a cited fact with an algorithmic construction, while the candidate offers a counting-plus-normal-form argument tying directly to κ-equivalence and SDS cycle-equivalence .