Failures of Theorem 1 and Conjecture 3 in Knudson's Persistence–Gradient Abstract, and a Corrected Form of Conjecture 2 after Kriebel

In an Oberwolfach abstract, Knudson considered the discrete vector field \(V_P\) formed by the incident pairs of the persistence pairing \(P\) of a simplexwise filtration. He stated that \(V_P\) is a discrete gradient (his Theorem 1), conjectured a single gradient path from \(\tau\) to \(\sigma\) in \(V_P\) for every non-incident pair \(\{\sigma,\tau\}\) of \(P\) (Conjecture 2), and conjectured that a filtration by a function decreasing along the modified Hasse diagram reproduces \(P\) (Conjecture 3). Conjecture 2 was first refuted publicly by Kriebel (Zenodo, September 2026) with a filtration of \(7\) simplices; we credit this and add that there are exactly two such filtrations with \(7\) simplices, none with fewer, that no filtration with at most \(13\) simplices has a pair joined by two or more paths, and examples with two and three paths on \(17\) and \(19\) simplices. The parts of this note that we did not find elsewhere are the following two. Theorem 1 is false, also in the abstract's own formulation by the modified Hasse diagram: a filtration of \(17\) simplices has a closed \(V\)-path, and for every filtration of every complex with at most \(16\) simplices \(V_P\) is a gradient (a short lemma and an exhaustive search over \(2{,}484{,}335{,}648\) filtrations). Conjecture 3 is false even when \(V_P\) is acyclic, for every choice of the function and of tie-breaking: a filtration of \(17\) simplices with a hand-checkable proof; it holds for all \(4{,}005{,}434\) filtrations with at most \(11\) simplices. Finally, we prove a corrected form of Conjecture 2, which is a consequence of algebraic Morse theory: for the earliest non-incident pair, if the field formed by the earlier pairs is a gradient, the number of gradient paths in it from \(\tau\) to \(\sigma\) is odd, and \(\sigma\) is the youngest critical simplex reached an odd number of times. This is an unrefereed note.