Conley-Morse persistence barcode: a homological signature of a combinatorial bifurcation

The paper proves that the Conley–Morse persistence module arising from the transition diagram TD is a finite-dimensional representation of a gentle bound quiver (Q, I), then uses the string/band classification for gentle algebras and a structural property of TD to exclude bands; hence the module decomposes uniquely into string modules (the barcode) as stated in Theorem 7.4 and supported by Proposition 7.3 and Remark 7.2. The candidate solution instead assumes the module is a zigzag over a type-A quiver A_{T+1} and invokes the interval-module classification for A-type quivers. That identification is generally false here because the indexing is a poset from TD (with relations from AR-splits), not a single fence; crucially, the model does not address the possibility of band indecomposables that must be ruled out in this setting. Therefore, its proof route is invalid even though the final conclusion matches the paper’s theorem. See Theorem 7.4 and Proposition 7.3 in the paper for the correct framework and argument, including the no-band argument tied to AR-splits and the gentle structure of (Q, I) ; background on the zigzag/interval case is recalled separately in Theorem 6.6 but is not the general setting for TD-based modules .