Persistent homology of Morse decomposition in Markov chains based on combinatorial multivector fields

The paper states a stability theorem for Morse-set persistence diagrams under sup-norm perturbations of transition matrices (Theorem 1/4: dB(D(P), D(P′)) < C ||P−P′||∞) and outlines a proof based on a “local stability” lemma for single-entry changes and a telescoping argument over changed entries . However, the local lemma merely asserts—without rigorous justification—that birth/death thresholds of all affected Morse sets shift by at most the magnitude of the single-entry perturbation, and then matches features accordingly; the proof does not address how global merges, SCC structure, and index changes can couple nonlocally, nor does it handle edge on/off toggling (skeleton changes) created by entries crossing zero . The construction of the multivector-field filtration in γ is specified (Algorithm 1) and the monotone coarsening intuition is present (Theorem 3), but formal functorial control of the SCCs and the induced persistence structure needed for stability is not established . By contrast, the candidate solution builds a standard 1-parameter persistence-module model (per index k), proves a δ-interleaving of the γ-filtrations from the sup-norm bound (under a fixed transition skeleton; otherwise with a structural margin hypothesis), and then invokes the algebraic stability/isometry theorem for q-tame modules to conclude dB ≤ δ (or ≤ Cδ with a structural C). This aligns with the accepted stability blueprint in TDA and fills the logical gaps the paper leaves open. Therefore, the paper’s proof is incomplete, while the model’s solution is substantively correct under clearly stated structural assumptions.