Comment on “Failure of the simultaneous block diagonalization technique applied to complete and cluster synchronization of random networks”

The paper’s comment asserts that SBD finds a finest simultaneous block diagonalization that guarantees the maximal, model-independent decoupling for undirected networks, and that the limited reduction seen on random networks reflects intrinsic irreducibility rather than a failure of SBD. This is stated explicitly in the paper as “SBD works by finding a finest simultaneous block diagonalization … This guarantees that, for any undirected networks, the variational equation is decoupled to the fullest extent possible, largely independent of the specifics of node dynamics and coupling functions” and that the observed lack of reduction in random networks “is an intrinsic property of the systems chosen … and has little to do with which technique is being applied.” The candidate solution provides a rigorous module-theoretic justification (via Krull–Schmidt) and a standard dimension-count for generic irreducibility, which align with the paper’s claims while giving more formal detail. Aside from a minor wording issue (“blocks End(V_j)” should read “restrictions to V_j”), the model’s reasoning matches the paper’s position. Thus both are correct, with the model offering a different, more formal proof than the paper’s high-level argument and citations.