Designing topological cluster synchronization patterns with the Dirac operator

The paper linearizes DESD around the target eigenstate and projects onto the Dirac eigenbasis, yielding modal ODEs ċE = ΩE − σ(E−Ē)² cE (their Eqs. 62–66), exactly the starting point used by the model . From this, the paper derives the roughness W² as a spectral sum/integral proportional to ∑ 1/(E−Ē)⁴ (Appendix A), establishing the δ-threshold δ>3 for finiteness and divergence otherwise, in full agreement with the model’s W² criterion . For entrainment, the paper shows V²(t) is controlled by ∫ ρ(E) e^{−2σ(E−Ē)² t} dE (Appendix C), which decays to 0 (exponentially if gapped, algebraically if gapless), consistent with the model’s asymptotics; the paper also highlights a δ≤1 threshold where the linearization fails (via divergence of C ~ ∫ ρ/(E−Ē)²), matching the model’s third regime and its non-uniform-in-N relaxation warning . The paper’s summary of the three regimes (gapped stable; 1<δ≤3 entrained but thermodynamically rough; δ≤1 potential failure of entrainment) is the same classification the model presents . Net: same diagonalization, same spectral criteria, same thresholds and interpretations.