Rigidity of saddle loops

The paper proves Theorem B (analytic conjugacy of Poincaré maps implies Difffib(C^2,Δ̄)-equivalence of loop germs) by lifting to Dulac coordinates, using the variation operator to identify conjugate holonomies, and then applying Mattei–Moussu’s path-lifting theorem to build a fibered equivalence and adjust transversals via holonomy (Theorem 3.37 and Theorem 3.15) . The candidate solution constructs a fiber-preserving conjugacy directly by splitting the conjugacy across R and D, transporting it along the base fibration via holomorphic holonomy families T_F(x), taking a Dulac limit to match boundary values, and then verifying that the constructed Φ maps R to R~ and D to D~. This is a valid approach under standard holomorphic dependence and Dulac-limit facts; it differs technically from the paper’s route via the variation operator, but relies on the same path-lifting/holonomy mechanism. Minor gaps in the candidate’s write-up (holomorphic extension across Σ via the x→0 limit) can be resolved by standard results (or by invoking Mattei–Moussu directly), so both are correct.