Natural levels in return maps of elementary polycycles

The paper proves the balanced–convergent case (Theorem 1.8) via an axiomatic/structural decomposition and a quantitative quasi-analyticity estimate, yielding the dichotomy that ∆ − id is either identically zero or has no small zeros (cf. Theorem 1.8 and the sufficiency-of-axioms scheme) . In the convergent setting it shows that all needed transition maps lie in a class of pointwise-convergent Dulac series and are closed under the compositions arising in monodromy (Proposition 3.5) , and it supplies a quasi-analyticity tool (Theorem 3.7) to force identity if the “difference from id” is smaller than any exponential . The candidate solution, in contrast, argues directly by composing analytic transitions and local Dulac maps, extracting a convergent Dulac-series expansion for ∆Γ − id and then using quasianalyticity/non-oscillation of the Dulac class to obtain the same dichotomy. One notable overclaim in the candidate is that the balanced condition forces the affine part in the logarithmic chart to be exactly the identity (product of hyperbolicity ratios 1 and zero shift). The paper does not assert this and it need not hold in general; nevertheless, the candidate’s conclusion still follows without that claim because any nontrivial leading term (affine or Dulac) enforces eventual sign and hence a zero-free punctured neighborhood. Thus, both reach the correct result, but by different routes, with the model requiring a small fix to remove an unnecessary and generally unjustified cancellation assertion.