Explicit linearization of multi-dimensional germs and vector fields through Ecalle’s tree expansions

The paper’s Theorems A–B give the composition operator of the linearizing map as a mould–coarmould contraction Ch = ΣF S•(spectrum) DF(a), and the component tree formulas; the proof uses the separativity of S•, the coseparativity of D•, homogeneity, and the identity (qn+∥F∥ − 1)S_{n◁F} = SF (and its vector-field analogue), plus a vanishing lemma. The candidate solution reconstructs the same operator, repackages the tree expansion as a Taylor-substitution operator, and verifies the conjugacy on coordinates using the same homogeneity and root-grafting identities. Aside from a small typo in the uniqueness denominator (should be “≠ 0,” not “≠ 1”), the arguments align closely and are mathematically equivalent to the paper’s method.