NON-DEGENERATE BIRKHOFF FUNCTIONS

The paper proves Theorem 2.3 by constructing the stabilized relative span W_∞ from the truncation correspondences V_n, then restricting f to U×W_∞, applying Whitney extension and the implicit function theorem to obtain a Birkhoff function-germ B whose Taylor derivatives generate W_∞, hence KMS-nondegenerate as a germ. These steps are explicitly laid out in the definitions of X-convergent series and Birkhoff function-germs, the existence of Whitney extensions, and the final proof of Theorem 2.3 via Lemmas 4.8–4.9 and the extension/implicit function argument (see the set-up in Definition 2.1 and Definition 2.2, the non-degeneracy aim and statement of Theorem 2.3, and the proof in §4.7) . By contrast, the model’s Steps 2–5 reproduce the standard Whitney-extension/implicit-function route to a Birkhoff germ consistent with the paper, but Step 6 (the KMS-nondegeneracy claim via finite-parameter jet-transversality) is flawed: transversality alone does not ensure that the degeneracy locus is empty at every point without a codimension argument; the model provides no such dimension count, yet claims the span condition holds “at every point.” The paper avoids this pitfall by the W_∞–span method and establishes nondegeneracy at the germ level (the relevant notion in the theorem) without a global pointwise genericity claim .