Unfolding globally resonant homoclinic tangencies.

The paper’s Theorem 6.1 proves that, for one‑parameter perturbations of a globally resonant homoclinic tangency in 2D maps, the saddle‑node and period‑doubling bifurcation values bounding stability intervals of single‑round period‑(k+m) orbits scale like |α|^{2k} when perturbing transversely to the tangency surface, and like |α|^{k}/k when perturbing tangent to that surface but transverse to the λσ=±1 surface. The proof constructs a first‑return map P_k = T0^k ∘ T1, uses an iterate expansion of T0^k (Lemma 7.1), inserts a k‑dependent ansatz x(k)=α^k(1+… ), y(k)=1+ψ_k α^k+…, derives a quadratic for ψ_k whose discriminant is Δ0, and then computes the Jacobian to locate SN and PD thresholds by δ−τ+1=0 and δ+τ+1=0, respectively, obtaining the announced scalings (equations (7.22)–(7.25)) . The candidate solution follows essentially the same structure: it builds P_{k,ε}, eliminates x by the implicit function theorem using the α^k contraction, rescales η=y−1 as η=α^k u, reduces to a quadratic normal form in u with coefficients expressed by the same Taylor data and parity condition, and shows monotone parameter motion of the constant term gives one SN and one PD boundary with the same scalings. Minor differences are not substantive (e.g., the candidate phrases the intermediate step as a 1D reduction by IFT rather than via the explicit ansatz and direct Jacobian calculation). Hence both are correct and substantially the same proof strategy.