A Proof of the Persistence of Anti-integrable States for Three-Dimensional Quadratic Diffeomorphisms

The paper’s Theorem 3.1 rigorously establishes persistence from a hyperbolic AI relation to a compact uniformly hyperbolic invariant set Ae for the 3D map L, with a topological conjugacy Ψe to the shift on ΣΛ, using an implicit function theorem (IFT) framework and a careful invertibility analysis of the sequence-space derivative DL(ξ; e†) (equations (3), (4)–(6), Lemma 3.2, and the commutative diagram (⋆)) . Hyperbolicity of the continued set is then deduced via the sequence-space criterion equivalent to uniform hyperbolicity of the 3D map (Theorem 5.5), after linking the 3D variational equation to DL(ξ; e) (eq. (27) → (9)) . By contrast, the model’s argument contains a wrong step: it asserts that “the product of the two stable eigenvalues is |det(DL)| = |δ|” and that shrinking ∥e − e†∥ forces |δ| to be uniformly small. Neither assertion is supported by the paper’s parameterization (where e = (ϵ, α1, σ1, δ1, a, c) with δ1 = 0 in the main theorem) nor by linear algebra: the product of the two stable singular values equals |det(DL)| divided by the unstable singular value, not |det(DL)| itself, and closeness to e† does not in general control δ in the 3D phase-space Jacobian; the paper explicitly avoids any small-|δ| hypothesis and treats the δ1 ≠ 0 (endomorphism) case separately as outside its scope . Hence the model’s proof is flawed at a critical step even though its overall goal matches the paper’s.