Conefield approach to identifying regions without flux surfaces for magnetic fields

The paper states the conefield Converse KAM nonexistence criterion (Theorem 3.1) using the same objects and hypotheses as the candidate solution: α0 = iB iξ Ω ≡ (ξ×B)♭, its pullbacks αt = ϕt*α0, the slope σt = αt(η)/αt(ξ), and the condition that there exist times t± with ±αt±(ξ)>0 and σt+ ≤ σt− implying no invariant torus transverse to ξ through x0. The paper summarizes the argument and procedure without a detailed proof, while the candidate solution gives a clean contradiction proof via the inequality αt(ξ)(σt−s*)>0 built from the torus’ horizontal tangent direction w0=η−s*ξ. This is the standard cone-field reasoning underpinning Theorem 3.1 and matches the paper’s construction and definitions, hence both are correct and essentially the same proof in content, with the model providing the missing detailed steps. See the theorem statement and setup in the paper (slope/order definitions, α0 choice, and Theorem 3.1) .