Nonexistence of Invariant Tori Transverse to Foliations: An Application of Converse KAM Theory

The paper’s Theorem II.4 asserts a nonexistence criterion based on K(t)=Ω(v_t,η_t,ξ_t) changing sign while ⟨η_t,ξ_t⟩<0. Its derivation leans on earlier orientation inequalities (3),(5),(6) and the observation that K=0 implies a linear dependence ξ_t=−cη_t+dv_t; however, the crucial step that ⟨η_t,ξ_t⟩<0 forces c>0 at the K-zero is not rigorously justified in the text (it tacitly conflates the inner product in 3D with the angle after projecting along v), and the authors themselves later treat the dot-product filter as a practical heuristic in numerics rather than a proved necessity-sufficiency bridge. Thus, the statement as written appears to be missing a hypothesis (e.g., using the inner product after projection along v, or an orthogonality condition) to make the logic watertight. The candidate model’s proof, on the other hand, assumes there exists a unit tangent n(t) to the foliation that is orthogonal to T_x S for the given inner product; this is generally false. That unjustified orthogonality is used to equate ⟨n(t),ξ_t⟩ with the n-component α(t), which drives the contradiction. Without that assumption, the moving-frame argument does not rule out S. Consequently, both the paper’s presentation (missing technical conditions) and the model’s proof (uses a false orthogonality premise) are incomplete.