Strict increase in the number of normally hyperbolic limit tori in 3D polynomial vector fields

The paper proves Theorem 1: if Nh(m) is finite, then Nh(m+1) ≥ Nh(m)+1, by a concrete construction that avoids the model’s purported ‘localization’ obstruction. The authors first isolate the existing Nh(m) tori inside a ball B and find a vertical plane Σ disjoint from B (Lemma 5). They then multiply the vector field by a linear factor vanishing on Σ to raise the degree by one without changing the dynamics on B (time reparametrization), and use a two-parameter perturbation to create a Hopf–Zero equilibrium at the origin outside B. Applying their explicit torus-bifurcation criterion (Theorems 2 and 4), they produce a new normally hyperbolic torus near the Hopf–Zero while Fenichel persistence keeps the original tori intact, yielding the strict +1 step (proof of Theorem 1). See the statement of Theorem 1 and its proof outline and ingredients in the paper’s introduction and Section 4 (Theorem 1 and its setup, torus-bifurcation criterion, the XL,δ construction and Jacobian, parameter tuning, and the conclusion) . The model’s claim that the result is likely open and that bounded-degree polynomials cannot be sharply localized (hence preventing an ‘add-one’ gadget) is moot here because the paper’s method does not rely on spatial localization; it uses time-rescaling by a linear factor and a local Hopf–Zero torus-bifurcation mechanism, thereby bypassing the alleged obstacle.