A NOTE ON THE PERIODIC ORBIT CONJECTURE

Cardona’s note proves that if a non-vanishing vector field on a closed manifold admits a strongly adapted one-form (ι_X dα exact with α(X) > 0), then the periodic orbit conjecture holds. The proof is by contradiction using currents and the Edwards–Millet–Sullivan moving-leaf machinery: from an assumed unbounded-length bad set, one builds tangent 2-chains whose boundaries converge to a positive foliation cycle and shows their dα-flux tends to zero, contradicting the characterization of Eulerisable/strongly-adapted flows (no zero-flux tangent 2-chain sequence can approximate a non-trivial foliation cycle) . The candidate’s argument correctly observes that on each regular level set S_b of the Bernoulli function B, the normalized flow is a geodesic foliation for a suitable auxiliary metric, so Wadsley applies locally. However, it then asserts a uniform bound over all b by appealing to “local C^k bounds” on a cover; this step is unsubstantiated. Wadsley’s local period bounds do not automatically yield a single constant uniform across a continuum of varying hypersurfaces S_b (including near critical levels), nor is the existence of approximating closed orbits on nearby regular levels established. These are precisely the subtle issues the paper resolves via the currents-based contradiction. Hence, the paper’s theorem and proof are correct, while the model’s proof is incomplete.