Contact Homology and the Strong Closing Lemma for Ellipsoids

The paper proves that the contact homology spectral gap vanishes for all ellipsoids (Theorem 6) and deduces the strong closing property via an abstract criterion (Theorem 3.19). The periodic (integer/rational) case is handled by a direct holomorphic-curve analysis of a constrained U-map (Theorem 7), and the general (irrational) case follows from a careful approximation and a quantitative continuity/monotonicity statement for spectral invariants under a controlled comparison of contact forms (Propositions 8–9; Lemma 5.6) . By contrast, the candidate solution relies on (i) a heuristic Morse–Bott argument that counts trivial cylinders with a point constraint without addressing transversality or the precise definition of the constrained U-map in this setting; (ii) an unsupported claim of upper semicontinuity of spectral invariants under arbitrary C^∞ convergence of contact forms, instead of the paper’s multiplicative comparison framework (α_k ≤ α ≤ (1+ε_k)α_k with ε_k s_η(α_k) → 0) that is crucial for passing to the irrational limit (Proposition 8) ; and (iii) a monotonicity-lemma-based lower bound in the closing argument that bypasses the paper’s capacity-based cobordism estimate used in Theorem 3.19 . These gaps are substantial enough that the candidate’s proof is not currently correct as written, despite reaching the same conclusion.