Periodic Orbits of Non-Degenerate Lacunary Contact Forms on Prequantization Bundles

The paper proves that every non-degenerate lacunary contact form on a suitable prequantization has precisely r_B simple contractible closed Reeb orbits (Theorem 1.6) using positive S^1-equivariant symplectic homology, the computation HC^0_*(M) ≅ ⊕_{m≥1} H^{*−2mc_B+n}(B;Λ), lacunarity ⇒ vanishing differential, positivity of the mean index, and an index-recurrence/CIJT window-counting argument . The candidate’s solution follows the same architecture: identify HC/CH with SH^{S^1,+}, use the prequantization computation, infer perfection from lacunarity, isolate one good iterate per simple orbit via index recurrence, and count to get r_B. Two technical oversimplifications appear in the candidate solution: (i) the claim that the graded rank in any index window of length <2c_B equals r_B is not established in the paper (which instead uses resonance relations and a carefully chosen truncation about d=2s c_B), and (ii) the uniqueness of one iterate per simple orbit in a wide window is asserted but in the paper is achieved via a narrow window and properties (ii)–(iii) of index recurrence. These are patchable and do not affect the conclusion. Overall, the statement and the proof strategy match; the paper’s version is the rigorous, detailed implementation of the model’s outline.