ALMOST EXISTENCE FROM THE FERAL PERSPECTIVE AND SOME QUESTIONS

The paper’s Theorem 2 exactly addresses the setup and proves almost-existence using feral curves and an adiabatic-degeneration/measure argument; see the statement of Theorem 2 and the proof outline reducing to local bands and invoking a localized result (Theorem 4) to obtain full-measure sets of energies with periodic orbits . The authors explicitly avoid capacities in this proof, contrasting with classical Hofer–Zehnder arguments . By contrast, the candidate solution’s Step 2 claims a nonstandard ‘suspension/graph’ argument in W×ℂ and a positivity-of-intersections/monotonicity bound max K ≤ ∫ u*Ω. This bound is not justified: positivity/monotonicity apply to J-holomorphic maps, not to general symplectic embeddings like the proposed suspension annulus; no reference or mechanism is given to compare its ℂ-area to the area of a J-holomorphic “barrier.” Hence the capacity upper bound—and the entire capacity-based proof—lacks a correct core argument. Step 1 (capacity blow-up from aperiodic levels) is standard, but without a valid Step 2 the model’s proof does not go through.