Multiplicity and stability of closed geodesics on positively curved Finsler 4-spheres

The paper proves the stated dichotomy on S^4 under the pinching 25/9·(λ/(1+λ))^2 < K ≤ 1 using (i) Rademacher’s quantitative lower bounds to obtain î(c) > 5 and i(c^m) ≥ 3⌊5m/3⌋ (Lemma 3.1), (ii) the enhanced common index jump theorem to produce a common index window, (iii) S^1-equivariant Morse inequalities with explicit Betti numbers of (ΛS^4/S^1, Λ^0S^4/S^1), and (iv) the mean index identity Σ χ̂(cj)/î(cj) = −2/3, culminating in Theorem 1.1; see , , , and . By contrast, the model’s Step 1 asserts an incorrect conjugate-point multiplicity claim (“each conjugate instant along a geodesic in S^4 has multiplicity at least 3”) to deduce î(c) > 5. This is not generally true and is precisely where the paper uses Rademacher’s results instead (). The remaining steps of the model follow the paper’s structure (ECIJT window and Morse–theoretic counting, mean index identity), but the proof hinges on the flawed Step 1 bound, so as written it has a critical gap even though the final conclusion matches the paper.