ON THE MINIMAL NUMBER OF CLOSED GEODESICS ON POSITIVELY CURVED FINSLER SPHERES

The paper proves that every Finsler metric on S^n (n ≥ 4) with reversibility λ and flag curvature K satisfying ((2n−3)/(n−1))^2(λ/(λ+1))^2 < K ≤ 1 and λ < (n−1)/(n−2) carries at least n prime closed geodesics (Theorem 1.1). The argument uses: (i) Rademacher’s length and index bounds to obtain i(cm) ≥ floor(((2n−3)m)/(n−1))(n−1) and mean index î(c) > 2n−3 (Lemma 4.1), (ii) the enhanced common index jump theorem to place iterates symmetrically around 2N (equalities (4.6)–(4.9)), (iii) an S^1–equivariant Morse/Fadell–Rabinowitz construction to get n−2 primes in a first window and one more elliptic prime c_{j0} with a critical module concentrated at degree 2N+n−1 (Lemma 4.3), and (iv) a second disjoint window to derive a contradiction unless an additional prime exists, giving at least n total. These steps and degree placements are set out explicitly in the paper’s outline and proofs (see the statement of (1.2) and Theorem 1.1, Lemma 4.1, (4.6)–(4.9), Claim 1 (4.42), and Step 2’s contradiction) . The candidate’s solution follows the same blueprint: the same pinching drives the same index lower bounds; ECIJT supplies the same symmetric index placement; S^1–equivariant Morse/FR index is used to populate degrees; and an elliptic/non-hyperbolicity input plus a second window yields the nth geodesic. The two main discrepancies are that the candidate (a) claims n−1 geodesics already arise in a single “left half-window,” whereas the paper first produces n−2 in G1 and then one more via the elliptic c_{j0} outside that window, and (b) locates the extra elliptic contribution “at the center 2N,” while the paper concentrates it at degree 2N+n−1; nonetheless, these are refinements of placement rather than essential changes of method. With these clarifications, the candidate’s proof aligns with the paper’s and reaches the same conclusion under the paper’s hypotheses .