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We simplify the proof of this statement by the following observation: If for some $N \\in \\mathbb{N}$ all closed geodesics of index $\\le N$ of a non-reversible and bumpy Finsler metric on $S^n$ are geometrically equivalent to the closed geodesic $c$ then there is a covering $c^r$ of minimal index growth, i.e. $${\\rm ind}(c^{rm})=m {\\rm ind}(c^r)-(m-1)(n-1)$$ for all $m \\ge 1$ with ${\\rm ind}\\"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1608.01937","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-08-05T17:00:17Z","cross_cats_sorted":[],"title_canon_sha256":"68013a88642cb59271d7496a20e6bfa1e00311e9159f564525192d121886e566","abstract_canon_sha256":"04a5fed5fd88fc6a140ff1b7901449053fc800362ab81943142be7199f0a23e2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:03:48.662194Z","signature_b64":"mSY3rLF+xSyUs+XAoq+1EWBYB+1bb9OhKJU3ySTpggg1EBzjht6a4fIw0hdJpsGtCd8xGceWeif4QbQT5SryCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2875380fec82565f7f1e7f8122423d0e0edfa53aabb26ac398068c746761e559","last_reissued_at":"2026-05-18T01:03:48.661467Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:03:48.661467Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bumpy metrics on spheres and minimal index growth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Hans-Bert Rademacher","submitted_at":"2016-08-05T17:00:17Z","abstract_excerpt":"The existence of two geometrically distinct closed geodesics on an $n$-dimensional sphere $S^n$ with a non-reversible and bumpy Finsler metric was shown independently by Duan--Long [7] and the author [27]. 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