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This follows from the expansion H(a) = sinh r(a)/K(a) = σ* cosh σ* + C0 (a - 1/2) + O((a-1/2)^2) with C0 > 0, combined with reductions (E), (F), (G)."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The reduction of φ_a > 0 to H'(a) > 0 under the assumption sinh r(a) > 2K(a) (condition G), whose analytic closure on (1/2,1] is claimed via strict concavity of a transcendental function; if this inequality fails or the concavity argument has a gap, the local positivity may not hold."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Local analytic resolution of the strong Medvedev conjecture: the critical hyperbolic catenoid has Morse index 4 and nullity 2 for a in (1/2, 1/2 + δ0)."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"The critical hyperbolic catenoid has Morse index 4 and nullity 2 for a slightly larger than 1/2."}],"snapshot_sha256":"8b0bf1158e67ec27dbb7782057614cf7d7c737be4739f141c2407e73ea401afd"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $\\Sigma_a\\subset B^3(r(a))\\subset\\mathbb{H}^3$ ($a>1/2$) be the critical hyperbolic catenoid of the Mori family, a free boundary minimal surface in the geodesic ball. The Medvedev conjecture [8] states ind$(\\Sigma_a)=4$ for all $a>1/2$. We study its strong form: ind$(\\Sigma_a)=4$ and nul$(\\Sigma_a)=2$. The nullity condition nul$(\\Sigma_a)=2$ combines the mode-$|k|=1$ result $\\text{nul}_R(\\Sigma_a)|_{|k|=1}=2$ of [10, Cor. 4.4] with vanishing kernel in modes $|k|=0,|k|\\ge2$; the latter, not in [10], is established here for $a\\in(1/2,1/2+\\delta_0)$. The main result is the analytic local reso","authors_text":"Alexander Pigazzini","cross_cats":["math.AP","math.SP"],"headline":"The critical hyperbolic catenoid has Morse index 4 and nullity 2 for a slightly larger than 1/2.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2026-05-13T14:03:12Z","title":"Analytic local resolution of Medvedev's Morse index conjecture for the critical hyperbolic catenoid in $\\mathbb{H}^3$"},"references":{"count":11,"internal_anchors":1,"resolved_work":11,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"E.A. Coddington and N. Levinson,Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955","work_id":"19869b66-ec41-4851-a6c0-26e37e3488b5","year":1955},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"T. H. Colding, W. P. Minicozzi,A course in minimal surfaces, Graduate Studies in Mathematics, 121, AMS, 2011","work_id":"fcb9d2fc-2898-4f34-a811-4c0959bbea3d","year":2011},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"Devyver,Index of the critical catenoid, Geom","work_id":"9b6d8474-acb3-45e8-954d-fd70b7dea3e1","year":2019},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"Eastham,The Spectral Theory of Periodic Differential Equations, Scottish Acad","work_id":"5dfb9610-e328-4b20-bf81-cc707481b61b","year":1973},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"A. Fraser, R. Schoen,Sharp eigenvalue bounds and minimal surfaces in the ball, Invent. 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This follows from the expansion H(a) = sinh r(a)/K(a) = σ* cosh σ* + C0 (a - 1/2) + O((a-1/2)^2) with C0 > 0, combined with reductions (E), (F), (G).","weakest_assumption":"The reduction of φ_a > 0 to H'(a) > 0 under the assumption sinh r(a) > 2K(a) (condition G), whose analytic closure on (1/2,1] is claimed via strict concavity of a transcendental function; if this inequality fails or the concavity argument has a gap, the local positivity may not hold."}},"verdict_id":"84968c77-9112-44c8-b1b0-6cdc8e00013c"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:49e8eee5ed003782924cfd41a5beb72ac9fc3f51629363fc5a2729305af08838","target":"record","created_at":"2026-05-18T02:44:23Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"7ea10ba4e56a679034026d9983a950e13bbfb369cc326886df04a86fa7949bb0","cross_cats_sorted":["math.AP","math.SP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2026-05-13T14:03:12Z","title_canon_sha256":"dd8e22edb5e553707221edc503030a7e8b7f2ba24fac627a95688018d1e931ad"},"schema_version":"1.0","source":{"id":"2605.13562","kind":"arxiv","version":1}},"canonical_sha256":"ab0829ddb952c0e4e529436590fed517dc0a2585e09f427b8ced41f9fcb2cb86","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ab0829ddb952c0e4e529436590fed517dc0a2585e09f427b8ced41f9fcb2cb86","first_computed_at":"2026-05-18T02:44:23.501428Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:44:23.501428Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CV+KrXLDW6EgfaTnMWttkhSVZ3RlwrUwUKsNFYUq7Eh/qIY14UxWLZkNbmaUyifWKG/4DVjKApYFehySIGxqDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:44:23.502015Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.13562","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:49e8eee5ed003782924cfd41a5beb72ac9fc3f51629363fc5a2729305af08838","sha256:dbbfe0d77d5759ffe51be2f2de52fc09354bbe23fb3def269a5ae0ceda010dce"],"state_sha256":"57f43fdf7941dde63a0f774090fbe4f190347381dbfc6e252007c8b767055f92"}