{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:VMECTXNZKLAOJZJJINSZB7WVC7","short_pith_number":"pith:VMECTXNZ","schema_version":"1.0","canonical_sha256":"ab0829ddb952c0e4e529436590fed517dc0a2585e09f427b8ced41f9fcb2cb86","source":{"kind":"arxiv","id":"2605.13562","version":1},"attestation_state":"computed","paper":{"title":"Analytic local resolution of Medvedev's Morse index conjecture for the critical hyperbolic catenoid in $\\mathbb{H}^3$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The critical hyperbolic catenoid has Morse index 4 and nullity 2 for a slightly larger than 1/2.","cross_cats":["math.AP","math.SP"],"primary_cat":"math.DG","authors_text":"Alexander Pigazzini","submitted_at":"2026-05-13T14:03:12Z","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"},"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":true,"formal_links_present":false},"canonical_record":{"source":{"id":"2605.13562","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2026-05-13T14:03:12Z","cross_cats_sorted":["math.AP","math.SP"],"title_canon_sha256":"dd8e22edb5e553707221edc503030a7e8b7f2ba24fac627a95688018d1e931ad","abstract_canon_sha256":"7ea10ba4e56a679034026d9983a950e13bbfb369cc326886df04a86fa7949bb0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:44:23.502015Z","signature_b64":"CV+KrXLDW6EgfaTnMWttkhSVZ3RlwrUwUKsNFYUq7Eh/qIY14UxWLZkNbmaUyifWKG/4DVjKApYFehySIGxqDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ab0829ddb952c0e4e529436590fed517dc0a2585e09f427b8ced41f9fcb2cb86","last_reissued_at":"2026-05-18T02:44:23.501428Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:44:23.501428Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Analytic local resolution of Medvedev's Morse index conjecture for the critical hyperbolic catenoid in $\\mathbb{H}^3$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The critical hyperbolic catenoid has Morse index 4 and nullity 2 for a slightly larger than 1/2.","cross_cats":["math.AP","math.SP"],"primary_cat":"math.DG","authors_text":"Alexander Pigazzini","submitted_at":"2026-05-13T14:03:12Z","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"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"There exists δ0 > 0 such that ind(Σ_a) = 4 and nul(Σ_a) = 2 for all a ∈ (1/2, 1/2 + δ0). 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).","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","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.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","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).","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The critical hyperbolic catenoid has Morse index 4 and nullity 2 for a slightly larger than 1/2.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"8b0bf1158e67ec27dbb7782057614cf7d7c737be4739f141c2407e73ea401afd"},"source":{"id":"2605.13562","kind":"arxiv","version":1},"verdict":{"id":"84968c77-9112-44c8-b1b0-6cdc8e00013c","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:55:46.818283Z","strongest_claim":"There exists δ0 > 0 such that ind(Σ_a) = 4 and nul(Σ_a) = 2 for all a ∈ (1/2, 1/2 + δ0). 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).","one_line_summary":"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).","pipeline_version":"pith-pipeline@v0.9.0","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.","pith_extraction_headline":"The critical hyperbolic catenoid has Morse index 4 and nullity 2 for a slightly larger than 1/2."},"references":{"count":11,"sample":[{"doi":"","year":1955,"title":"E.A. Coddington and N. 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