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We prove that $R$ is real-analytic, tends to $\\pi/2$ at both ends, and therefore folds: it has an interior maximum $R_*>\\pi/2$ and is not injective. Hence each $B(\\rho)$ with $\\pi/2<\\rho<R_*$ contains at least two, and only finitely many, mutually non-congruent annuli of the family. At every critical point of $R$ the annulus is degenerate modulo ball-preserving isometries: its Jacobi--Robin kernel contains a rotationally"},"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":"2607.23534","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2026-07-26T08:05:08Z","cross_cats_sorted":[],"title_canon_sha256":"640ae994bd4d87ed98f53938dd828d74132c6b6e9a474f6506b84c09f82a234b","abstract_canon_sha256":"60e621d6b0a0c2c4378a6227309eeab53a3badb2f0e39278ccab8955ae3680e8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-28T01:22:58.063083Z","signature_b64":"Fap6Ih2md8cWc7ilZXRf3l6ZrDx+Ij/5RR0hrWrcpp61w89wiNmIor4Rp2imdbuQ4HFhT8FaFuEZO8uHWuFRCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d376616cb62ba3ebc1dc8a04cd54ca8ca0141b721578a08ec8098fd9d76244f1","last_reissued_at":"2026-07-28T01:22:58.062314Z","signature_status":"signed_v1","first_computed_at":"2026-07-28T01:22:58.062314Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A degenerate free boundary minimal annulus and non-uniqueness in spherical caps beyond the hemisphere","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Alexander Pigazzini","submitted_at":"2026-07-26T08:05:08Z","abstract_excerpt":"Let $\\{\\Sigma_a\\}$, $a\\in(0,1/2)$, be de Oliveira's family of embedded free boundary minimal annuli of revolution in geodesic balls $B(R(a))\\subset\\mathbb{S}^3$, $R(a)>\\pi/2$. We prove that $R$ is real-analytic, tends to $\\pi/2$ at both ends, and therefore folds: it has an interior maximum $R_*>\\pi/2$ and is not injective. Hence each $B(\\rho)$ with $\\pi/2<\\rho<R_*$ contains at least two, and only finitely many, mutually non-congruent annuli of the family. At every critical point of $R$ the annulus is degenerate modulo ball-preserving isometries: its Jacobi--Robin kernel contains a rotationally"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.23534","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.23534/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2607.23534","created_at":"2026-07-28T01:22:58.062744+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.23534v1","created_at":"2026-07-28T01:22:58.062744+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.23534","created_at":"2026-07-28T01:22:58.062744+00:00"},{"alias_kind":"pith_short_12","alias_value":"2N3GC3FWFOR6","created_at":"2026-07-28T01:22:58.062744+00:00"},{"alias_kind":"pith_short_16","alias_value":"2N3GC3FWFOR6XQO4","created_at":"2026-07-28T01:22:58.062744+00:00"},{"alias_kind":"pith_short_8","alias_value":"2N3GC3FW","created_at":"2026-07-28T01:22:58.062744+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/2N3GC3FWFOR6XQO4RICM2VGKRS","json":"https://pith.science/pith/2N3GC3FWFOR6XQO4RICM2VGKRS.json","graph_json":"https://pith.science/api/pith-number/2N3GC3FWFOR6XQO4RICM2VGKRS/graph.json","events_json":"https://pith.science/api/pith-number/2N3GC3FWFOR6XQO4RICM2VGKRS/events.json","paper":"https://pith.science/paper/2N3GC3FW"},"agent_actions":{"view_html":"https://pith.science/pith/2N3GC3FWFOR6XQO4RICM2VGKRS","download_json":"https://pith.science/pith/2N3GC3FWFOR6XQO4RICM2VGKRS.json","view_paper":"https://pith.science/paper/2N3GC3FW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.23534&json=true","fetch_graph":"https://pith.science/api/pith-number/2N3GC3FWFOR6XQO4RICM2VGKRS/graph.json","fetch_events":"https://pith.science/api/pith-number/2N3GC3FWFOR6XQO4RICM2VGKRS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2N3GC3FWFOR6XQO4RICM2VGKRS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2N3GC3FWFOR6XQO4RICM2VGKRS/action/storage_attestation","attest_author":"https://pith.science/pith/2N3GC3FWFOR6XQO4RICM2VGKRS/action/author_attestation","sign_citation":"https://pith.science/pith/2N3GC3FWFOR6XQO4RICM2VGKRS/action/citation_signature","submit_replication":"https://pith.science/pith/2N3GC3FWFOR6XQO4RICM2VGKRS/action/replication_record"}},"created_at":"2026-07-28T01:22:58.062744+00:00","updated_at":"2026-07-28T01:22:58.062744+00:00"}