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More precisely, if $S$ is constant and \\[ 0\\leqslant S\\leqslant n+\\delta, \\] where $\\delta$ is an explicit constant satisfying $\\delta\\geqslant \\frac{n}{87}$, then either $S\\equiv0$ and $M$ is a totally geodesic sphere, or $S\\equiv n$ and $M$ is a Clifford torus contained in a totally geodesic $\\mathbb S^{n+1}\\subset\\mathbb S^{n+m}$. %W"},"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":"2607.10733","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-07-12T12:34:23Z","cross_cats_sorted":[],"title_canon_sha256":"576293257ae10583da7288161d5b1d9598e6d810d4a87128bda72f706f9ac2c0","abstract_canon_sha256":"c37d2a513308f6e41c7d2b627b75a38416a003e4217ae1fbf3de3114e5a7c7d2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-14T01:21:37.316528Z","signature_b64":"IeRRiCBpkJCjKuqexVNBF5kWB07NC7fBFq1nhuEIQYASHzp3x1+X3j1SqHZUd4bdscg3xGpm3cD1Un0cZjaqCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"50eaa186339a0c8292c6176a1e62cfb52ecca7145722a162da88e062545b01b6","last_reissued_at":"2026-07-14T01:21:37.315699Z","signature_status":"signed_v1","first_computed_at":"2026-07-14T01:21:37.315699Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Chern's Conjecture for Minimal Submanifolds with Flat Normal Bundle in Spheres","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Fagui Li, Jianquan Ge, Yunheng Zhang","submitted_at":"2026-07-12T12:34:23Z","abstract_excerpt":"Let $M^n$ $(n\\geqslant3)$ be a closed minimal submanifold in the unit sphere $\\mathbb S^{n+m}$ $(m\\geqslant2)$ with flat normal bundle, and let $S$ denote the squared norm of its second fundamental form. We prove an explicit second-gap rigidity theorem for $S$. More precisely, if $S$ is constant and \\[ 0\\leqslant S\\leqslant n+\\delta, \\] where $\\delta$ is an explicit constant satisfying $\\delta\\geqslant \\frac{n}{87}$, then either $S\\equiv0$ and $M$ is a totally geodesic sphere, or $S\\equiv n$ and $M$ is a Clifford torus contained in a totally geodesic $\\mathbb S^{n+1}\\subset\\mathbb S^{n+m}$. %W"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.10733","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/2607.10733/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":"2607.10733","created_at":"2026-07-14T01:21:37.316128+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.10733v1","created_at":"2026-07-14T01:21:37.316128+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.10733","created_at":"2026-07-14T01:21:37.316128+00:00"},{"alias_kind":"pith_short_12","alias_value":"KDVKDBRTTIGI","created_at":"2026-07-14T01:21:37.316128+00:00"},{"alias_kind":"pith_short_16","alias_value":"KDVKDBRTTIGIFEWG","created_at":"2026-07-14T01:21:37.316128+00:00"},{"alias_kind":"pith_short_8","alias_value":"KDVKDBRT","created_at":"2026-07-14T01:21:37.316128+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/KDVKDBRTTIGIFEWGC5VB4YWPWU","json":"https://pith.science/pith/KDVKDBRTTIGIFEWGC5VB4YWPWU.json","graph_json":"https://pith.science/api/pith-number/KDVKDBRTTIGIFEWGC5VB4YWPWU/graph.json","events_json":"https://pith.science/api/pith-number/KDVKDBRTTIGIFEWGC5VB4YWPWU/events.json","paper":"https://pith.science/paper/KDVKDBRT"},"agent_actions":{"view_html":"https://pith.science/pith/KDVKDBRTTIGIFEWGC5VB4YWPWU","download_json":"https://pith.science/pith/KDVKDBRTTIGIFEWGC5VB4YWPWU.json","view_paper":"https://pith.science/paper/KDVKDBRT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.10733&json=true","fetch_graph":"https://pith.science/api/pith-number/KDVKDBRTTIGIFEWGC5VB4YWPWU/graph.json","fetch_events":"https://pith.science/api/pith-number/KDVKDBRTTIGIFEWGC5VB4YWPWU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KDVKDBRTTIGIFEWGC5VB4YWPWU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KDVKDBRTTIGIFEWGC5VB4YWPWU/action/storage_attestation","attest_author":"https://pith.science/pith/KDVKDBRTTIGIFEWGC5VB4YWPWU/action/author_attestation","sign_citation":"https://pith.science/pith/KDVKDBRTTIGIFEWGC5VB4YWPWU/action/citation_signature","submit_replication":"https://pith.science/pith/KDVKDBRTTIGIFEWGC5VB4YWPWU/action/replication_record"}},"created_at":"2026-07-14T01:21:37.316128+00:00","updated_at":"2026-07-14T01:21:37.316128+00:00"}