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Thus Lu's second-gap conjecture holds for minimal surfaces in codimension two.\n  Together with the hypersurface result of Peng--Terng and the counterexamples of Li--Zhao in every codimension $m\\geq3$, our theorem completes the codimension picture of Lu's second-gap conjecture for minimal surfaces: it holds precisel"},"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":"2607.21336","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-07-23T14:05:10Z","cross_cats_sorted":[],"title_canon_sha256":"07396daa8b9eb2860e571ddc6f9785c1895aa42495dac75c9f0bbcf6fa5d2425","abstract_canon_sha256":"93081ce769220b77bb8ed57107b01751bb6c24de9f02eeac6b893ccb200ac0f4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-24T01:24:26.447658Z","signature_b64":"bkN1i3TPqlP+dEfEL75dySLXaVTBORzqeQFDUgO04zzvQQZuCLL5/qymk1uZEeDYTlDaAM9vMJvJOyUPd91FBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"99f5caa505f5f2afc28719fd22ab502e571bfc242d3089253b5d7c6481dceb54","last_reissued_at":"2026-07-24T01:24:26.446759Z","signature_status":"signed_v1","first_computed_at":"2026-07-24T01:24:26.446759Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Lu's conjecture for minimal surfaces in codimension two","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":"2026-07-23T14:05:10Z","abstract_excerpt":"Let $M^2\\to\\Sph^4$ be a closed minimal immersion, let $S$ be the squared norm of its second fundamental form, and let $\\lambda_1\\geq\\lambda_2\\geq0$ be the eigenvalues of Lu's fundamental matrix. We prove that if $S+\\lambda_2$ is constant and larger than $2$, then $S+\\lambda_2\\geq3$. Thus Lu's second-gap conjecture holds for minimal surfaces in codimension two.\n  Together with the hypersurface result of Peng--Terng and the counterexamples of Li--Zhao in every codimension $m\\geq3$, our theorem completes the codimension picture of Lu's second-gap conjecture for minimal surfaces: it holds precisel"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.21336","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/2607.21336/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":"2607.21336","created_at":"2026-07-24T01:24:26.447219+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.21336v1","created_at":"2026-07-24T01:24:26.447219+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.21336","created_at":"2026-07-24T01:24:26.447219+00:00"},{"alias_kind":"pith_short_12","alias_value":"TH24VJIF6XZK","created_at":"2026-07-24T01:24:26.447219+00:00"},{"alias_kind":"pith_short_16","alias_value":"TH24VJIF6XZK7QUH","created_at":"2026-07-24T01:24:26.447219+00:00"},{"alias_kind":"pith_short_8","alias_value":"TH24VJIF","created_at":"2026-07-24T01:24:26.447219+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/TH24VJIF6XZK7QUHDH6SFK2QFZ","json":"https://pith.science/pith/TH24VJIF6XZK7QUHDH6SFK2QFZ.json","graph_json":"https://pith.science/api/pith-number/TH24VJIF6XZK7QUHDH6SFK2QFZ/graph.json","events_json":"https://pith.science/api/pith-number/TH24VJIF6XZK7QUHDH6SFK2QFZ/events.json","paper":"https://pith.science/paper/TH24VJIF"},"agent_actions":{"view_html":"https://pith.science/pith/TH24VJIF6XZK7QUHDH6SFK2QFZ","download_json":"https://pith.science/pith/TH24VJIF6XZK7QUHDH6SFK2QFZ.json","view_paper":"https://pith.science/paper/TH24VJIF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.21336&json=true","fetch_graph":"https://pith.science/api/pith-number/TH24VJIF6XZK7QUHDH6SFK2QFZ/graph.json","fetch_events":"https://pith.science/api/pith-number/TH24VJIF6XZK7QUHDH6SFK2QFZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TH24VJIF6XZK7QUHDH6SFK2QFZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TH24VJIF6XZK7QUHDH6SFK2QFZ/action/storage_attestation","attest_author":"https://pith.science/pith/TH24VJIF6XZK7QUHDH6SFK2QFZ/action/author_attestation","sign_citation":"https://pith.science/pith/TH24VJIF6XZK7QUHDH6SFK2QFZ/action/citation_signature","submit_replication":"https://pith.science/pith/TH24VJIF6XZK7QUHDH6SFK2QFZ/action/replication_record"}},"created_at":"2026-07-24T01:24:26.447219+00:00","updated_at":"2026-07-24T01:24:26.447219+00:00"}