{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:TH24VJIF6XZK7QUHDH6SFK2QFZ","short_pith_number":"pith:TH24VJIF","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"},"canonical_sha256":"99f5caa505f5f2afc28719fd22ab502e571bfc242d3089253b5d7c6481dceb54","source":{"kind":"arxiv","id":"2607.21336","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.21336","created_at":"2026-07-24T01:24:26Z"},{"alias_kind":"arxiv_version","alias_value":"2607.21336v1","created_at":"2026-07-24T01:24:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.21336","created_at":"2026-07-24T01:24:26Z"},{"alias_kind":"pith_short_12","alias_value":"TH24VJIF6XZK","created_at":"2026-07-24T01:24:26Z"},{"alias_kind":"pith_short_16","alias_value":"TH24VJIF6XZK7QUH","created_at":"2026-07-24T01:24:26Z"},{"alias_kind":"pith_short_8","alias_value":"TH24VJIF","created_at":"2026-07-24T01:24:26Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:TH24VJIF6XZK7QUHDH6SFK2QFZ","target":"record","payload":{"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"},"canonical_sha256":"99f5caa505f5f2afc28719fd22ab502e571bfc242d3089253b5d7c6481dceb54","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"},"source_kind":"arxiv","source_id":"2607.21336","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-24T01:24:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JzPUKzKyIdh2E3wVzHaUU1RDVFI9T10P4QwUTP8AGkGP04JITW030AFbeuUFOiU9OIl0ITtsxpFtNgZsv1KsCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T17:07:59.744027Z"},"content_sha256":"62f719cef139adcd379ca835b8b51c99fe48b51bfdd2a93e823084773700ce9c","schema_version":"1.0","event_id":"sha256:62f719cef139adcd379ca835b8b51c99fe48b51bfdd2a93e823084773700ce9c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:TH24VJIF6XZK7QUHDH6SFK2QFZ","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-24T01:24:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jtmQwsdnpXUYrxDgAH5JqvVR6SSHcodZVNjLh101nQ+N/Fup64FtTqjmwvdjjgK1a/DChuQMAtDj30z/3RNvCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T17:07:59.744539Z"},"content_sha256":"01643fc59fc02c34359e8a67a6cf32593dfb63c955e87bcb9900d1af8e029710","schema_version":"1.0","event_id":"sha256:01643fc59fc02c34359e8a67a6cf32593dfb63c955e87bcb9900d1af8e029710"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TH24VJIF6XZK7QUHDH6SFK2QFZ/bundle.json","state_url":"https://pith.science/pith/TH24VJIF6XZK7QUHDH6SFK2QFZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TH24VJIF6XZK7QUHDH6SFK2QFZ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-07T17:07:59Z","links":{"resolver":"https://pith.science/pith/TH24VJIF6XZK7QUHDH6SFK2QFZ","bundle":"https://pith.science/pith/TH24VJIF6XZK7QUHDH6SFK2QFZ/bundle.json","state":"https://pith.science/pith/TH24VJIF6XZK7QUHDH6SFK2QFZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TH24VJIF6XZK7QUHDH6SFK2QFZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:TH24VJIF6XZK7QUHDH6SFK2QFZ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"93081ce769220b77bb8ed57107b01751bb6c24de9f02eeac6b893ccb200ac0f4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-07-23T14:05:10Z","title_canon_sha256":"07396daa8b9eb2860e571ddc6f9785c1895aa42495dac75c9f0bbcf6fa5d2425"},"schema_version":"1.0","source":{"id":"2607.21336","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.21336","created_at":"2026-07-24T01:24:26Z"},{"alias_kind":"arxiv_version","alias_value":"2607.21336v1","created_at":"2026-07-24T01:24:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.21336","created_at":"2026-07-24T01:24:26Z"},{"alias_kind":"pith_short_12","alias_value":"TH24VJIF6XZK","created_at":"2026-07-24T01:24:26Z"},{"alias_kind":"pith_short_16","alias_value":"TH24VJIF6XZK7QUH","created_at":"2026-07-24T01:24:26Z"},{"alias_kind":"pith_short_8","alias_value":"TH24VJIF","created_at":"2026-07-24T01:24:26Z"}],"graph_snapshots":[{"event_id":"sha256:01643fc59fc02c34359e8a67a6cf32593dfb63c955e87bcb9900d1af8e029710","target":"graph","created_at":"2026-07-24T01:24:26Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2607.21336/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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","authors_text":"Fagui Li, Jianquan Ge, Yunheng Zhang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-07-23T14:05:10Z","title":"Lu's conjecture for minimal surfaces in codimension two"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.21336","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:62f719cef139adcd379ca835b8b51c99fe48b51bfdd2a93e823084773700ce9c","target":"record","created_at":"2026-07-24T01:24:26Z","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":"93081ce769220b77bb8ed57107b01751bb6c24de9f02eeac6b893ccb200ac0f4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-07-23T14:05:10Z","title_canon_sha256":"07396daa8b9eb2860e571ddc6f9785c1895aa42495dac75c9f0bbcf6fa5d2425"},"schema_version":"1.0","source":{"id":"2607.21336","kind":"arxiv","version":1}},"canonical_sha256":"99f5caa505f5f2afc28719fd22ab502e571bfc242d3089253b5d7c6481dceb54","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"99f5caa505f5f2afc28719fd22ab502e571bfc242d3089253b5d7c6481dceb54","first_computed_at":"2026-07-24T01:24:26.446759Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-24T01:24:26.446759Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bkN1i3TPqlP+dEfEL75dySLXaVTBORzqeQFDUgO04zzvQQZuCLL5/qymk1uZEeDYTlDaAM9vMJvJOyUPd91FBA==","signature_status":"signed_v1","signed_at":"2026-07-24T01:24:26.447658Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.21336","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:62f719cef139adcd379ca835b8b51c99fe48b51bfdd2a93e823084773700ce9c","sha256:01643fc59fc02c34359e8a67a6cf32593dfb63c955e87bcb9900d1af8e029710"],"state_sha256":"74432acd954bd0b93b72f20dbf057a9024accf2ff58eb11a59a9736424d2adf9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vXfvszfGxPUAOvQzfY5Agvmfz17222oG1/7UeX3EJ1aFNX5ZtaidMZjhK7FrkcllMm8VjEBv5uC3cg8JO+l+DQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-07T17:07:59.749698Z","bundle_sha256":"d9e6276e97b857d084dacae70e6d4bcb4f20fb60302c2176c871690c3ccf1d58"}}