{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:UZZ3CHJP4GDTLNPWMQGEKALF45","short_pith_number":"pith:UZZ3CHJP","canonical_record":{"source":{"id":"2606.21816","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2026-06-20T00:41:15Z","cross_cats_sorted":[],"title_canon_sha256":"d6539ff977917daab777d54d3a8899c0053b3247a578868d3b8851e3fe0d4643","abstract_canon_sha256":"c159d4cb156281258ffb96b1585ec529b0d5654392687611cd45811fb4a6ee3b"},"schema_version":"1.0"},"canonical_sha256":"a673b11d2fe18735b5f6640c450165e74a00a77ecbf4662e4553b24f5e859e80","source":{"kind":"arxiv","id":"2606.21816","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.21816","created_at":"2026-06-23T01:13:23Z"},{"alias_kind":"arxiv_version","alias_value":"2606.21816v1","created_at":"2026-06-23T01:13:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.21816","created_at":"2026-06-23T01:13:23Z"},{"alias_kind":"pith_short_12","alias_value":"UZZ3CHJP4GDT","created_at":"2026-06-23T01:13:23Z"},{"alias_kind":"pith_short_16","alias_value":"UZZ3CHJP4GDTLNPW","created_at":"2026-06-23T01:13:23Z"},{"alias_kind":"pith_short_8","alias_value":"UZZ3CHJP","created_at":"2026-06-23T01:13:23Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:UZZ3CHJP4GDTLNPWMQGEKALF45","target":"record","payload":{"canonical_record":{"source":{"id":"2606.21816","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2026-06-20T00:41:15Z","cross_cats_sorted":[],"title_canon_sha256":"d6539ff977917daab777d54d3a8899c0053b3247a578868d3b8851e3fe0d4643","abstract_canon_sha256":"c159d4cb156281258ffb96b1585ec529b0d5654392687611cd45811fb4a6ee3b"},"schema_version":"1.0"},"canonical_sha256":"a673b11d2fe18735b5f6640c450165e74a00a77ecbf4662e4553b24f5e859e80","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-23T01:13:23.553991Z","signature_b64":"DQi1WuuOlhv+qLyOGonKKolisIMC6Tz7Urn9Yuu5+B354IhJbbkLV8DNlF6x+LQQKWgmR74Q6lTLD9pbEEPgDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a673b11d2fe18735b5f6640c450165e74a00a77ecbf4662e4553b24f5e859e80","last_reissued_at":"2026-06-23T01:13:23.553540Z","signature_status":"signed_v1","first_computed_at":"2026-06-23T01:13:23.553540Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.21816","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":"2026-06-23T01:13:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"q6XHsqNKZt5BRmXBGNbjidJARXWmAt1CqVud6tH6yE3zhm0gIprXawT5zs5wNAtq/WF8UXjJW35EPXbdla17Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T17:13:34.544233Z"},"content_sha256":"bcd2300663ff63df06324e8dbe65c33bb0dd2fb2859f4eb9842e1edd319d9258","schema_version":"1.0","event_id":"sha256:bcd2300663ff63df06324e8dbe65c33bb0dd2fb2859f4eb9842e1edd319d9258"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:UZZ3CHJP4GDTLNPWMQGEKALF45","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Bartnik Mass of CMC surfaces under a Spectral non-negativity condition","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Albert Chau, Luke Kuo Han","submitted_at":"2026-06-20T00:41:15Z","abstract_excerpt":"Let \\(g\\) be a smooth Riemannian metric and $H$ a function on $\\mathbb{S}^2$ and let $\\lambda_1(g)$ be the first eigenvalue of the operator $(-\\Delta_g+K_g)$. We prove that the Bartnik mass of the triple $(\\mathbb{S}^2,g,H)$ is bounded above by $\\sqrt{|\\mathbb{S}^2|_g/16\\pi}$ provided either (a) $\\lambda_1(g)=0$ and $H>0$; or (b) $\\lambda_1(g)>0$ and $H\\geq 0$. The eigenvalue condition, in particular, imposes no lower bound on $K_g$ (even under an area constraint) and thereby extends previous results which assume $K_g\\geq 0$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.21816","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/2606.21816/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":"2026-06-23T01:13:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8AYsvGYBcy/wyaVnkTa25aEpTJmg2YY2EQy2xcHF1EKbSCcwcby+3/mcsipH6LnBfRqy30eRMvXqziLJ5QDcAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T17:13:34.544623Z"},"content_sha256":"ab47876f2112b72e9d6c45fd147a5dc3caccf37baba759dd5d790234610a9585","schema_version":"1.0","event_id":"sha256:ab47876f2112b72e9d6c45fd147a5dc3caccf37baba759dd5d790234610a9585"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UZZ3CHJP4GDTLNPWMQGEKALF45/bundle.json","state_url":"https://pith.science/pith/UZZ3CHJP4GDTLNPWMQGEKALF45/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UZZ3CHJP4GDTLNPWMQGEKALF45/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":"2026-06-27T17:13:34Z","links":{"resolver":"https://pith.science/pith/UZZ3CHJP4GDTLNPWMQGEKALF45","bundle":"https://pith.science/pith/UZZ3CHJP4GDTLNPWMQGEKALF45/bundle.json","state":"https://pith.science/pith/UZZ3CHJP4GDTLNPWMQGEKALF45/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UZZ3CHJP4GDTLNPWMQGEKALF45/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:UZZ3CHJP4GDTLNPWMQGEKALF45","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":"c159d4cb156281258ffb96b1585ec529b0d5654392687611cd45811fb4a6ee3b","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2026-06-20T00:41:15Z","title_canon_sha256":"d6539ff977917daab777d54d3a8899c0053b3247a578868d3b8851e3fe0d4643"},"schema_version":"1.0","source":{"id":"2606.21816","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.21816","created_at":"2026-06-23T01:13:23Z"},{"alias_kind":"arxiv_version","alias_value":"2606.21816v1","created_at":"2026-06-23T01:13:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.21816","created_at":"2026-06-23T01:13:23Z"},{"alias_kind":"pith_short_12","alias_value":"UZZ3CHJP4GDT","created_at":"2026-06-23T01:13:23Z"},{"alias_kind":"pith_short_16","alias_value":"UZZ3CHJP4GDTLNPW","created_at":"2026-06-23T01:13:23Z"},{"alias_kind":"pith_short_8","alias_value":"UZZ3CHJP","created_at":"2026-06-23T01:13:23Z"}],"graph_snapshots":[{"event_id":"sha256:ab47876f2112b72e9d6c45fd147a5dc3caccf37baba759dd5d790234610a9585","target":"graph","created_at":"2026-06-23T01:13:23Z","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/2606.21816/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let \\(g\\) be a smooth Riemannian metric and $H$ a function on $\\mathbb{S}^2$ and let $\\lambda_1(g)$ be the first eigenvalue of the operator $(-\\Delta_g+K_g)$. We prove that the Bartnik mass of the triple $(\\mathbb{S}^2,g,H)$ is bounded above by $\\sqrt{|\\mathbb{S}^2|_g/16\\pi}$ provided either (a) $\\lambda_1(g)=0$ and $H>0$; or (b) $\\lambda_1(g)>0$ and $H\\geq 0$. The eigenvalue condition, in particular, imposes no lower bound on $K_g$ (even under an area constraint) and thereby extends previous results which assume $K_g\\geq 0$.","authors_text":"Albert Chau, Luke Kuo Han","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2026-06-20T00:41:15Z","title":"Bartnik Mass of CMC surfaces under a Spectral non-negativity condition"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.21816","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:bcd2300663ff63df06324e8dbe65c33bb0dd2fb2859f4eb9842e1edd319d9258","target":"record","created_at":"2026-06-23T01:13:23Z","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":"c159d4cb156281258ffb96b1585ec529b0d5654392687611cd45811fb4a6ee3b","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2026-06-20T00:41:15Z","title_canon_sha256":"d6539ff977917daab777d54d3a8899c0053b3247a578868d3b8851e3fe0d4643"},"schema_version":"1.0","source":{"id":"2606.21816","kind":"arxiv","version":1}},"canonical_sha256":"a673b11d2fe18735b5f6640c450165e74a00a77ecbf4662e4553b24f5e859e80","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a673b11d2fe18735b5f6640c450165e74a00a77ecbf4662e4553b24f5e859e80","first_computed_at":"2026-06-23T01:13:23.553540Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-23T01:13:23.553540Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"DQi1WuuOlhv+qLyOGonKKolisIMC6Tz7Urn9Yuu5+B354IhJbbkLV8DNlF6x+LQQKWgmR74Q6lTLD9pbEEPgDA==","signature_status":"signed_v1","signed_at":"2026-06-23T01:13:23.553991Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.21816","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bcd2300663ff63df06324e8dbe65c33bb0dd2fb2859f4eb9842e1edd319d9258","sha256:ab47876f2112b72e9d6c45fd147a5dc3caccf37baba759dd5d790234610a9585"],"state_sha256":"64637cd8e787781cb4d4f3145f984e33283671aab25604acbaad37cb47ed755f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZJMqZ1HHIwH8MwG4qPaXkPcHbMKkyAzsCLvB+lAeky6sVSVJQyvSQe7CRyxkNKjVrGpWYv05M9TVae6ZPGH2BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T17:13:34.546540Z","bundle_sha256":"ed54cca788887718e40c584fae14763f4d89317fd5ecb580f720f7b2fbff8fe8"}}