{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:2X77CWTTKWTA4Z3ZZIIZEXBCAO","short_pith_number":"pith:2X77CWTT","canonical_record":{"source":{"id":"2309.10143","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2023-09-18T20:46:02Z","cross_cats_sorted":["math.PR","math.ST","stat.TH"],"title_canon_sha256":"ab6db3f4139e3e57bc2f28b23c5ed3300e7fa4cf4de8ad0e55cfc7fa87cad534","abstract_canon_sha256":"0b71ded868e51eb39a8b298b31e97a0407957d3f2b3adea567500d85d5ebd089"},"schema_version":"1.0"},"canonical_sha256":"d5fff15a7355a60e6779ca11925c2203813838c52b3a606bb394570a3713d58c","source":{"kind":"arxiv","id":"2309.10143","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2309.10143","created_at":"2026-07-05T06:51:51Z"},{"alias_kind":"arxiv_version","alias_value":"2309.10143v1","created_at":"2026-07-05T06:51:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2309.10143","created_at":"2026-07-05T06:51:51Z"},{"alias_kind":"pith_short_12","alias_value":"2X77CWTTKWTA","created_at":"2026-07-05T06:51:51Z"},{"alias_kind":"pith_short_16","alias_value":"2X77CWTTKWTA4Z3Z","created_at":"2026-07-05T06:51:51Z"},{"alias_kind":"pith_short_8","alias_value":"2X77CWTT","created_at":"2026-07-05T06:51:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:2X77CWTTKWTA4Z3ZZIIZEXBCAO","target":"record","payload":{"canonical_record":{"source":{"id":"2309.10143","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2023-09-18T20:46:02Z","cross_cats_sorted":["math.PR","math.ST","stat.TH"],"title_canon_sha256":"ab6db3f4139e3e57bc2f28b23c5ed3300e7fa4cf4de8ad0e55cfc7fa87cad534","abstract_canon_sha256":"0b71ded868e51eb39a8b298b31e97a0407957d3f2b3adea567500d85d5ebd089"},"schema_version":"1.0"},"canonical_sha256":"d5fff15a7355a60e6779ca11925c2203813838c52b3a606bb394570a3713d58c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:51:51.551656Z","signature_b64":"Ujc4F7+v9ZhXzka9X/1ysT3sK+13fArH5QyOdn6PVXLnoLSe64dxDqCumle+16wJrfrW7a9kGmhyBZdX2CjqAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d5fff15a7355a60e6779ca11925c2203813838c52b3a606bb394570a3713d58c","last_reissued_at":"2026-07-05T06:51:51.551201Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:51:51.551201Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2309.10143","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":"2026-07-05T06:51:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8G2p2IXrhtZyLBKlzyLyWLLKCMq3jhBg/LRtwVqsxP4cZaDRrFt+DfIMyWdHv7XpldBvUjO1kV4klyin6+1TDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T17:41:13.819286Z"},"content_sha256":"789f62f2f2c8c66c4cadbc1a6e3a519871a4d3265513c1ff40a9c573f54b9c13","schema_version":"1.0","event_id":"sha256:789f62f2f2c8c66c4cadbc1a6e3a519871a4d3265513c1ff40a9c573f54b9c13"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:2X77CWTTKWTA4Z3ZZIIZEXBCAO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Positive-Definite Completion Problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR","math.ST","stat.TH"],"primary_cat":"math.FA","authors_text":"Kartik G. Waghmare, Victor M. Panaretos","submitted_at":"2023-09-18T20:46:02Z","abstract_excerpt":"We study the positive-definite completion problem for kernels on a variety of domains and prove results concerning the existence, uniqueness, and characterization of solutions. In particular, we study a special solution called the canonical completion which is the reproducing kernel analogue of the determinant-maximizing completion known to exist for matrices. We establish several results concerning its existence and uniqueness, which include algebraic and variational characterizations. Notably, we prove the existence of a canonical completion for domains which are equivalent to the band conta"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2309.10143","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/2309.10143/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":"2026-07-05T06:51:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UB2xoXOcrfqi/N73m0Qh0iOgRLwmUkw5IRh1Bxoh5qWFyJDZ0oAx/PTgPhrYO9Vo/sS/DJ9Y6NLptkD8CDBOAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T17:41:13.819812Z"},"content_sha256":"1f7718983f8589626f7fb522c01997e070b73e82bb1fbb7ce1ac7f6c65457911","schema_version":"1.0","event_id":"sha256:1f7718983f8589626f7fb522c01997e070b73e82bb1fbb7ce1ac7f6c65457911"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2X77CWTTKWTA4Z3ZZIIZEXBCAO/bundle.json","state_url":"https://pith.science/pith/2X77CWTTKWTA4Z3ZZIIZEXBCAO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2X77CWTTKWTA4Z3ZZIIZEXBCAO/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":"2026-08-08T17:41:13Z","links":{"resolver":"https://pith.science/pith/2X77CWTTKWTA4Z3ZZIIZEXBCAO","bundle":"https://pith.science/pith/2X77CWTTKWTA4Z3ZZIIZEXBCAO/bundle.json","state":"https://pith.science/pith/2X77CWTTKWTA4Z3ZZIIZEXBCAO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2X77CWTTKWTA4Z3ZZIIZEXBCAO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:2X77CWTTKWTA4Z3ZZIIZEXBCAO","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":"0b71ded868e51eb39a8b298b31e97a0407957d3f2b3adea567500d85d5ebd089","cross_cats_sorted":["math.PR","math.ST","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2023-09-18T20:46:02Z","title_canon_sha256":"ab6db3f4139e3e57bc2f28b23c5ed3300e7fa4cf4de8ad0e55cfc7fa87cad534"},"schema_version":"1.0","source":{"id":"2309.10143","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2309.10143","created_at":"2026-07-05T06:51:51Z"},{"alias_kind":"arxiv_version","alias_value":"2309.10143v1","created_at":"2026-07-05T06:51:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2309.10143","created_at":"2026-07-05T06:51:51Z"},{"alias_kind":"pith_short_12","alias_value":"2X77CWTTKWTA","created_at":"2026-07-05T06:51:51Z"},{"alias_kind":"pith_short_16","alias_value":"2X77CWTTKWTA4Z3Z","created_at":"2026-07-05T06:51:51Z"},{"alias_kind":"pith_short_8","alias_value":"2X77CWTT","created_at":"2026-07-05T06:51:51Z"}],"graph_snapshots":[{"event_id":"sha256:1f7718983f8589626f7fb522c01997e070b73e82bb1fbb7ce1ac7f6c65457911","target":"graph","created_at":"2026-07-05T06:51:51Z","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/2309.10143/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the positive-definite completion problem for kernels on a variety of domains and prove results concerning the existence, uniqueness, and characterization of solutions. In particular, we study a special solution called the canonical completion which is the reproducing kernel analogue of the determinant-maximizing completion known to exist for matrices. We establish several results concerning its existence and uniqueness, which include algebraic and variational characterizations. Notably, we prove the existence of a canonical completion for domains which are equivalent to the band conta","authors_text":"Kartik G. Waghmare, Victor M. Panaretos","cross_cats":["math.PR","math.ST","stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2023-09-18T20:46:02Z","title":"The Positive-Definite Completion Problem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2309.10143","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":"sha256:789f62f2f2c8c66c4cadbc1a6e3a519871a4d3265513c1ff40a9c573f54b9c13","target":"record","created_at":"2026-07-05T06:51:51Z","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":"0b71ded868e51eb39a8b298b31e97a0407957d3f2b3adea567500d85d5ebd089","cross_cats_sorted":["math.PR","math.ST","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2023-09-18T20:46:02Z","title_canon_sha256":"ab6db3f4139e3e57bc2f28b23c5ed3300e7fa4cf4de8ad0e55cfc7fa87cad534"},"schema_version":"1.0","source":{"id":"2309.10143","kind":"arxiv","version":1}},"canonical_sha256":"d5fff15a7355a60e6779ca11925c2203813838c52b3a606bb394570a3713d58c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d5fff15a7355a60e6779ca11925c2203813838c52b3a606bb394570a3713d58c","first_computed_at":"2026-07-05T06:51:51.551201Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T06:51:51.551201Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ujc4F7+v9ZhXzka9X/1ysT3sK+13fArH5QyOdn6PVXLnoLSe64dxDqCumle+16wJrfrW7a9kGmhyBZdX2CjqAA==","signature_status":"signed_v1","signed_at":"2026-07-05T06:51:51.551656Z","signed_message":"canonical_sha256_bytes"},"source_id":"2309.10143","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:789f62f2f2c8c66c4cadbc1a6e3a519871a4d3265513c1ff40a9c573f54b9c13","sha256:1f7718983f8589626f7fb522c01997e070b73e82bb1fbb7ce1ac7f6c65457911"],"state_sha256":"99a5da3164d164a3a8af96584d9b33fb60cffc14ff910c4f501a93e6d363116a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nH4bjbUrTNpluk7oKHZTb9plx2J42x40rKQMJc0khxk57erEqKK/0xu4g7ehSGoRCeuMIy2mkO5EZIePJ2ZxAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T17:41:13.824832Z","bundle_sha256":"c14ed7af4c4ac6bff17c9b259ef7f656650f9e78c7dd6c21adf91861087cab28"}}