{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:MR6JRAV6XL6TJY35HDDX7SVGBR","short_pith_number":"pith:MR6JRAV6","canonical_record":{"source":{"id":"1211.6448","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-11-27T21:05:18Z","cross_cats_sorted":[],"title_canon_sha256":"29075b8d842d2dee8e6b674aad4c5745f0aa1941cfc912e501805905fa9ea963","abstract_canon_sha256":"20d221a0a5a620b04d54b746ce26fa9e427680377a36dd3ea95c0444c0d255f6"},"schema_version":"1.0"},"canonical_sha256":"647c9882bebafd34e37d38c77fcaa60c59ab7a26eea3b3d89c84c8845ea61314","source":{"kind":"arxiv","id":"1211.6448","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.6448","created_at":"2026-05-18T01:29:59Z"},{"alias_kind":"arxiv_version","alias_value":"1211.6448v1","created_at":"2026-05-18T01:29:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.6448","created_at":"2026-05-18T01:29:59Z"},{"alias_kind":"pith_short_12","alias_value":"MR6JRAV6XL6T","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_16","alias_value":"MR6JRAV6XL6TJY35","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_8","alias_value":"MR6JRAV6","created_at":"2026-05-18T12:27:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:MR6JRAV6XL6TJY35HDDX7SVGBR","target":"record","payload":{"canonical_record":{"source":{"id":"1211.6448","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-11-27T21:05:18Z","cross_cats_sorted":[],"title_canon_sha256":"29075b8d842d2dee8e6b674aad4c5745f0aa1941cfc912e501805905fa9ea963","abstract_canon_sha256":"20d221a0a5a620b04d54b746ce26fa9e427680377a36dd3ea95c0444c0d255f6"},"schema_version":"1.0"},"canonical_sha256":"647c9882bebafd34e37d38c77fcaa60c59ab7a26eea3b3d89c84c8845ea61314","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:29:59.902343Z","signature_b64":"5C9jMW6J35roAHQ5eaWmd96i8YpIa/MNsgsnnXzwiGcFxY6P77GIMDAHWaft5h6SF7XsIWP+N1GcNU/oF4pnAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"647c9882bebafd34e37d38c77fcaa60c59ab7a26eea3b3d89c84c8845ea61314","last_reissued_at":"2026-05-18T01:29:59.901824Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:29:59.901824Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1211.6448","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":"2026-05-18T01:29:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8JhXcPTX0K3jyMNaV7HJcAqVf414NphhAhS/h7qjcfEOXl9VCtj1yc+5x86uZAbiHYIeIvUmoM0xbfqkO0scBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T14:48:57.484219Z"},"content_sha256":"3b38ab5f4f3ce164e720d912db578d24fc58a8fa906973d07fd67660e410ae8b","schema_version":"1.0","event_id":"sha256:3b38ab5f4f3ce164e720d912db578d24fc58a8fa906973d07fd67660e410ae8b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:MR6JRAV6XL6TJY35HDDX7SVGBR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Harnack Estimates for Ricci Flow on a Warped Product","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Hung Tran","submitted_at":"2012-11-27T21:05:18Z","abstract_excerpt":"In this paper, we study the Ricci flow on closed manifolds equipped with warped product metric $(N\\times F,g_{N}+f^2 g_{F})$ with $(F,g_{F})$ Ricci flat. Using the framework of monotone formulas, we derive several estimates for the adapted heat conjugate fundamental solution which include an analog of G. Perelman's differential Harnack inequality."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.6448","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":""},"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":"2026-05-18T01:29:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9HLtMbloJgi2JCbvgX1nSMZ3LRpaWnwMOMk8mIU/PC2z+K2gME9KOlEe6MIF619MOxL4XFW9VHNpLP0Ff0U2Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T14:48:57.484578Z"},"content_sha256":"63b14ac5fdf72c2786895f6100f8f00d57e4304454a2fdbaec166935192b6a79","schema_version":"1.0","event_id":"sha256:63b14ac5fdf72c2786895f6100f8f00d57e4304454a2fdbaec166935192b6a79"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MR6JRAV6XL6TJY35HDDX7SVGBR/bundle.json","state_url":"https://pith.science/pith/MR6JRAV6XL6TJY35HDDX7SVGBR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MR6JRAV6XL6TJY35HDDX7SVGBR/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":"2026-06-08T14:48:57Z","links":{"resolver":"https://pith.science/pith/MR6JRAV6XL6TJY35HDDX7SVGBR","bundle":"https://pith.science/pith/MR6JRAV6XL6TJY35HDDX7SVGBR/bundle.json","state":"https://pith.science/pith/MR6JRAV6XL6TJY35HDDX7SVGBR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MR6JRAV6XL6TJY35HDDX7SVGBR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:MR6JRAV6XL6TJY35HDDX7SVGBR","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":"20d221a0a5a620b04d54b746ce26fa9e427680377a36dd3ea95c0444c0d255f6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-11-27T21:05:18Z","title_canon_sha256":"29075b8d842d2dee8e6b674aad4c5745f0aa1941cfc912e501805905fa9ea963"},"schema_version":"1.0","source":{"id":"1211.6448","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.6448","created_at":"2026-05-18T01:29:59Z"},{"alias_kind":"arxiv_version","alias_value":"1211.6448v1","created_at":"2026-05-18T01:29:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.6448","created_at":"2026-05-18T01:29:59Z"},{"alias_kind":"pith_short_12","alias_value":"MR6JRAV6XL6T","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_16","alias_value":"MR6JRAV6XL6TJY35","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_8","alias_value":"MR6JRAV6","created_at":"2026-05-18T12:27:14Z"}],"graph_snapshots":[{"event_id":"sha256:63b14ac5fdf72c2786895f6100f8f00d57e4304454a2fdbaec166935192b6a79","target":"graph","created_at":"2026-05-18T01:29:59Z","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"},"paper":{"abstract_excerpt":"In this paper, we study the Ricci flow on closed manifolds equipped with warped product metric $(N\\times F,g_{N}+f^2 g_{F})$ with $(F,g_{F})$ Ricci flat. Using the framework of monotone formulas, we derive several estimates for the adapted heat conjugate fundamental solution which include an analog of G. Perelman's differential Harnack inequality.","authors_text":"Hung Tran","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-11-27T21:05:18Z","title":"Harnack Estimates for Ricci Flow on a Warped Product"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.6448","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":"sha256:3b38ab5f4f3ce164e720d912db578d24fc58a8fa906973d07fd67660e410ae8b","target":"record","created_at":"2026-05-18T01:29:59Z","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":"20d221a0a5a620b04d54b746ce26fa9e427680377a36dd3ea95c0444c0d255f6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-11-27T21:05:18Z","title_canon_sha256":"29075b8d842d2dee8e6b674aad4c5745f0aa1941cfc912e501805905fa9ea963"},"schema_version":"1.0","source":{"id":"1211.6448","kind":"arxiv","version":1}},"canonical_sha256":"647c9882bebafd34e37d38c77fcaa60c59ab7a26eea3b3d89c84c8845ea61314","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"647c9882bebafd34e37d38c77fcaa60c59ab7a26eea3b3d89c84c8845ea61314","first_computed_at":"2026-05-18T01:29:59.901824Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:29:59.901824Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5C9jMW6J35roAHQ5eaWmd96i8YpIa/MNsgsnnXzwiGcFxY6P77GIMDAHWaft5h6SF7XsIWP+N1GcNU/oF4pnAw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:29:59.902343Z","signed_message":"canonical_sha256_bytes"},"source_id":"1211.6448","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3b38ab5f4f3ce164e720d912db578d24fc58a8fa906973d07fd67660e410ae8b","sha256:63b14ac5fdf72c2786895f6100f8f00d57e4304454a2fdbaec166935192b6a79"],"state_sha256":"91633dd6b6202cf6fa60085eeaf5b03cd46bd4502f3a2269fac6d60339d8dfa1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ewNFxKkYrJEtbIeS+v6SvQAFgzsAuBQ3tarw9uxmWdbd+nc6vUDhT4U/HxbXTvoA+y45y6Gf644i2S+iPfn3Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T14:48:57.486445Z","bundle_sha256":"9d90639dd01aff52eae848b6c477af784a4941c8151a7b713b3fa5e0e0262d05"}}