{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:DFZCGC3ZP5PYXZ6UT7PANU7YFJ","short_pith_number":"pith:DFZCGC3Z","canonical_record":{"source":{"id":"1209.4150","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-09-19T05:14:13Z","cross_cats_sorted":["math.AP","math.DG"],"title_canon_sha256":"fee212d0c7d6ec848461603cc948361318e6acf82238a985efa1ed2d11809f6b","abstract_canon_sha256":"19935c47c4d826c4dc64d54dec89e4cd7bc27b9e03c6b40a6764a0d2af8b610a"},"schema_version":"1.0"},"canonical_sha256":"1972230b797f5f8be7d49fde06d3f82a5ccdcdc93220bfd6538599014f1534be","source":{"kind":"arxiv","id":"1209.4150","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.4150","created_at":"2026-05-18T01:18:04Z"},{"alias_kind":"arxiv_version","alias_value":"1209.4150v3","created_at":"2026-05-18T01:18:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.4150","created_at":"2026-05-18T01:18:04Z"},{"alias_kind":"pith_short_12","alias_value":"DFZCGC3ZP5PY","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_16","alias_value":"DFZCGC3ZP5PYXZ6U","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_8","alias_value":"DFZCGC3Z","created_at":"2026-05-18T12:27:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:DFZCGC3ZP5PYXZ6UT7PANU7YFJ","target":"record","payload":{"canonical_record":{"source":{"id":"1209.4150","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-09-19T05:14:13Z","cross_cats_sorted":["math.AP","math.DG"],"title_canon_sha256":"fee212d0c7d6ec848461603cc948361318e6acf82238a985efa1ed2d11809f6b","abstract_canon_sha256":"19935c47c4d826c4dc64d54dec89e4cd7bc27b9e03c6b40a6764a0d2af8b610a"},"schema_version":"1.0"},"canonical_sha256":"1972230b797f5f8be7d49fde06d3f82a5ccdcdc93220bfd6538599014f1534be","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:18:04.862794Z","signature_b64":"BFSmINImnUyn49edu6Z+tuo7BOic4CL0WszsIvwWLzZ3hBInu1aXKHdpQC9SuaaoWTNjP2mowOnq8Rlf9gOpAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1972230b797f5f8be7d49fde06d3f82a5ccdcdc93220bfd6538599014f1534be","last_reissued_at":"2026-05-18T01:18:04.862190Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:18:04.862190Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1209.4150","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:18:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Zvsr+hYYUnP9ZKhv8XxpZ9R3LOuRDr1hHr2+w1fP62OXrWyoQoWZfdmquee0R256A7UruRuDgDHVsjVYp2EOCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T23:39:11.635112Z"},"content_sha256":"ec4bec4842efd2f95aa469447297c1334a18a75f769c731669911914b53477e8","schema_version":"1.0","event_id":"sha256:ec4bec4842efd2f95aa469447297c1334a18a75f769c731669911914b53477e8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:DFZCGC3ZP5PYXZ6UT7PANU7YFJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A stochastic target approach to Ricci flow on surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.DG"],"primary_cat":"math.PR","authors_text":"Ionel Popescu, Robert W. Neel","submitted_at":"2012-09-19T05:14:13Z","abstract_excerpt":"We develop a stochastic target representation for Ricci flow and normalized Ricci flow on smooth, compact surfaces, analogous to Soner and Touzi's representation of mean curvature flow. We prove a verification/uniqueness theorem, and then consider geometric consequences of this stochastic representation. Based on this stochastic approach, we give a proof that, for surfaces of nonpositive Euler characteristic, the normalized Ricci flow converges to a constant curvature metric exponentially quickly in every $C^k$-norm. In the case of $C^0$ and $C^1$-convergence, we achieve this by coupling two p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.4150","kind":"arxiv","version":3},"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":"2026-05-18T01:18:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IZZACSOKBJE2vUjta4ZGU9zpjuw1ZHw+FoEZhrtFu9Pd2nKX30tISXAbfqwOwI+r9mo6YKdnBHbnHxYswf+GDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T23:39:11.635489Z"},"content_sha256":"acb76c356139cee9d939e335b21cb280ebddb89fe5c291341a47b012e55f5487","schema_version":"1.0","event_id":"sha256:acb76c356139cee9d939e335b21cb280ebddb89fe5c291341a47b012e55f5487"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DFZCGC3ZP5PYXZ6UT7PANU7YFJ/bundle.json","state_url":"https://pith.science/pith/DFZCGC3ZP5PYXZ6UT7PANU7YFJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DFZCGC3ZP5PYXZ6UT7PANU7YFJ/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":"2026-06-20T23:39:11Z","links":{"resolver":"https://pith.science/pith/DFZCGC3ZP5PYXZ6UT7PANU7YFJ","bundle":"https://pith.science/pith/DFZCGC3ZP5PYXZ6UT7PANU7YFJ/bundle.json","state":"https://pith.science/pith/DFZCGC3ZP5PYXZ6UT7PANU7YFJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DFZCGC3ZP5PYXZ6UT7PANU7YFJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:DFZCGC3ZP5PYXZ6UT7PANU7YFJ","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":"19935c47c4d826c4dc64d54dec89e4cd7bc27b9e03c6b40a6764a0d2af8b610a","cross_cats_sorted":["math.AP","math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-09-19T05:14:13Z","title_canon_sha256":"fee212d0c7d6ec848461603cc948361318e6acf82238a985efa1ed2d11809f6b"},"schema_version":"1.0","source":{"id":"1209.4150","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.4150","created_at":"2026-05-18T01:18:04Z"},{"alias_kind":"arxiv_version","alias_value":"1209.4150v3","created_at":"2026-05-18T01:18:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.4150","created_at":"2026-05-18T01:18:04Z"},{"alias_kind":"pith_short_12","alias_value":"DFZCGC3ZP5PY","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_16","alias_value":"DFZCGC3ZP5PYXZ6U","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_8","alias_value":"DFZCGC3Z","created_at":"2026-05-18T12:27:04Z"}],"graph_snapshots":[{"event_id":"sha256:acb76c356139cee9d939e335b21cb280ebddb89fe5c291341a47b012e55f5487","target":"graph","created_at":"2026-05-18T01:18:04Z","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":"We develop a stochastic target representation for Ricci flow and normalized Ricci flow on smooth, compact surfaces, analogous to Soner and Touzi's representation of mean curvature flow. We prove a verification/uniqueness theorem, and then consider geometric consequences of this stochastic representation. Based on this stochastic approach, we give a proof that, for surfaces of nonpositive Euler characteristic, the normalized Ricci flow converges to a constant curvature metric exponentially quickly in every $C^k$-norm. In the case of $C^0$ and $C^1$-convergence, we achieve this by coupling two p","authors_text":"Ionel Popescu, Robert W. Neel","cross_cats":["math.AP","math.DG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-09-19T05:14:13Z","title":"A stochastic target approach to Ricci flow on surfaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.4150","kind":"arxiv","version":3},"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:ec4bec4842efd2f95aa469447297c1334a18a75f769c731669911914b53477e8","target":"record","created_at":"2026-05-18T01:18:04Z","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":"19935c47c4d826c4dc64d54dec89e4cd7bc27b9e03c6b40a6764a0d2af8b610a","cross_cats_sorted":["math.AP","math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-09-19T05:14:13Z","title_canon_sha256":"fee212d0c7d6ec848461603cc948361318e6acf82238a985efa1ed2d11809f6b"},"schema_version":"1.0","source":{"id":"1209.4150","kind":"arxiv","version":3}},"canonical_sha256":"1972230b797f5f8be7d49fde06d3f82a5ccdcdc93220bfd6538599014f1534be","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1972230b797f5f8be7d49fde06d3f82a5ccdcdc93220bfd6538599014f1534be","first_computed_at":"2026-05-18T01:18:04.862190Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:18:04.862190Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BFSmINImnUyn49edu6Z+tuo7BOic4CL0WszsIvwWLzZ3hBInu1aXKHdpQC9SuaaoWTNjP2mowOnq8Rlf9gOpAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:18:04.862794Z","signed_message":"canonical_sha256_bytes"},"source_id":"1209.4150","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ec4bec4842efd2f95aa469447297c1334a18a75f769c731669911914b53477e8","sha256:acb76c356139cee9d939e335b21cb280ebddb89fe5c291341a47b012e55f5487"],"state_sha256":"046cf1a9a6b60aab45eb5fdb0cc8cd6dc4c74469c625a600a1fa45a5dc1ae0ae"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"b3G52d8HUOcHSOnTIZslyBBwgJFs/znE2t48kQrjyIHWR7kPgZDrRElqueudu/1CZuEW+K4m9QAaZ3u1cuD1DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T23:39:11.637475Z","bundle_sha256":"1cd3192c0b250e0c98dc05693ef75ca6cf7a602134dd1a54960a82f9714f3021"}}