{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:3GDWFYHABJZ3O3EE4QQQ6E55DM","short_pith_number":"pith:3GDWFYHA","canonical_record":{"source":{"id":"1510.00581","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-10-02T12:42:34Z","cross_cats_sorted":[],"title_canon_sha256":"68a182760f183e22c59fedcc5932aabb46588308f6e25aeb456dd3723453efb9","abstract_canon_sha256":"bdf847f506fd45358a2df4fa35e21d9679f9a7f9f4697d8e4e78836f22eca66d"},"schema_version":"1.0"},"canonical_sha256":"d98762e0e00a73b76c84e4210f13bd1b0fed83b3440deb55ecee949e1363efeb","source":{"kind":"arxiv","id":"1510.00581","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.00581","created_at":"2026-05-18T01:00:12Z"},{"alias_kind":"arxiv_version","alias_value":"1510.00581v2","created_at":"2026-05-18T01:00:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.00581","created_at":"2026-05-18T01:00:12Z"},{"alias_kind":"pith_short_12","alias_value":"3GDWFYHABJZ3","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_16","alias_value":"3GDWFYHABJZ3O3EE","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_8","alias_value":"3GDWFYHA","created_at":"2026-05-18T12:29:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:3GDWFYHABJZ3O3EE4QQQ6E55DM","target":"record","payload":{"canonical_record":{"source":{"id":"1510.00581","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-10-02T12:42:34Z","cross_cats_sorted":[],"title_canon_sha256":"68a182760f183e22c59fedcc5932aabb46588308f6e25aeb456dd3723453efb9","abstract_canon_sha256":"bdf847f506fd45358a2df4fa35e21d9679f9a7f9f4697d8e4e78836f22eca66d"},"schema_version":"1.0"},"canonical_sha256":"d98762e0e00a73b76c84e4210f13bd1b0fed83b3440deb55ecee949e1363efeb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:00:12.724198Z","signature_b64":"hOi4MrJQuGlsVdNJ2pJawoPkI9012KQ7KyVR+HLFCrsA7iyLoAn1EKuxPVlmn5n+IOTxPUg64JwtcEfhD8GaAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d98762e0e00a73b76c84e4210f13bd1b0fed83b3440deb55ecee949e1363efeb","last_reissued_at":"2026-05-18T01:00:12.723586Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:00:12.723586Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1510.00581","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:00:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7iztrhzrctyskodEkblvAjdLiHD1srGqqm3PFAOPEVzGeQhvy2E5ATgomQFR+arDQP5HQdyFjCXb9zzLHJGQDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T17:05:42.768798Z"},"content_sha256":"5a39b95c6929ca0f90cf1f8b8c2c777eec6da3ed70eefc0a9bbc18276dc06f04","schema_version":"1.0","event_id":"sha256:5a39b95c6929ca0f90cf1f8b8c2c777eec6da3ed70eefc0a9bbc18276dc06f04"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:3GDWFYHABJZ3O3EE4QQQ6E55DM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The size of the singular set of a Type I Ricci flow","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Panagiotis Gianniotis","submitted_at":"2015-10-02T12:42:34Z","abstract_excerpt":"In a singular Type I Ricci flow, we consider a stratification of the set where there is curvature blow-up, according to the number of the Euclidean factors split by the tangent flows. We then show that the strata are characterized roughly in terms of the decay rate of their volume, which in our context plays the role of a dimension estimate."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.00581","kind":"arxiv","version":2},"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":"2026-05-18T01:00:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MU5Kx7PIZoY5D69w65xl/cybe/rbfZ/tWs0yqe0fSj5eTEZEPBl+NHnkfSFxrte2XyyqVpRu+nocqRhmlBfVBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T17:05:42.769131Z"},"content_sha256":"222e299656321984b48aad43c600bca6ac5fa9024fa0092e856bda1dae6fd654","schema_version":"1.0","event_id":"sha256:222e299656321984b48aad43c600bca6ac5fa9024fa0092e856bda1dae6fd654"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3GDWFYHABJZ3O3EE4QQQ6E55DM/bundle.json","state_url":"https://pith.science/pith/3GDWFYHABJZ3O3EE4QQQ6E55DM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3GDWFYHABJZ3O3EE4QQQ6E55DM/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":"2026-06-23T17:05:42Z","links":{"resolver":"https://pith.science/pith/3GDWFYHABJZ3O3EE4QQQ6E55DM","bundle":"https://pith.science/pith/3GDWFYHABJZ3O3EE4QQQ6E55DM/bundle.json","state":"https://pith.science/pith/3GDWFYHABJZ3O3EE4QQQ6E55DM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3GDWFYHABJZ3O3EE4QQQ6E55DM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:3GDWFYHABJZ3O3EE4QQQ6E55DM","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":"bdf847f506fd45358a2df4fa35e21d9679f9a7f9f4697d8e4e78836f22eca66d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-10-02T12:42:34Z","title_canon_sha256":"68a182760f183e22c59fedcc5932aabb46588308f6e25aeb456dd3723453efb9"},"schema_version":"1.0","source":{"id":"1510.00581","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.00581","created_at":"2026-05-18T01:00:12Z"},{"alias_kind":"arxiv_version","alias_value":"1510.00581v2","created_at":"2026-05-18T01:00:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.00581","created_at":"2026-05-18T01:00:12Z"},{"alias_kind":"pith_short_12","alias_value":"3GDWFYHABJZ3","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_16","alias_value":"3GDWFYHABJZ3O3EE","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_8","alias_value":"3GDWFYHA","created_at":"2026-05-18T12:29:02Z"}],"graph_snapshots":[{"event_id":"sha256:222e299656321984b48aad43c600bca6ac5fa9024fa0092e856bda1dae6fd654","target":"graph","created_at":"2026-05-18T01:00:12Z","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 a singular Type I Ricci flow, we consider a stratification of the set where there is curvature blow-up, according to the number of the Euclidean factors split by the tangent flows. We then show that the strata are characterized roughly in terms of the decay rate of their volume, which in our context plays the role of a dimension estimate.","authors_text":"Panagiotis Gianniotis","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-10-02T12:42:34Z","title":"The size of the singular set of a Type I Ricci flow"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.00581","kind":"arxiv","version":2},"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":"sha256:5a39b95c6929ca0f90cf1f8b8c2c777eec6da3ed70eefc0a9bbc18276dc06f04","target":"record","created_at":"2026-05-18T01:00:12Z","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":"bdf847f506fd45358a2df4fa35e21d9679f9a7f9f4697d8e4e78836f22eca66d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-10-02T12:42:34Z","title_canon_sha256":"68a182760f183e22c59fedcc5932aabb46588308f6e25aeb456dd3723453efb9"},"schema_version":"1.0","source":{"id":"1510.00581","kind":"arxiv","version":2}},"canonical_sha256":"d98762e0e00a73b76c84e4210f13bd1b0fed83b3440deb55ecee949e1363efeb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d98762e0e00a73b76c84e4210f13bd1b0fed83b3440deb55ecee949e1363efeb","first_computed_at":"2026-05-18T01:00:12.723586Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:00:12.723586Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hOi4MrJQuGlsVdNJ2pJawoPkI9012KQ7KyVR+HLFCrsA7iyLoAn1EKuxPVlmn5n+IOTxPUg64JwtcEfhD8GaAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:00:12.724198Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.00581","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5a39b95c6929ca0f90cf1f8b8c2c777eec6da3ed70eefc0a9bbc18276dc06f04","sha256:222e299656321984b48aad43c600bca6ac5fa9024fa0092e856bda1dae6fd654"],"state_sha256":"dc4925438f1ee11fa1bd3ab076409d33929be54f67477649693cf24e17607edb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9cU96BJtq8AiHfsRzTFCrH0DmYXak6/OZKXPfwR0ftUFLDVL0XFpYMN5C5QOKDG40mmGbH298z0xbv+H3jeUAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T17:05:42.770923Z","bundle_sha256":"b070781778af7debb6c6619c1b82c6f4e2b788c5243b1e45c6d97c5be4d5ae23"}}