{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:XNI7WV5MBECKU5WOBWCAXDCUJP","short_pith_number":"pith:XNI7WV5M","canonical_record":{"source":{"id":"1511.02572","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-11-09T05:42:20Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"d594db63c827ccc8b3d5350f3335bfc9ec1ed4c7dd8ed51c63a55a3245e5b304","abstract_canon_sha256":"ad92957ac7ce1027911610080caad0221c451c51a5a73f6bd87fa9f31cb3caa9"},"schema_version":"1.0"},"canonical_sha256":"bb51fb57ac0904aa76ce0d840b8c544bc7eaac359ce92dca33a05eac72cda860","source":{"kind":"arxiv","id":"1511.02572","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.02572","created_at":"2026-05-18T00:06:31Z"},{"alias_kind":"arxiv_version","alias_value":"1511.02572v2","created_at":"2026-05-18T00:06:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.02572","created_at":"2026-05-18T00:06:31Z"},{"alias_kind":"pith_short_12","alias_value":"XNI7WV5MBECK","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_16","alias_value":"XNI7WV5MBECKU5WO","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_8","alias_value":"XNI7WV5M","created_at":"2026-05-18T12:29:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:XNI7WV5MBECKU5WOBWCAXDCUJP","target":"record","payload":{"canonical_record":{"source":{"id":"1511.02572","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-11-09T05:42:20Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"d594db63c827ccc8b3d5350f3335bfc9ec1ed4c7dd8ed51c63a55a3245e5b304","abstract_canon_sha256":"ad92957ac7ce1027911610080caad0221c451c51a5a73f6bd87fa9f31cb3caa9"},"schema_version":"1.0"},"canonical_sha256":"bb51fb57ac0904aa76ce0d840b8c544bc7eaac359ce92dca33a05eac72cda860","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:06:31.217894Z","signature_b64":"NWPSvwmNAwn17zcGBe2M9z8JYvd21lUBPnqhjZ3yShZdk+aHZtFRvnlKIU73te41QvuXKVg0CXHaFBnqPwzuCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bb51fb57ac0904aa76ce0d840b8c544bc7eaac359ce92dca33a05eac72cda860","last_reissued_at":"2026-05-18T00:06:31.217351Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:06:31.217351Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1511.02572","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":"2026-05-18T00:06:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6OE3MNckunJXe4WNNLZBOSJ7daGO46ueICSuiUCD/e7sHAKTyPhWH+keuuTlizo6SoRPqU3lxGkzkzQL0/vkDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T01:14:23.290116Z"},"content_sha256":"b72fd6ad0f811385202f2b625daad153f559fdfb63ec6bc594edabfbc859c48f","schema_version":"1.0","event_id":"sha256:b72fd6ad0f811385202f2b625daad153f559fdfb63ec6bc594edabfbc859c48f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:XNI7WV5MBECKU5WOBWCAXDCUJP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the mean curvature flow of grain boundaries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Lami Kim, Yoshihiro Tonegawa","submitted_at":"2015-11-09T05:42:20Z","abstract_excerpt":"Suppose that $\\Gamma_0\\subset\\mathbb R^{n+1}$ is a closed countably $n$-rectifiable set whose complement $\\mathbb R^{n+1}\\setminus \\Gamma_0$ consists of more than one connected component. Assume that the $n$-dimensional Hausdorff measure of $\\Gamma_0$ is finite or grows at most exponentially near infinity. Under these assumptions, we prove a global-in-time existence of mean curvature flow in the sense of Brakke starting from $\\Gamma_0$. There exists a finite family of open sets which move continuously with respect to the Lebesgue measure, and whose boundaries coincide with the space-time suppo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.02572","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":"2026-05-18T00:06:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qxqILvKBL6/Ue58mylvdBcEPjBiw0n62SabOJFqdpHA9lu277ZBcWz8NvoI/7vZOfgwoHAUbT3zJQMhHmo2qDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T01:14:23.290827Z"},"content_sha256":"1182d6f2720a0ccca1fcfb03bea66408064360afa91c161502fd3779bb7dc9bd","schema_version":"1.0","event_id":"sha256:1182d6f2720a0ccca1fcfb03bea66408064360afa91c161502fd3779bb7dc9bd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XNI7WV5MBECKU5WOBWCAXDCUJP/bundle.json","state_url":"https://pith.science/pith/XNI7WV5MBECKU5WOBWCAXDCUJP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XNI7WV5MBECKU5WOBWCAXDCUJP/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":"2026-06-10T01:14:23Z","links":{"resolver":"https://pith.science/pith/XNI7WV5MBECKU5WOBWCAXDCUJP","bundle":"https://pith.science/pith/XNI7WV5MBECKU5WOBWCAXDCUJP/bundle.json","state":"https://pith.science/pith/XNI7WV5MBECKU5WOBWCAXDCUJP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XNI7WV5MBECKU5WOBWCAXDCUJP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:XNI7WV5MBECKU5WOBWCAXDCUJP","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":"ad92957ac7ce1027911610080caad0221c451c51a5a73f6bd87fa9f31cb3caa9","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-11-09T05:42:20Z","title_canon_sha256":"d594db63c827ccc8b3d5350f3335bfc9ec1ed4c7dd8ed51c63a55a3245e5b304"},"schema_version":"1.0","source":{"id":"1511.02572","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.02572","created_at":"2026-05-18T00:06:31Z"},{"alias_kind":"arxiv_version","alias_value":"1511.02572v2","created_at":"2026-05-18T00:06:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.02572","created_at":"2026-05-18T00:06:31Z"},{"alias_kind":"pith_short_12","alias_value":"XNI7WV5MBECK","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_16","alias_value":"XNI7WV5MBECKU5WO","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_8","alias_value":"XNI7WV5M","created_at":"2026-05-18T12:29:50Z"}],"graph_snapshots":[{"event_id":"sha256:1182d6f2720a0ccca1fcfb03bea66408064360afa91c161502fd3779bb7dc9bd","target":"graph","created_at":"2026-05-18T00:06:31Z","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":"Suppose that $\\Gamma_0\\subset\\mathbb R^{n+1}$ is a closed countably $n$-rectifiable set whose complement $\\mathbb R^{n+1}\\setminus \\Gamma_0$ consists of more than one connected component. Assume that the $n$-dimensional Hausdorff measure of $\\Gamma_0$ is finite or grows at most exponentially near infinity. Under these assumptions, we prove a global-in-time existence of mean curvature flow in the sense of Brakke starting from $\\Gamma_0$. There exists a finite family of open sets which move continuously with respect to the Lebesgue measure, and whose boundaries coincide with the space-time suppo","authors_text":"Lami Kim, Yoshihiro Tonegawa","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-11-09T05:42:20Z","title":"On the mean curvature flow of grain boundaries"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.02572","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:b72fd6ad0f811385202f2b625daad153f559fdfb63ec6bc594edabfbc859c48f","target":"record","created_at":"2026-05-18T00:06:31Z","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":"ad92957ac7ce1027911610080caad0221c451c51a5a73f6bd87fa9f31cb3caa9","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-11-09T05:42:20Z","title_canon_sha256":"d594db63c827ccc8b3d5350f3335bfc9ec1ed4c7dd8ed51c63a55a3245e5b304"},"schema_version":"1.0","source":{"id":"1511.02572","kind":"arxiv","version":2}},"canonical_sha256":"bb51fb57ac0904aa76ce0d840b8c544bc7eaac359ce92dca33a05eac72cda860","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bb51fb57ac0904aa76ce0d840b8c544bc7eaac359ce92dca33a05eac72cda860","first_computed_at":"2026-05-18T00:06:31.217351Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:06:31.217351Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NWPSvwmNAwn17zcGBe2M9z8JYvd21lUBPnqhjZ3yShZdk+aHZtFRvnlKIU73te41QvuXKVg0CXHaFBnqPwzuCg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:06:31.217894Z","signed_message":"canonical_sha256_bytes"},"source_id":"1511.02572","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b72fd6ad0f811385202f2b625daad153f559fdfb63ec6bc594edabfbc859c48f","sha256:1182d6f2720a0ccca1fcfb03bea66408064360afa91c161502fd3779bb7dc9bd"],"state_sha256":"da2d77334359261fd1c5842296defe28d306bdfe519f63b917b65cf04deacf11"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pscr8hYWwVIEYVPjOfqykRjvayRPE1P+aJHKkf3t0KljBs5cEMAqJ+37REpf/yCq4S0AMacFGqjTcluvQ9xFBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T01:14:23.295132Z","bundle_sha256":"8bb887822523759e5773a6e74dcc38b7c3a627a1d9ba65100aafc96db9568a6d"}}