{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:WWMNZ6R7TDPWP7IFMO22BCYXZG","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":"f0d60f1f0024fd1fdee6e8da8db19be22d379510a2451b56dbc381c64be08c04","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-11-27T12:48:12Z","title_canon_sha256":"68f6fd57a8be552628a1c90e01f338188fa1ef57b98dddb5d85d0a128e947ff1"},"schema_version":"1.0","source":{"id":"1611.08835","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.08835","created_at":"2026-05-17T23:57:21Z"},{"alias_kind":"arxiv_version","alias_value":"1611.08835v4","created_at":"2026-05-17T23:57:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.08835","created_at":"2026-05-17T23:57:21Z"},{"alias_kind":"pith_short_12","alias_value":"WWMNZ6R7TDPW","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"WWMNZ6R7TDPWP7IF","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"WWMNZ6R7","created_at":"2026-05-18T12:30:51Z"}],"graph_snapshots":[{"event_id":"sha256:4669fc65d5da571235426378a23ce4a89c94d4581e04f4c7eadea6fa3bcb4016","target":"graph","created_at":"2026-05-17T23:57:21Z","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 prove the global rigidity of sphere packings on 3-dimensional manifolds. This is a 3-dimensional analogue of the rigidity theorem of Andreev-Thurston and was conjectured by Cooper and Rivin. We also prove a global rigidity result using a combinatorial scalar curvature introduced by Ge and the author.","authors_text":"Xu Xu","cross_cats":["math.DG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-11-27T12:48:12Z","title":"On the global rigidity of sphere packings on 3-dimensional manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.08835","kind":"arxiv","version":4},"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":"sha256:372c561ddd4720cf5ce977e5caf634312064dc5c762f3b8c2c6afac93f4f5b7c","target":"record","created_at":"2026-05-17T23:57:21Z","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":"f0d60f1f0024fd1fdee6e8da8db19be22d379510a2451b56dbc381c64be08c04","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-11-27T12:48:12Z","title_canon_sha256":"68f6fd57a8be552628a1c90e01f338188fa1ef57b98dddb5d85d0a128e947ff1"},"schema_version":"1.0","source":{"id":"1611.08835","kind":"arxiv","version":4}},"canonical_sha256":"b598dcfa3f98df67fd0563b5a08b17c9b8882605f9e181776e404720abb40b17","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b598dcfa3f98df67fd0563b5a08b17c9b8882605f9e181776e404720abb40b17","first_computed_at":"2026-05-17T23:57:21.873163Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:57:21.873163Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vkVrg1XWNij8p9Wlx2ki3MLCOgxyvXD8ei0iucspovwxf9iCFhVL3f6Jh01Q4dZHlESerKfz2tFH9lLkGi9+Dw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:57:21.873675Z","signed_message":"canonical_sha256_bytes"},"source_id":"1611.08835","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:372c561ddd4720cf5ce977e5caf634312064dc5c762f3b8c2c6afac93f4f5b7c","sha256:4669fc65d5da571235426378a23ce4a89c94d4581e04f4c7eadea6fa3bcb4016"],"state_sha256":"def53d1b627263003c91c729ab4bc4328ba33f892782c705624deacf8563c30c"}