{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:43OHRBQIUCP3HHRUOIHFBSMAO5","short_pith_number":"pith:43OHRBQI","canonical_record":{"source":{"id":"2009.13647","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2020-09-28T21:48:49Z","cross_cats_sorted":["math.GT","math.MG"],"title_canon_sha256":"0d028edc4b1793913e2641aaae82dcafc737c9aef070b3fa051f62a0321a8b31","abstract_canon_sha256":"169852b7cbce9a13c01cd53b2d892b94d38139855c9c4db12331e239061ef87a"},"schema_version":"1.0"},"canonical_sha256":"e6dc788608a09fb39e34720e50c980775e18ce1ef9cc63d63d10a0088a05a470","source":{"kind":"arxiv","id":"2009.13647","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2009.13647","created_at":"2026-07-05T06:47:50Z"},{"alias_kind":"arxiv_version","alias_value":"2009.13647v1","created_at":"2026-07-05T06:47:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2009.13647","created_at":"2026-07-05T06:47:50Z"},{"alias_kind":"pith_short_12","alias_value":"43OHRBQIUCP3","created_at":"2026-07-05T06:47:50Z"},{"alias_kind":"pith_short_16","alias_value":"43OHRBQIUCP3HHRU","created_at":"2026-07-05T06:47:50Z"},{"alias_kind":"pith_short_8","alias_value":"43OHRBQI","created_at":"2026-07-05T06:47:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:43OHRBQIUCP3HHRUOIHFBSMAO5","target":"record","payload":{"canonical_record":{"source":{"id":"2009.13647","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2020-09-28T21:48:49Z","cross_cats_sorted":["math.GT","math.MG"],"title_canon_sha256":"0d028edc4b1793913e2641aaae82dcafc737c9aef070b3fa051f62a0321a8b31","abstract_canon_sha256":"169852b7cbce9a13c01cd53b2d892b94d38139855c9c4db12331e239061ef87a"},"schema_version":"1.0"},"canonical_sha256":"e6dc788608a09fb39e34720e50c980775e18ce1ef9cc63d63d10a0088a05a470","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:47:50.200799Z","signature_b64":"5fj3CZI9IwyXnvWGB56OHHv5eeKUb2bY43QgzF+lz7BTL5+XrxFXnNU0I8meF3PtF8bMRXbdn0MuCclvrq0/Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e6dc788608a09fb39e34720e50c980775e18ce1ef9cc63d63d10a0088a05a470","last_reissued_at":"2026-07-05T06:47:50.200389Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:47:50.200389Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2009.13647","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":"2026-07-05T06:47:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tLrs0qEzfgWYtXm46jeRgfXPWGwLYAgTSphFxmjJOzCbsfvFiKDOt2Kw8Qf4yy9yaDbVmD58uDN2w2YLJ7MoAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T09:14:38.787765Z"},"content_sha256":"e6d42ddefbb4ede08f8b625912bcf0cc3e83239357abd2504a4e962703b191aa","schema_version":"1.0","event_id":"sha256:e6d42ddefbb4ede08f8b625912bcf0cc3e83239357abd2504a4e962703b191aa"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:43OHRBQIUCP3HHRUOIHFBSMAO5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Stable cubulations, bicombings, and barycenters","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT","math.MG"],"primary_cat":"math.GR","authors_text":"Alessandro Sisto, Matthew G. Durham, Yair N. Minsky","submitted_at":"2020-09-28T21:48:49Z","abstract_excerpt":"We prove that the hierarchical hulls of finite sets of points in mapping class groups and Teichm\\\"uller spaces are stably approximated by a CAT(0) cube complexes, strengthening a result of Behrstock-Hagen-Sisto. As applications, we prove that mapping class groups are semihyperbolic and Teichm\\\"uller spaces are coarsely equivariantly bicombable, and both admit stable coarse barycenters. Our results apply to the broader class of \"colorable\" hierarchically hyperbolic spaces and groups."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2009.13647","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2009.13647/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T06:47:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GkJIFuwCpu7uXMadu7BQoYUSZqe8y9SmdzuWX1l0VSpz6fpg6aiFgp2kyYTbIGjHWPRsYu7F0/5ogVvN7HAADA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T09:14:38.788298Z"},"content_sha256":"8cc4eca91bd854107bed191d0f0f6ff64deaa7bbafbf6bde73977221ca2f2f5a","schema_version":"1.0","event_id":"sha256:8cc4eca91bd854107bed191d0f0f6ff64deaa7bbafbf6bde73977221ca2f2f5a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/43OHRBQIUCP3HHRUOIHFBSMAO5/bundle.json","state_url":"https://pith.science/pith/43OHRBQIUCP3HHRUOIHFBSMAO5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/43OHRBQIUCP3HHRUOIHFBSMAO5/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":"2026-08-12T09:14:38Z","links":{"resolver":"https://pith.science/pith/43OHRBQIUCP3HHRUOIHFBSMAO5","bundle":"https://pith.science/pith/43OHRBQIUCP3HHRUOIHFBSMAO5/bundle.json","state":"https://pith.science/pith/43OHRBQIUCP3HHRUOIHFBSMAO5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/43OHRBQIUCP3HHRUOIHFBSMAO5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:43OHRBQIUCP3HHRUOIHFBSMAO5","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":"169852b7cbce9a13c01cd53b2d892b94d38139855c9c4db12331e239061ef87a","cross_cats_sorted":["math.GT","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2020-09-28T21:48:49Z","title_canon_sha256":"0d028edc4b1793913e2641aaae82dcafc737c9aef070b3fa051f62a0321a8b31"},"schema_version":"1.0","source":{"id":"2009.13647","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2009.13647","created_at":"2026-07-05T06:47:50Z"},{"alias_kind":"arxiv_version","alias_value":"2009.13647v1","created_at":"2026-07-05T06:47:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2009.13647","created_at":"2026-07-05T06:47:50Z"},{"alias_kind":"pith_short_12","alias_value":"43OHRBQIUCP3","created_at":"2026-07-05T06:47:50Z"},{"alias_kind":"pith_short_16","alias_value":"43OHRBQIUCP3HHRU","created_at":"2026-07-05T06:47:50Z"},{"alias_kind":"pith_short_8","alias_value":"43OHRBQI","created_at":"2026-07-05T06:47:50Z"}],"graph_snapshots":[{"event_id":"sha256:8cc4eca91bd854107bed191d0f0f6ff64deaa7bbafbf6bde73977221ca2f2f5a","target":"graph","created_at":"2026-07-05T06:47:50Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2009.13647/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove that the hierarchical hulls of finite sets of points in mapping class groups and Teichm\\\"uller spaces are stably approximated by a CAT(0) cube complexes, strengthening a result of Behrstock-Hagen-Sisto. As applications, we prove that mapping class groups are semihyperbolic and Teichm\\\"uller spaces are coarsely equivariantly bicombable, and both admit stable coarse barycenters. Our results apply to the broader class of \"colorable\" hierarchically hyperbolic spaces and groups.","authors_text":"Alessandro Sisto, Matthew G. Durham, Yair N. Minsky","cross_cats":["math.GT","math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2020-09-28T21:48:49Z","title":"Stable cubulations, bicombings, and barycenters"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2009.13647","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:e6d42ddefbb4ede08f8b625912bcf0cc3e83239357abd2504a4e962703b191aa","target":"record","created_at":"2026-07-05T06:47:50Z","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":"169852b7cbce9a13c01cd53b2d892b94d38139855c9c4db12331e239061ef87a","cross_cats_sorted":["math.GT","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2020-09-28T21:48:49Z","title_canon_sha256":"0d028edc4b1793913e2641aaae82dcafc737c9aef070b3fa051f62a0321a8b31"},"schema_version":"1.0","source":{"id":"2009.13647","kind":"arxiv","version":1}},"canonical_sha256":"e6dc788608a09fb39e34720e50c980775e18ce1ef9cc63d63d10a0088a05a470","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e6dc788608a09fb39e34720e50c980775e18ce1ef9cc63d63d10a0088a05a470","first_computed_at":"2026-07-05T06:47:50.200389Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T06:47:50.200389Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5fj3CZI9IwyXnvWGB56OHHv5eeKUb2bY43QgzF+lz7BTL5+XrxFXnNU0I8meF3PtF8bMRXbdn0MuCclvrq0/Ag==","signature_status":"signed_v1","signed_at":"2026-07-05T06:47:50.200799Z","signed_message":"canonical_sha256_bytes"},"source_id":"2009.13647","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e6d42ddefbb4ede08f8b625912bcf0cc3e83239357abd2504a4e962703b191aa","sha256:8cc4eca91bd854107bed191d0f0f6ff64deaa7bbafbf6bde73977221ca2f2f5a"],"state_sha256":"ee640f61c1f5c671df0232cba889ae80b92a523b3185788caf997283fed78d65"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mJOsjANzrn4XmmctKU36FnG/S6/VRBitSVrb4w+d0QmZ2COe2uNULEFpMpunhcwc5FWqz+6a6D4znnGGlhD4CA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T09:14:38.794140Z","bundle_sha256":"45bfbefb4208c04eab17715fa4abb75ee2ce35786c0681cbbcb17bcbecd6ce03"}}