{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2022:Y7W574ES6BTEEXTMZ3OCNA5OZK","short_pith_number":"pith:Y7W574ES","canonical_record":{"source":{"id":"2210.12951","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2022-10-24T04:54:49Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"1f8f9acace00cad1b3d550e9911124f4e9b9a6054be3d1854a718bbf98f08dd6","abstract_canon_sha256":"60e7d84822a6434199308bf47a5a1e9d6b52f7162fefeb9eb342c30498e865ad"},"schema_version":"1.0"},"canonical_sha256":"c7eddff092f066425e6ccedc2683aecaa94fd1e6a23ba76f3234a485f98e8147","source":{"kind":"arxiv","id":"2210.12951","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2210.12951","created_at":"2026-07-05T05:09:34Z"},{"alias_kind":"arxiv_version","alias_value":"2210.12951v1","created_at":"2026-07-05T05:09:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2210.12951","created_at":"2026-07-05T05:09:34Z"},{"alias_kind":"pith_short_12","alias_value":"Y7W574ES6BTE","created_at":"2026-07-05T05:09:34Z"},{"alias_kind":"pith_short_16","alias_value":"Y7W574ES6BTEEXTM","created_at":"2026-07-05T05:09:34Z"},{"alias_kind":"pith_short_8","alias_value":"Y7W574ES","created_at":"2026-07-05T05:09:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2022:Y7W574ES6BTEEXTMZ3OCNA5OZK","target":"record","payload":{"canonical_record":{"source":{"id":"2210.12951","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2022-10-24T04:54:49Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"1f8f9acace00cad1b3d550e9911124f4e9b9a6054be3d1854a718bbf98f08dd6","abstract_canon_sha256":"60e7d84822a6434199308bf47a5a1e9d6b52f7162fefeb9eb342c30498e865ad"},"schema_version":"1.0"},"canonical_sha256":"c7eddff092f066425e6ccedc2683aecaa94fd1e6a23ba76f3234a485f98e8147","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:09:34.686229Z","signature_b64":"xWLR2aBbBtJtctmAhw9H8VSBnFBkh2N8AvdhwJCYj8mcZtyvmUoSUm16n3fTPSHoRZUSyoUxgSxqWSSvC8RIDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c7eddff092f066425e6ccedc2683aecaa94fd1e6a23ba76f3234a485f98e8147","last_reissued_at":"2026-07-05T05:09:34.685832Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:09:34.685832Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2210.12951","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":"2026-07-05T05:09:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"91PksCOggTPeNa0qeJvut8bXUeE7eVwbsLHynt8vn60wOS7wbBeVYlRdlDpXhQEHnkCtqG2WkW4dUigEq+DADA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-23T18:13:37.070266Z"},"content_sha256":"8c4a85071862aada85860706ae443070ecb3e30fafa7ef58108945d4da5afb7c","schema_version":"1.0","event_id":"sha256:8c4a85071862aada85860706ae443070ecb3e30fafa7ef58108945d4da5afb7c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2022:Y7W574ES6BTEEXTMZ3OCNA5OZK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Distances between fixed-point sets in 2-dimensional Euclidean buildings are realised","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.GR","authors_text":"Anne Thomas, Harris Leung, Jeroen Schillewaert","submitted_at":"2022-10-24T04:54:49Z","abstract_excerpt":"We prove that if two finitely generated groups act on a metrically complete 2-dimensional Euclidean building, then the distance between their fixed-point sets is realised. Our proof uses the geometry of Euclidean buildings, which we view as CAT(0) spaces, and properties of ultrapowers of Euclidean buildings."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2210.12951","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/2210.12951/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":"2026-07-05T05:09:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"j9g4qTZ7iCxMTp2ma994asrfgE3NjTY5+MkwHWNDtUD1JVETb6q+LSY3FCMVU4ctk74rcXNSYm7+QjDwfoWhCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-23T18:13:37.070760Z"},"content_sha256":"6fed629b1b30a9ca4521a8041fce66c907b5487178ba54cced211d14c8e2a9a2","schema_version":"1.0","event_id":"sha256:6fed629b1b30a9ca4521a8041fce66c907b5487178ba54cced211d14c8e2a9a2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Y7W574ES6BTEEXTMZ3OCNA5OZK/bundle.json","state_url":"https://pith.science/pith/Y7W574ES6BTEEXTMZ3OCNA5OZK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Y7W574ES6BTEEXTMZ3OCNA5OZK/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":"2026-08-23T18:13:37Z","links":{"resolver":"https://pith.science/pith/Y7W574ES6BTEEXTMZ3OCNA5OZK","bundle":"https://pith.science/pith/Y7W574ES6BTEEXTMZ3OCNA5OZK/bundle.json","state":"https://pith.science/pith/Y7W574ES6BTEEXTMZ3OCNA5OZK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Y7W574ES6BTEEXTMZ3OCNA5OZK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:Y7W574ES6BTEEXTMZ3OCNA5OZK","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":"60e7d84822a6434199308bf47a5a1e9d6b52f7162fefeb9eb342c30498e865ad","cross_cats_sorted":["math.MG"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2022-10-24T04:54:49Z","title_canon_sha256":"1f8f9acace00cad1b3d550e9911124f4e9b9a6054be3d1854a718bbf98f08dd6"},"schema_version":"1.0","source":{"id":"2210.12951","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2210.12951","created_at":"2026-07-05T05:09:34Z"},{"alias_kind":"arxiv_version","alias_value":"2210.12951v1","created_at":"2026-07-05T05:09:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2210.12951","created_at":"2026-07-05T05:09:34Z"},{"alias_kind":"pith_short_12","alias_value":"Y7W574ES6BTE","created_at":"2026-07-05T05:09:34Z"},{"alias_kind":"pith_short_16","alias_value":"Y7W574ES6BTEEXTM","created_at":"2026-07-05T05:09:34Z"},{"alias_kind":"pith_short_8","alias_value":"Y7W574ES","created_at":"2026-07-05T05:09:34Z"}],"graph_snapshots":[{"event_id":"sha256:6fed629b1b30a9ca4521a8041fce66c907b5487178ba54cced211d14c8e2a9a2","target":"graph","created_at":"2026-07-05T05:09:34Z","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/2210.12951/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove that if two finitely generated groups act on a metrically complete 2-dimensional Euclidean building, then the distance between their fixed-point sets is realised. Our proof uses the geometry of Euclidean buildings, which we view as CAT(0) spaces, and properties of ultrapowers of Euclidean buildings.","authors_text":"Anne Thomas, Harris Leung, Jeroen Schillewaert","cross_cats":["math.MG"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2022-10-24T04:54:49Z","title":"Distances between fixed-point sets in 2-dimensional Euclidean buildings are realised"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2210.12951","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":"sha256:8c4a85071862aada85860706ae443070ecb3e30fafa7ef58108945d4da5afb7c","target":"record","created_at":"2026-07-05T05:09:34Z","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":"60e7d84822a6434199308bf47a5a1e9d6b52f7162fefeb9eb342c30498e865ad","cross_cats_sorted":["math.MG"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2022-10-24T04:54:49Z","title_canon_sha256":"1f8f9acace00cad1b3d550e9911124f4e9b9a6054be3d1854a718bbf98f08dd6"},"schema_version":"1.0","source":{"id":"2210.12951","kind":"arxiv","version":1}},"canonical_sha256":"c7eddff092f066425e6ccedc2683aecaa94fd1e6a23ba76f3234a485f98e8147","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c7eddff092f066425e6ccedc2683aecaa94fd1e6a23ba76f3234a485f98e8147","first_computed_at":"2026-07-05T05:09:34.685832Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:09:34.685832Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xWLR2aBbBtJtctmAhw9H8VSBnFBkh2N8AvdhwJCYj8mcZtyvmUoSUm16n3fTPSHoRZUSyoUxgSxqWSSvC8RIDw==","signature_status":"signed_v1","signed_at":"2026-07-05T05:09:34.686229Z","signed_message":"canonical_sha256_bytes"},"source_id":"2210.12951","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8c4a85071862aada85860706ae443070ecb3e30fafa7ef58108945d4da5afb7c","sha256:6fed629b1b30a9ca4521a8041fce66c907b5487178ba54cced211d14c8e2a9a2"],"state_sha256":"c5afebfc67d16492055dddd24e191a3e88ecc939d74bffc87648e8b482275248"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1JxCKomuucGvvCH3IR4OFZSVdHShIM3DXrIY+3K8gSURmrAU1Q3BKSYqhn7nEmVHRnfaX34ennVDn6JcqNRSBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-23T18:13:37.075325Z","bundle_sha256":"395b5bc6c22675e83272e3f4a4a072e221c0435f26a7d7dc1d4b3433d5b7fb46"}}