{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:A46PWVJ225PEGAKQSM5BMXIWV2","short_pith_number":"pith:A46PWVJ2","canonical_record":{"source":{"id":"2606.27085","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2026-06-25T14:22:34Z","cross_cats_sorted":[],"title_canon_sha256":"133b202a32669537de4bd093bced6c371d6c8eae17beb21105bc94a8709efc23","abstract_canon_sha256":"250f5b82097146629da5899e411c6b17cc7525ab2235b6e44dfa042295280d11"},"schema_version":"1.0"},"canonical_sha256":"073cfb553ad75e430150933a165d16aeb9992d53f25d9269618010a7fb29a174","source":{"kind":"arxiv","id":"2606.27085","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.27085","created_at":"2026-06-26T01:16:09Z"},{"alias_kind":"arxiv_version","alias_value":"2606.27085v1","created_at":"2026-06-26T01:16:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.27085","created_at":"2026-06-26T01:16:09Z"},{"alias_kind":"pith_short_12","alias_value":"A46PWVJ225PE","created_at":"2026-06-26T01:16:09Z"},{"alias_kind":"pith_short_16","alias_value":"A46PWVJ225PEGAKQ","created_at":"2026-06-26T01:16:09Z"},{"alias_kind":"pith_short_8","alias_value":"A46PWVJ2","created_at":"2026-06-26T01:16:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:A46PWVJ225PEGAKQSM5BMXIWV2","target":"record","payload":{"canonical_record":{"source":{"id":"2606.27085","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2026-06-25T14:22:34Z","cross_cats_sorted":[],"title_canon_sha256":"133b202a32669537de4bd093bced6c371d6c8eae17beb21105bc94a8709efc23","abstract_canon_sha256":"250f5b82097146629da5899e411c6b17cc7525ab2235b6e44dfa042295280d11"},"schema_version":"1.0"},"canonical_sha256":"073cfb553ad75e430150933a165d16aeb9992d53f25d9269618010a7fb29a174","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-26T01:16:09.029862Z","signature_b64":"TJ0VENfJdmuK8M8DkVILJWtKfjGit+pVBF6oBv1neoGu62GPTAT+we/1JDJj92vypow0+Dceo9CT0bj5otKvDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"073cfb553ad75e430150933a165d16aeb9992d53f25d9269618010a7fb29a174","last_reissued_at":"2026-06-26T01:16:09.029531Z","signature_status":"signed_v1","first_computed_at":"2026-06-26T01:16:09.029531Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.27085","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":"2026-06-26T01:16:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QYMJ0Twb2gHxATqb1NxzMbKdCBruROhNgS1Mhhwl0/FfxMqaIsK/sTI6djawWwEIGtUfbfSSis8s7PG6YNrVCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T11:50:15.750146Z"},"content_sha256":"36be88ab644848e33cccfe8590ac73ea7c98d2b764d63b5451709398565104c4","schema_version":"1.0","event_id":"sha256:36be88ab644848e33cccfe8590ac73ea7c98d2b764d63b5451709398565104c4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:A46PWVJ225PEGAKQSM5BMXIWV2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Average Distortion of Commensurators of Hyperbolic Groups","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Mahan Mj, Nir Lazarovich, Suraj Krishna M S","submitted_at":"2026-06-25T14:22:34Z","abstract_excerpt":"We prove that commensurators of a geometrically rigid residually finite hyperbolic group have bounded average distortion."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.27085","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/2606.27085/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":"2026-06-26T01:16:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"M9DLfGtIWRRMLlQPog4w8t3XdwGMB44b70FvNn2+CXDG9aOYtbib/G0QRne2Ox/elhgzvKzwsoS8xVi8/UAoCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T11:50:15.750521Z"},"content_sha256":"67b006f8ab695ea80555a769d8eb0ba3af7b844e1a66282470fb386718a065f1","schema_version":"1.0","event_id":"sha256:67b006f8ab695ea80555a769d8eb0ba3af7b844e1a66282470fb386718a065f1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/A46PWVJ225PEGAKQSM5BMXIWV2/bundle.json","state_url":"https://pith.science/pith/A46PWVJ225PEGAKQSM5BMXIWV2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/A46PWVJ225PEGAKQSM5BMXIWV2/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":"2026-06-27T11:50:15Z","links":{"resolver":"https://pith.science/pith/A46PWVJ225PEGAKQSM5BMXIWV2","bundle":"https://pith.science/pith/A46PWVJ225PEGAKQSM5BMXIWV2/bundle.json","state":"https://pith.science/pith/A46PWVJ225PEGAKQSM5BMXIWV2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/A46PWVJ225PEGAKQSM5BMXIWV2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:A46PWVJ225PEGAKQSM5BMXIWV2","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":"250f5b82097146629da5899e411c6b17cc7525ab2235b6e44dfa042295280d11","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2026-06-25T14:22:34Z","title_canon_sha256":"133b202a32669537de4bd093bced6c371d6c8eae17beb21105bc94a8709efc23"},"schema_version":"1.0","source":{"id":"2606.27085","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.27085","created_at":"2026-06-26T01:16:09Z"},{"alias_kind":"arxiv_version","alias_value":"2606.27085v1","created_at":"2026-06-26T01:16:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.27085","created_at":"2026-06-26T01:16:09Z"},{"alias_kind":"pith_short_12","alias_value":"A46PWVJ225PE","created_at":"2026-06-26T01:16:09Z"},{"alias_kind":"pith_short_16","alias_value":"A46PWVJ225PEGAKQ","created_at":"2026-06-26T01:16:09Z"},{"alias_kind":"pith_short_8","alias_value":"A46PWVJ2","created_at":"2026-06-26T01:16:09Z"}],"graph_snapshots":[{"event_id":"sha256:67b006f8ab695ea80555a769d8eb0ba3af7b844e1a66282470fb386718a065f1","target":"graph","created_at":"2026-06-26T01:16:09Z","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/2606.27085/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove that commensurators of a geometrically rigid residually finite hyperbolic group have bounded average distortion.","authors_text":"Mahan Mj, Nir Lazarovich, Suraj Krishna M S","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2026-06-25T14:22:34Z","title":"Average Distortion of Commensurators of Hyperbolic Groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.27085","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":"sha256:36be88ab644848e33cccfe8590ac73ea7c98d2b764d63b5451709398565104c4","target":"record","created_at":"2026-06-26T01:16:09Z","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":"250f5b82097146629da5899e411c6b17cc7525ab2235b6e44dfa042295280d11","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2026-06-25T14:22:34Z","title_canon_sha256":"133b202a32669537de4bd093bced6c371d6c8eae17beb21105bc94a8709efc23"},"schema_version":"1.0","source":{"id":"2606.27085","kind":"arxiv","version":1}},"canonical_sha256":"073cfb553ad75e430150933a165d16aeb9992d53f25d9269618010a7fb29a174","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"073cfb553ad75e430150933a165d16aeb9992d53f25d9269618010a7fb29a174","first_computed_at":"2026-06-26T01:16:09.029531Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-26T01:16:09.029531Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TJ0VENfJdmuK8M8DkVILJWtKfjGit+pVBF6oBv1neoGu62GPTAT+we/1JDJj92vypow0+Dceo9CT0bj5otKvDw==","signature_status":"signed_v1","signed_at":"2026-06-26T01:16:09.029862Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.27085","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:36be88ab644848e33cccfe8590ac73ea7c98d2b764d63b5451709398565104c4","sha256:67b006f8ab695ea80555a769d8eb0ba3af7b844e1a66282470fb386718a065f1"],"state_sha256":"f43ea40c4e0964ecbb74d1ac8c7ac46826c855b069263698d5cb80c5d6a87c30"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZfI1oA8QUzCM3wAiLFVKoTPGIN2ApPybdpaMVgp070wwvfcA3D3xD0QzG3NivZtEcyT1F6u9WF3jJkR+pMnLBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T11:50:15.752401Z","bundle_sha256":"2ec6c15a4503651877f878005fa68ae588d8a4071ab237e447ed40ec3656e9d6"}}