{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:ES7ES6WB4DQMQUXUWM2WWP55M2","short_pith_number":"pith:ES7ES6WB","canonical_record":{"source":{"id":"2605.20397","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2026-05-19T18:52:16Z","cross_cats_sorted":["math.LO","math.OA"],"title_canon_sha256":"9a91a33b625d289a8faf3c4096830e608c1eb5b681f0b26ada1f2d77329c90bb","abstract_canon_sha256":"90c576b5c76e6ea87eb6e4878e3157d0cd8c5bcecdf4b27f43241aaf8c3a694c"},"schema_version":"1.0"},"canonical_sha256":"24be497ac1e0e0c852f4b3356b3fbd668a93468da2ed9cef1974f8f510c098c9","source":{"kind":"arxiv","id":"2605.20397","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.20397","created_at":"2026-05-21T01:04:36Z"},{"alias_kind":"arxiv_version","alias_value":"2605.20397v1","created_at":"2026-05-21T01:04:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.20397","created_at":"2026-05-21T01:04:36Z"},{"alias_kind":"pith_short_12","alias_value":"ES7ES6WB4DQM","created_at":"2026-05-21T01:04:36Z"},{"alias_kind":"pith_short_16","alias_value":"ES7ES6WB4DQMQUXU","created_at":"2026-05-21T01:04:36Z"},{"alias_kind":"pith_short_8","alias_value":"ES7ES6WB","created_at":"2026-05-21T01:04:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:ES7ES6WB4DQMQUXUWM2WWP55M2","target":"record","payload":{"canonical_record":{"source":{"id":"2605.20397","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2026-05-19T18:52:16Z","cross_cats_sorted":["math.LO","math.OA"],"title_canon_sha256":"9a91a33b625d289a8faf3c4096830e608c1eb5b681f0b26ada1f2d77329c90bb","abstract_canon_sha256":"90c576b5c76e6ea87eb6e4878e3157d0cd8c5bcecdf4b27f43241aaf8c3a694c"},"schema_version":"1.0"},"canonical_sha256":"24be497ac1e0e0c852f4b3356b3fbd668a93468da2ed9cef1974f8f510c098c9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-21T01:04:36.489752Z","signature_b64":"N8TY7naMb8PhfmeIH5P8lRV36wQ061XEy7VlDXbfjXvq16mts33SXRv+/JpS1+YL7DwLUI3PXeiJXKwj9BsaAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"24be497ac1e0e0c852f4b3356b3fbd668a93468da2ed9cef1974f8f510c098c9","last_reissued_at":"2026-05-21T01:04:36.489114Z","signature_status":"signed_v1","first_computed_at":"2026-05-21T01:04:36.489114Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.20397","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":"2026-05-21T01:04:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pB3IB9CG/9oQEmLmcotPmBS/3+YPFXnDcVVphMStUZ/K5aymSV3OVTx042qHx4CS2LXITWQndbYtGsXeAwecCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T16:43:00.248728Z"},"content_sha256":"1534f61b2dc750a47594f3b5968e8c141bbaed80d9d64dfa347a203dfc5b5cb1","schema_version":"1.0","event_id":"sha256:1534f61b2dc750a47594f3b5968e8c141bbaed80d9d64dfa347a203dfc5b5cb1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:ES7ES6WB4DQMQUXUWM2WWP55M2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An $\\ell^2$ Obstruction for Elementary Embeddings of Hyperbolic Groups","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.LO","math.OA"],"primary_cat":"math.GR","authors_text":"Connor MacMahon","submitted_at":"2026-05-19T18:52:16Z","abstract_excerpt":"The first $\\ell^2$ Betti number of a group is non-decreasing under various embeddings arising from first order logic. Strict inequality is proved for elementary embeddings of non-abelian proper subgroups within torsion free hyperbolic groups using Perin's classification of such inclusions. The monotonicity is further demonstrated for existential embeddings of arbitrary finitely generated groups."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.20397","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/2605.20397/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":"2026-05-21T01:04:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PXAghn1Y4yDx1fQpwMbiaSQZPrAmI9BLMhiH9oK71CAZUEA4CoVio0jMj5yKksDg2CH6YZbtg27ZSPe6OsnKCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T16:43:00.249507Z"},"content_sha256":"5ab70b55c8b21beb9a8cc6831cf28e49739b2604dbf71540b4b423c3b4fe431e","schema_version":"1.0","event_id":"sha256:5ab70b55c8b21beb9a8cc6831cf28e49739b2604dbf71540b4b423c3b4fe431e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ES7ES6WB4DQMQUXUWM2WWP55M2/bundle.json","state_url":"https://pith.science/pith/ES7ES6WB4DQMQUXUWM2WWP55M2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ES7ES6WB4DQMQUXUWM2WWP55M2/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":"2026-05-25T16:43:00Z","links":{"resolver":"https://pith.science/pith/ES7ES6WB4DQMQUXUWM2WWP55M2","bundle":"https://pith.science/pith/ES7ES6WB4DQMQUXUWM2WWP55M2/bundle.json","state":"https://pith.science/pith/ES7ES6WB4DQMQUXUWM2WWP55M2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ES7ES6WB4DQMQUXUWM2WWP55M2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:ES7ES6WB4DQMQUXUWM2WWP55M2","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":"90c576b5c76e6ea87eb6e4878e3157d0cd8c5bcecdf4b27f43241aaf8c3a694c","cross_cats_sorted":["math.LO","math.OA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2026-05-19T18:52:16Z","title_canon_sha256":"9a91a33b625d289a8faf3c4096830e608c1eb5b681f0b26ada1f2d77329c90bb"},"schema_version":"1.0","source":{"id":"2605.20397","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.20397","created_at":"2026-05-21T01:04:36Z"},{"alias_kind":"arxiv_version","alias_value":"2605.20397v1","created_at":"2026-05-21T01:04:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.20397","created_at":"2026-05-21T01:04:36Z"},{"alias_kind":"pith_short_12","alias_value":"ES7ES6WB4DQM","created_at":"2026-05-21T01:04:36Z"},{"alias_kind":"pith_short_16","alias_value":"ES7ES6WB4DQMQUXU","created_at":"2026-05-21T01:04:36Z"},{"alias_kind":"pith_short_8","alias_value":"ES7ES6WB","created_at":"2026-05-21T01:04:36Z"}],"graph_snapshots":[{"event_id":"sha256:5ab70b55c8b21beb9a8cc6831cf28e49739b2604dbf71540b4b423c3b4fe431e","target":"graph","created_at":"2026-05-21T01:04:36Z","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/2605.20397/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The first $\\ell^2$ Betti number of a group is non-decreasing under various embeddings arising from first order logic. Strict inequality is proved for elementary embeddings of non-abelian proper subgroups within torsion free hyperbolic groups using Perin's classification of such inclusions. The monotonicity is further demonstrated for existential embeddings of arbitrary finitely generated groups.","authors_text":"Connor MacMahon","cross_cats":["math.LO","math.OA"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2026-05-19T18:52:16Z","title":"An $\\ell^2$ Obstruction for Elementary Embeddings of Hyperbolic Groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.20397","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":"sha256:1534f61b2dc750a47594f3b5968e8c141bbaed80d9d64dfa347a203dfc5b5cb1","target":"record","created_at":"2026-05-21T01:04:36Z","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":"90c576b5c76e6ea87eb6e4878e3157d0cd8c5bcecdf4b27f43241aaf8c3a694c","cross_cats_sorted":["math.LO","math.OA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2026-05-19T18:52:16Z","title_canon_sha256":"9a91a33b625d289a8faf3c4096830e608c1eb5b681f0b26ada1f2d77329c90bb"},"schema_version":"1.0","source":{"id":"2605.20397","kind":"arxiv","version":1}},"canonical_sha256":"24be497ac1e0e0c852f4b3356b3fbd668a93468da2ed9cef1974f8f510c098c9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"24be497ac1e0e0c852f4b3356b3fbd668a93468da2ed9cef1974f8f510c098c9","first_computed_at":"2026-05-21T01:04:36.489114Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-21T01:04:36.489114Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"N8TY7naMb8PhfmeIH5P8lRV36wQ061XEy7VlDXbfjXvq16mts33SXRv+/JpS1+YL7DwLUI3PXeiJXKwj9BsaAA==","signature_status":"signed_v1","signed_at":"2026-05-21T01:04:36.489752Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.20397","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1534f61b2dc750a47594f3b5968e8c141bbaed80d9d64dfa347a203dfc5b5cb1","sha256:5ab70b55c8b21beb9a8cc6831cf28e49739b2604dbf71540b4b423c3b4fe431e"],"state_sha256":"ada39cf562779a53f390ad1d0299c4653ede70ab797e864c18dfe5ca1c508435"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cohg/CV0FopPyVi/7jaFAY9OXftBGLuOHWdV2BruidJTL1gMP9YsqUymwe1C9Kz7MRVIC7wT5X+F8t9pE/gBDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T16:43:00.253095Z","bundle_sha256":"f3cdcdbece605808e395adad04f5b81030c433f77ce35dd353ec22aa911192a8"}}