{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:PDRKXS5BC7KTTNUBLWHI7A2V4X","short_pith_number":"pith:PDRKXS5B","canonical_record":{"source":{"id":"1212.2855","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-12-12T16:02:52Z","cross_cats_sorted":["math.LO"],"title_canon_sha256":"7c9e5d96a5fbd715ac95b6489d5276513d3b7ad0e63f0849a97030d29a0434ef","abstract_canon_sha256":"7f824394f62390e7646b26daa3114a10ed562c7d2c184eadd4383a61367da548"},"schema_version":"1.0"},"canonical_sha256":"78e2abcba117d539b6815d8e8f8355e5c0067c616fb9ea6d9ebc0b696e9c7eb3","source":{"kind":"arxiv","id":"1212.2855","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.2855","created_at":"2026-05-18T03:10:47Z"},{"alias_kind":"arxiv_version","alias_value":"1212.2855v2","created_at":"2026-05-18T03:10:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.2855","created_at":"2026-05-18T03:10:47Z"},{"alias_kind":"pith_short_12","alias_value":"PDRKXS5BC7KT","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"PDRKXS5BC7KTTNUB","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"PDRKXS5B","created_at":"2026-05-18T12:27:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:PDRKXS5BC7KTTNUBLWHI7A2V4X","target":"record","payload":{"canonical_record":{"source":{"id":"1212.2855","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-12-12T16:02:52Z","cross_cats_sorted":["math.LO"],"title_canon_sha256":"7c9e5d96a5fbd715ac95b6489d5276513d3b7ad0e63f0849a97030d29a0434ef","abstract_canon_sha256":"7f824394f62390e7646b26daa3114a10ed562c7d2c184eadd4383a61367da548"},"schema_version":"1.0"},"canonical_sha256":"78e2abcba117d539b6815d8e8f8355e5c0067c616fb9ea6d9ebc0b696e9c7eb3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:10:47.962307Z","signature_b64":"GpD5mdOIYCA0QXcKuxMlGIgyVV87UwD2GQ/WGGDWTGFNpZR1wQHl2fRulnD7ZA3VFTc9x3aBvnICXVQV3n9TCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"78e2abcba117d539b6815d8e8f8355e5c0067c616fb9ea6d9ebc0b696e9c7eb3","last_reissued_at":"2026-05-18T03:10:47.961604Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:10:47.961604Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1212.2855","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:10:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AMUo4w3vC9SE3tnL618SMRyGBm40t8F/qQHB7TpFo+RVfM3ZY2AgAjRvzXcPL0Xfbvj0PB0sXQITBZif9+Q4Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T10:45:24.792606Z"},"content_sha256":"680919c2a3037d40a2c3aee12a2fbf8ecb03a96615e2ecafcb53599bcb94e128","schema_version":"1.0","event_id":"sha256:680919c2a3037d40a2c3aee12a2fbf8ecb03a96615e2ecafcb53599bcb94e128"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:PDRKXS5BC7KTTNUBLWHI7A2V4X","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Graev ultrametrics and free products of Polish groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.LO"],"primary_cat":"math.GR","authors_text":"Konstantin Slutsky","submitted_at":"2012-12-12T16:02:52Z","abstract_excerpt":"We construct Graev ultrametrics on free products of groups with two-sided invariant ultrametrics and HNN extensions of such groups. We also introduce a notion of a free product of general Polish groups and prove, in particular, that two Polish groups G and H can be embedded into a Polish group T in such a way that the subgroup of T generated by G and H is isomorphic to the free product G*H."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.2855","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"2026-05-18T03:10:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nt22CgNChVVH++nYAt5K20VnByv8PHFiLv4hWUt55uNVZFML4xFNSPcq2ydL+UWP9bjGdM5OZjD1nuz9o4iKBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T10:45:24.792947Z"},"content_sha256":"2650a7d044f62008e5f37d454500a1d468efbcd674a172f253481a929a6cb84b","schema_version":"1.0","event_id":"sha256:2650a7d044f62008e5f37d454500a1d468efbcd674a172f253481a929a6cb84b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PDRKXS5BC7KTTNUBLWHI7A2V4X/bundle.json","state_url":"https://pith.science/pith/PDRKXS5BC7KTTNUBLWHI7A2V4X/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PDRKXS5BC7KTTNUBLWHI7A2V4X/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":"2026-06-08T10:45:24Z","links":{"resolver":"https://pith.science/pith/PDRKXS5BC7KTTNUBLWHI7A2V4X","bundle":"https://pith.science/pith/PDRKXS5BC7KTTNUBLWHI7A2V4X/bundle.json","state":"https://pith.science/pith/PDRKXS5BC7KTTNUBLWHI7A2V4X/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PDRKXS5BC7KTTNUBLWHI7A2V4X/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:PDRKXS5BC7KTTNUBLWHI7A2V4X","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":"7f824394f62390e7646b26daa3114a10ed562c7d2c184eadd4383a61367da548","cross_cats_sorted":["math.LO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-12-12T16:02:52Z","title_canon_sha256":"7c9e5d96a5fbd715ac95b6489d5276513d3b7ad0e63f0849a97030d29a0434ef"},"schema_version":"1.0","source":{"id":"1212.2855","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.2855","created_at":"2026-05-18T03:10:47Z"},{"alias_kind":"arxiv_version","alias_value":"1212.2855v2","created_at":"2026-05-18T03:10:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.2855","created_at":"2026-05-18T03:10:47Z"},{"alias_kind":"pith_short_12","alias_value":"PDRKXS5BC7KT","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"PDRKXS5BC7KTTNUB","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"PDRKXS5B","created_at":"2026-05-18T12:27:18Z"}],"graph_snapshots":[{"event_id":"sha256:2650a7d044f62008e5f37d454500a1d468efbcd674a172f253481a929a6cb84b","target":"graph","created_at":"2026-05-18T03:10:47Z","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":"We construct Graev ultrametrics on free products of groups with two-sided invariant ultrametrics and HNN extensions of such groups. We also introduce a notion of a free product of general Polish groups and prove, in particular, that two Polish groups G and H can be embedded into a Polish group T in such a way that the subgroup of T generated by G and H is isomorphic to the free product G*H.","authors_text":"Konstantin Slutsky","cross_cats":["math.LO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-12-12T16:02:52Z","title":"Graev ultrametrics and free products of Polish groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.2855","kind":"arxiv","version":2},"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":"sha256:680919c2a3037d40a2c3aee12a2fbf8ecb03a96615e2ecafcb53599bcb94e128","target":"record","created_at":"2026-05-18T03:10:47Z","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":"7f824394f62390e7646b26daa3114a10ed562c7d2c184eadd4383a61367da548","cross_cats_sorted":["math.LO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-12-12T16:02:52Z","title_canon_sha256":"7c9e5d96a5fbd715ac95b6489d5276513d3b7ad0e63f0849a97030d29a0434ef"},"schema_version":"1.0","source":{"id":"1212.2855","kind":"arxiv","version":2}},"canonical_sha256":"78e2abcba117d539b6815d8e8f8355e5c0067c616fb9ea6d9ebc0b696e9c7eb3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"78e2abcba117d539b6815d8e8f8355e5c0067c616fb9ea6d9ebc0b696e9c7eb3","first_computed_at":"2026-05-18T03:10:47.961604Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:10:47.961604Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GpD5mdOIYCA0QXcKuxMlGIgyVV87UwD2GQ/WGGDWTGFNpZR1wQHl2fRulnD7ZA3VFTc9x3aBvnICXVQV3n9TCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:10:47.962307Z","signed_message":"canonical_sha256_bytes"},"source_id":"1212.2855","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:680919c2a3037d40a2c3aee12a2fbf8ecb03a96615e2ecafcb53599bcb94e128","sha256:2650a7d044f62008e5f37d454500a1d468efbcd674a172f253481a929a6cb84b"],"state_sha256":"2056cd0a7b86ed5652ad73f116297315fe984fa33e54a1aca28655c96228353c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0CjYJ1wp1+U0ONSdVvZgkZg4o/TuyjAs0COIq9gMQothWK7dhqsHzfqU+/22Miuswq8wDYGwyMbEgXHrkYOxDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T10:45:24.794810Z","bundle_sha256":"1bdc5579f322ea8c03c9ba53b2ac4bf6d66e93ac48605dae730d9cf8329b0200"}}