{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:N4PYPPIGCUQQNOQBSICOW3T5OH","short_pith_number":"pith:N4PYPPIG","canonical_record":{"source":{"id":"2603.15864","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-03-16T19:53:44Z","cross_cats_sorted":[],"title_canon_sha256":"7488fae094b3a78cd4ed3410443bbcc461933458b3ea6c919008b96c9bcd998c","abstract_canon_sha256":"9704ed9ce18c177142c978ff99c3770773d70b9475e3f662c24a5c463f4ee037"},"schema_version":"1.0"},"canonical_sha256":"6f1f87bd06152106ba019204eb6e7d71dd244222ff10185d2944ee5e7a998310","source":{"kind":"arxiv","id":"2603.15864","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2603.15864","created_at":"2026-06-11T01:09:33Z"},{"alias_kind":"arxiv_version","alias_value":"2603.15864v3","created_at":"2026-06-11T01:09:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2603.15864","created_at":"2026-06-11T01:09:33Z"},{"alias_kind":"pith_short_12","alias_value":"N4PYPPIGCUQQ","created_at":"2026-06-11T01:09:33Z"},{"alias_kind":"pith_short_16","alias_value":"N4PYPPIGCUQQNOQB","created_at":"2026-06-11T01:09:33Z"},{"alias_kind":"pith_short_8","alias_value":"N4PYPPIG","created_at":"2026-06-11T01:09:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:N4PYPPIGCUQQNOQBSICOW3T5OH","target":"record","payload":{"canonical_record":{"source":{"id":"2603.15864","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-03-16T19:53:44Z","cross_cats_sorted":[],"title_canon_sha256":"7488fae094b3a78cd4ed3410443bbcc461933458b3ea6c919008b96c9bcd998c","abstract_canon_sha256":"9704ed9ce18c177142c978ff99c3770773d70b9475e3f662c24a5c463f4ee037"},"schema_version":"1.0"},"canonical_sha256":"6f1f87bd06152106ba019204eb6e7d71dd244222ff10185d2944ee5e7a998310","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-11T01:09:33.358894Z","signature_b64":"o6ZxrM5zpDCYF6mGGoBCV8mbKB7w901WUUPwS0svp69GekJKfRULnHXfkH7M6RapbDwKxJiQ5CeGZ6bzUmTgCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6f1f87bd06152106ba019204eb6e7d71dd244222ff10185d2944ee5e7a998310","last_reissued_at":"2026-06-11T01:09:33.357973Z","signature_status":"signed_v1","first_computed_at":"2026-06-11T01:09:33.357973Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2603.15864","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-11T01:09:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UPMyBFYiIqhY1GHlNidG40y+X8HrolcnuCyn0hsDZuv60pkYRP+XD44hf/eINa1f1bG/xXe5lIDm42eVDrYWAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T02:21:33.203331Z"},"content_sha256":"3aa2314caba0eb6865188b28690d509c5fda60fc252239fae1127d9837948f1d","schema_version":"1.0","event_id":"sha256:3aa2314caba0eb6865188b28690d509c5fda60fc252239fae1127d9837948f1d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:N4PYPPIGCUQQNOQBSICOW3T5OH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the first-order genus of wreath products and their central extensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Alexei Miasnikov, Denis Osin, Olga Kharlampovich","submitted_at":"2026-03-16T19:53:44Z","abstract_excerpt":"We prove that groups of the form $\\mathbb Z^m {\\,\\rm wr\\,} \\mathbb Z^n$, where $m,n \\in \\mathbb N$, are regularly bi-interpretable with $\\mathbb Z$ and therefore are first-order rigid: every finitely generated group elementarily equivalent to $\\mathbb Z^m {\\,\\rm wr\\,} \\mathbb Z^n$ is isomorphic to $\\mathbb Z^m {\\,\\rm wr\\,} \\mathbb Z^n$. On the other hand, we show that $\\mathbb Z^2 {\\,\\rm wr\\,} \\mathbb Z$ admits $2^{\\aleph_0}$ elementarily equivalent, pairwise non-isomorphic central extensions with finite kernel."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2603.15864","kind":"arxiv","version":3},"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/2603.15864/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":"2026-06-11T01:09:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c1aU2Vhws8Fv0FiI9xhJv0tqnaoRHNOz1DNb36rn/ho35pGO7AgROq7abgv+mtDP0cv/bI1EGh4QkO8C0xgtDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T02:21:33.203700Z"},"content_sha256":"63251c12696f3d062d63d7c5dc066576d6998c7b5c26a8b04323dc37df77407b","schema_version":"1.0","event_id":"sha256:63251c12696f3d062d63d7c5dc066576d6998c7b5c26a8b04323dc37df77407b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/N4PYPPIGCUQQNOQBSICOW3T5OH/bundle.json","state_url":"https://pith.science/pith/N4PYPPIGCUQQNOQBSICOW3T5OH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/N4PYPPIGCUQQNOQBSICOW3T5OH/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":"2026-07-03T02:21:33Z","links":{"resolver":"https://pith.science/pith/N4PYPPIGCUQQNOQBSICOW3T5OH","bundle":"https://pith.science/pith/N4PYPPIGCUQQNOQBSICOW3T5OH/bundle.json","state":"https://pith.science/pith/N4PYPPIGCUQQNOQBSICOW3T5OH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/N4PYPPIGCUQQNOQBSICOW3T5OH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:N4PYPPIGCUQQNOQBSICOW3T5OH","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":"9704ed9ce18c177142c978ff99c3770773d70b9475e3f662c24a5c463f4ee037","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-03-16T19:53:44Z","title_canon_sha256":"7488fae094b3a78cd4ed3410443bbcc461933458b3ea6c919008b96c9bcd998c"},"schema_version":"1.0","source":{"id":"2603.15864","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2603.15864","created_at":"2026-06-11T01:09:33Z"},{"alias_kind":"arxiv_version","alias_value":"2603.15864v3","created_at":"2026-06-11T01:09:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2603.15864","created_at":"2026-06-11T01:09:33Z"},{"alias_kind":"pith_short_12","alias_value":"N4PYPPIGCUQQ","created_at":"2026-06-11T01:09:33Z"},{"alias_kind":"pith_short_16","alias_value":"N4PYPPIGCUQQNOQB","created_at":"2026-06-11T01:09:33Z"},{"alias_kind":"pith_short_8","alias_value":"N4PYPPIG","created_at":"2026-06-11T01:09:33Z"}],"graph_snapshots":[{"event_id":"sha256:63251c12696f3d062d63d7c5dc066576d6998c7b5c26a8b04323dc37df77407b","target":"graph","created_at":"2026-06-11T01:09:33Z","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/2603.15864/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove that groups of the form $\\mathbb Z^m {\\,\\rm wr\\,} \\mathbb Z^n$, where $m,n \\in \\mathbb N$, are regularly bi-interpretable with $\\mathbb Z$ and therefore are first-order rigid: every finitely generated group elementarily equivalent to $\\mathbb Z^m {\\,\\rm wr\\,} \\mathbb Z^n$ is isomorphic to $\\mathbb Z^m {\\,\\rm wr\\,} \\mathbb Z^n$. On the other hand, we show that $\\mathbb Z^2 {\\,\\rm wr\\,} \\mathbb Z$ admits $2^{\\aleph_0}$ elementarily equivalent, pairwise non-isomorphic central extensions with finite kernel.","authors_text":"Alexei Miasnikov, Denis Osin, Olga Kharlampovich","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-03-16T19:53:44Z","title":"On the first-order genus of wreath products and their central extensions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2603.15864","kind":"arxiv","version":3},"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":"sha256:3aa2314caba0eb6865188b28690d509c5fda60fc252239fae1127d9837948f1d","target":"record","created_at":"2026-06-11T01:09:33Z","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":"9704ed9ce18c177142c978ff99c3770773d70b9475e3f662c24a5c463f4ee037","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-03-16T19:53:44Z","title_canon_sha256":"7488fae094b3a78cd4ed3410443bbcc461933458b3ea6c919008b96c9bcd998c"},"schema_version":"1.0","source":{"id":"2603.15864","kind":"arxiv","version":3}},"canonical_sha256":"6f1f87bd06152106ba019204eb6e7d71dd244222ff10185d2944ee5e7a998310","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6f1f87bd06152106ba019204eb6e7d71dd244222ff10185d2944ee5e7a998310","first_computed_at":"2026-06-11T01:09:33.357973Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-11T01:09:33.357973Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"o6ZxrM5zpDCYF6mGGoBCV8mbKB7w901WUUPwS0svp69GekJKfRULnHXfkH7M6RapbDwKxJiQ5CeGZ6bzUmTgCg==","signature_status":"signed_v1","signed_at":"2026-06-11T01:09:33.358894Z","signed_message":"canonical_sha256_bytes"},"source_id":"2603.15864","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3aa2314caba0eb6865188b28690d509c5fda60fc252239fae1127d9837948f1d","sha256:63251c12696f3d062d63d7c5dc066576d6998c7b5c26a8b04323dc37df77407b"],"state_sha256":"fd4b48e6e16ff42cf9144f41e04654f72d788eb82cd379cfabd92d0c7853b781"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RJetvcvdcwkCjSQeC2Lzo0wL5TR5aVI+EK4u4dpOvkLtQEzjNuoKrWCimEzS/9B+fvwpq2TRxqaJtYaMvnqaCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-03T02:21:33.205593Z","bundle_sha256":"41d634594c08a0c6fb67180e4981d190c275c4e96034197f02b93c1d85b8cd3b"}}