{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:ZS6WAI2I6E7V4ZXR36JJ63VDVK","short_pith_number":"pith:ZS6WAI2I","canonical_record":{"source":{"id":"2607.19193","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.GR","submitted_at":"2026-07-21T15:29:14Z","cross_cats_sorted":[],"title_canon_sha256":"01deee6405e848c57f00bf5709478a2c5c36456c3418f2e2bd8bb94eb539b1cc","abstract_canon_sha256":"4c82ba69e2503b024602c1aa3100360a06e95a5d847cbb1d732e2893239a8a0e"},"schema_version":"1.0"},"canonical_sha256":"ccbd602348f13f5e66f1df929f6ea3aa8460d0d96f5c7a3ea49c99055266e938","source":{"kind":"arxiv","id":"2607.19193","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.19193","created_at":"2026-07-22T01:24:13Z"},{"alias_kind":"arxiv_version","alias_value":"2607.19193v1","created_at":"2026-07-22T01:24:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.19193","created_at":"2026-07-22T01:24:13Z"},{"alias_kind":"pith_short_12","alias_value":"ZS6WAI2I6E7V","created_at":"2026-07-22T01:24:13Z"},{"alias_kind":"pith_short_16","alias_value":"ZS6WAI2I6E7V4ZXR","created_at":"2026-07-22T01:24:13Z"},{"alias_kind":"pith_short_8","alias_value":"ZS6WAI2I","created_at":"2026-07-22T01:24:13Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:ZS6WAI2I6E7V4ZXR36JJ63VDVK","target":"record","payload":{"canonical_record":{"source":{"id":"2607.19193","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.GR","submitted_at":"2026-07-21T15:29:14Z","cross_cats_sorted":[],"title_canon_sha256":"01deee6405e848c57f00bf5709478a2c5c36456c3418f2e2bd8bb94eb539b1cc","abstract_canon_sha256":"4c82ba69e2503b024602c1aa3100360a06e95a5d847cbb1d732e2893239a8a0e"},"schema_version":"1.0"},"canonical_sha256":"ccbd602348f13f5e66f1df929f6ea3aa8460d0d96f5c7a3ea49c99055266e938","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-22T01:24:13.597584Z","signature_b64":"TcKIzumuGzrMp058KCHFcVhSiTuu5YSeStMxQO3FBNsqtAmB8GKowh4iBjzDyFClqjYtDWd8G94Y8qev6YGnDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ccbd602348f13f5e66f1df929f6ea3aa8460d0d96f5c7a3ea49c99055266e938","last_reissued_at":"2026-07-22T01:24:13.596745Z","signature_status":"signed_v1","first_computed_at":"2026-07-22T01:24:13.596745Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.19193","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":"2026-07-22T01:24:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iMaMC2ZccayZwYwG10nevqsH1hHqCBpmoLaJGzf/mp9v/+HReh0Gnb02UqpfkUzpqpTii97w8ZIGZyGcmgt3Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T21:18:22.589784Z"},"content_sha256":"af1fe7f7532862f1dca58279a37110e241e17e6449385a127fe467307310e4ec","schema_version":"1.0","event_id":"sha256:af1fe7f7532862f1dca58279a37110e241e17e6449385a127fe467307310e4ec"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:ZS6WAI2I6E7V4ZXR36JJ63VDVK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Hopficity of profinite completions of abelian groups","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"E. Ingrosso, M. Brescia, M. Trombetti","submitted_at":"2026-07-21T15:29:14Z","abstract_excerpt":"We determine exactly when the profinite completion of an arbitrary abelian\n  group is topologically Hopfian. For an abelian group $A$, we prove that \\[\n  \\widehat A \\text{ is topologically Hopfian}\n  \\quad\\Longleftrightarrow\\quad\n  A/pA \\text{ is finite for every prime }p. \\] As a byproduct, we answer Problem 6.30 of the Kourovka Notebook in the negative: for pairwise distinct odd primes $q_i$, the group $\\bigoplus_{i\\geq1}\\Z[1/q_i]$ is residually finite and Hopfian, whereas its profinite completion is not topologically Hopfian."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.19193","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/2607.19193/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":"2026-07-22T01:24:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1RnRU5KrsqISXhOmk2r1Sl5It9aynC7rsUKQ60DKoxDcs6oCsMC5CjFC+Omq0pe+9sESjVV1/BM0QCDzdm9xCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T21:18:22.590323Z"},"content_sha256":"567893252c26c496d4970bcdcfa857535da0c78074cd7ba9fc7c25e2cdf2f9ec","schema_version":"1.0","event_id":"sha256:567893252c26c496d4970bcdcfa857535da0c78074cd7ba9fc7c25e2cdf2f9ec"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZS6WAI2I6E7V4ZXR36JJ63VDVK/bundle.json","state_url":"https://pith.science/pith/ZS6WAI2I6E7V4ZXR36JJ63VDVK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZS6WAI2I6E7V4ZXR36JJ63VDVK/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":"2026-08-21T21:18:22Z","links":{"resolver":"https://pith.science/pith/ZS6WAI2I6E7V4ZXR36JJ63VDVK","bundle":"https://pith.science/pith/ZS6WAI2I6E7V4ZXR36JJ63VDVK/bundle.json","state":"https://pith.science/pith/ZS6WAI2I6E7V4ZXR36JJ63VDVK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZS6WAI2I6E7V4ZXR36JJ63VDVK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:ZS6WAI2I6E7V4ZXR36JJ63VDVK","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":"4c82ba69e2503b024602c1aa3100360a06e95a5d847cbb1d732e2893239a8a0e","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.GR","submitted_at":"2026-07-21T15:29:14Z","title_canon_sha256":"01deee6405e848c57f00bf5709478a2c5c36456c3418f2e2bd8bb94eb539b1cc"},"schema_version":"1.0","source":{"id":"2607.19193","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.19193","created_at":"2026-07-22T01:24:13Z"},{"alias_kind":"arxiv_version","alias_value":"2607.19193v1","created_at":"2026-07-22T01:24:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.19193","created_at":"2026-07-22T01:24:13Z"},{"alias_kind":"pith_short_12","alias_value":"ZS6WAI2I6E7V","created_at":"2026-07-22T01:24:13Z"},{"alias_kind":"pith_short_16","alias_value":"ZS6WAI2I6E7V4ZXR","created_at":"2026-07-22T01:24:13Z"},{"alias_kind":"pith_short_8","alias_value":"ZS6WAI2I","created_at":"2026-07-22T01:24:13Z"}],"graph_snapshots":[{"event_id":"sha256:567893252c26c496d4970bcdcfa857535da0c78074cd7ba9fc7c25e2cdf2f9ec","target":"graph","created_at":"2026-07-22T01:24:13Z","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/2607.19193/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We determine exactly when the profinite completion of an arbitrary abelian\n  group is topologically Hopfian. For an abelian group $A$, we prove that \\[\n  \\widehat A \\text{ is topologically Hopfian}\n  \\quad\\Longleftrightarrow\\quad\n  A/pA \\text{ is finite for every prime }p. \\] As a byproduct, we answer Problem 6.30 of the Kourovka Notebook in the negative: for pairwise distinct odd primes $q_i$, the group $\\bigoplus_{i\\geq1}\\Z[1/q_i]$ is residually finite and Hopfian, whereas its profinite completion is not topologically Hopfian.","authors_text":"E. Ingrosso, M. Brescia, M. Trombetti","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.GR","submitted_at":"2026-07-21T15:29:14Z","title":"Hopficity of profinite completions of abelian groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.19193","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:af1fe7f7532862f1dca58279a37110e241e17e6449385a127fe467307310e4ec","target":"record","created_at":"2026-07-22T01:24:13Z","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":"4c82ba69e2503b024602c1aa3100360a06e95a5d847cbb1d732e2893239a8a0e","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.GR","submitted_at":"2026-07-21T15:29:14Z","title_canon_sha256":"01deee6405e848c57f00bf5709478a2c5c36456c3418f2e2bd8bb94eb539b1cc"},"schema_version":"1.0","source":{"id":"2607.19193","kind":"arxiv","version":1}},"canonical_sha256":"ccbd602348f13f5e66f1df929f6ea3aa8460d0d96f5c7a3ea49c99055266e938","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ccbd602348f13f5e66f1df929f6ea3aa8460d0d96f5c7a3ea49c99055266e938","first_computed_at":"2026-07-22T01:24:13.596745Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-22T01:24:13.596745Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TcKIzumuGzrMp058KCHFcVhSiTuu5YSeStMxQO3FBNsqtAmB8GKowh4iBjzDyFClqjYtDWd8G94Y8qev6YGnDw==","signature_status":"signed_v1","signed_at":"2026-07-22T01:24:13.597584Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.19193","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:af1fe7f7532862f1dca58279a37110e241e17e6449385a127fe467307310e4ec","sha256:567893252c26c496d4970bcdcfa857535da0c78074cd7ba9fc7c25e2cdf2f9ec"],"state_sha256":"e0c82f23b538508d6a2802c7eb46d121963bebe1b9b324fa6972c69c86cd863b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AeRrCs0P7mb7EKpIGeA7vtlSt+OA/Fr5YXgZUcpANNKFgk0SmLz68HJsT/64p1JYa41oDNMowBiqHR2hh7l2Aw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-21T21:18:22.594684Z","bundle_sha256":"2081c175c04ad3fae0a85ac3edf9ec85eb8b04c9321e04e120b3f3b5d533e7c9"}}