{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:XK3XUJFN7HJBYFGQKCECYK2UVL","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":"b9f2bd96b3fee607e14659de78ecaa11d30958726827dccfc5ccb423c41210dc","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2014-10-07T11:40:17Z","title_canon_sha256":"574d5cf067fc7b55655d96e01faf442140e7740bda6cdc96b33271fb5d6941fd"},"schema_version":"1.0","source":{"id":"1410.1693","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.1693","created_at":"2026-05-18T01:09:37Z"},{"alias_kind":"arxiv_version","alias_value":"1410.1693v3","created_at":"2026-05-18T01:09:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.1693","created_at":"2026-05-18T01:09:37Z"},{"alias_kind":"pith_short_12","alias_value":"XK3XUJFN7HJB","created_at":"2026-05-18T12:28:57Z"},{"alias_kind":"pith_short_16","alias_value":"XK3XUJFN7HJBYFGQ","created_at":"2026-05-18T12:28:57Z"},{"alias_kind":"pith_short_8","alias_value":"XK3XUJFN","created_at":"2026-05-18T12:28:57Z"}],"graph_snapshots":[{"event_id":"sha256:9798c7f86104d6a4755ddbd905475486b8d555c14580e1d6993bfb035411d50f","target":"graph","created_at":"2026-05-18T01:09:37Z","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":"Recently the so-called Atiyah conjecture about l^2-Betti numbers has been disproved. The counterexamples were found using a specific method of computing the spectral measure of a matrix over a complex group ring. We show that in many situations the same method allows to compute homology gradients, i.e. generalizations of l^2-Betti numbers to fields of arbitrary characteristic. As an application we point out that (i) the homology gradient over any field of characteristic different than 2 can be an irrational number, and (ii) there exists a finite CW-complex with the property that the homology g","authors_text":"{\\L}ukasz Grabowski, Thomas Schick","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2014-10-07T11:40:17Z","title":"On computing homology gradients over finite fields"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.1693","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:be69718828f018007dea7bdffeb53648419b8e3f1d622bdd2662dbfecd4cfaa1","target":"record","created_at":"2026-05-18T01:09:37Z","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":"b9f2bd96b3fee607e14659de78ecaa11d30958726827dccfc5ccb423c41210dc","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2014-10-07T11:40:17Z","title_canon_sha256":"574d5cf067fc7b55655d96e01faf442140e7740bda6cdc96b33271fb5d6941fd"},"schema_version":"1.0","source":{"id":"1410.1693","kind":"arxiv","version":3}},"canonical_sha256":"bab77a24adf9d21c14d050882c2b54aac315cf81202720fe8fd7a44fdf43e9bb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bab77a24adf9d21c14d050882c2b54aac315cf81202720fe8fd7a44fdf43e9bb","first_computed_at":"2026-05-18T01:09:37.372380Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:09:37.372380Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ha5mPmWNlI3fC/Wq9qKGTQZTsOs8CuojqCV6MFI5sw71stpuNpiD3UNWZUP21MyzZYQIJJMRQbuxTgkua7ABBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:09:37.372741Z","signed_message":"canonical_sha256_bytes"},"source_id":"1410.1693","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:be69718828f018007dea7bdffeb53648419b8e3f1d622bdd2662dbfecd4cfaa1","sha256:9798c7f86104d6a4755ddbd905475486b8d555c14580e1d6993bfb035411d50f"],"state_sha256":"7cd1c48416b106b9b2262a08d6c2851d560378ab8107a558e7b8860c51bc8c8e"}