{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:6TQGEYNBGKAXQOPSRFLI6BHAA4","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":"4cd697be36a83b333ad885bb39abca5d034b091d4e1726292941470f9cc3b081","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2025-04-08T18:00:27Z","title_canon_sha256":"e54d32ba7b48f5410835d193dc5c17947c50d5157ae2d609d7f8946359847f79"},"schema_version":"1.0","source":{"id":"2504.07142","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2504.07142","created_at":"2026-07-05T10:47:04Z"},{"alias_kind":"arxiv_version","alias_value":"2504.07142v1","created_at":"2026-07-05T10:47:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2504.07142","created_at":"2026-07-05T10:47:04Z"},{"alias_kind":"pith_short_12","alias_value":"6TQGEYNBGKAX","created_at":"2026-07-05T10:47:04Z"},{"alias_kind":"pith_short_16","alias_value":"6TQGEYNBGKAXQOPS","created_at":"2026-07-05T10:47:04Z"},{"alias_kind":"pith_short_8","alias_value":"6TQGEYNB","created_at":"2026-07-05T10:47:04Z"}],"graph_snapshots":[{"event_id":"sha256:26ff50e9b33c60021c362be425bb832891cd7e0904a1fb606db3df761bde2fc9","target":"graph","created_at":"2026-07-05T10:47:04Z","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/2504.07142/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We consider a particular generalized Lambert function, $y(x)$, defined by the implicit equation $y^\\beta = 1 - e^{-xy}$, with $x>0$ and $ \\beta > 1$. Solutions to this equation can be found in terms of a certain continued exponential. Asymptotic and structural properties of a non-trivial solution, $y_\\beta(x)$, and its connection to the extinction probability of related branching processes are discussed. We demonstrate that this function constitutes a cumulative distribution function of a previously unknown non-negative absolutely continuous random variable.","authors_text":"Alexander Kreinin, Andrey Marchenko, Vladimir Vinogradov","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2025-04-08T18:00:27Z","title":"On generalized Lambert function"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.07142","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":"sha256:02980a4d6a7c719b93a3579f95b9fb80f1f0a7af6d6bc20d754d59396379000d","target":"record","created_at":"2026-07-05T10:47:04Z","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":"4cd697be36a83b333ad885bb39abca5d034b091d4e1726292941470f9cc3b081","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2025-04-08T18:00:27Z","title_canon_sha256":"e54d32ba7b48f5410835d193dc5c17947c50d5157ae2d609d7f8946359847f79"},"schema_version":"1.0","source":{"id":"2504.07142","kind":"arxiv","version":1}},"canonical_sha256":"f4e06261a132817839f289568f04e007122f2dcf4962c9a74b33183648c7050e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f4e06261a132817839f289568f04e007122f2dcf4962c9a74b33183648c7050e","first_computed_at":"2026-07-05T10:47:04.020084Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:47:04.020084Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2Xxi/InCBgHnJx+un6CbHJTQQXetiTmWCL8Q6BQ7pzYjG1+loXM/0UdsCQby/g24PktuB9vC4QYkJMJyRHM4BQ==","signature_status":"signed_v1","signed_at":"2026-07-05T10:47:04.020632Z","signed_message":"canonical_sha256_bytes"},"source_id":"2504.07142","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:02980a4d6a7c719b93a3579f95b9fb80f1f0a7af6d6bc20d754d59396379000d","sha256:26ff50e9b33c60021c362be425bb832891cd7e0904a1fb606db3df761bde2fc9"],"state_sha256":"6e67ad38a0d5e8db7ef7aeb54b3b936a61350a6cd312f068290c445e66087b2e"}