{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:U2GZED6T7U6Q4YQJO3GOIGPW3M","short_pith_number":"pith:U2GZED6T","canonical_record":{"source":{"id":"2506.08503","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2025-06-10T06:58:38Z","cross_cats_sorted":["math.PR","math.ST","stat.TH"],"title_canon_sha256":"94537a4292e123a8cdba9a4496977865c5aac6e72f1c1bee02476b9177265467","abstract_canon_sha256":"d55368c4dc555c14f6538b5cf24618a4686062f826f0c84d3695d87abaa492ef"},"schema_version":"1.0"},"canonical_sha256":"a68d920fd3fd3d0e620976cce419f6db396df15fe563bbb21a7fc906e5caf818","source":{"kind":"arxiv","id":"2506.08503","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.08503","created_at":"2026-07-05T11:18:50Z"},{"alias_kind":"arxiv_version","alias_value":"2506.08503v1","created_at":"2026-07-05T11:18:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.08503","created_at":"2026-07-05T11:18:50Z"},{"alias_kind":"pith_short_12","alias_value":"U2GZED6T7U6Q","created_at":"2026-07-05T11:18:50Z"},{"alias_kind":"pith_short_16","alias_value":"U2GZED6T7U6Q4YQJ","created_at":"2026-07-05T11:18:50Z"},{"alias_kind":"pith_short_8","alias_value":"U2GZED6T","created_at":"2026-07-05T11:18:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:U2GZED6T7U6Q4YQJO3GOIGPW3M","target":"record","payload":{"canonical_record":{"source":{"id":"2506.08503","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2025-06-10T06:58:38Z","cross_cats_sorted":["math.PR","math.ST","stat.TH"],"title_canon_sha256":"94537a4292e123a8cdba9a4496977865c5aac6e72f1c1bee02476b9177265467","abstract_canon_sha256":"d55368c4dc555c14f6538b5cf24618a4686062f826f0c84d3695d87abaa492ef"},"schema_version":"1.0"},"canonical_sha256":"a68d920fd3fd3d0e620976cce419f6db396df15fe563bbb21a7fc906e5caf818","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:18:50.698212Z","signature_b64":"UyInLcpKs2tqlUJqdqX5L8zvVAjFnHJBq5uZ2u9zXeTGXaqtoEB13y3+i412jedAKoXCBldUaBBQ2iP/kduYBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a68d920fd3fd3d0e620976cce419f6db396df15fe563bbb21a7fc906e5caf818","last_reissued_at":"2026-07-05T11:18:50.697742Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:18:50.697742Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2506.08503","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":"2026-07-05T11:18:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Le3dHV+MtrFyPtZLc/qL0yPCJYAWE+JkZX6VgnOhVKpPxIo8sHYe0FFQNjC/EURDkCthgwTB971l3xtPQGj4Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T19:34:29.404322Z"},"content_sha256":"ad36161dec921c29b1b22bf00096494d3892388c12c6cad62606e397c4f3e783","schema_version":"1.0","event_id":"sha256:ad36161dec921c29b1b22bf00096494d3892388c12c6cad62606e397c4f3e783"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:U2GZED6T7U6Q4YQJO3GOIGPW3M","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Transforming the Erd\\H{o}s-Kac theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR","math.ST","stat.TH"],"primary_cat":"math.NT","authors_text":"Kimihiro Noguchi","submitted_at":"2025-06-10T06:58:38Z","abstract_excerpt":"Transforming the Erd\\H{o}s-Kac theorem provides more flexibility in how the theorem can be utilized as an interval estimate for the prime omega function, which counts the number of distinct prime divisors. Here, we consider a direct transformation by the delta method. Then, we demonstrate that the square-root and three-quarters power asymptotically achieve variance stabilization and an optimal width, respectively. Furthermore, by adjusting the denominator of the theorem, we derive the score interval estimate. To make these interval estimates reliable for small positive integers, we examine per"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.08503","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/2506.08503/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":"2026-07-05T11:18:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IgSuQPGTPWaXAFA3sQwHnop61GVH/tjJNEDmEhU32QLcHqGt5UNV/xtz4On1tAcB/Et2QRp33e0cngCO7E7lAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T19:34:29.405257Z"},"content_sha256":"22ce326896ec02da287029b68db72a927d096ba8ad0fbf83d17b7444a7c19fd0","schema_version":"1.0","event_id":"sha256:22ce326896ec02da287029b68db72a927d096ba8ad0fbf83d17b7444a7c19fd0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/U2GZED6T7U6Q4YQJO3GOIGPW3M/bundle.json","state_url":"https://pith.science/pith/U2GZED6T7U6Q4YQJO3GOIGPW3M/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/U2GZED6T7U6Q4YQJO3GOIGPW3M/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":"2026-08-09T19:34:29Z","links":{"resolver":"https://pith.science/pith/U2GZED6T7U6Q4YQJO3GOIGPW3M","bundle":"https://pith.science/pith/U2GZED6T7U6Q4YQJO3GOIGPW3M/bundle.json","state":"https://pith.science/pith/U2GZED6T7U6Q4YQJO3GOIGPW3M/state.json","well_known_bundle":"https://pith.science/.well-known/pith/U2GZED6T7U6Q4YQJO3GOIGPW3M/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:U2GZED6T7U6Q4YQJO3GOIGPW3M","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":"d55368c4dc555c14f6538b5cf24618a4686062f826f0c84d3695d87abaa492ef","cross_cats_sorted":["math.PR","math.ST","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2025-06-10T06:58:38Z","title_canon_sha256":"94537a4292e123a8cdba9a4496977865c5aac6e72f1c1bee02476b9177265467"},"schema_version":"1.0","source":{"id":"2506.08503","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.08503","created_at":"2026-07-05T11:18:50Z"},{"alias_kind":"arxiv_version","alias_value":"2506.08503v1","created_at":"2026-07-05T11:18:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.08503","created_at":"2026-07-05T11:18:50Z"},{"alias_kind":"pith_short_12","alias_value":"U2GZED6T7U6Q","created_at":"2026-07-05T11:18:50Z"},{"alias_kind":"pith_short_16","alias_value":"U2GZED6T7U6Q4YQJ","created_at":"2026-07-05T11:18:50Z"},{"alias_kind":"pith_short_8","alias_value":"U2GZED6T","created_at":"2026-07-05T11:18:50Z"}],"graph_snapshots":[{"event_id":"sha256:22ce326896ec02da287029b68db72a927d096ba8ad0fbf83d17b7444a7c19fd0","target":"graph","created_at":"2026-07-05T11:18:50Z","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/2506.08503/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Transforming the Erd\\H{o}s-Kac theorem provides more flexibility in how the theorem can be utilized as an interval estimate for the prime omega function, which counts the number of distinct prime divisors. Here, we consider a direct transformation by the delta method. Then, we demonstrate that the square-root and three-quarters power asymptotically achieve variance stabilization and an optimal width, respectively. Furthermore, by adjusting the denominator of the theorem, we derive the score interval estimate. To make these interval estimates reliable for small positive integers, we examine per","authors_text":"Kimihiro Noguchi","cross_cats":["math.PR","math.ST","stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2025-06-10T06:58:38Z","title":"Transforming the Erd\\H{o}s-Kac theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.08503","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:ad36161dec921c29b1b22bf00096494d3892388c12c6cad62606e397c4f3e783","target":"record","created_at":"2026-07-05T11:18:50Z","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":"d55368c4dc555c14f6538b5cf24618a4686062f826f0c84d3695d87abaa492ef","cross_cats_sorted":["math.PR","math.ST","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2025-06-10T06:58:38Z","title_canon_sha256":"94537a4292e123a8cdba9a4496977865c5aac6e72f1c1bee02476b9177265467"},"schema_version":"1.0","source":{"id":"2506.08503","kind":"arxiv","version":1}},"canonical_sha256":"a68d920fd3fd3d0e620976cce419f6db396df15fe563bbb21a7fc906e5caf818","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a68d920fd3fd3d0e620976cce419f6db396df15fe563bbb21a7fc906e5caf818","first_computed_at":"2026-07-05T11:18:50.697742Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:18:50.697742Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UyInLcpKs2tqlUJqdqX5L8zvVAjFnHJBq5uZ2u9zXeTGXaqtoEB13y3+i412jedAKoXCBldUaBBQ2iP/kduYBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T11:18:50.698212Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.08503","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ad36161dec921c29b1b22bf00096494d3892388c12c6cad62606e397c4f3e783","sha256:22ce326896ec02da287029b68db72a927d096ba8ad0fbf83d17b7444a7c19fd0"],"state_sha256":"177d4ec5fb21578bbff8c7b3e869a7a538f08f1891963cc682836ba4c0add495"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YAAsZKZeND2G8IVotsmIMpB/tNISn/YHZyA9sDGwHettS/6GeBGuqPseN9G6VagNLD98HSJ4D3tzqa+UVaumBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T19:34:29.411123Z","bundle_sha256":"c317da4e524116b3bba20506f0bd4b3fe5b4e988e6fdce5390027106097adc5c"}}