{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:ZVR7TX7DXIPBUPDYNAF7D2WTFW","short_pith_number":"pith:ZVR7TX7D","canonical_record":{"source":{"id":"2105.11020","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2021-05-23T21:13:10Z","cross_cats_sorted":[],"title_canon_sha256":"db031d063cde745965a9a605d2d289c64214aa4ab1d36e40be0ecd0fb00a6eee","abstract_canon_sha256":"68230e6e49b0c4fc2da6fa909f2507c6275d10a6bdfcd732037ab9a19b1b4386"},"schema_version":"1.0"},"canonical_sha256":"cd63f9dfe3ba1e1a3c78680bf1ead32d8a64fd52e97aa8d98bb869b97f8dd729","source":{"kind":"arxiv","id":"2105.11020","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2105.11020","created_at":"2026-05-22T01:03:38Z"},{"alias_kind":"arxiv_version","alias_value":"2105.11020v1","created_at":"2026-05-22T01:03:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2105.11020","created_at":"2026-05-22T01:03:38Z"},{"alias_kind":"pith_short_12","alias_value":"ZVR7TX7DXIPB","created_at":"2026-05-22T01:03:38Z"},{"alias_kind":"pith_short_16","alias_value":"ZVR7TX7DXIPBUPDY","created_at":"2026-05-22T01:03:38Z"},{"alias_kind":"pith_short_8","alias_value":"ZVR7TX7D","created_at":"2026-05-22T01:03:38Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:ZVR7TX7DXIPBUPDYNAF7D2WTFW","target":"record","payload":{"canonical_record":{"source":{"id":"2105.11020","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2021-05-23T21:13:10Z","cross_cats_sorted":[],"title_canon_sha256":"db031d063cde745965a9a605d2d289c64214aa4ab1d36e40be0ecd0fb00a6eee","abstract_canon_sha256":"68230e6e49b0c4fc2da6fa909f2507c6275d10a6bdfcd732037ab9a19b1b4386"},"schema_version":"1.0"},"canonical_sha256":"cd63f9dfe3ba1e1a3c78680bf1ead32d8a64fd52e97aa8d98bb869b97f8dd729","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-22T01:03:38.415467Z","signature_b64":"X3U3XKy8OFtB9OgCHxU+6akDYDSUAJrs3eBiJN0HD5LJvaFry1yhwt+beBBusRNnuRsmyYnN91mjJHwFcCNcAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cd63f9dfe3ba1e1a3c78680bf1ead32d8a64fd52e97aa8d98bb869b97f8dd729","last_reissued_at":"2026-05-22T01:03:38.414280Z","signature_status":"signed_v1","first_computed_at":"2026-05-22T01:03:38.414280Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2105.11020","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":"2026-05-22T01:03:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2A2+///IPbbwqjksNaulSv9/cyRrTYn8nuTNLQPOpU9XS172YoFOo1NWqCGMmGSvmJzraeUGtkyivUjjYPJKAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T03:52:31.246529Z"},"content_sha256":"8c2dde995cf3b02ae766a94c22635d070554ad2529ff0a3a0bf54a7f8725c9ec","schema_version":"1.0","event_id":"sha256:8c2dde995cf3b02ae766a94c22635d070554ad2529ff0a3a0bf54a7f8725c9ec"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:ZVR7TX7DXIPBUPDYNAF7D2WTFW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Critical probabilistic characteristics of the Cram\\'er model for primes and arithmetical properties","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Michel Weber","submitted_at":"2021-05-23T21:13:10Z","abstract_excerpt":"This work is a probabilistic study of the 'primes' of the Cram\\'er model. We prove that there exists a set of integers $\\mathcal S$ of density 1 such that \\begin{equation}\\liminf_{ \\mathcal S\\ni n\\to\\infty} (\\log n)\\mathbb{P} \\{S_n\\ \\hbox{prime} \\} \\ge \\frac{1}{\\sqrt{2\\pi e}\\, }, \\end{equation} and that for $b>\\frac12$, the formula \\begin{equation} \\mathbb{P} \\{S_n\\ \\text{prime}\\, \\} \\, =\\, \\frac{ (1+ o( 1) )}{ \\sqrt{2\\pi B_n } } \\int_{m_n-\\sqrt{ 2bB_n\\log n}}^{m_n+\\sqrt{ 2bB_n\\log n}} \\, e^{-\\frac{(t - m_n)^2}{ 2 B_n } }\\, {\\rm d}\\pi(t), \\end{equation} in which $m_n=\\mathbb{E} S_n,B_n={\\rm Va"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2105.11020","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/2105.11020/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":"2026-05-22T01:03:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+O3hlVzBAX4EcYhbuIgRJTlzwp0qqVmL6qK8mG86ao6ftkKQbSMEjpO5D+TLVYZCxDdzxgIHpK/9cj4HTvB3Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T03:52:31.246910Z"},"content_sha256":"aa7d915dc52944d80a5766123fd45d00daabe7b0277fe47061af925c11543f23","schema_version":"1.0","event_id":"sha256:aa7d915dc52944d80a5766123fd45d00daabe7b0277fe47061af925c11543f23"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZVR7TX7DXIPBUPDYNAF7D2WTFW/bundle.json","state_url":"https://pith.science/pith/ZVR7TX7DXIPBUPDYNAF7D2WTFW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZVR7TX7DXIPBUPDYNAF7D2WTFW/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":"2026-06-23T03:52:31Z","links":{"resolver":"https://pith.science/pith/ZVR7TX7DXIPBUPDYNAF7D2WTFW","bundle":"https://pith.science/pith/ZVR7TX7DXIPBUPDYNAF7D2WTFW/bundle.json","state":"https://pith.science/pith/ZVR7TX7DXIPBUPDYNAF7D2WTFW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZVR7TX7DXIPBUPDYNAF7D2WTFW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:ZVR7TX7DXIPBUPDYNAF7D2WTFW","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":"68230e6e49b0c4fc2da6fa909f2507c6275d10a6bdfcd732037ab9a19b1b4386","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2021-05-23T21:13:10Z","title_canon_sha256":"db031d063cde745965a9a605d2d289c64214aa4ab1d36e40be0ecd0fb00a6eee"},"schema_version":"1.0","source":{"id":"2105.11020","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2105.11020","created_at":"2026-05-22T01:03:38Z"},{"alias_kind":"arxiv_version","alias_value":"2105.11020v1","created_at":"2026-05-22T01:03:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2105.11020","created_at":"2026-05-22T01:03:38Z"},{"alias_kind":"pith_short_12","alias_value":"ZVR7TX7DXIPB","created_at":"2026-05-22T01:03:38Z"},{"alias_kind":"pith_short_16","alias_value":"ZVR7TX7DXIPBUPDY","created_at":"2026-05-22T01:03:38Z"},{"alias_kind":"pith_short_8","alias_value":"ZVR7TX7D","created_at":"2026-05-22T01:03:38Z"}],"graph_snapshots":[{"event_id":"sha256:aa7d915dc52944d80a5766123fd45d00daabe7b0277fe47061af925c11543f23","target":"graph","created_at":"2026-05-22T01:03:38Z","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/2105.11020/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This work is a probabilistic study of the 'primes' of the Cram\\'er model. We prove that there exists a set of integers $\\mathcal S$ of density 1 such that \\begin{equation}\\liminf_{ \\mathcal S\\ni n\\to\\infty} (\\log n)\\mathbb{P} \\{S_n\\ \\hbox{prime} \\} \\ge \\frac{1}{\\sqrt{2\\pi e}\\, }, \\end{equation} and that for $b>\\frac12$, the formula \\begin{equation} \\mathbb{P} \\{S_n\\ \\text{prime}\\, \\} \\, =\\, \\frac{ (1+ o( 1) )}{ \\sqrt{2\\pi B_n } } \\int_{m_n-\\sqrt{ 2bB_n\\log n}}^{m_n+\\sqrt{ 2bB_n\\log n}} \\, e^{-\\frac{(t - m_n)^2}{ 2 B_n } }\\, {\\rm d}\\pi(t), \\end{equation} in which $m_n=\\mathbb{E} S_n,B_n={\\rm Va","authors_text":"Michel Weber","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2021-05-23T21:13:10Z","title":"Critical probabilistic characteristics of the Cram\\'er model for primes and arithmetical properties"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2105.11020","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":"sha256:8c2dde995cf3b02ae766a94c22635d070554ad2529ff0a3a0bf54a7f8725c9ec","target":"record","created_at":"2026-05-22T01:03:38Z","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":"68230e6e49b0c4fc2da6fa909f2507c6275d10a6bdfcd732037ab9a19b1b4386","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2021-05-23T21:13:10Z","title_canon_sha256":"db031d063cde745965a9a605d2d289c64214aa4ab1d36e40be0ecd0fb00a6eee"},"schema_version":"1.0","source":{"id":"2105.11020","kind":"arxiv","version":1}},"canonical_sha256":"cd63f9dfe3ba1e1a3c78680bf1ead32d8a64fd52e97aa8d98bb869b97f8dd729","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cd63f9dfe3ba1e1a3c78680bf1ead32d8a64fd52e97aa8d98bb869b97f8dd729","first_computed_at":"2026-05-22T01:03:38.414280Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-22T01:03:38.414280Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"X3U3XKy8OFtB9OgCHxU+6akDYDSUAJrs3eBiJN0HD5LJvaFry1yhwt+beBBusRNnuRsmyYnN91mjJHwFcCNcAQ==","signature_status":"signed_v1","signed_at":"2026-05-22T01:03:38.415467Z","signed_message":"canonical_sha256_bytes"},"source_id":"2105.11020","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8c2dde995cf3b02ae766a94c22635d070554ad2529ff0a3a0bf54a7f8725c9ec","sha256:aa7d915dc52944d80a5766123fd45d00daabe7b0277fe47061af925c11543f23"],"state_sha256":"24fa0987147e632aae17a15da702a81436a19a2291b7e9feaabf0076f996ee00"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8XxD/a6mmhQEQDtMlFh6Jm1YF5Rytibu2980kYj75uYuzlGOp+T2S9yDvQ394pO2kryYn9QyEv+cqz0C7GUbBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T03:52:31.248971Z","bundle_sha256":"f97f4dd54ed4623be334eae7f078b290d79d52f8c6293fb7712d0686f73c411b"}}