{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:5B6OMON2YHRK7FNQXFXIRVTFUN","short_pith_number":"pith:5B6OMON2","canonical_record":{"source":{"id":"2402.03276","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2024-02-05T18:34:02Z","cross_cats_sorted":["math.CO","math.NT","math.PR"],"title_canon_sha256":"69324be9d7df9ca577e76a83527de1222b340faf0be5859e538dcae1b3ea0dec","abstract_canon_sha256":"61ecdb902e3e3c8f11c3b67418031d05dff6458b0b7429d82f17bd1f0a11284c"},"schema_version":"1.0"},"canonical_sha256":"e87ce639bac1e2af95b0b96e88d665a351ab0298d0abe8b9b46b0d8e2e91d06d","source":{"kind":"arxiv","id":"2402.03276","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2402.03276","created_at":"2026-07-05T08:54:46Z"},{"alias_kind":"arxiv_version","alias_value":"2402.03276v3","created_at":"2026-07-05T08:54:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2402.03276","created_at":"2026-07-05T08:54:46Z"},{"alias_kind":"pith_short_12","alias_value":"5B6OMON2YHRK","created_at":"2026-07-05T08:54:46Z"},{"alias_kind":"pith_short_16","alias_value":"5B6OMON2YHRK7FNQ","created_at":"2026-07-05T08:54:46Z"},{"alias_kind":"pith_short_8","alias_value":"5B6OMON2","created_at":"2026-07-05T08:54:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:5B6OMON2YHRK7FNQXFXIRVTFUN","target":"record","payload":{"canonical_record":{"source":{"id":"2402.03276","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2024-02-05T18:34:02Z","cross_cats_sorted":["math.CO","math.NT","math.PR"],"title_canon_sha256":"69324be9d7df9ca577e76a83527de1222b340faf0be5859e538dcae1b3ea0dec","abstract_canon_sha256":"61ecdb902e3e3c8f11c3b67418031d05dff6458b0b7429d82f17bd1f0a11284c"},"schema_version":"1.0"},"canonical_sha256":"e87ce639bac1e2af95b0b96e88d665a351ab0298d0abe8b9b46b0d8e2e91d06d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:54:46.978537Z","signature_b64":"Xv1iEQOeJL7vf+TCiL6eET7GEIk+AOi5bA4GcGrB57Y2PqMlPx+8kDaQie7og/WdFmCeunzfM/ML1hRJIu9/Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e87ce639bac1e2af95b0b96e88d665a351ab0298d0abe8b9b46b0d8e2e91d06d","last_reissued_at":"2026-07-05T08:54:46.978114Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:54:46.978114Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2402.03276","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T08:54:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"T7YA3UjoCaAH1EUwRoA25tSE0GOugmH53bI0E4qgc0kL53/Ulo+Q0D1gek7z5oh6KjwtFt9UXdhFuQMJZMenBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T22:11:34.335310Z"},"content_sha256":"3e3aef7e9d17edd049c1b889c7bde4d7cb05f5234436aadb8cd4a2b1a292dc35","schema_version":"1.0","event_id":"sha256:3e3aef7e9d17edd049c1b889c7bde4d7cb05f5234436aadb8cd4a2b1a292dc35"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:5B6OMON2YHRK7FNQXFXIRVTFUN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An approximation of the Collatz map and a lower bound for the average total stopping time","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.NT","math.PR"],"primary_cat":"math.DS","authors_text":"Manuel Inselmann","submitted_at":"2024-02-05T18:34:02Z","abstract_excerpt":"Define the map $\\mathsf{T}$ on the positive integers by $\\mathsf{T}(m)=\\frac{m}{2}$ if $m$ is even and by $\\mathsf{T}(m)=\\frac{3m+1}{2}$ if $m$ is odd. Results of Terras and Everett imply that, given any $\\epsilon>0$, almost all $m\\in\\mathbb{Z}^+$ (in the sense of natural density) fulfill $(\\frac{\\sqrt{3}}{2})^km^{1-\\epsilon}\\leq \\mathsf{T}^k(m)\\leq (\\frac{\\sqrt{3}}{2})^km^{1+\\epsilon}$ simultaneously for all $0\\leq k\\leq \\alpha\\log m$ with $\\alpha=(\\log 2)^{-1}\\approx 1.443$. We extend this result to $\\alpha=2(\\log\\frac{4}{3})^{-1}\\approx 6.952$, which is the maximally possible value. Set $\\m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2402.03276","kind":"arxiv","version":3},"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/2402.03276/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":"2026-07-05T08:54:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NtvaF3hA4YlDNslwIzM6RA2Y0jrQmmK6OuyfHje9RJdUAgE0bIKVmKvRT5KiDWg5N5wPj2qF8fZY47k7aH6IBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T22:11:34.336015Z"},"content_sha256":"aedea7ec4f0000b41f7959a20c1375ce48ee64d879313f1c6a366ca6f302e6a3","schema_version":"1.0","event_id":"sha256:aedea7ec4f0000b41f7959a20c1375ce48ee64d879313f1c6a366ca6f302e6a3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5B6OMON2YHRK7FNQXFXIRVTFUN/bundle.json","state_url":"https://pith.science/pith/5B6OMON2YHRK7FNQXFXIRVTFUN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5B6OMON2YHRK7FNQXFXIRVTFUN/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":"2026-08-11T22:11:34Z","links":{"resolver":"https://pith.science/pith/5B6OMON2YHRK7FNQXFXIRVTFUN","bundle":"https://pith.science/pith/5B6OMON2YHRK7FNQXFXIRVTFUN/bundle.json","state":"https://pith.science/pith/5B6OMON2YHRK7FNQXFXIRVTFUN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5B6OMON2YHRK7FNQXFXIRVTFUN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:5B6OMON2YHRK7FNQXFXIRVTFUN","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":"61ecdb902e3e3c8f11c3b67418031d05dff6458b0b7429d82f17bd1f0a11284c","cross_cats_sorted":["math.CO","math.NT","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2024-02-05T18:34:02Z","title_canon_sha256":"69324be9d7df9ca577e76a83527de1222b340faf0be5859e538dcae1b3ea0dec"},"schema_version":"1.0","source":{"id":"2402.03276","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2402.03276","created_at":"2026-07-05T08:54:46Z"},{"alias_kind":"arxiv_version","alias_value":"2402.03276v3","created_at":"2026-07-05T08:54:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2402.03276","created_at":"2026-07-05T08:54:46Z"},{"alias_kind":"pith_short_12","alias_value":"5B6OMON2YHRK","created_at":"2026-07-05T08:54:46Z"},{"alias_kind":"pith_short_16","alias_value":"5B6OMON2YHRK7FNQ","created_at":"2026-07-05T08:54:46Z"},{"alias_kind":"pith_short_8","alias_value":"5B6OMON2","created_at":"2026-07-05T08:54:46Z"}],"graph_snapshots":[{"event_id":"sha256:aedea7ec4f0000b41f7959a20c1375ce48ee64d879313f1c6a366ca6f302e6a3","target":"graph","created_at":"2026-07-05T08:54:46Z","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/2402.03276/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Define the map $\\mathsf{T}$ on the positive integers by $\\mathsf{T}(m)=\\frac{m}{2}$ if $m$ is even and by $\\mathsf{T}(m)=\\frac{3m+1}{2}$ if $m$ is odd. Results of Terras and Everett imply that, given any $\\epsilon>0$, almost all $m\\in\\mathbb{Z}^+$ (in the sense of natural density) fulfill $(\\frac{\\sqrt{3}}{2})^km^{1-\\epsilon}\\leq \\mathsf{T}^k(m)\\leq (\\frac{\\sqrt{3}}{2})^km^{1+\\epsilon}$ simultaneously for all $0\\leq k\\leq \\alpha\\log m$ with $\\alpha=(\\log 2)^{-1}\\approx 1.443$. We extend this result to $\\alpha=2(\\log\\frac{4}{3})^{-1}\\approx 6.952$, which is the maximally possible value. Set $\\m","authors_text":"Manuel Inselmann","cross_cats":["math.CO","math.NT","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2024-02-05T18:34:02Z","title":"An approximation of the Collatz map and a lower bound for the average total stopping time"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2402.03276","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":"sha256:3e3aef7e9d17edd049c1b889c7bde4d7cb05f5234436aadb8cd4a2b1a292dc35","target":"record","created_at":"2026-07-05T08:54:46Z","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":"61ecdb902e3e3c8f11c3b67418031d05dff6458b0b7429d82f17bd1f0a11284c","cross_cats_sorted":["math.CO","math.NT","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2024-02-05T18:34:02Z","title_canon_sha256":"69324be9d7df9ca577e76a83527de1222b340faf0be5859e538dcae1b3ea0dec"},"schema_version":"1.0","source":{"id":"2402.03276","kind":"arxiv","version":3}},"canonical_sha256":"e87ce639bac1e2af95b0b96e88d665a351ab0298d0abe8b9b46b0d8e2e91d06d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e87ce639bac1e2af95b0b96e88d665a351ab0298d0abe8b9b46b0d8e2e91d06d","first_computed_at":"2026-07-05T08:54:46.978114Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:54:46.978114Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Xv1iEQOeJL7vf+TCiL6eET7GEIk+AOi5bA4GcGrB57Y2PqMlPx+8kDaQie7og/WdFmCeunzfM/ML1hRJIu9/Aw==","signature_status":"signed_v1","signed_at":"2026-07-05T08:54:46.978537Z","signed_message":"canonical_sha256_bytes"},"source_id":"2402.03276","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3e3aef7e9d17edd049c1b889c7bde4d7cb05f5234436aadb8cd4a2b1a292dc35","sha256:aedea7ec4f0000b41f7959a20c1375ce48ee64d879313f1c6a366ca6f302e6a3"],"state_sha256":"b1f0173965a435d4ac931d0e357e8fda24a9cc4f830dc2bebfe2f3b217d69eca"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wRrEdv5CYTZT1qNcR750aOeN4wldMvNBsUKy+D3fozf3KeO2FZqovbPyogHkL2sr/ixOAadZG8QehlVBwkcAAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-11T22:11:34.343114Z","bundle_sha256":"d76942826abcabaec13a4bfe2ae0a1c95a037d69b0adb59673998962632ecb17"}}