{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:JX2XRG3Z2CM3NJLOUFWBRPXAUO","short_pith_number":"pith:JX2XRG3Z","canonical_record":{"source":{"id":"2402.00001","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GM","submitted_at":"2023-09-20T23:40:40Z","cross_cats_sorted":[],"title_canon_sha256":"4c1941b12d8969f2ab07c661cd00bab4f7609e41dbed87d2dc2d01422b7f3161","abstract_canon_sha256":"fb876a558e49bbaac2f4bbeed350f2eeb63fbef1c90eecce5e4d28f164fcdf39"},"schema_version":"1.0"},"canonical_sha256":"4df5789b79d099b6a56ea16c18bee0a3a5d4564a48165b2d1a94f34363620fdd","source":{"kind":"arxiv","id":"2402.00001","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2402.00001","created_at":"2026-07-05T08:30:07Z"},{"alias_kind":"arxiv_version","alias_value":"2402.00001v3","created_at":"2026-07-05T08:30:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2402.00001","created_at":"2026-07-05T08:30:07Z"},{"alias_kind":"pith_short_12","alias_value":"JX2XRG3Z2CM3","created_at":"2026-07-05T08:30:07Z"},{"alias_kind":"pith_short_16","alias_value":"JX2XRG3Z2CM3NJLO","created_at":"2026-07-05T08:30:07Z"},{"alias_kind":"pith_short_8","alias_value":"JX2XRG3Z","created_at":"2026-07-05T08:30:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:JX2XRG3Z2CM3NJLOUFWBRPXAUO","target":"record","payload":{"canonical_record":{"source":{"id":"2402.00001","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GM","submitted_at":"2023-09-20T23:40:40Z","cross_cats_sorted":[],"title_canon_sha256":"4c1941b12d8969f2ab07c661cd00bab4f7609e41dbed87d2dc2d01422b7f3161","abstract_canon_sha256":"fb876a558e49bbaac2f4bbeed350f2eeb63fbef1c90eecce5e4d28f164fcdf39"},"schema_version":"1.0"},"canonical_sha256":"4df5789b79d099b6a56ea16c18bee0a3a5d4564a48165b2d1a94f34363620fdd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:30:07.984256Z","signature_b64":"Ct+NmRk9r1sOEf/z3YMfYAl50HMq0DKH1Qis5aUfjEi8HyfjevddrqNn8a5goblpCdZxYndVGju1H0OvUmV5Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4df5789b79d099b6a56ea16c18bee0a3a5d4564a48165b2d1a94f34363620fdd","last_reissued_at":"2026-07-05T08:30:07.983786Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:30:07.983786Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2402.00001","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":"2026-07-05T08:30:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aPNVu5tdrR/S3NBfPkRrvDuLCjSiJ5wWN+xFfr7mktHZCO1cMy9WGC0GYBVq9PBHyb5PCeycSsRW5M+Uk1RyAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T19:40:56.930427Z"},"content_sha256":"f2b887a9541c129f67a344b71d9b5d4ac41fe426f19a548b12096c3d347db610","schema_version":"1.0","event_id":"sha256:f2b887a9541c129f67a344b71d9b5d4ac41fe426f19a548b12096c3d347db610"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:JX2XRG3Z2CM3NJLOUFWBRPXAUO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Using binary string to prove the Collatz conjecture","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Jishe Feng","submitted_at":"2023-09-20T23:40:40Z","abstract_excerpt":"We introduce a full binary directed tree structure to represent the set of natural numbers, further categorizing them into three distinct subsets: pure odd numbers, pure even numbers, and mixed numbers. We adopt a binary string representation for natural numbers and elaborate on the composite methodology encompassing odd- and even-number functions. Our analysis focuses on examining the iteration sequence (or composition) of the Collatz function and its reduced variant, which serves as an analog to the inverse function, to scrutinize the validity of the Collatz conjecture. To substantiate this "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2402.00001","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/2402.00001/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":"2026-07-05T08:30:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"W0tYeAiz/3i08su7cunFws01PvVu95Yrzyw/7oQY3MoxZf6ovRnIJ1CIUC95PIOw/1jEjTGDp9FgxBlMHMH1Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T19:40:56.931081Z"},"content_sha256":"bb81a29cd47c84fe14cf20d640b4af7f04bbba5f6ef8ed6a211f4de0f474df20","schema_version":"1.0","event_id":"sha256:bb81a29cd47c84fe14cf20d640b4af7f04bbba5f6ef8ed6a211f4de0f474df20"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JX2XRG3Z2CM3NJLOUFWBRPXAUO/bundle.json","state_url":"https://pith.science/pith/JX2XRG3Z2CM3NJLOUFWBRPXAUO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JX2XRG3Z2CM3NJLOUFWBRPXAUO/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":"2026-08-11T19:40:56Z","links":{"resolver":"https://pith.science/pith/JX2XRG3Z2CM3NJLOUFWBRPXAUO","bundle":"https://pith.science/pith/JX2XRG3Z2CM3NJLOUFWBRPXAUO/bundle.json","state":"https://pith.science/pith/JX2XRG3Z2CM3NJLOUFWBRPXAUO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JX2XRG3Z2CM3NJLOUFWBRPXAUO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:JX2XRG3Z2CM3NJLOUFWBRPXAUO","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":"fb876a558e49bbaac2f4bbeed350f2eeb63fbef1c90eecce5e4d28f164fcdf39","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GM","submitted_at":"2023-09-20T23:40:40Z","title_canon_sha256":"4c1941b12d8969f2ab07c661cd00bab4f7609e41dbed87d2dc2d01422b7f3161"},"schema_version":"1.0","source":{"id":"2402.00001","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2402.00001","created_at":"2026-07-05T08:30:07Z"},{"alias_kind":"arxiv_version","alias_value":"2402.00001v3","created_at":"2026-07-05T08:30:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2402.00001","created_at":"2026-07-05T08:30:07Z"},{"alias_kind":"pith_short_12","alias_value":"JX2XRG3Z2CM3","created_at":"2026-07-05T08:30:07Z"},{"alias_kind":"pith_short_16","alias_value":"JX2XRG3Z2CM3NJLO","created_at":"2026-07-05T08:30:07Z"},{"alias_kind":"pith_short_8","alias_value":"JX2XRG3Z","created_at":"2026-07-05T08:30:07Z"}],"graph_snapshots":[{"event_id":"sha256:bb81a29cd47c84fe14cf20d640b4af7f04bbba5f6ef8ed6a211f4de0f474df20","target":"graph","created_at":"2026-07-05T08:30:07Z","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/2402.00001/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We introduce a full binary directed tree structure to represent the set of natural numbers, further categorizing them into three distinct subsets: pure odd numbers, pure even numbers, and mixed numbers. We adopt a binary string representation for natural numbers and elaborate on the composite methodology encompassing odd- and even-number functions. Our analysis focuses on examining the iteration sequence (or composition) of the Collatz function and its reduced variant, which serves as an analog to the inverse function, to scrutinize the validity of the Collatz conjecture. To substantiate this ","authors_text":"Jishe Feng","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GM","submitted_at":"2023-09-20T23:40:40Z","title":"Using binary string to prove the Collatz conjecture"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2402.00001","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:f2b887a9541c129f67a344b71d9b5d4ac41fe426f19a548b12096c3d347db610","target":"record","created_at":"2026-07-05T08:30:07Z","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":"fb876a558e49bbaac2f4bbeed350f2eeb63fbef1c90eecce5e4d28f164fcdf39","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GM","submitted_at":"2023-09-20T23:40:40Z","title_canon_sha256":"4c1941b12d8969f2ab07c661cd00bab4f7609e41dbed87d2dc2d01422b7f3161"},"schema_version":"1.0","source":{"id":"2402.00001","kind":"arxiv","version":3}},"canonical_sha256":"4df5789b79d099b6a56ea16c18bee0a3a5d4564a48165b2d1a94f34363620fdd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4df5789b79d099b6a56ea16c18bee0a3a5d4564a48165b2d1a94f34363620fdd","first_computed_at":"2026-07-05T08:30:07.983786Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:30:07.983786Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ct+NmRk9r1sOEf/z3YMfYAl50HMq0DKH1Qis5aUfjEi8HyfjevddrqNn8a5goblpCdZxYndVGju1H0OvUmV5Cw==","signature_status":"signed_v1","signed_at":"2026-07-05T08:30:07.984256Z","signed_message":"canonical_sha256_bytes"},"source_id":"2402.00001","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f2b887a9541c129f67a344b71d9b5d4ac41fe426f19a548b12096c3d347db610","sha256:bb81a29cd47c84fe14cf20d640b4af7f04bbba5f6ef8ed6a211f4de0f474df20"],"state_sha256":"4d8e5adf33ebd453c8a602c6a04c7bb7e2221acdecc6299068fa88130f586150"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1J4M/8hHN0XRZGCELhEHrYHVsGGU0D7wCoVzWWJ/mtJ557CMzRs5Flh5flKdrLGT18fyRRUjhmvNRwlDW/ilDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-11T19:40:56.937210Z","bundle_sha256":"9369a099492cf91a3b66a5e9f8f70aef69b93b0f6a0cb0d022abad18cf04a24d"}}