{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:E232VXYLS4V247N74556PCEK7Q","short_pith_number":"pith:E232VXYL","canonical_record":{"source":{"id":"1109.6820","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.GM","submitted_at":"2011-09-29T18:19:29Z","cross_cats_sorted":[],"title_canon_sha256":"794d159e953835430dfd734be754b5364c5690f7025257ddcf3c660a9836a053","abstract_canon_sha256":"ab405897560a91080cec449b54222125f5b748a399026d776984b45cb8db6ffc"},"schema_version":"1.0"},"canonical_sha256":"26b7aadf0b972bae7dbfe77be7888afc0395b40007e583844313022041af8ed6","source":{"kind":"arxiv","id":"1109.6820","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.6820","created_at":"2026-05-18T04:11:52Z"},{"alias_kind":"arxiv_version","alias_value":"1109.6820v1","created_at":"2026-05-18T04:11:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.6820","created_at":"2026-05-18T04:11:52Z"},{"alias_kind":"pith_short_12","alias_value":"E232VXYLS4V2","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_16","alias_value":"E232VXYLS4V247N7","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_8","alias_value":"E232VXYL","created_at":"2026-05-18T12:26:26Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:E232VXYLS4V247N74556PCEK7Q","target":"record","payload":{"canonical_record":{"source":{"id":"1109.6820","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.GM","submitted_at":"2011-09-29T18:19:29Z","cross_cats_sorted":[],"title_canon_sha256":"794d159e953835430dfd734be754b5364c5690f7025257ddcf3c660a9836a053","abstract_canon_sha256":"ab405897560a91080cec449b54222125f5b748a399026d776984b45cb8db6ffc"},"schema_version":"1.0"},"canonical_sha256":"26b7aadf0b972bae7dbfe77be7888afc0395b40007e583844313022041af8ed6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:11:52.476866Z","signature_b64":"iWtA9tHdMJDwjAlGiB6hNgWYVF/Vv5FKB7HBIfrDasGy3bZ2vBS2t3kmCsPfPZwOODNPajioZYTrVh5NIkIOAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"26b7aadf0b972bae7dbfe77be7888afc0395b40007e583844313022041af8ed6","last_reissued_at":"2026-05-18T04:11:52.476239Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:11:52.476239Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1109.6820","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":"2026-05-18T04:11:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"weCJtEigD1B7hTjNw/SKOMY+xWupZOuFt4mHMG2ldP39E1XfLV49l7vveo+XI1F2jLHbKFjQ5L9GQ0BYwfaODw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T02:49:51.968199Z"},"content_sha256":"aaa8765bf492e5cfd2ff1cf4d55f6adc32bc0fefe3b782d912416594d0e636ac","schema_version":"1.0","event_id":"sha256:aaa8765bf492e5cfd2ff1cf4d55f6adc32bc0fefe3b782d912416594d0e636ac"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:E232VXYLS4V247N74556PCEK7Q","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Properties of proper rational numbers","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Konstantine Zelator","submitted_at":"2011-09-29T18:19:29Z","abstract_excerpt":"This short article is aimed at educators and teachers of mathematics.Its goal is simple and direct:to explore some of the basic/elementary properties of proper rational numbers.A proper rational number is a rational which is not an integer. A proper rational r can be written in standard form: r=c/b,where c and b are relatively prime integers; and with b greater than or equal to 2. There are seven theorems, one proposition, and one lemma; Lemma1, in this paper. Lemma1 is a very well known result, commonly known as Euclid's lemma.It is used repeatedly throughout this paper, and its proof can be "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.6820","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":""},"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":"2026-05-18T04:11:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Hxn2b1GrNfaBKmWiW4zpPfGGu9jFiwKgom+DTV9hFdEBCnbijq699TJogkB+cP416mxCkUzh/56LMBzXLkRzAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T02:49:51.968854Z"},"content_sha256":"6ef01d6bce1d1db28ed5b203b258b432b83e443c8dcff02aafb4f086f27ed739","schema_version":"1.0","event_id":"sha256:6ef01d6bce1d1db28ed5b203b258b432b83e443c8dcff02aafb4f086f27ed739"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/E232VXYLS4V247N74556PCEK7Q/bundle.json","state_url":"https://pith.science/pith/E232VXYLS4V247N74556PCEK7Q/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/E232VXYLS4V247N74556PCEK7Q/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":"2026-06-08T02:49:51Z","links":{"resolver":"https://pith.science/pith/E232VXYLS4V247N74556PCEK7Q","bundle":"https://pith.science/pith/E232VXYLS4V247N74556PCEK7Q/bundle.json","state":"https://pith.science/pith/E232VXYLS4V247N74556PCEK7Q/state.json","well_known_bundle":"https://pith.science/.well-known/pith/E232VXYLS4V247N74556PCEK7Q/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:E232VXYLS4V247N74556PCEK7Q","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":"ab405897560a91080cec449b54222125f5b748a399026d776984b45cb8db6ffc","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.GM","submitted_at":"2011-09-29T18:19:29Z","title_canon_sha256":"794d159e953835430dfd734be754b5364c5690f7025257ddcf3c660a9836a053"},"schema_version":"1.0","source":{"id":"1109.6820","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.6820","created_at":"2026-05-18T04:11:52Z"},{"alias_kind":"arxiv_version","alias_value":"1109.6820v1","created_at":"2026-05-18T04:11:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.6820","created_at":"2026-05-18T04:11:52Z"},{"alias_kind":"pith_short_12","alias_value":"E232VXYLS4V2","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_16","alias_value":"E232VXYLS4V247N7","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_8","alias_value":"E232VXYL","created_at":"2026-05-18T12:26:26Z"}],"graph_snapshots":[{"event_id":"sha256:6ef01d6bce1d1db28ed5b203b258b432b83e443c8dcff02aafb4f086f27ed739","target":"graph","created_at":"2026-05-18T04:11:52Z","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"},"paper":{"abstract_excerpt":"This short article is aimed at educators and teachers of mathematics.Its goal is simple and direct:to explore some of the basic/elementary properties of proper rational numbers.A proper rational number is a rational which is not an integer. A proper rational r can be written in standard form: r=c/b,where c and b are relatively prime integers; and with b greater than or equal to 2. There are seven theorems, one proposition, and one lemma; Lemma1, in this paper. Lemma1 is a very well known result, commonly known as Euclid's lemma.It is used repeatedly throughout this paper, and its proof can be ","authors_text":"Konstantine Zelator","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.GM","submitted_at":"2011-09-29T18:19:29Z","title":"Properties of proper rational numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.6820","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:aaa8765bf492e5cfd2ff1cf4d55f6adc32bc0fefe3b782d912416594d0e636ac","target":"record","created_at":"2026-05-18T04:11:52Z","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":"ab405897560a91080cec449b54222125f5b748a399026d776984b45cb8db6ffc","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.GM","submitted_at":"2011-09-29T18:19:29Z","title_canon_sha256":"794d159e953835430dfd734be754b5364c5690f7025257ddcf3c660a9836a053"},"schema_version":"1.0","source":{"id":"1109.6820","kind":"arxiv","version":1}},"canonical_sha256":"26b7aadf0b972bae7dbfe77be7888afc0395b40007e583844313022041af8ed6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"26b7aadf0b972bae7dbfe77be7888afc0395b40007e583844313022041af8ed6","first_computed_at":"2026-05-18T04:11:52.476239Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:11:52.476239Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"iWtA9tHdMJDwjAlGiB6hNgWYVF/Vv5FKB7HBIfrDasGy3bZ2vBS2t3kmCsPfPZwOODNPajioZYTrVh5NIkIOAA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:11:52.476866Z","signed_message":"canonical_sha256_bytes"},"source_id":"1109.6820","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:aaa8765bf492e5cfd2ff1cf4d55f6adc32bc0fefe3b782d912416594d0e636ac","sha256:6ef01d6bce1d1db28ed5b203b258b432b83e443c8dcff02aafb4f086f27ed739"],"state_sha256":"3ee49208eac477ebcf55a2aebc2c1b897720bcd55df38fbb8dfaef4930f635c4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wTEriWjg7hqnreDuZnGzYWR5DMfcAfA5AY/Y2am2t/9Z5p/niezMURHDJBHZn+zhwAikWgIq+nZyAX8IJmo7Dg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T02:49:51.972976Z","bundle_sha256":"5587584a722ef3ace41ef9456c43a2598df0c1d80050651837d328c7b168964e"}}