{"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"}