{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:HY4J34B5PCYGVW2Z36LO2B2VNA","short_pith_number":"pith:HY4J34B5","canonical_record":{"source":{"id":"1606.07878","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-06-25T07:47:33Z","cross_cats_sorted":[],"title_canon_sha256":"e6cf055c2e77022980d43efe912f6c7babc6474153370f94dbc44ee8b86d86dc","abstract_canon_sha256":"d9501eb032664e40bad5143c662cb1a1debe0b86dc9bddfbedb5039d097b2455"},"schema_version":"1.0"},"canonical_sha256":"3e389df03d78b06adb59df96ed0755681614c5e37166dc0063f38358876d857d","source":{"kind":"arxiv","id":"1606.07878","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.07878","created_at":"2026-05-18T01:11:54Z"},{"alias_kind":"arxiv_version","alias_value":"1606.07878v1","created_at":"2026-05-18T01:11:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.07878","created_at":"2026-05-18T01:11:54Z"},{"alias_kind":"pith_short_12","alias_value":"HY4J34B5PCYG","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_16","alias_value":"HY4J34B5PCYGVW2Z","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_8","alias_value":"HY4J34B5","created_at":"2026-05-18T12:30:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:HY4J34B5PCYGVW2Z36LO2B2VNA","target":"record","payload":{"canonical_record":{"source":{"id":"1606.07878","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-06-25T07:47:33Z","cross_cats_sorted":[],"title_canon_sha256":"e6cf055c2e77022980d43efe912f6c7babc6474153370f94dbc44ee8b86d86dc","abstract_canon_sha256":"d9501eb032664e40bad5143c662cb1a1debe0b86dc9bddfbedb5039d097b2455"},"schema_version":"1.0"},"canonical_sha256":"3e389df03d78b06adb59df96ed0755681614c5e37166dc0063f38358876d857d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:11:54.235058Z","signature_b64":"HARu+Nlw7O+oQiGqUSRZFlJYqNim+I9YmafRcfYRZzFcIXYwsUI6AMMPosowco+ixkbapgu7g+1qH+96HuogBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3e389df03d78b06adb59df96ed0755681614c5e37166dc0063f38358876d857d","last_reissued_at":"2026-05-18T01:11:54.234707Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:11:54.234707Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1606.07878","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":"2026-05-18T01:11:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"00vf7V/U0bQo5r3CKDECCS69lGP38EGsvVRlv6tOqk9EfNHyi7moJOd0YsLy/bqXHA+fSIHzOWdNTB/ZhHZeCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T15:56:17.280580Z"},"content_sha256":"e02592d0f1df78833d42aad60d9545248054c84382d6e268bf9e64fd28b7f39c","schema_version":"1.0","event_id":"sha256:e02592d0f1df78833d42aad60d9545248054c84382d6e268bf9e64fd28b7f39c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:HY4J34B5PCYGVW2Z36LO2B2VNA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On linear equations arising in Combinatorics (Part III)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Masood Aryapoor","submitted_at":"2016-06-25T07:47:33Z","abstract_excerpt":"In the first two papers, the author embarked on a study of classes of linear equations over integers satisfying a \"Farkas-type\" property. As the third paper in this study, the present paper deals with another class of linear equations over integers that has a similar \"Farkas-type\" property. Furthermore it is shown that if an arbitrary system of equations over integers satisfies the conditions imposed by Farkas' lemma then it has rational solutions of a special type."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.07878","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":"2026-05-18T01:11:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3RDDnH+6io2b12BHQS1qNQGoIy9/2G0Z/30wzwoy11Jz+DRTlhxATKAzap0bGRfdw3+htFdLei2s/5Mr4y2FAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T15:56:17.280913Z"},"content_sha256":"df8d17acb788f3c3c9dce883a9535516e9ed86fe9aa264e0587cefbf0a285410","schema_version":"1.0","event_id":"sha256:df8d17acb788f3c3c9dce883a9535516e9ed86fe9aa264e0587cefbf0a285410"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HY4J34B5PCYGVW2Z36LO2B2VNA/bundle.json","state_url":"https://pith.science/pith/HY4J34B5PCYGVW2Z36LO2B2VNA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HY4J34B5PCYGVW2Z36LO2B2VNA/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":"2026-06-20T15:56:17Z","links":{"resolver":"https://pith.science/pith/HY4J34B5PCYGVW2Z36LO2B2VNA","bundle":"https://pith.science/pith/HY4J34B5PCYGVW2Z36LO2B2VNA/bundle.json","state":"https://pith.science/pith/HY4J34B5PCYGVW2Z36LO2B2VNA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HY4J34B5PCYGVW2Z36LO2B2VNA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:HY4J34B5PCYGVW2Z36LO2B2VNA","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":"d9501eb032664e40bad5143c662cb1a1debe0b86dc9bddfbedb5039d097b2455","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-06-25T07:47:33Z","title_canon_sha256":"e6cf055c2e77022980d43efe912f6c7babc6474153370f94dbc44ee8b86d86dc"},"schema_version":"1.0","source":{"id":"1606.07878","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.07878","created_at":"2026-05-18T01:11:54Z"},{"alias_kind":"arxiv_version","alias_value":"1606.07878v1","created_at":"2026-05-18T01:11:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.07878","created_at":"2026-05-18T01:11:54Z"},{"alias_kind":"pith_short_12","alias_value":"HY4J34B5PCYG","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_16","alias_value":"HY4J34B5PCYGVW2Z","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_8","alias_value":"HY4J34B5","created_at":"2026-05-18T12:30:22Z"}],"graph_snapshots":[{"event_id":"sha256:df8d17acb788f3c3c9dce883a9535516e9ed86fe9aa264e0587cefbf0a285410","target":"graph","created_at":"2026-05-18T01:11:54Z","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":"In the first two papers, the author embarked on a study of classes of linear equations over integers satisfying a \"Farkas-type\" property. As the third paper in this study, the present paper deals with another class of linear equations over integers that has a similar \"Farkas-type\" property. Furthermore it is shown that if an arbitrary system of equations over integers satisfies the conditions imposed by Farkas' lemma then it has rational solutions of a special type.","authors_text":"Masood Aryapoor","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-06-25T07:47:33Z","title":"On linear equations arising in Combinatorics (Part III)"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.07878","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:e02592d0f1df78833d42aad60d9545248054c84382d6e268bf9e64fd28b7f39c","target":"record","created_at":"2026-05-18T01:11:54Z","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":"d9501eb032664e40bad5143c662cb1a1debe0b86dc9bddfbedb5039d097b2455","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-06-25T07:47:33Z","title_canon_sha256":"e6cf055c2e77022980d43efe912f6c7babc6474153370f94dbc44ee8b86d86dc"},"schema_version":"1.0","source":{"id":"1606.07878","kind":"arxiv","version":1}},"canonical_sha256":"3e389df03d78b06adb59df96ed0755681614c5e37166dc0063f38358876d857d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3e389df03d78b06adb59df96ed0755681614c5e37166dc0063f38358876d857d","first_computed_at":"2026-05-18T01:11:54.234707Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:11:54.234707Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HARu+Nlw7O+oQiGqUSRZFlJYqNim+I9YmafRcfYRZzFcIXYwsUI6AMMPosowco+ixkbapgu7g+1qH+96HuogBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:11:54.235058Z","signed_message":"canonical_sha256_bytes"},"source_id":"1606.07878","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e02592d0f1df78833d42aad60d9545248054c84382d6e268bf9e64fd28b7f39c","sha256:df8d17acb788f3c3c9dce883a9535516e9ed86fe9aa264e0587cefbf0a285410"],"state_sha256":"0258153dcdce4f93cd613928f125561c2d10a2d821e6c7187f5e194a9687e779"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lVAX93w6DDukBrBk+DIm5oEhSxd3vChEC/LDaxNNzrUTX4CS5pMgcNRXhrbfI5ev6whdKGNt4MYPadF87aLiDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T15:56:17.282812Z","bundle_sha256":"32e43b371c7f52c0edc05fad5ffdc8f7028ef9dc22edd5b36a25dbee58d2f13f"}}