{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:7S7WHL7Y5GTKECKMWROO4UJM7X","short_pith_number":"pith:7S7WHL7Y","canonical_record":{"source":{"id":"1206.5114","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.HO","submitted_at":"2012-06-22T11:14:09Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"1d178062a5bd3dbed7af7ae75c5b07474c77367b5378b21bbb8c57e9024ae1f8","abstract_canon_sha256":"2d5458550fc61fe7ffc4f02d53903a077b35099c5492a4779dbbb4a08790c603"},"schema_version":"1.0"},"canonical_sha256":"fcbf63aff8e9a6a2094cb45cee512cfdd0c45e17d0d53e6ed8779b89a4495a11","source":{"kind":"arxiv","id":"1206.5114","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.5114","created_at":"2026-05-18T03:52:52Z"},{"alias_kind":"arxiv_version","alias_value":"1206.5114v1","created_at":"2026-05-18T03:52:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.5114","created_at":"2026-05-18T03:52:52Z"},{"alias_kind":"pith_short_12","alias_value":"7S7WHL7Y5GTK","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"7S7WHL7Y5GTKECKM","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"7S7WHL7Y","created_at":"2026-05-18T12:26:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:7S7WHL7Y5GTKECKMWROO4UJM7X","target":"record","payload":{"canonical_record":{"source":{"id":"1206.5114","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.HO","submitted_at":"2012-06-22T11:14:09Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"1d178062a5bd3dbed7af7ae75c5b07474c77367b5378b21bbb8c57e9024ae1f8","abstract_canon_sha256":"2d5458550fc61fe7ffc4f02d53903a077b35099c5492a4779dbbb4a08790c603"},"schema_version":"1.0"},"canonical_sha256":"fcbf63aff8e9a6a2094cb45cee512cfdd0c45e17d0d53e6ed8779b89a4495a11","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:52:52.212396Z","signature_b64":"wUy6zksMjqeWYAc90QmDaf33Q8rcW4P1K+6EH8cxu/7Qc3D+c5pwBZTxPWqkwF/mui2srKn3dwGqJqWRRkzvAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fcbf63aff8e9a6a2094cb45cee512cfdd0c45e17d0d53e6ed8779b89a4495a11","last_reissued_at":"2026-05-18T03:52:52.211551Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:52:52.211551Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1206.5114","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":"2026-05-18T03:52:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uNiNTWQiklWzRP/avdZX3yd4rF228D/RfCui+gm00I/q9hgEZKC9NYcT8uav/A5ToiDuYbXZT266JcUOcituBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T05:14:46.383647Z"},"content_sha256":"4c66f16a90a678261ce4e44693c2ef329f5c11a84aa59f871af06eb344084985","schema_version":"1.0","event_id":"sha256:4c66f16a90a678261ce4e44693c2ef329f5c11a84aa59f871af06eb344084985"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:7S7WHL7Y5GTKECKMWROO4UJM7X","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Multivariable Chinese Remainder Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.HO","authors_text":"Oliver Knill","submitted_at":"2012-06-22T11:14:09Z","abstract_excerpt":"Using an adaptation of Qin Jiushao's method from the 13th century, it is possible to prove that a system of linear modular equations\na(i,1) x(i) + ... + a(i,n) x(n) = b(i) mod m(i), i=1, ..., n has integer solutions if m(i)>1 are pairwise relatively prime and in each row, at least one matrix element a(i,j) is relatively prime to m(i). The Chinese remainder theorem is the special case, where A has only one column."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.5114","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":"2026-05-18T03:52:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DnmmPGPCLoBLauM3WbH1RZuOVDbCwU4rcyEoYtKGAO14MJ8Wtas9TmaSwDO0j838V9SjR9+YBY1MAOTNUqPJCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T05:14:46.384133Z"},"content_sha256":"5d0bf014cb2dc512e6156ceb73b34fd4f6e256c1a705a50a237693053661d74e","schema_version":"1.0","event_id":"sha256:5d0bf014cb2dc512e6156ceb73b34fd4f6e256c1a705a50a237693053661d74e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7S7WHL7Y5GTKECKMWROO4UJM7X/bundle.json","state_url":"https://pith.science/pith/7S7WHL7Y5GTKECKMWROO4UJM7X/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7S7WHL7Y5GTKECKMWROO4UJM7X/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":"2026-05-26T05:14:46Z","links":{"resolver":"https://pith.science/pith/7S7WHL7Y5GTKECKMWROO4UJM7X","bundle":"https://pith.science/pith/7S7WHL7Y5GTKECKMWROO4UJM7X/bundle.json","state":"https://pith.science/pith/7S7WHL7Y5GTKECKMWROO4UJM7X/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7S7WHL7Y5GTKECKMWROO4UJM7X/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:7S7WHL7Y5GTKECKMWROO4UJM7X","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":"2d5458550fc61fe7ffc4f02d53903a077b35099c5492a4779dbbb4a08790c603","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.HO","submitted_at":"2012-06-22T11:14:09Z","title_canon_sha256":"1d178062a5bd3dbed7af7ae75c5b07474c77367b5378b21bbb8c57e9024ae1f8"},"schema_version":"1.0","source":{"id":"1206.5114","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.5114","created_at":"2026-05-18T03:52:52Z"},{"alias_kind":"arxiv_version","alias_value":"1206.5114v1","created_at":"2026-05-18T03:52:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.5114","created_at":"2026-05-18T03:52:52Z"},{"alias_kind":"pith_short_12","alias_value":"7S7WHL7Y5GTK","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"7S7WHL7Y5GTKECKM","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"7S7WHL7Y","created_at":"2026-05-18T12:26:58Z"}],"graph_snapshots":[{"event_id":"sha256:5d0bf014cb2dc512e6156ceb73b34fd4f6e256c1a705a50a237693053661d74e","target":"graph","created_at":"2026-05-18T03:52:52Z","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":"Using an adaptation of Qin Jiushao's method from the 13th century, it is possible to prove that a system of linear modular equations\na(i,1) x(i) + ... + a(i,n) x(n) = b(i) mod m(i), i=1, ..., n has integer solutions if m(i)>1 are pairwise relatively prime and in each row, at least one matrix element a(i,j) is relatively prime to m(i). The Chinese remainder theorem is the special case, where A has only one column.","authors_text":"Oliver Knill","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.HO","submitted_at":"2012-06-22T11:14:09Z","title":"A Multivariable Chinese Remainder Theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.5114","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":"sha256:4c66f16a90a678261ce4e44693c2ef329f5c11a84aa59f871af06eb344084985","target":"record","created_at":"2026-05-18T03:52:52Z","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":"2d5458550fc61fe7ffc4f02d53903a077b35099c5492a4779dbbb4a08790c603","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.HO","submitted_at":"2012-06-22T11:14:09Z","title_canon_sha256":"1d178062a5bd3dbed7af7ae75c5b07474c77367b5378b21bbb8c57e9024ae1f8"},"schema_version":"1.0","source":{"id":"1206.5114","kind":"arxiv","version":1}},"canonical_sha256":"fcbf63aff8e9a6a2094cb45cee512cfdd0c45e17d0d53e6ed8779b89a4495a11","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fcbf63aff8e9a6a2094cb45cee512cfdd0c45e17d0d53e6ed8779b89a4495a11","first_computed_at":"2026-05-18T03:52:52.211551Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:52:52.211551Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wUy6zksMjqeWYAc90QmDaf33Q8rcW4P1K+6EH8cxu/7Qc3D+c5pwBZTxPWqkwF/mui2srKn3dwGqJqWRRkzvAg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:52:52.212396Z","signed_message":"canonical_sha256_bytes"},"source_id":"1206.5114","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4c66f16a90a678261ce4e44693c2ef329f5c11a84aa59f871af06eb344084985","sha256:5d0bf014cb2dc512e6156ceb73b34fd4f6e256c1a705a50a237693053661d74e"],"state_sha256":"921cda2c86b1dcd4360de9fc3ef3173b3da4b18216c2e319619912dc169b8953"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nKA0SpX3PM5QsHq8V/EPPUsBOX9deFEwX3VepUKitAfUsEzm/waYZIOHPouSQFNqI9iMjmkfR/dm4bm5F4GpDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T05:14:46.387357Z","bundle_sha256":"5793e21411649a4191ee29d0ff666cb3a320efe33b7a5066b9ba8b5201c886c7"}}