{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:AT5RIQKLMXCZQISKVX3OHU6IKT","short_pith_number":"pith:AT5RIQKL","canonical_record":{"source":{"id":"1201.3223","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-01-16T11:45:22Z","cross_cats_sorted":["math.AP","math.MP"],"title_canon_sha256":"1b080a37ae3fd200770997260920c30abd484105ccda4890c3a904b674ae8baf","abstract_canon_sha256":"fd1994eb054736b25dc3def9f3601121fa344fcf5a0c2cf52ae4508effabb4b9"},"schema_version":"1.0"},"canonical_sha256":"04fb14414b65c598224aadf6e3d3c854e37f8db3c5583b5911c0831332826df6","source":{"kind":"arxiv","id":"1201.3223","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.3223","created_at":"2026-05-18T00:29:02Z"},{"alias_kind":"arxiv_version","alias_value":"1201.3223v4","created_at":"2026-05-18T00:29:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.3223","created_at":"2026-05-18T00:29:02Z"},{"alias_kind":"pith_short_12","alias_value":"AT5RIQKLMXCZ","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"AT5RIQKLMXCZQISK","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"AT5RIQKL","created_at":"2026-05-18T12:26:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:AT5RIQKLMXCZQISKVX3OHU6IKT","target":"record","payload":{"canonical_record":{"source":{"id":"1201.3223","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-01-16T11:45:22Z","cross_cats_sorted":["math.AP","math.MP"],"title_canon_sha256":"1b080a37ae3fd200770997260920c30abd484105ccda4890c3a904b674ae8baf","abstract_canon_sha256":"fd1994eb054736b25dc3def9f3601121fa344fcf5a0c2cf52ae4508effabb4b9"},"schema_version":"1.0"},"canonical_sha256":"04fb14414b65c598224aadf6e3d3c854e37f8db3c5583b5911c0831332826df6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:29:02.148367Z","signature_b64":"nzsr4PtOf1pZUdR43zC8jCBukvuw2bM3uThcvX1prCe7HO919zglOHNmCnTcSclgM5E7+gCNV4r27KYJPOdcDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"04fb14414b65c598224aadf6e3d3c854e37f8db3c5583b5911c0831332826df6","last_reissued_at":"2026-05-18T00:29:02.147919Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:29:02.147919Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1201.3223","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:29:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5PiK6KaLWWa9C8HNlVgud8VaC3llufCTgGHsA3kpvZQPB0gH1vijGLynMolqo2g5ii2884T6fBc7VJuMl5jsBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T13:54:11.193326Z"},"content_sha256":"4db06b69c1aaa1084ad81884cfeb1845694736a04138044fb86e4b1c99af5536","schema_version":"1.0","event_id":"sha256:4db06b69c1aaa1084ad81884cfeb1845694736a04138044fb86e4b1c99af5536"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:AT5RIQKLMXCZQISKVX3OHU6IKT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Singular reduction modules of differential equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MP"],"primary_cat":"math-ph","authors_text":"Michael Kunzinger, Roman O. Popovych, Vaycheslav M. Boyko","submitted_at":"2012-01-16T11:45:22Z","abstract_excerpt":"The notion of singular reduction modules, i.e., of singular modules of nonclassical (conditional) symmetry, of differential equations is introduced. It is shown that the derivation of nonclassical symmetries for differential equations can be improved by an in-depth prior study of the associated singular modules of vector fields. The form of differential functions and differential equations possessing parameterized families of singular modules is described up to point transformations. Singular cases of finding reduction modules are related to lowering the order of the corresponding reduced equa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.3223","kind":"arxiv","version":4},"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":"2026-05-18T00:29:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wDdIVM3smxTGZD5eyrcH7BejCYEMGIlP3H7+UBZRfGoYVhUN/251xNlFQzUh/t6JJu+54PaWwr9RhNP4xOJJCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T13:54:11.194002Z"},"content_sha256":"aea8b3b0fbfb37de36ec89bacb403229a7f06e851f9ba9b7774314517993763d","schema_version":"1.0","event_id":"sha256:aea8b3b0fbfb37de36ec89bacb403229a7f06e851f9ba9b7774314517993763d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AT5RIQKLMXCZQISKVX3OHU6IKT/bundle.json","state_url":"https://pith.science/pith/AT5RIQKLMXCZQISKVX3OHU6IKT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AT5RIQKLMXCZQISKVX3OHU6IKT/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":"2026-05-27T13:54:11Z","links":{"resolver":"https://pith.science/pith/AT5RIQKLMXCZQISKVX3OHU6IKT","bundle":"https://pith.science/pith/AT5RIQKLMXCZQISKVX3OHU6IKT/bundle.json","state":"https://pith.science/pith/AT5RIQKLMXCZQISKVX3OHU6IKT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AT5RIQKLMXCZQISKVX3OHU6IKT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:AT5RIQKLMXCZQISKVX3OHU6IKT","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":"fd1994eb054736b25dc3def9f3601121fa344fcf5a0c2cf52ae4508effabb4b9","cross_cats_sorted":["math.AP","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-01-16T11:45:22Z","title_canon_sha256":"1b080a37ae3fd200770997260920c30abd484105ccda4890c3a904b674ae8baf"},"schema_version":"1.0","source":{"id":"1201.3223","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.3223","created_at":"2026-05-18T00:29:02Z"},{"alias_kind":"arxiv_version","alias_value":"1201.3223v4","created_at":"2026-05-18T00:29:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.3223","created_at":"2026-05-18T00:29:02Z"},{"alias_kind":"pith_short_12","alias_value":"AT5RIQKLMXCZ","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"AT5RIQKLMXCZQISK","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"AT5RIQKL","created_at":"2026-05-18T12:26:58Z"}],"graph_snapshots":[{"event_id":"sha256:aea8b3b0fbfb37de36ec89bacb403229a7f06e851f9ba9b7774314517993763d","target":"graph","created_at":"2026-05-18T00:29:02Z","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":"The notion of singular reduction modules, i.e., of singular modules of nonclassical (conditional) symmetry, of differential equations is introduced. It is shown that the derivation of nonclassical symmetries for differential equations can be improved by an in-depth prior study of the associated singular modules of vector fields. The form of differential functions and differential equations possessing parameterized families of singular modules is described up to point transformations. Singular cases of finding reduction modules are related to lowering the order of the corresponding reduced equa","authors_text":"Michael Kunzinger, Roman O. Popovych, Vaycheslav M. Boyko","cross_cats":["math.AP","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-01-16T11:45:22Z","title":"Singular reduction modules of differential equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.3223","kind":"arxiv","version":4},"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":"sha256:4db06b69c1aaa1084ad81884cfeb1845694736a04138044fb86e4b1c99af5536","target":"record","created_at":"2026-05-18T00:29:02Z","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":"fd1994eb054736b25dc3def9f3601121fa344fcf5a0c2cf52ae4508effabb4b9","cross_cats_sorted":["math.AP","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-01-16T11:45:22Z","title_canon_sha256":"1b080a37ae3fd200770997260920c30abd484105ccda4890c3a904b674ae8baf"},"schema_version":"1.0","source":{"id":"1201.3223","kind":"arxiv","version":4}},"canonical_sha256":"04fb14414b65c598224aadf6e3d3c854e37f8db3c5583b5911c0831332826df6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"04fb14414b65c598224aadf6e3d3c854e37f8db3c5583b5911c0831332826df6","first_computed_at":"2026-05-18T00:29:02.147919Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:29:02.147919Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nzsr4PtOf1pZUdR43zC8jCBukvuw2bM3uThcvX1prCe7HO919zglOHNmCnTcSclgM5E7+gCNV4r27KYJPOdcDw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:29:02.148367Z","signed_message":"canonical_sha256_bytes"},"source_id":"1201.3223","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4db06b69c1aaa1084ad81884cfeb1845694736a04138044fb86e4b1c99af5536","sha256:aea8b3b0fbfb37de36ec89bacb403229a7f06e851f9ba9b7774314517993763d"],"state_sha256":"a4a9b5de44e8bf4aa077e7bc871a7aa38ae45559d8d7ab6b7bd6fa687cb8b17c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"og5ZkUQXePF4TDzRQKSpYTEX6B4QTUMzzpvcgw8SBYVZ2it7ZPLbtJ1RKFUQDntu5lTKCoCpAq70DO/nqFYcCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T13:54:11.198223Z","bundle_sha256":"c84ba6ee3b29e36392a5224bedacf5b586e12d5bfe0269ee3820e08dc601036f"}}