{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:JJGSBNETDCPFZPLYQPNYVYUB2Z","short_pith_number":"pith:JJGSBNET","canonical_record":{"source":{"id":"1511.08142","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-11-25T18:29:02Z","cross_cats_sorted":["math-ph","math.MP","math.OA"],"title_canon_sha256":"08fef35c35dadd4344403669f697da0fb581c9e427046d2ad1f35c37cdfee18c","abstract_canon_sha256":"d5319b890ecfea12059efaa4c996f6a97daabc1f292271b0ffdacc34f411d8a7"},"schema_version":"1.0"},"canonical_sha256":"4a4d20b493189e5cbd7883db8ae281d64ae9598ff6457cd3347fcad2046df576","source":{"kind":"arxiv","id":"1511.08142","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.08142","created_at":"2026-05-18T01:25:57Z"},{"alias_kind":"arxiv_version","alias_value":"1511.08142v1","created_at":"2026-05-18T01:25:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.08142","created_at":"2026-05-18T01:25:57Z"},{"alias_kind":"pith_short_12","alias_value":"JJGSBNETDCPF","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JJGSBNETDCPFZPLY","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JJGSBNET","created_at":"2026-05-18T12:29:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:JJGSBNETDCPFZPLYQPNYVYUB2Z","target":"record","payload":{"canonical_record":{"source":{"id":"1511.08142","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-11-25T18:29:02Z","cross_cats_sorted":["math-ph","math.MP","math.OA"],"title_canon_sha256":"08fef35c35dadd4344403669f697da0fb581c9e427046d2ad1f35c37cdfee18c","abstract_canon_sha256":"d5319b890ecfea12059efaa4c996f6a97daabc1f292271b0ffdacc34f411d8a7"},"schema_version":"1.0"},"canonical_sha256":"4a4d20b493189e5cbd7883db8ae281d64ae9598ff6457cd3347fcad2046df576","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:25:57.251551Z","signature_b64":"GZ+35xWybMxpvsxRPz6ZqLKdVuwc1Zol9gz+dmWsQBNoOkjCxKnjtdUhA8BjBufgmLVrGmpPUR4qb0t3IW/NDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4a4d20b493189e5cbd7883db8ae281d64ae9598ff6457cd3347fcad2046df576","last_reissued_at":"2026-05-18T01:25:57.250838Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:25:57.250838Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1511.08142","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":"2026-05-18T01:25:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tccW3zDzM/rmeSyW1iP9a/YftHJuA9DimW1hKsvFRSZd0Cj28EliJVXsNjKW28GpnXe1MC5qX5RmDZp38Ze8DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T06:46:37.390943Z"},"content_sha256":"7ab981cbeb95d0ebbcb6a582fbc5e45a73b060a43f04ede7a5b1e08471dcca93","schema_version":"1.0","event_id":"sha256:7ab981cbeb95d0ebbcb6a582fbc5e45a73b060a43f04ede7a5b1e08471dcca93"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:JJGSBNETDCPFZPLYQPNYVYUB2Z","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Factoring an Operator Using Elements of its Kernel","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","math.OA"],"primary_cat":"math.RA","authors_text":"Alex Kasman","submitted_at":"2015-11-25T18:29:02Z","abstract_excerpt":"A well-known theorem factors a scalar coefficient differential operator given a linearly independent set of functions in its kernel. The goal of this paper is to generalize this useful result to other types of operators. In place of the derivation $\\partial$ acting on some ring of functions, this paper considers the more general situation of an endomorphism $\\mathfrak{D}$ acting on a unital associative algebra. The operators considered, analogous to differential operators, are those which can be written as a finite sum of powers of $\\mathfrak{D}$ followed by left multiplication by elements of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.08142","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":"2026-05-18T01:25:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"inzARWtYj8fzsMyP4y7MSS6yAOCBRCCJy6CsAis22PhhiOIehpdHNErkDITey+nxJbj+OfQg2qI1X1riLntmBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T06:46:37.391563Z"},"content_sha256":"ae782768a4a15091d9c97df807fa4232859ee80b6907507bd48b97f4a0d7f9f1","schema_version":"1.0","event_id":"sha256:ae782768a4a15091d9c97df807fa4232859ee80b6907507bd48b97f4a0d7f9f1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JJGSBNETDCPFZPLYQPNYVYUB2Z/bundle.json","state_url":"https://pith.science/pith/JJGSBNETDCPFZPLYQPNYVYUB2Z/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JJGSBNETDCPFZPLYQPNYVYUB2Z/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":"2026-06-05T06:46:37Z","links":{"resolver":"https://pith.science/pith/JJGSBNETDCPFZPLYQPNYVYUB2Z","bundle":"https://pith.science/pith/JJGSBNETDCPFZPLYQPNYVYUB2Z/bundle.json","state":"https://pith.science/pith/JJGSBNETDCPFZPLYQPNYVYUB2Z/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JJGSBNETDCPFZPLYQPNYVYUB2Z/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:JJGSBNETDCPFZPLYQPNYVYUB2Z","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":"d5319b890ecfea12059efaa4c996f6a97daabc1f292271b0ffdacc34f411d8a7","cross_cats_sorted":["math-ph","math.MP","math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-11-25T18:29:02Z","title_canon_sha256":"08fef35c35dadd4344403669f697da0fb581c9e427046d2ad1f35c37cdfee18c"},"schema_version":"1.0","source":{"id":"1511.08142","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.08142","created_at":"2026-05-18T01:25:57Z"},{"alias_kind":"arxiv_version","alias_value":"1511.08142v1","created_at":"2026-05-18T01:25:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.08142","created_at":"2026-05-18T01:25:57Z"},{"alias_kind":"pith_short_12","alias_value":"JJGSBNETDCPF","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JJGSBNETDCPFZPLY","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JJGSBNET","created_at":"2026-05-18T12:29:27Z"}],"graph_snapshots":[{"event_id":"sha256:ae782768a4a15091d9c97df807fa4232859ee80b6907507bd48b97f4a0d7f9f1","target":"graph","created_at":"2026-05-18T01:25:57Z","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":"A well-known theorem factors a scalar coefficient differential operator given a linearly independent set of functions in its kernel. The goal of this paper is to generalize this useful result to other types of operators. In place of the derivation $\\partial$ acting on some ring of functions, this paper considers the more general situation of an endomorphism $\\mathfrak{D}$ acting on a unital associative algebra. The operators considered, analogous to differential operators, are those which can be written as a finite sum of powers of $\\mathfrak{D}$ followed by left multiplication by elements of ","authors_text":"Alex Kasman","cross_cats":["math-ph","math.MP","math.OA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-11-25T18:29:02Z","title":"On Factoring an Operator Using Elements of its Kernel"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.08142","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":"sha256:7ab981cbeb95d0ebbcb6a582fbc5e45a73b060a43f04ede7a5b1e08471dcca93","target":"record","created_at":"2026-05-18T01:25:57Z","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":"d5319b890ecfea12059efaa4c996f6a97daabc1f292271b0ffdacc34f411d8a7","cross_cats_sorted":["math-ph","math.MP","math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-11-25T18:29:02Z","title_canon_sha256":"08fef35c35dadd4344403669f697da0fb581c9e427046d2ad1f35c37cdfee18c"},"schema_version":"1.0","source":{"id":"1511.08142","kind":"arxiv","version":1}},"canonical_sha256":"4a4d20b493189e5cbd7883db8ae281d64ae9598ff6457cd3347fcad2046df576","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4a4d20b493189e5cbd7883db8ae281d64ae9598ff6457cd3347fcad2046df576","first_computed_at":"2026-05-18T01:25:57.250838Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:25:57.250838Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GZ+35xWybMxpvsxRPz6ZqLKdVuwc1Zol9gz+dmWsQBNoOkjCxKnjtdUhA8BjBufgmLVrGmpPUR4qb0t3IW/NDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:25:57.251551Z","signed_message":"canonical_sha256_bytes"},"source_id":"1511.08142","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7ab981cbeb95d0ebbcb6a582fbc5e45a73b060a43f04ede7a5b1e08471dcca93","sha256:ae782768a4a15091d9c97df807fa4232859ee80b6907507bd48b97f4a0d7f9f1"],"state_sha256":"ed9ca033c5187a902d2082bf4536670458dc9b5cf76a04eaf5e62558e8a47191"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EPMNvjaNq+5gZ4TtB43uMkK/242uEFeqqNNchMkviC/8OaewPR7PEH5Qwf0h4lLC7cbkOMLpiTEkfgHMzLhsDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T06:46:37.394741Z","bundle_sha256":"d29b247fbf2b321057e3fdc4a4867d1d9e18a55da64571d472bdf0e158921914"}}