{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:NJTT3CWVUSAJ7PVAYCKOWCEVGC","short_pith_number":"pith:NJTT3CWV","canonical_record":{"source":{"id":"1504.02805","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-04-10T22:22:30Z","cross_cats_sorted":[],"title_canon_sha256":"4c2f33f025610a1a6fd9e75f5b9146cd33c8b4906262b354ef60c6dc4027145c","abstract_canon_sha256":"7eea121598fa609391e6937a50a0d7629d2905af4a3a6770fdf4e068b9b9a92d"},"schema_version":"1.0"},"canonical_sha256":"6a673d8ad5a4809fbea0c094eb08953097f07d3e5674912001aa085f1b735cbc","source":{"kind":"arxiv","id":"1504.02805","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.02805","created_at":"2026-05-18T02:18:59Z"},{"alias_kind":"arxiv_version","alias_value":"1504.02805v1","created_at":"2026-05-18T02:18:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.02805","created_at":"2026-05-18T02:18:59Z"},{"alias_kind":"pith_short_12","alias_value":"NJTT3CWVUSAJ","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_16","alias_value":"NJTT3CWVUSAJ7PVA","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_8","alias_value":"NJTT3CWV","created_at":"2026-05-18T12:29:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:NJTT3CWVUSAJ7PVAYCKOWCEVGC","target":"record","payload":{"canonical_record":{"source":{"id":"1504.02805","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-04-10T22:22:30Z","cross_cats_sorted":[],"title_canon_sha256":"4c2f33f025610a1a6fd9e75f5b9146cd33c8b4906262b354ef60c6dc4027145c","abstract_canon_sha256":"7eea121598fa609391e6937a50a0d7629d2905af4a3a6770fdf4e068b9b9a92d"},"schema_version":"1.0"},"canonical_sha256":"6a673d8ad5a4809fbea0c094eb08953097f07d3e5674912001aa085f1b735cbc","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:18:59.935920Z","signature_b64":"A269cLzjPR1ecV8Tac5ttnKqq5eDVX8RzHJMjtQblG8kZ8bp/Z+gHMKES+uj6jecprszvtKuMgiZmVmVKUu2DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6a673d8ad5a4809fbea0c094eb08953097f07d3e5674912001aa085f1b735cbc","last_reissued_at":"2026-05-18T02:18:59.935273Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:18:59.935273Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1504.02805","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":"2026-05-18T02:18:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pACyV4ZXBUZgB1mQUKkTDmDn8EO6BsCw8u3Fhy+zJHso09I7cRJ7SEIXLouMi7ue/63tJIFjyxJpqmKhDiWpDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T14:38:20.157333Z"},"content_sha256":"cc45f64ce7fa4deaef9c3ca29b34a52a92cdc7af4ebbcfda63398b9b87d0a4ba","schema_version":"1.0","event_id":"sha256:cc45f64ce7fa4deaef9c3ca29b34a52a92cdc7af4ebbcfda63398b9b87d0a4ba"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:NJTT3CWVUSAJ7PVAYCKOWCEVGC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"DNR and incomparable Turing degrees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Michael McInerney, Mingzhong Cai, Noam Greenberg","submitted_at":"2015-04-10T22:22:30Z","abstract_excerpt":"We construct an increasing $\\omega$-sequence $(a_n)$ of Turing degrees which forms an initial segment of the Turing degrees, and such that each~$a_{n+1}$ is diagonally noncomputable relative to $a_n$. It follows that the~$\\mathsf{DNR}$ principle of reverse mathematics does not imply the existence of Turing incomparable degrees."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.02805","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":"2026-05-18T02:18:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"33rA3hiMOvpGA5DHI5+yodQv5+GW3ywRb3C76+Q8ck6sLO1YETYmB5DhmjX/GOXwYa36bwDTF7SGYEcCVu34DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T14:38:20.157693Z"},"content_sha256":"40ab435860f62317f51445b2515f09786b17a24f8ae9992cb6ed2a9b5ba8d6a7","schema_version":"1.0","event_id":"sha256:40ab435860f62317f51445b2515f09786b17a24f8ae9992cb6ed2a9b5ba8d6a7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NJTT3CWVUSAJ7PVAYCKOWCEVGC/bundle.json","state_url":"https://pith.science/pith/NJTT3CWVUSAJ7PVAYCKOWCEVGC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NJTT3CWVUSAJ7PVAYCKOWCEVGC/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":"2026-06-01T14:38:20Z","links":{"resolver":"https://pith.science/pith/NJTT3CWVUSAJ7PVAYCKOWCEVGC","bundle":"https://pith.science/pith/NJTT3CWVUSAJ7PVAYCKOWCEVGC/bundle.json","state":"https://pith.science/pith/NJTT3CWVUSAJ7PVAYCKOWCEVGC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NJTT3CWVUSAJ7PVAYCKOWCEVGC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:NJTT3CWVUSAJ7PVAYCKOWCEVGC","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":"7eea121598fa609391e6937a50a0d7629d2905af4a3a6770fdf4e068b9b9a92d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-04-10T22:22:30Z","title_canon_sha256":"4c2f33f025610a1a6fd9e75f5b9146cd33c8b4906262b354ef60c6dc4027145c"},"schema_version":"1.0","source":{"id":"1504.02805","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.02805","created_at":"2026-05-18T02:18:59Z"},{"alias_kind":"arxiv_version","alias_value":"1504.02805v1","created_at":"2026-05-18T02:18:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.02805","created_at":"2026-05-18T02:18:59Z"},{"alias_kind":"pith_short_12","alias_value":"NJTT3CWVUSAJ","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_16","alias_value":"NJTT3CWVUSAJ7PVA","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_8","alias_value":"NJTT3CWV","created_at":"2026-05-18T12:29:32Z"}],"graph_snapshots":[{"event_id":"sha256:40ab435860f62317f51445b2515f09786b17a24f8ae9992cb6ed2a9b5ba8d6a7","target":"graph","created_at":"2026-05-18T02:18:59Z","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":"We construct an increasing $\\omega$-sequence $(a_n)$ of Turing degrees which forms an initial segment of the Turing degrees, and such that each~$a_{n+1}$ is diagonally noncomputable relative to $a_n$. It follows that the~$\\mathsf{DNR}$ principle of reverse mathematics does not imply the existence of Turing incomparable degrees.","authors_text":"Michael McInerney, Mingzhong Cai, Noam Greenberg","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-04-10T22:22:30Z","title":"DNR and incomparable Turing degrees"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.02805","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:cc45f64ce7fa4deaef9c3ca29b34a52a92cdc7af4ebbcfda63398b9b87d0a4ba","target":"record","created_at":"2026-05-18T02:18:59Z","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":"7eea121598fa609391e6937a50a0d7629d2905af4a3a6770fdf4e068b9b9a92d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-04-10T22:22:30Z","title_canon_sha256":"4c2f33f025610a1a6fd9e75f5b9146cd33c8b4906262b354ef60c6dc4027145c"},"schema_version":"1.0","source":{"id":"1504.02805","kind":"arxiv","version":1}},"canonical_sha256":"6a673d8ad5a4809fbea0c094eb08953097f07d3e5674912001aa085f1b735cbc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6a673d8ad5a4809fbea0c094eb08953097f07d3e5674912001aa085f1b735cbc","first_computed_at":"2026-05-18T02:18:59.935273Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:18:59.935273Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"A269cLzjPR1ecV8Tac5ttnKqq5eDVX8RzHJMjtQblG8kZ8bp/Z+gHMKES+uj6jecprszvtKuMgiZmVmVKUu2DA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:18:59.935920Z","signed_message":"canonical_sha256_bytes"},"source_id":"1504.02805","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cc45f64ce7fa4deaef9c3ca29b34a52a92cdc7af4ebbcfda63398b9b87d0a4ba","sha256:40ab435860f62317f51445b2515f09786b17a24f8ae9992cb6ed2a9b5ba8d6a7"],"state_sha256":"37d0744f6f4edbf9cf990c3e3bbca61e5e207d054c318311c62c0b113b85f96d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wLQtWTFT5Ae/AIbsPnopdk40B68CXPOGgafWW4Ck9KbavZplwROB1osX/9LTcpjqNz8VYZwJ3dRfD414C0GUCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T14:38:20.159737Z","bundle_sha256":"f87803365cfb68ca9d551444938d3b079d35893ce763a3e030b03dacf74e7f16"}}