{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:XDGRABV7FDX5EIGRDSVRQA6VJD","short_pith_number":"pith:XDGRABV7","canonical_record":{"source":{"id":"1209.3282","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2012-09-14T18:48:35Z","cross_cats_sorted":[],"title_canon_sha256":"fb1c8849dc48c57d49c6abc7bfdc434237733c7a7fd47f17fb4bb9007f780f3d","abstract_canon_sha256":"b66698d284124f702061151b2eb89e3cbeb1813a318bcd952bcfad4c0dd6e078"},"schema_version":"1.0"},"canonical_sha256":"b8cd1006bf28efd220d11cab1803d548ec90f5372ab6005818e6aced2b552690","source":{"kind":"arxiv","id":"1209.3282","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.3282","created_at":"2026-05-18T03:45:32Z"},{"alias_kind":"arxiv_version","alias_value":"1209.3282v1","created_at":"2026-05-18T03:45:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.3282","created_at":"2026-05-18T03:45:32Z"},{"alias_kind":"pith_short_12","alias_value":"XDGRABV7FDX5","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"XDGRABV7FDX5EIGR","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"XDGRABV7","created_at":"2026-05-18T12:27:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:XDGRABV7FDX5EIGRDSVRQA6VJD","target":"record","payload":{"canonical_record":{"source":{"id":"1209.3282","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2012-09-14T18:48:35Z","cross_cats_sorted":[],"title_canon_sha256":"fb1c8849dc48c57d49c6abc7bfdc434237733c7a7fd47f17fb4bb9007f780f3d","abstract_canon_sha256":"b66698d284124f702061151b2eb89e3cbeb1813a318bcd952bcfad4c0dd6e078"},"schema_version":"1.0"},"canonical_sha256":"b8cd1006bf28efd220d11cab1803d548ec90f5372ab6005818e6aced2b552690","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:45:32.353391Z","signature_b64":"oqePTmo/HKjKP9F03+IvD1OB+Yvwu6+/if/4FJy44A0bYvj461wtTKTmLynr2DwAva83XB4GkpBla7kLQk77DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b8cd1006bf28efd220d11cab1803d548ec90f5372ab6005818e6aced2b552690","last_reissued_at":"2026-05-18T03:45:32.352958Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:45:32.352958Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1209.3282","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":"2026-05-18T03:45:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5csGfxQTXm3QG0QjWmFIpSZIpaQdqYSLtCMzZ/HcH90t4BuE3xAddqlXcCA63/17F2VUakLmRxOF8RZTiPTDDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T12:42:06.859041Z"},"content_sha256":"b2207346d7c9a57ffe9006c640d7015d503f9951219ac7f4298d0a9182d57c3f","schema_version":"1.0","event_id":"sha256:b2207346d7c9a57ffe9006c640d7015d503f9951219ac7f4298d0a9182d57c3f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:XDGRABV7FDX5EIGRDSVRQA6VJD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Limits to joining with generics and randoms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Adam R. Day, Damir D. Dzhafarov","submitted_at":"2012-09-14T18:48:35Z","abstract_excerpt":"Posner and Robinson (1981) proved that if $S \\subseteq \\omega$ is non-computable, then there exists a $G \\subseteq \\omega$ such that $S \\oplus G \\geq_T G'$. Shore and Slaman (1999) extended this result to all $n \\in \\omega$, by showing that if $S \\nleq_T \\emptyset^{(n-1)}$ then there exists a $G$ such that $S \\oplus G \\geq_T G^{(n)}$. Their argument employs Kumabe-Slaman forcing, and so the set they obtain, unlike that of the Posner-Robinson theorem, is not generic for Cohen forcing in any way. We answer the question of whether this is a necessary complication by showing that for all $n \\geq 1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.3282","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":"2026-05-18T03:45:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ke7UJLXZ/G2m9WeosY9nFLm0NISyCFfrKRkGMzcmwKJbIjusJjdSvUmeZcm9szo3ByuWAsQSHmIVE/huOBJeBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T12:42:06.859399Z"},"content_sha256":"23d30fdce086caa160f5d6c5a46fbb71e513b255caee2f4fb21237f36ba23787","schema_version":"1.0","event_id":"sha256:23d30fdce086caa160f5d6c5a46fbb71e513b255caee2f4fb21237f36ba23787"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XDGRABV7FDX5EIGRDSVRQA6VJD/bundle.json","state_url":"https://pith.science/pith/XDGRABV7FDX5EIGRDSVRQA6VJD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XDGRABV7FDX5EIGRDSVRQA6VJD/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":"2026-06-01T12:42:06Z","links":{"resolver":"https://pith.science/pith/XDGRABV7FDX5EIGRDSVRQA6VJD","bundle":"https://pith.science/pith/XDGRABV7FDX5EIGRDSVRQA6VJD/bundle.json","state":"https://pith.science/pith/XDGRABV7FDX5EIGRDSVRQA6VJD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XDGRABV7FDX5EIGRDSVRQA6VJD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:XDGRABV7FDX5EIGRDSVRQA6VJD","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":"b66698d284124f702061151b2eb89e3cbeb1813a318bcd952bcfad4c0dd6e078","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2012-09-14T18:48:35Z","title_canon_sha256":"fb1c8849dc48c57d49c6abc7bfdc434237733c7a7fd47f17fb4bb9007f780f3d"},"schema_version":"1.0","source":{"id":"1209.3282","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.3282","created_at":"2026-05-18T03:45:32Z"},{"alias_kind":"arxiv_version","alias_value":"1209.3282v1","created_at":"2026-05-18T03:45:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.3282","created_at":"2026-05-18T03:45:32Z"},{"alias_kind":"pith_short_12","alias_value":"XDGRABV7FDX5","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"XDGRABV7FDX5EIGR","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"XDGRABV7","created_at":"2026-05-18T12:27:27Z"}],"graph_snapshots":[{"event_id":"sha256:23d30fdce086caa160f5d6c5a46fbb71e513b255caee2f4fb21237f36ba23787","target":"graph","created_at":"2026-05-18T03:45:32Z","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":"Posner and Robinson (1981) proved that if $S \\subseteq \\omega$ is non-computable, then there exists a $G \\subseteq \\omega$ such that $S \\oplus G \\geq_T G'$. Shore and Slaman (1999) extended this result to all $n \\in \\omega$, by showing that if $S \\nleq_T \\emptyset^{(n-1)}$ then there exists a $G$ such that $S \\oplus G \\geq_T G^{(n)}$. Their argument employs Kumabe-Slaman forcing, and so the set they obtain, unlike that of the Posner-Robinson theorem, is not generic for Cohen forcing in any way. We answer the question of whether this is a necessary complication by showing that for all $n \\geq 1","authors_text":"Adam R. Day, Damir D. Dzhafarov","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2012-09-14T18:48:35Z","title":"Limits to joining with generics and randoms"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.3282","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:b2207346d7c9a57ffe9006c640d7015d503f9951219ac7f4298d0a9182d57c3f","target":"record","created_at":"2026-05-18T03:45:32Z","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":"b66698d284124f702061151b2eb89e3cbeb1813a318bcd952bcfad4c0dd6e078","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2012-09-14T18:48:35Z","title_canon_sha256":"fb1c8849dc48c57d49c6abc7bfdc434237733c7a7fd47f17fb4bb9007f780f3d"},"schema_version":"1.0","source":{"id":"1209.3282","kind":"arxiv","version":1}},"canonical_sha256":"b8cd1006bf28efd220d11cab1803d548ec90f5372ab6005818e6aced2b552690","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b8cd1006bf28efd220d11cab1803d548ec90f5372ab6005818e6aced2b552690","first_computed_at":"2026-05-18T03:45:32.352958Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:45:32.352958Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"oqePTmo/HKjKP9F03+IvD1OB+Yvwu6+/if/4FJy44A0bYvj461wtTKTmLynr2DwAva83XB4GkpBla7kLQk77DQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:45:32.353391Z","signed_message":"canonical_sha256_bytes"},"source_id":"1209.3282","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b2207346d7c9a57ffe9006c640d7015d503f9951219ac7f4298d0a9182d57c3f","sha256:23d30fdce086caa160f5d6c5a46fbb71e513b255caee2f4fb21237f36ba23787"],"state_sha256":"825646dd9cc98b038df2e52a5a9378aedb041941be09a1d1a16f2fda8772c5df"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BDN0aCPiTdcEacsFMfPr6jjQWYPOuZWBbjnksacHxBZs9FlB2MAKvHuxxPH1i4usrX6XecuC9cTJcqL8OlCRCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T12:42:06.861340Z","bundle_sha256":"5952a43be32feb978d6bc32f2e155d4c109d70239d8b51b6af410ea273ec8ba6"}}