{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:4YRRDHAO7DEHUZQMSPGX7XGMZW","short_pith_number":"pith:4YRRDHAO","canonical_record":{"source":{"id":"1207.2533","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2012-07-11T02:54:21Z","cross_cats_sorted":[],"title_canon_sha256":"0a60e2fe8ac44f4c145ed2562457367fbcdca7d36f82130d976e38d687a170f7","abstract_canon_sha256":"40b05ad95049d5f3512e1e19b930cdf038be9d602ba87369f2cb490c3fb33809"},"schema_version":"1.0"},"canonical_sha256":"e623119c0ef8c87a660c93cd7fdccccda7fac987dcdf977b4618c7393d60a987","source":{"kind":"arxiv","id":"1207.2533","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.2533","created_at":"2026-05-18T01:21:05Z"},{"alias_kind":"arxiv_version","alias_value":"1207.2533v1","created_at":"2026-05-18T01:21:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.2533","created_at":"2026-05-18T01:21:05Z"},{"alias_kind":"pith_short_12","alias_value":"4YRRDHAO7DEH","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"4YRRDHAO7DEHUZQM","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"4YRRDHAO","created_at":"2026-05-18T12:26:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:4YRRDHAO7DEHUZQMSPGX7XGMZW","target":"record","payload":{"canonical_record":{"source":{"id":"1207.2533","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2012-07-11T02:54:21Z","cross_cats_sorted":[],"title_canon_sha256":"0a60e2fe8ac44f4c145ed2562457367fbcdca7d36f82130d976e38d687a170f7","abstract_canon_sha256":"40b05ad95049d5f3512e1e19b930cdf038be9d602ba87369f2cb490c3fb33809"},"schema_version":"1.0"},"canonical_sha256":"e623119c0ef8c87a660c93cd7fdccccda7fac987dcdf977b4618c7393d60a987","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:21:05.739115Z","signature_b64":"VdOP9GPz2m0Bu04ySiAeAZFULF4tB0Z9enyMZzB0N1mgcCUdpp6bZ9Rra9IMdMLvB3B68aeTZymfZP56LU3DAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e623119c0ef8c87a660c93cd7fdccccda7fac987dcdf977b4618c7393d60a987","last_reissued_at":"2026-05-18T01:21:05.738472Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:21:05.738472Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1207.2533","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":"2026-05-18T01:21:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HDB6KivermJ9vZj9QYE/uF3733g1viqWV2Z4jC8bb7wdbEq9uX8ayBww+Y0dgNH2wZlBPm7YFMIOMPSIYExtBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T13:47:32.064503Z"},"content_sha256":"77ae86615fb0915477898bc26663b0ceff705e35d2fb86ae3055c2ed85ef90a2","schema_version":"1.0","event_id":"sha256:77ae86615fb0915477898bc26663b0ceff705e35d2fb86ae3055c2ed85ef90a2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:4YRRDHAO7DEHUZQMSPGX7XGMZW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Independence, Relative Randomness, and PA Degrees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Adam R. Day, Jan Reimann","submitted_at":"2012-07-11T02:54:21Z","abstract_excerpt":"We study pairs of reals that are mutually Martin-L\\\"{o}f random with respect to a common, not necessarily computable probability measure. We show that a generalized version of van Lambalgen's Theorem holds for non-computable probability measures, too. We study, for a given real $A$, the \\emph{independence spectrum} of $A$, the set of all $B$ so that there exists a probability measure $\\mu$ so that $\\mu\\{A,B\\} = 0$ and $(A,B)$ is $\\mu\\times\\mu$-random. We prove that if $A$ is r.e., then no $\\Delta^0_2$ set is in the independence spectrum of $A$. We obtain applications of this fact to PA degrees"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.2533","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":"2026-05-18T01:21:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"30SvrFjYzhey/duLDTYVdL1V3ZWi0ZrK9/sIjQ9pncT3VuQZCRWFv/awU6OPMzvGT6FeICP77No3u3x5ngyQCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T13:47:32.065205Z"},"content_sha256":"29790e6a6bc90f27abc6c2e71c49f75cf5b7b97a82601949609eba4539c6ccc1","schema_version":"1.0","event_id":"sha256:29790e6a6bc90f27abc6c2e71c49f75cf5b7b97a82601949609eba4539c6ccc1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4YRRDHAO7DEHUZQMSPGX7XGMZW/bundle.json","state_url":"https://pith.science/pith/4YRRDHAO7DEHUZQMSPGX7XGMZW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4YRRDHAO7DEHUZQMSPGX7XGMZW/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":"2026-06-07T13:47:32Z","links":{"resolver":"https://pith.science/pith/4YRRDHAO7DEHUZQMSPGX7XGMZW","bundle":"https://pith.science/pith/4YRRDHAO7DEHUZQMSPGX7XGMZW/bundle.json","state":"https://pith.science/pith/4YRRDHAO7DEHUZQMSPGX7XGMZW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4YRRDHAO7DEHUZQMSPGX7XGMZW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:4YRRDHAO7DEHUZQMSPGX7XGMZW","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":"40b05ad95049d5f3512e1e19b930cdf038be9d602ba87369f2cb490c3fb33809","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2012-07-11T02:54:21Z","title_canon_sha256":"0a60e2fe8ac44f4c145ed2562457367fbcdca7d36f82130d976e38d687a170f7"},"schema_version":"1.0","source":{"id":"1207.2533","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.2533","created_at":"2026-05-18T01:21:05Z"},{"alias_kind":"arxiv_version","alias_value":"1207.2533v1","created_at":"2026-05-18T01:21:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.2533","created_at":"2026-05-18T01:21:05Z"},{"alias_kind":"pith_short_12","alias_value":"4YRRDHAO7DEH","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"4YRRDHAO7DEHUZQM","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"4YRRDHAO","created_at":"2026-05-18T12:26:53Z"}],"graph_snapshots":[{"event_id":"sha256:29790e6a6bc90f27abc6c2e71c49f75cf5b7b97a82601949609eba4539c6ccc1","target":"graph","created_at":"2026-05-18T01:21:05Z","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 study pairs of reals that are mutually Martin-L\\\"{o}f random with respect to a common, not necessarily computable probability measure. We show that a generalized version of van Lambalgen's Theorem holds for non-computable probability measures, too. We study, for a given real $A$, the \\emph{independence spectrum} of $A$, the set of all $B$ so that there exists a probability measure $\\mu$ so that $\\mu\\{A,B\\} = 0$ and $(A,B)$ is $\\mu\\times\\mu$-random. We prove that if $A$ is r.e., then no $\\Delta^0_2$ set is in the independence spectrum of $A$. We obtain applications of this fact to PA degrees","authors_text":"Adam R. Day, Jan Reimann","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2012-07-11T02:54:21Z","title":"Independence, Relative Randomness, and PA Degrees"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.2533","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":"sha256:77ae86615fb0915477898bc26663b0ceff705e35d2fb86ae3055c2ed85ef90a2","target":"record","created_at":"2026-05-18T01:21:05Z","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":"40b05ad95049d5f3512e1e19b930cdf038be9d602ba87369f2cb490c3fb33809","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2012-07-11T02:54:21Z","title_canon_sha256":"0a60e2fe8ac44f4c145ed2562457367fbcdca7d36f82130d976e38d687a170f7"},"schema_version":"1.0","source":{"id":"1207.2533","kind":"arxiv","version":1}},"canonical_sha256":"e623119c0ef8c87a660c93cd7fdccccda7fac987dcdf977b4618c7393d60a987","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e623119c0ef8c87a660c93cd7fdccccda7fac987dcdf977b4618c7393d60a987","first_computed_at":"2026-05-18T01:21:05.738472Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:21:05.738472Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VdOP9GPz2m0Bu04ySiAeAZFULF4tB0Z9enyMZzB0N1mgcCUdpp6bZ9Rra9IMdMLvB3B68aeTZymfZP56LU3DAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:21:05.739115Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.2533","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:77ae86615fb0915477898bc26663b0ceff705e35d2fb86ae3055c2ed85ef90a2","sha256:29790e6a6bc90f27abc6c2e71c49f75cf5b7b97a82601949609eba4539c6ccc1"],"state_sha256":"742fb651609cd9ef343204ff6028b8b248d83b597c65ba1abf95b1ebb322ef44"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Yy+dOHATKo2SDzBuP5bL8u4/qS/UPfcZRct4Z2Kpq6SvKfHpWZI/LVTjtBZ14XP7un6XPvqvPBBp3nLJpEadCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T13:47:32.069644Z","bundle_sha256":"b10fdcc0524bd052fc800207fcb7864d1a681d85ddc631ee4a9be6eda7e42cac"}}