{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:RX6AXAK3PGDSSGMK6SY6YJRKVE","short_pith_number":"pith:RX6AXAK3","canonical_record":{"source":{"id":"1607.04679","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2016-07-15T23:00:23Z","cross_cats_sorted":[],"title_canon_sha256":"b7c544caa09ed7e21aae8355e8e37a07fa9bf011e0e3dda6455cb672cf2e1d02","abstract_canon_sha256":"c60daef30cb1eb0749bbeb1518452ae9fae6c24378396c8dbfe18d586e826257"},"schema_version":"1.0"},"canonical_sha256":"8dfc0b815b798729198af4b1ec262aa920f4184b63d7e80735a7cc51dab34c45","source":{"kind":"arxiv","id":"1607.04679","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.04679","created_at":"2026-05-18T00:38:36Z"},{"alias_kind":"arxiv_version","alias_value":"1607.04679v3","created_at":"2026-05-18T00:38:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.04679","created_at":"2026-05-18T00:38:36Z"},{"alias_kind":"pith_short_12","alias_value":"RX6AXAK3PGDS","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_16","alias_value":"RX6AXAK3PGDSSGMK","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_8","alias_value":"RX6AXAK3","created_at":"2026-05-18T12:30:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:RX6AXAK3PGDSSGMK6SY6YJRKVE","target":"record","payload":{"canonical_record":{"source":{"id":"1607.04679","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2016-07-15T23:00:23Z","cross_cats_sorted":[],"title_canon_sha256":"b7c544caa09ed7e21aae8355e8e37a07fa9bf011e0e3dda6455cb672cf2e1d02","abstract_canon_sha256":"c60daef30cb1eb0749bbeb1518452ae9fae6c24378396c8dbfe18d586e826257"},"schema_version":"1.0"},"canonical_sha256":"8dfc0b815b798729198af4b1ec262aa920f4184b63d7e80735a7cc51dab34c45","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:38:36.418600Z","signature_b64":"y8F4tZQ9kOqKHBTCs/+L4gJUocQ0bb/FW3gLz7AJPxJ1c4pitgjY1pbPWP2KM5A0+uU7Yp64W6lEV0kUEPUfCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8dfc0b815b798729198af4b1ec262aa920f4184b63d7e80735a7cc51dab34c45","last_reissued_at":"2026-05-18T00:38:36.418180Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:38:36.418180Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1607.04679","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:38:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8g0gvRXDxzUtdPfl9XYQ8dsIGfXhDYtVwt2KJ+8eWZOZ/4Nql/vHhDBAfkR/87Bp4MHOl9ze2YzI2Zx4tbNbAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T06:21:57.711908Z"},"content_sha256":"bcd2289719d949e69ded2ddfc1ad72ffcd93e7190df734e1a030953e7b5d31bf","schema_version":"1.0","event_id":"sha256:bcd2289719d949e69ded2ddfc1ad72ffcd93e7190df734e1a030953e7b5d31bf"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:RX6AXAK3PGDSSGMK6SY6YJRKVE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Schnorr randomness for noncomputable measures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Jason Rute","submitted_at":"2016-07-15T23:00:23Z","abstract_excerpt":"This paper explores a novel definition of Schnorr randomness for noncomputable measures. We say $x$ is uniformly Schnorr $\\mu$-random if $t(\\mu,x)<\\infty$ for all lower semicomputable functions $t(\\mu,x)$ such that $\\mu\\mapsto\\int t(\\mu,x)\\,d\\mu(x)$ is computable. We prove a number of theorems demonstrating that this is the correct definition which enjoys many of the same properties as Martin-L\\\"of randomness for noncomputable measures. Nonetheless, a number of our proofs significantly differ from the Martin-L\\\"of case, requiring new ideas from computable analysis."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.04679","kind":"arxiv","version":3},"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":"2026-05-18T00:38:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7XR4HSvdfjdbJwmhg9HgEhgOL6An4dEbFulx42gNc6jcg8xVMWbDfuSgWp7eTnxJSX5UN+wx5hoeXR9vlt3HDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T06:21:57.712605Z"},"content_sha256":"67a5113abe5e527ed72a34151165c99a902b74ba727fc50f13c60a51761520ff","schema_version":"1.0","event_id":"sha256:67a5113abe5e527ed72a34151165c99a902b74ba727fc50f13c60a51761520ff"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RX6AXAK3PGDSSGMK6SY6YJRKVE/bundle.json","state_url":"https://pith.science/pith/RX6AXAK3PGDSSGMK6SY6YJRKVE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RX6AXAK3PGDSSGMK6SY6YJRKVE/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":"2026-06-11T06:21:57Z","links":{"resolver":"https://pith.science/pith/RX6AXAK3PGDSSGMK6SY6YJRKVE","bundle":"https://pith.science/pith/RX6AXAK3PGDSSGMK6SY6YJRKVE/bundle.json","state":"https://pith.science/pith/RX6AXAK3PGDSSGMK6SY6YJRKVE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RX6AXAK3PGDSSGMK6SY6YJRKVE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:RX6AXAK3PGDSSGMK6SY6YJRKVE","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":"c60daef30cb1eb0749bbeb1518452ae9fae6c24378396c8dbfe18d586e826257","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2016-07-15T23:00:23Z","title_canon_sha256":"b7c544caa09ed7e21aae8355e8e37a07fa9bf011e0e3dda6455cb672cf2e1d02"},"schema_version":"1.0","source":{"id":"1607.04679","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.04679","created_at":"2026-05-18T00:38:36Z"},{"alias_kind":"arxiv_version","alias_value":"1607.04679v3","created_at":"2026-05-18T00:38:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.04679","created_at":"2026-05-18T00:38:36Z"},{"alias_kind":"pith_short_12","alias_value":"RX6AXAK3PGDS","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_16","alias_value":"RX6AXAK3PGDSSGMK","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_8","alias_value":"RX6AXAK3","created_at":"2026-05-18T12:30:41Z"}],"graph_snapshots":[{"event_id":"sha256:67a5113abe5e527ed72a34151165c99a902b74ba727fc50f13c60a51761520ff","target":"graph","created_at":"2026-05-18T00:38:36Z","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":"This paper explores a novel definition of Schnorr randomness for noncomputable measures. We say $x$ is uniformly Schnorr $\\mu$-random if $t(\\mu,x)<\\infty$ for all lower semicomputable functions $t(\\mu,x)$ such that $\\mu\\mapsto\\int t(\\mu,x)\\,d\\mu(x)$ is computable. We prove a number of theorems demonstrating that this is the correct definition which enjoys many of the same properties as Martin-L\\\"of randomness for noncomputable measures. Nonetheless, a number of our proofs significantly differ from the Martin-L\\\"of case, requiring new ideas from computable analysis.","authors_text":"Jason Rute","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2016-07-15T23:00:23Z","title":"Schnorr randomness for noncomputable measures"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.04679","kind":"arxiv","version":3},"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:bcd2289719d949e69ded2ddfc1ad72ffcd93e7190df734e1a030953e7b5d31bf","target":"record","created_at":"2026-05-18T00:38:36Z","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":"c60daef30cb1eb0749bbeb1518452ae9fae6c24378396c8dbfe18d586e826257","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2016-07-15T23:00:23Z","title_canon_sha256":"b7c544caa09ed7e21aae8355e8e37a07fa9bf011e0e3dda6455cb672cf2e1d02"},"schema_version":"1.0","source":{"id":"1607.04679","kind":"arxiv","version":3}},"canonical_sha256":"8dfc0b815b798729198af4b1ec262aa920f4184b63d7e80735a7cc51dab34c45","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8dfc0b815b798729198af4b1ec262aa920f4184b63d7e80735a7cc51dab34c45","first_computed_at":"2026-05-18T00:38:36.418180Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:38:36.418180Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"y8F4tZQ9kOqKHBTCs/+L4gJUocQ0bb/FW3gLz7AJPxJ1c4pitgjY1pbPWP2KM5A0+uU7Yp64W6lEV0kUEPUfCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:38:36.418600Z","signed_message":"canonical_sha256_bytes"},"source_id":"1607.04679","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bcd2289719d949e69ded2ddfc1ad72ffcd93e7190df734e1a030953e7b5d31bf","sha256:67a5113abe5e527ed72a34151165c99a902b74ba727fc50f13c60a51761520ff"],"state_sha256":"eb0c8961328451a5679577d2b98acbaa5176390e890ad99af59a5a76ab61e661"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1KV8qfaTFT4z2DPeLDcPOGIDcD66SZfPgYWybBFFijHb6LYXzH4aEdcVcEbgDtAsyntxy7HOcE3c4XGPAX2mDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T06:21:57.716359Z","bundle_sha256":"2cd2572fe6ff08b47e5e1e1992eb49caaa7c4fc421a68b3a87b0961c7c85c4d0"}}