{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:FW3ZODCQHSBGFINRK5MBLSCMFA","short_pith_number":"pith:FW3ZODCQ","canonical_record":{"source":{"id":"1608.01799","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-08-05T08:22:08Z","cross_cats_sorted":["math.DS","math.MP","math.SP"],"title_canon_sha256":"a417227e242dbd0f21948d8041807fc592e33e1d6c1e8cd891fb6f94a35f86c5","abstract_canon_sha256":"82716fd690a1b9ed753e490eb0d6b26fd974ec6b434887062525225f6e265cd4"},"schema_version":"1.0"},"canonical_sha256":"2db7970c503c8262a1b1575815c84c28357f2190280ad7deba5a76c093398edc","source":{"kind":"arxiv","id":"1608.01799","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.01799","created_at":"2026-05-18T01:09:44Z"},{"alias_kind":"arxiv_version","alias_value":"1608.01799v1","created_at":"2026-05-18T01:09:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.01799","created_at":"2026-05-18T01:09:44Z"},{"alias_kind":"pith_short_12","alias_value":"FW3ZODCQHSBG","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"FW3ZODCQHSBGFINR","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"FW3ZODCQ","created_at":"2026-05-18T12:30:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:FW3ZODCQHSBGFINRK5MBLSCMFA","target":"record","payload":{"canonical_record":{"source":{"id":"1608.01799","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-08-05T08:22:08Z","cross_cats_sorted":["math.DS","math.MP","math.SP"],"title_canon_sha256":"a417227e242dbd0f21948d8041807fc592e33e1d6c1e8cd891fb6f94a35f86c5","abstract_canon_sha256":"82716fd690a1b9ed753e490eb0d6b26fd974ec6b434887062525225f6e265cd4"},"schema_version":"1.0"},"canonical_sha256":"2db7970c503c8262a1b1575815c84c28357f2190280ad7deba5a76c093398edc","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:09:44.559500Z","signature_b64":"KFTvOzPj/b5xKVDeGkzaTmsV0hz9sFNftD7BN9FsQbf/TZMIowtrilnoaf8v8ciDpmGT851y2p+jTxsWfFgDCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2db7970c503c8262a1b1575815c84c28357f2190280ad7deba5a76c093398edc","last_reissued_at":"2026-05-18T01:09:44.558973Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:09:44.558973Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1608.01799","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":"2026-05-18T01:09:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0nZ+ilQKlt3wJPfq0SX8FXpiRSRdejfkubb8nJ+W4Gh4xxH8eD89IpPzGRBcsS1I44VFzRKmllKcqBnrgjXeAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T06:49:15.191850Z"},"content_sha256":"57a863137ed73f526540ec66b281335e6b6a38a8eb824846325493d3bdd14908","schema_version":"1.0","event_id":"sha256:57a863137ed73f526540ec66b281335e6b6a38a8eb824846325493d3bdd14908"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:FW3ZODCQHSBGFINRK5MBLSCMFA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Second Phase transition line","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","math.MP","math.SP"],"primary_cat":"math-ph","authors_text":"Artur Avila, Qi Zhou, Svetlana Jitomirskaya","submitted_at":"2016-08-05T08:22:08Z","abstract_excerpt":"We study the phase transion line of the almost Mathieu operator, that separates arithmetic regions corresponding to singular continuous and a.e. pure point regimes, and prove that both purely singular continuous and a.e. pure point spectrum occur for dense sets of frequencies."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.01799","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":"2026-05-18T01:09:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mYWVyXtB8gdFhqoY+ZbkOYxDTtroeRxGK/dOVbeTi7hyAPa1+Pfpl5uQ+ltvniV6Fw8qYXnf90OOk/7qI5DNCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T06:49:15.192197Z"},"content_sha256":"65685fbabd2385f7ca5a06e9d4c7257e76ee3e600c8e801e8b2daf3d59647461","schema_version":"1.0","event_id":"sha256:65685fbabd2385f7ca5a06e9d4c7257e76ee3e600c8e801e8b2daf3d59647461"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FW3ZODCQHSBGFINRK5MBLSCMFA/bundle.json","state_url":"https://pith.science/pith/FW3ZODCQHSBGFINRK5MBLSCMFA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FW3ZODCQHSBGFINRK5MBLSCMFA/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":"2026-06-02T06:49:15Z","links":{"resolver":"https://pith.science/pith/FW3ZODCQHSBGFINRK5MBLSCMFA","bundle":"https://pith.science/pith/FW3ZODCQHSBGFINRK5MBLSCMFA/bundle.json","state":"https://pith.science/pith/FW3ZODCQHSBGFINRK5MBLSCMFA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FW3ZODCQHSBGFINRK5MBLSCMFA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:FW3ZODCQHSBGFINRK5MBLSCMFA","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":"82716fd690a1b9ed753e490eb0d6b26fd974ec6b434887062525225f6e265cd4","cross_cats_sorted":["math.DS","math.MP","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-08-05T08:22:08Z","title_canon_sha256":"a417227e242dbd0f21948d8041807fc592e33e1d6c1e8cd891fb6f94a35f86c5"},"schema_version":"1.0","source":{"id":"1608.01799","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.01799","created_at":"2026-05-18T01:09:44Z"},{"alias_kind":"arxiv_version","alias_value":"1608.01799v1","created_at":"2026-05-18T01:09:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.01799","created_at":"2026-05-18T01:09:44Z"},{"alias_kind":"pith_short_12","alias_value":"FW3ZODCQHSBG","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"FW3ZODCQHSBGFINR","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"FW3ZODCQ","created_at":"2026-05-18T12:30:15Z"}],"graph_snapshots":[{"event_id":"sha256:65685fbabd2385f7ca5a06e9d4c7257e76ee3e600c8e801e8b2daf3d59647461","target":"graph","created_at":"2026-05-18T01:09:44Z","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 the phase transion line of the almost Mathieu operator, that separates arithmetic regions corresponding to singular continuous and a.e. pure point regimes, and prove that both purely singular continuous and a.e. pure point spectrum occur for dense sets of frequencies.","authors_text":"Artur Avila, Qi Zhou, Svetlana Jitomirskaya","cross_cats":["math.DS","math.MP","math.SP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-08-05T08:22:08Z","title":"Second Phase transition line"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.01799","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":"sha256:57a863137ed73f526540ec66b281335e6b6a38a8eb824846325493d3bdd14908","target":"record","created_at":"2026-05-18T01:09:44Z","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":"82716fd690a1b9ed753e490eb0d6b26fd974ec6b434887062525225f6e265cd4","cross_cats_sorted":["math.DS","math.MP","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-08-05T08:22:08Z","title_canon_sha256":"a417227e242dbd0f21948d8041807fc592e33e1d6c1e8cd891fb6f94a35f86c5"},"schema_version":"1.0","source":{"id":"1608.01799","kind":"arxiv","version":1}},"canonical_sha256":"2db7970c503c8262a1b1575815c84c28357f2190280ad7deba5a76c093398edc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2db7970c503c8262a1b1575815c84c28357f2190280ad7deba5a76c093398edc","first_computed_at":"2026-05-18T01:09:44.558973Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:09:44.558973Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KFTvOzPj/b5xKVDeGkzaTmsV0hz9sFNftD7BN9FsQbf/TZMIowtrilnoaf8v8ciDpmGT851y2p+jTxsWfFgDCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:09:44.559500Z","signed_message":"canonical_sha256_bytes"},"source_id":"1608.01799","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:57a863137ed73f526540ec66b281335e6b6a38a8eb824846325493d3bdd14908","sha256:65685fbabd2385f7ca5a06e9d4c7257e76ee3e600c8e801e8b2daf3d59647461"],"state_sha256":"3667084c4f0eb1d43ba45fa91b1491b30f4cb3d4623be22cb529927f6b808e31"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ip75MAPBxNr6tiRGs23A8bjmNmMXVQcLBN+LgmdL0MDqzJpfYrWdVj1sIJCHuO84M6RpxuWSg3uqzz7/4v1aBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T06:49:15.194163Z","bundle_sha256":"78219c5ae926c2c2b1183585a02823b0759491e3d299638f7b94d1f8bac452ac"}}