{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2007:Z3JAEPPZ7YLUCJ34G2BP3PBQRK","short_pith_number":"pith:Z3JAEPPZ","canonical_record":{"source":{"id":"0711.4291","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.DS","submitted_at":"2007-11-27T16:08:40Z","cross_cats_sorted":["math-ph","math.MP","math.SP"],"title_canon_sha256":"baad49e2a84ecdca8beb63417ac3b97919c44794401c77c7265cc75b46b353be","abstract_canon_sha256":"e8627685584f6f0f27a8c3fc96b8d181b8d713db396d699a48430d915cc8595a"},"schema_version":"1.0"},"canonical_sha256":"ced2023df9fe1741277c3682fdbc308ab21f4948e770b601457284180c383ba4","source":{"kind":"arxiv","id":"0711.4291","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0711.4291","created_at":"2026-05-18T02:26:39Z"},{"alias_kind":"arxiv_version","alias_value":"0711.4291v1","created_at":"2026-05-18T02:26:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0711.4291","created_at":"2026-05-18T02:26:39Z"},{"alias_kind":"pith_short_12","alias_value":"Z3JAEPPZ7YLU","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_16","alias_value":"Z3JAEPPZ7YLUCJ34","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_8","alias_value":"Z3JAEPPZ","created_at":"2026-05-18T12:25:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2007:Z3JAEPPZ7YLUCJ34G2BP3PBQRK","target":"record","payload":{"canonical_record":{"source":{"id":"0711.4291","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.DS","submitted_at":"2007-11-27T16:08:40Z","cross_cats_sorted":["math-ph","math.MP","math.SP"],"title_canon_sha256":"baad49e2a84ecdca8beb63417ac3b97919c44794401c77c7265cc75b46b353be","abstract_canon_sha256":"e8627685584f6f0f27a8c3fc96b8d181b8d713db396d699a48430d915cc8595a"},"schema_version":"1.0"},"canonical_sha256":"ced2023df9fe1741277c3682fdbc308ab21f4948e770b601457284180c383ba4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:26:39.782410Z","signature_b64":"aPVkdK3FUUxGmO25eBzIApNtUZGfD2x3ZI6Ma8SNwIgemA+Bp/XEQzRmDgm0SCHQ6Mjd+F9Su4jdOk61iO21BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ced2023df9fe1741277c3682fdbc308ab21f4948e770b601457284180c383ba4","last_reissued_at":"2026-05-18T02:26:39.781890Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:26:39.781890Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0711.4291","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":"2026-05-18T02:26:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6u6GhYj2nksknFOW4BPcpSvenQxAU5cVGUjI8ludv5WvMu+Di3oU5GmJRmrROJxriTvjB/k0Mdgm22sQ7jnkCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T02:50:27.011817Z"},"content_sha256":"8e54ce04455bfdb48705c29a4eadf1b10480e14c8b1003b0da71211bb590b9d2","schema_version":"1.0","event_id":"sha256:8e54ce04455bfdb48705c29a4eadf1b10480e14c8b1003b0da71211bb590b9d2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2007:Z3JAEPPZ7YLUCJ34G2BP3PBQRK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Absolute Continuity of the Integrated Density of States for the Almost Mathieu Operator with Non-Critical Coupling","license":"","headline":"","cross_cats":["math-ph","math.MP","math.SP"],"primary_cat":"math.DS","authors_text":"Artur Avila, David Damanik","submitted_at":"2007-11-27T16:08:40Z","abstract_excerpt":"We show that the integrated density of states of the almost Mathieu operator is absolutely continuous if and only if the coupling is non-critical. We deduce for subcritical coupling that the spectrum is purely absolutely continuous for almost every phase, settling the measure-theoretical case of Problem 6 of Barry Simon's list of Schr\\\"odinger operator problems for the twenty-first century."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0711.4291","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":"2026-05-18T02:26:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iWzJhjEG1fMZQA1O63bLmcpWuwrf4pbXI9ddaPs1GMstv6wfa183QCaamBXgQc3nMA1BYln4yK/Fkvt0Y7V2CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T02:50:27.012681Z"},"content_sha256":"d508c7cb4ff8b8d1a5003402714e7a699ec2844757e3dd0edd6c2bb0bfcf0f6a","schema_version":"1.0","event_id":"sha256:d508c7cb4ff8b8d1a5003402714e7a699ec2844757e3dd0edd6c2bb0bfcf0f6a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Z3JAEPPZ7YLUCJ34G2BP3PBQRK/bundle.json","state_url":"https://pith.science/pith/Z3JAEPPZ7YLUCJ34G2BP3PBQRK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Z3JAEPPZ7YLUCJ34G2BP3PBQRK/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":"2026-06-06T02:50:27Z","links":{"resolver":"https://pith.science/pith/Z3JAEPPZ7YLUCJ34G2BP3PBQRK","bundle":"https://pith.science/pith/Z3JAEPPZ7YLUCJ34G2BP3PBQRK/bundle.json","state":"https://pith.science/pith/Z3JAEPPZ7YLUCJ34G2BP3PBQRK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Z3JAEPPZ7YLUCJ34G2BP3PBQRK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:Z3JAEPPZ7YLUCJ34G2BP3PBQRK","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":"e8627685584f6f0f27a8c3fc96b8d181b8d713db396d699a48430d915cc8595a","cross_cats_sorted":["math-ph","math.MP","math.SP"],"license":"","primary_cat":"math.DS","submitted_at":"2007-11-27T16:08:40Z","title_canon_sha256":"baad49e2a84ecdca8beb63417ac3b97919c44794401c77c7265cc75b46b353be"},"schema_version":"1.0","source":{"id":"0711.4291","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0711.4291","created_at":"2026-05-18T02:26:39Z"},{"alias_kind":"arxiv_version","alias_value":"0711.4291v1","created_at":"2026-05-18T02:26:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0711.4291","created_at":"2026-05-18T02:26:39Z"},{"alias_kind":"pith_short_12","alias_value":"Z3JAEPPZ7YLU","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_16","alias_value":"Z3JAEPPZ7YLUCJ34","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_8","alias_value":"Z3JAEPPZ","created_at":"2026-05-18T12:25:56Z"}],"graph_snapshots":[{"event_id":"sha256:d508c7cb4ff8b8d1a5003402714e7a699ec2844757e3dd0edd6c2bb0bfcf0f6a","target":"graph","created_at":"2026-05-18T02:26:39Z","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 show that the integrated density of states of the almost Mathieu operator is absolutely continuous if and only if the coupling is non-critical. We deduce for subcritical coupling that the spectrum is purely absolutely continuous for almost every phase, settling the measure-theoretical case of Problem 6 of Barry Simon's list of Schr\\\"odinger operator problems for the twenty-first century.","authors_text":"Artur Avila, David Damanik","cross_cats":["math-ph","math.MP","math.SP"],"headline":"","license":"","primary_cat":"math.DS","submitted_at":"2007-11-27T16:08:40Z","title":"Absolute Continuity of the Integrated Density of States for the Almost Mathieu Operator with Non-Critical Coupling"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0711.4291","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":"sha256:8e54ce04455bfdb48705c29a4eadf1b10480e14c8b1003b0da71211bb590b9d2","target":"record","created_at":"2026-05-18T02:26:39Z","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":"e8627685584f6f0f27a8c3fc96b8d181b8d713db396d699a48430d915cc8595a","cross_cats_sorted":["math-ph","math.MP","math.SP"],"license":"","primary_cat":"math.DS","submitted_at":"2007-11-27T16:08:40Z","title_canon_sha256":"baad49e2a84ecdca8beb63417ac3b97919c44794401c77c7265cc75b46b353be"},"schema_version":"1.0","source":{"id":"0711.4291","kind":"arxiv","version":1}},"canonical_sha256":"ced2023df9fe1741277c3682fdbc308ab21f4948e770b601457284180c383ba4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ced2023df9fe1741277c3682fdbc308ab21f4948e770b601457284180c383ba4","first_computed_at":"2026-05-18T02:26:39.781890Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:26:39.781890Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aPVkdK3FUUxGmO25eBzIApNtUZGfD2x3ZI6Ma8SNwIgemA+Bp/XEQzRmDgm0SCHQ6Mjd+F9Su4jdOk61iO21BA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:26:39.782410Z","signed_message":"canonical_sha256_bytes"},"source_id":"0711.4291","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8e54ce04455bfdb48705c29a4eadf1b10480e14c8b1003b0da71211bb590b9d2","sha256:d508c7cb4ff8b8d1a5003402714e7a699ec2844757e3dd0edd6c2bb0bfcf0f6a"],"state_sha256":"09ea72713b138782601c2870ae983ee7563bfdf3175f0d9daf470f9553a14db6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xeR6mdWfbCpE6/a63yg/NAKyRimtLlXPB2+eb4zBy9ECGPR7MGvcIRBvFLYph4BxC3Cl30azzgDUd1h2b70YBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T02:50:27.017511Z","bundle_sha256":"346cc001de38cbe600d89c4be5d02466781c97f88855e006b911320c8373620a"}}