{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:EENOVKG7BCGHENOM5NHJZMFVQW","short_pith_number":"pith:EENOVKG7","canonical_record":{"source":{"id":"1804.01658","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2018-04-05T03:00:31Z","cross_cats_sorted":["math.MP","math.SP"],"title_canon_sha256":"7f497331825fc99f02882f403023895a1eed9078d8d84f8111d9fca5f94f31fa","abstract_canon_sha256":"dac1ded7f702ec52bee50082ff2cd2523c7691ff147df5096a3bc23824fa1dd6"},"schema_version":"1.0"},"canonical_sha256":"211aeaa8df088c7235cceb4e9cb0b585965eab6dc9987e06fa9b2288457f93a6","source":{"kind":"arxiv","id":"1804.01658","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.01658","created_at":"2026-05-18T00:00:56Z"},{"alias_kind":"arxiv_version","alias_value":"1804.01658v1","created_at":"2026-05-18T00:00:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.01658","created_at":"2026-05-18T00:00:56Z"},{"alias_kind":"pith_short_12","alias_value":"EENOVKG7BCGH","created_at":"2026-05-18T12:32:22Z"},{"alias_kind":"pith_short_16","alias_value":"EENOVKG7BCGHENOM","created_at":"2026-05-18T12:32:22Z"},{"alias_kind":"pith_short_8","alias_value":"EENOVKG7","created_at":"2026-05-18T12:32:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:EENOVKG7BCGHENOM5NHJZMFVQW","target":"record","payload":{"canonical_record":{"source":{"id":"1804.01658","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2018-04-05T03:00:31Z","cross_cats_sorted":["math.MP","math.SP"],"title_canon_sha256":"7f497331825fc99f02882f403023895a1eed9078d8d84f8111d9fca5f94f31fa","abstract_canon_sha256":"dac1ded7f702ec52bee50082ff2cd2523c7691ff147df5096a3bc23824fa1dd6"},"schema_version":"1.0"},"canonical_sha256":"211aeaa8df088c7235cceb4e9cb0b585965eab6dc9987e06fa9b2288457f93a6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:00:56.607838Z","signature_b64":"un1ULcc5BwT+d8itZGr2kXV1JR+WybyRyXwkFnpkDZBx5SCbzZbH2f5vv1PIGcjEc5W8coCdN4CX0nrTk97ADQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"211aeaa8df088c7235cceb4e9cb0b585965eab6dc9987e06fa9b2288457f93a6","last_reissued_at":"2026-05-18T00:00:56.607344Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:00:56.607344Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1804.01658","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":"2026-05-18T00:00:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"w99X6iWQ4dgxgQgGq0K8mWA6HKCEVrUVp4S0DwtDqa/3sdx/BOeDgVulcwF5Ceaduj/csHBk0UZmz3qrETxaBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T22:19:06.061394Z"},"content_sha256":"519003865dce4e6b2ee6619f77058ab59c0f964040ced176243ca422cc8da4e2","schema_version":"1.0","event_id":"sha256:519003865dce4e6b2ee6619f77058ab59c0f964040ced176243ca422cc8da4e2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:EENOVKG7BCGHENOM5NHJZMFVQW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Positive Hausdorff dimensional spectrum for the critical almost Mathieu operator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP","math.SP"],"primary_cat":"math-ph","authors_text":"Bernard Helffer, Qinghui Liu, Qi Zhou, Yanhui Qu","submitted_at":"2018-04-05T03:00:31Z","abstract_excerpt":"We show that there exists a dense set of frequencies with positive Hausdorff dimension for which the Hausdorff dimension of the spectrum of the critical almost Mathieu operator is positive."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.01658","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":"2026-05-18T00:00:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OvXcC3LBCPS8rAdNufot1ygPpTALlGmT/WShW9ZYBzJtnSnkR6Sh2SY8QLzPIdbS8L/qN/v/AMHwqy+39HgjCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T22:19:06.061750Z"},"content_sha256":"1c8c9100f744684f41b40eba3f1979c668ae503b51e2c6fd1ec23a82caf891c6","schema_version":"1.0","event_id":"sha256:1c8c9100f744684f41b40eba3f1979c668ae503b51e2c6fd1ec23a82caf891c6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EENOVKG7BCGHENOM5NHJZMFVQW/bundle.json","state_url":"https://pith.science/pith/EENOVKG7BCGHENOM5NHJZMFVQW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EENOVKG7BCGHENOM5NHJZMFVQW/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":"2026-06-02T22:19:06Z","links":{"resolver":"https://pith.science/pith/EENOVKG7BCGHENOM5NHJZMFVQW","bundle":"https://pith.science/pith/EENOVKG7BCGHENOM5NHJZMFVQW/bundle.json","state":"https://pith.science/pith/EENOVKG7BCGHENOM5NHJZMFVQW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EENOVKG7BCGHENOM5NHJZMFVQW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:EENOVKG7BCGHENOM5NHJZMFVQW","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":"dac1ded7f702ec52bee50082ff2cd2523c7691ff147df5096a3bc23824fa1dd6","cross_cats_sorted":["math.MP","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2018-04-05T03:00:31Z","title_canon_sha256":"7f497331825fc99f02882f403023895a1eed9078d8d84f8111d9fca5f94f31fa"},"schema_version":"1.0","source":{"id":"1804.01658","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.01658","created_at":"2026-05-18T00:00:56Z"},{"alias_kind":"arxiv_version","alias_value":"1804.01658v1","created_at":"2026-05-18T00:00:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.01658","created_at":"2026-05-18T00:00:56Z"},{"alias_kind":"pith_short_12","alias_value":"EENOVKG7BCGH","created_at":"2026-05-18T12:32:22Z"},{"alias_kind":"pith_short_16","alias_value":"EENOVKG7BCGHENOM","created_at":"2026-05-18T12:32:22Z"},{"alias_kind":"pith_short_8","alias_value":"EENOVKG7","created_at":"2026-05-18T12:32:22Z"}],"graph_snapshots":[{"event_id":"sha256:1c8c9100f744684f41b40eba3f1979c668ae503b51e2c6fd1ec23a82caf891c6","target":"graph","created_at":"2026-05-18T00:00:56Z","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 there exists a dense set of frequencies with positive Hausdorff dimension for which the Hausdorff dimension of the spectrum of the critical almost Mathieu operator is positive.","authors_text":"Bernard Helffer, Qinghui Liu, Qi Zhou, Yanhui Qu","cross_cats":["math.MP","math.SP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2018-04-05T03:00:31Z","title":"Positive Hausdorff dimensional spectrum for the critical almost Mathieu operator"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.01658","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":"sha256:519003865dce4e6b2ee6619f77058ab59c0f964040ced176243ca422cc8da4e2","target":"record","created_at":"2026-05-18T00:00:56Z","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":"dac1ded7f702ec52bee50082ff2cd2523c7691ff147df5096a3bc23824fa1dd6","cross_cats_sorted":["math.MP","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2018-04-05T03:00:31Z","title_canon_sha256":"7f497331825fc99f02882f403023895a1eed9078d8d84f8111d9fca5f94f31fa"},"schema_version":"1.0","source":{"id":"1804.01658","kind":"arxiv","version":1}},"canonical_sha256":"211aeaa8df088c7235cceb4e9cb0b585965eab6dc9987e06fa9b2288457f93a6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"211aeaa8df088c7235cceb4e9cb0b585965eab6dc9987e06fa9b2288457f93a6","first_computed_at":"2026-05-18T00:00:56.607344Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:00:56.607344Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"un1ULcc5BwT+d8itZGr2kXV1JR+WybyRyXwkFnpkDZBx5SCbzZbH2f5vv1PIGcjEc5W8coCdN4CX0nrTk97ADQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:00:56.607838Z","signed_message":"canonical_sha256_bytes"},"source_id":"1804.01658","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:519003865dce4e6b2ee6619f77058ab59c0f964040ced176243ca422cc8da4e2","sha256:1c8c9100f744684f41b40eba3f1979c668ae503b51e2c6fd1ec23a82caf891c6"],"state_sha256":"30a5f4cf0cdb520a13a531f597f35afab2377fa145ff0d5a9ff37dd5ebd48e3f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"spYT7ycXQhKAbYTndqq7DZO9BRw7Uy3Bg4MWGFeqG2IWKgyaXSvkJICMM5Vp1i4cISqgGhPV7fQNa8oAiduvBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T22:19:06.063700Z","bundle_sha256":"c40bb4b0a5be09533a1119e77b806418203c8468a164b8c28de7adc4122bf308"}}