{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:7QJEJPCTUD6RF6ZACCMC3AJ7UI","short_pith_number":"pith:7QJEJPCT","canonical_record":{"source":{"id":"1612.03208","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2016-12-09T22:56:03Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"a40da16f46e5c1a6bf84b25a183dedce7a25a28680814b92198d4790793aa8bb","abstract_canon_sha256":"fedc256f4605bc92dc00f2b47082f82d1f09f240524d3e4967c88d648d0d96d5"},"schema_version":"1.0"},"canonical_sha256":"fc1244bc53a0fd12fb2010982d813fa231c8f08ffa38a459619be103da8858ec","source":{"kind":"arxiv","id":"1612.03208","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.03208","created_at":"2026-05-18T00:55:19Z"},{"alias_kind":"arxiv_version","alias_value":"1612.03208v1","created_at":"2026-05-18T00:55:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.03208","created_at":"2026-05-18T00:55:19Z"},{"alias_kind":"pith_short_12","alias_value":"7QJEJPCTUD6R","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"7QJEJPCTUD6RF6ZA","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"7QJEJPCT","created_at":"2026-05-18T12:30:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:7QJEJPCTUD6RF6ZACCMC3AJ7UI","target":"record","payload":{"canonical_record":{"source":{"id":"1612.03208","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2016-12-09T22:56:03Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"a40da16f46e5c1a6bf84b25a183dedce7a25a28680814b92198d4790793aa8bb","abstract_canon_sha256":"fedc256f4605bc92dc00f2b47082f82d1f09f240524d3e4967c88d648d0d96d5"},"schema_version":"1.0"},"canonical_sha256":"fc1244bc53a0fd12fb2010982d813fa231c8f08ffa38a459619be103da8858ec","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:55:19.968501Z","signature_b64":"6yUzFMjYnZyp8a46kMa89Xmk/cgipdWW1/zv4KhmndIIZWAZENIymlrcViSoNnZigwgKjUYEWB6KexIe4QTqAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fc1244bc53a0fd12fb2010982d813fa231c8f08ffa38a459619be103da8858ec","last_reissued_at":"2026-05-18T00:55:19.967859Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:55:19.967859Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1612.03208","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":"2026-05-18T00:55:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Apb7I5fqOs7cDxCmZCARrkf68tuksy8CSxBJhj/efgmQmSyDZGoF59L908hnzhNL5gRnwmoptSvmsOK6i2YxBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T13:13:44.357204Z"},"content_sha256":"94d543c40f4cc8818cf4287ae22a08f1dd09f583afd86c8f07630a3c6be50c7a","schema_version":"1.0","event_id":"sha256:94d543c40f4cc8818cf4287ae22a08f1dd09f583afd86c8f07630a3c6be50c7a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:7QJEJPCTUD6RF6ZACCMC3AJ7UI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A condition for purely absolutely continuous spectrum for CMV operators using the density of states","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.SP","authors_text":"Darren C. Ong, Jake Fillman","submitted_at":"2016-12-09T22:56:03Z","abstract_excerpt":"We prove an averaging formula for the derivative of the absolutely continuous part of the density of states measure for an ergodic family of CMV matrices. As a consequence, we show that the spectral type of such a family is almost surely purely absolutely continuous if and only if the density of states is absolutely continuous and the Lyapunov exponent vanishes almost everywhere with respect to the same. Both of these results are CMV operator analogues of theorems obtained by Kotani for Schr\\\"odinger operators."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.03208","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":"2026-05-18T00:55:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"B0meqUCfc/WH+kf11X/3NbgPjuzTPrLvn8pxbaJVpSFt8614kG1Vtvljo9Rr1ZIEVjI9kL4bzb0UXGBrw5m8Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T13:13:44.357567Z"},"content_sha256":"68fbe78563aade3ea529b026b0ea396f73c8eed6c044cb0e361b94fc5f363fe3","schema_version":"1.0","event_id":"sha256:68fbe78563aade3ea529b026b0ea396f73c8eed6c044cb0e361b94fc5f363fe3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7QJEJPCTUD6RF6ZACCMC3AJ7UI/bundle.json","state_url":"https://pith.science/pith/7QJEJPCTUD6RF6ZACCMC3AJ7UI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7QJEJPCTUD6RF6ZACCMC3AJ7UI/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":"2026-06-09T13:13:44Z","links":{"resolver":"https://pith.science/pith/7QJEJPCTUD6RF6ZACCMC3AJ7UI","bundle":"https://pith.science/pith/7QJEJPCTUD6RF6ZACCMC3AJ7UI/bundle.json","state":"https://pith.science/pith/7QJEJPCTUD6RF6ZACCMC3AJ7UI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7QJEJPCTUD6RF6ZACCMC3AJ7UI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:7QJEJPCTUD6RF6ZACCMC3AJ7UI","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":"fedc256f4605bc92dc00f2b47082f82d1f09f240524d3e4967c88d648d0d96d5","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2016-12-09T22:56:03Z","title_canon_sha256":"a40da16f46e5c1a6bf84b25a183dedce7a25a28680814b92198d4790793aa8bb"},"schema_version":"1.0","source":{"id":"1612.03208","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.03208","created_at":"2026-05-18T00:55:19Z"},{"alias_kind":"arxiv_version","alias_value":"1612.03208v1","created_at":"2026-05-18T00:55:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.03208","created_at":"2026-05-18T00:55:19Z"},{"alias_kind":"pith_short_12","alias_value":"7QJEJPCTUD6R","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"7QJEJPCTUD6RF6ZA","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"7QJEJPCT","created_at":"2026-05-18T12:30:04Z"}],"graph_snapshots":[{"event_id":"sha256:68fbe78563aade3ea529b026b0ea396f73c8eed6c044cb0e361b94fc5f363fe3","target":"graph","created_at":"2026-05-18T00:55:19Z","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 prove an averaging formula for the derivative of the absolutely continuous part of the density of states measure for an ergodic family of CMV matrices. As a consequence, we show that the spectral type of such a family is almost surely purely absolutely continuous if and only if the density of states is absolutely continuous and the Lyapunov exponent vanishes almost everywhere with respect to the same. Both of these results are CMV operator analogues of theorems obtained by Kotani for Schr\\\"odinger operators.","authors_text":"Darren C. Ong, Jake Fillman","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2016-12-09T22:56:03Z","title":"A condition for purely absolutely continuous spectrum for CMV operators using the density of states"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.03208","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":"sha256:94d543c40f4cc8818cf4287ae22a08f1dd09f583afd86c8f07630a3c6be50c7a","target":"record","created_at":"2026-05-18T00:55:19Z","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":"fedc256f4605bc92dc00f2b47082f82d1f09f240524d3e4967c88d648d0d96d5","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2016-12-09T22:56:03Z","title_canon_sha256":"a40da16f46e5c1a6bf84b25a183dedce7a25a28680814b92198d4790793aa8bb"},"schema_version":"1.0","source":{"id":"1612.03208","kind":"arxiv","version":1}},"canonical_sha256":"fc1244bc53a0fd12fb2010982d813fa231c8f08ffa38a459619be103da8858ec","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fc1244bc53a0fd12fb2010982d813fa231c8f08ffa38a459619be103da8858ec","first_computed_at":"2026-05-18T00:55:19.967859Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:55:19.967859Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6yUzFMjYnZyp8a46kMa89Xmk/cgipdWW1/zv4KhmndIIZWAZENIymlrcViSoNnZigwgKjUYEWB6KexIe4QTqAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:55:19.968501Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.03208","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:94d543c40f4cc8818cf4287ae22a08f1dd09f583afd86c8f07630a3c6be50c7a","sha256:68fbe78563aade3ea529b026b0ea396f73c8eed6c044cb0e361b94fc5f363fe3"],"state_sha256":"f993d1e5e9b1a927ca420f4f1542b336964dacbec1ea0c371e854960cbb811f5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GwQoio2v71zTO6KgP6rsp3j0sRprwBh0Dzrh/9/WG40M62mHDcyf2t7ajweZSHoinmRK0eY2LkBJVOFVZlORAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T13:13:44.359678Z","bundle_sha256":"c92ef94bd9b23d961d556618cbff2906ac94b69c5d8e7d9a63d955b8c708317d"}}