{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2007:Q2C7TDO63XA65FKQGKCJZP34XA","short_pith_number":"pith:Q2C7TDO6","canonical_record":{"source":{"id":"math-ph/0702052","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math-ph","submitted_at":"2007-02-15T14:07:55Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"5c4aff37fe8556c7d1aa6d8134f72de397d86c6fe1889d4b46e1de72bcf42ccf","abstract_canon_sha256":"adcfe29eb75098d09eac0cb318c58822e3ea1f1321ee0a4194510854bfae5720"},"schema_version":"1.0"},"canonical_sha256":"8685f98ddeddc1ee955032849cbf7cb8099d0a6eab356364ab1e6699874cfa6c","source":{"kind":"arxiv","id":"math-ph/0702052","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math-ph/0702052","created_at":"2026-05-18T04:31:00Z"},{"alias_kind":"arxiv_version","alias_value":"math-ph/0702052v1","created_at":"2026-05-18T04:31:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math-ph/0702052","created_at":"2026-05-18T04:31:00Z"},{"alias_kind":"pith_short_12","alias_value":"Q2C7TDO63XA6","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_16","alias_value":"Q2C7TDO63XA65FKQ","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_8","alias_value":"Q2C7TDO6","created_at":"2026-05-18T12:25:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2007:Q2C7TDO63XA65FKQGKCJZP34XA","target":"record","payload":{"canonical_record":{"source":{"id":"math-ph/0702052","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math-ph","submitted_at":"2007-02-15T14:07:55Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"5c4aff37fe8556c7d1aa6d8134f72de397d86c6fe1889d4b46e1de72bcf42ccf","abstract_canon_sha256":"adcfe29eb75098d09eac0cb318c58822e3ea1f1321ee0a4194510854bfae5720"},"schema_version":"1.0"},"canonical_sha256":"8685f98ddeddc1ee955032849cbf7cb8099d0a6eab356364ab1e6699874cfa6c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:31:00.346342Z","signature_b64":"8y4tjUs5XOdk4KUZtwFU9XQceAgxQGWQIDbAAOfpRbF/o5CDq0FdovETGIFYMqnpdBuwV2OrvhOnj7fmh16VCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8685f98ddeddc1ee955032849cbf7cb8099d0a6eab356364ab1e6699874cfa6c","last_reissued_at":"2026-05-18T04:31:00.345463Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:31:00.345463Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math-ph/0702052","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":"2026-05-18T04:31:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0ZD1dTG1dM+0VUE+hTgydyNeo2B828nE+4oCDfI08Yxov67lZemTG8Kp9wvRFABKwLtFJvpsHa/+gfhZ0p5HDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T18:15:08.322919Z"},"content_sha256":"754bdfa8224f3f84c064660174c5b448683893000026c05c9ebf0368b704f46a","schema_version":"1.0","event_id":"sha256:754bdfa8224f3f84c064660174c5b448683893000026c05c9ebf0368b704f46a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2007:Q2C7TDO63XA65FKQGKCJZP34XA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Positive Lyapunov exponents and localization bounds for strongly mixing potentials","license":"","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Christian Sadel, Hermann Schulz-Baldes","submitted_at":"2007-02-15T14:07:55Z","abstract_excerpt":"For a one-dimensional discrete Schr\\\"odinger operator with a weakly coupled potential given by a strongly mixing dynamical system with power law decay of correlations, we derive for all energies including the band edges and the band center a perturbative formula for the Lyapunov exponent. Under adequate hypothesis, this shows that the Lyapunov exponent is positive on the whole spectrum. This in turn implies that the Hausdorff dimension of the spectral measure is zero and that the associated quantum dynamics grows at most logarithmically in time."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math-ph/0702052","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":"2026-05-18T04:31:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3BuyVI3ftZ8yhLF+eGR6dlB2guubcn16HpltbxnqEEicntYJLlOSS4Lb6L8oKC47+xIST9Cv9wp7tbV0JduhCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T18:15:08.323561Z"},"content_sha256":"28083d48056e5a28fa1bd824cac947c54ee082603b5d6f639140ec17f3d32f54","schema_version":"1.0","event_id":"sha256:28083d48056e5a28fa1bd824cac947c54ee082603b5d6f639140ec17f3d32f54"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Q2C7TDO63XA65FKQGKCJZP34XA/bundle.json","state_url":"https://pith.science/pith/Q2C7TDO63XA65FKQGKCJZP34XA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Q2C7TDO63XA65FKQGKCJZP34XA/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":"2026-05-31T18:15:08Z","links":{"resolver":"https://pith.science/pith/Q2C7TDO63XA65FKQGKCJZP34XA","bundle":"https://pith.science/pith/Q2C7TDO63XA65FKQGKCJZP34XA/bundle.json","state":"https://pith.science/pith/Q2C7TDO63XA65FKQGKCJZP34XA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Q2C7TDO63XA65FKQGKCJZP34XA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:Q2C7TDO63XA65FKQGKCJZP34XA","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":"adcfe29eb75098d09eac0cb318c58822e3ea1f1321ee0a4194510854bfae5720","cross_cats_sorted":["math.MP"],"license":"","primary_cat":"math-ph","submitted_at":"2007-02-15T14:07:55Z","title_canon_sha256":"5c4aff37fe8556c7d1aa6d8134f72de397d86c6fe1889d4b46e1de72bcf42ccf"},"schema_version":"1.0","source":{"id":"math-ph/0702052","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math-ph/0702052","created_at":"2026-05-18T04:31:00Z"},{"alias_kind":"arxiv_version","alias_value":"math-ph/0702052v1","created_at":"2026-05-18T04:31:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math-ph/0702052","created_at":"2026-05-18T04:31:00Z"},{"alias_kind":"pith_short_12","alias_value":"Q2C7TDO63XA6","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_16","alias_value":"Q2C7TDO63XA65FKQ","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_8","alias_value":"Q2C7TDO6","created_at":"2026-05-18T12:25:55Z"}],"graph_snapshots":[{"event_id":"sha256:28083d48056e5a28fa1bd824cac947c54ee082603b5d6f639140ec17f3d32f54","target":"graph","created_at":"2026-05-18T04:31:00Z","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":"For a one-dimensional discrete Schr\\\"odinger operator with a weakly coupled potential given by a strongly mixing dynamical system with power law decay of correlations, we derive for all energies including the band edges and the band center a perturbative formula for the Lyapunov exponent. Under adequate hypothesis, this shows that the Lyapunov exponent is positive on the whole spectrum. This in turn implies that the Hausdorff dimension of the spectral measure is zero and that the associated quantum dynamics grows at most logarithmically in time.","authors_text":"Christian Sadel, Hermann Schulz-Baldes","cross_cats":["math.MP"],"headline":"","license":"","primary_cat":"math-ph","submitted_at":"2007-02-15T14:07:55Z","title":"Positive Lyapunov exponents and localization bounds for strongly mixing potentials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math-ph/0702052","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":"sha256:754bdfa8224f3f84c064660174c5b448683893000026c05c9ebf0368b704f46a","target":"record","created_at":"2026-05-18T04:31:00Z","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":"adcfe29eb75098d09eac0cb318c58822e3ea1f1321ee0a4194510854bfae5720","cross_cats_sorted":["math.MP"],"license":"","primary_cat":"math-ph","submitted_at":"2007-02-15T14:07:55Z","title_canon_sha256":"5c4aff37fe8556c7d1aa6d8134f72de397d86c6fe1889d4b46e1de72bcf42ccf"},"schema_version":"1.0","source":{"id":"math-ph/0702052","kind":"arxiv","version":1}},"canonical_sha256":"8685f98ddeddc1ee955032849cbf7cb8099d0a6eab356364ab1e6699874cfa6c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8685f98ddeddc1ee955032849cbf7cb8099d0a6eab356364ab1e6699874cfa6c","first_computed_at":"2026-05-18T04:31:00.345463Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:31:00.345463Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8y4tjUs5XOdk4KUZtwFU9XQceAgxQGWQIDbAAOfpRbF/o5CDq0FdovETGIFYMqnpdBuwV2OrvhOnj7fmh16VCw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:31:00.346342Z","signed_message":"canonical_sha256_bytes"},"source_id":"math-ph/0702052","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:754bdfa8224f3f84c064660174c5b448683893000026c05c9ebf0368b704f46a","sha256:28083d48056e5a28fa1bd824cac947c54ee082603b5d6f639140ec17f3d32f54"],"state_sha256":"5de75d96791c73dbded2b74f0d649d2baab330e884377d2864116969bb6cd3d7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BmJWeq5BruRnw9bvMAtFHjutCx5MBRhcwb9rvAKTdkgDb3Z4dMBKAPC+FfX/IUn/guqPOwQ2KyqTD6HV96cyDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T18:15:08.327864Z","bundle_sha256":"8072a06ed9ae13e9dcfc70a961efe3f52d3025b1295613e081289c53f86029a7"}}