{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:I7TSNRIL7ME24N4QFTM2EXAAJM","short_pith_number":"pith:I7TSNRIL","canonical_record":{"source":{"id":"1102.4136","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-02-21T05:30:04Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"b0ec1a055a81e2f4abcead950a0ceb9d8b9c2dfd1f47a5cb564f90b66bc7c831","abstract_canon_sha256":"ee8b50515b5736deab3ae00f77eebe995698ca5c84d8c2207f35ddbe11c294e1"},"schema_version":"1.0"},"canonical_sha256":"47e726c50bfb09ae37902cd9a25c004b31f51eff7e080b83941f6bebc4abbad1","source":{"kind":"arxiv","id":"1102.4136","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1102.4136","created_at":"2026-05-18T02:37:59Z"},{"alias_kind":"arxiv_version","alias_value":"1102.4136v2","created_at":"2026-05-18T02:37:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.4136","created_at":"2026-05-18T02:37:59Z"},{"alias_kind":"pith_short_12","alias_value":"I7TSNRIL7ME2","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_16","alias_value":"I7TSNRIL7ME24N4Q","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_8","alias_value":"I7TSNRIL","created_at":"2026-05-18T12:26:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:I7TSNRIL7ME24N4QFTM2EXAAJM","target":"record","payload":{"canonical_record":{"source":{"id":"1102.4136","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-02-21T05:30:04Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"b0ec1a055a81e2f4abcead950a0ceb9d8b9c2dfd1f47a5cb564f90b66bc7c831","abstract_canon_sha256":"ee8b50515b5736deab3ae00f77eebe995698ca5c84d8c2207f35ddbe11c294e1"},"schema_version":"1.0"},"canonical_sha256":"47e726c50bfb09ae37902cd9a25c004b31f51eff7e080b83941f6bebc4abbad1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:37:59.108791Z","signature_b64":"UndIs6FNf8pHcoLG90lcCUM+LzrUZcnh3NreyzJs6DcQjLVn8fKkIKfzsa4R07YW9ZOVpzxiUR5K93IYnSUQCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"47e726c50bfb09ae37902cd9a25c004b31f51eff7e080b83941f6bebc4abbad1","last_reissued_at":"2026-05-18T02:37:59.108407Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:37:59.108407Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1102.4136","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:37:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"o2IkGUDkEXlCICx4H4p1B0kw97kMY+KcP3TmZhM5H4yrS/U7y36Zh7TcZswSn4vPBbHB+FMztLxIzYgefP9aDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T19:52:40.291197Z"},"content_sha256":"af66daee5f7e5e679f594d0b8e24d09879dd62b57fa100cb5cef1a396dd51d62","schema_version":"1.0","event_id":"sha256:af66daee5f7e5e679f594d0b8e24d09879dd62b57fa100cb5cef1a396dd51d62"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:I7TSNRIL7ME24N4QFTM2EXAAJM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The algebraic geometry of Harper operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Dan Li","submitted_at":"2011-02-21T05:30:04Z","abstract_excerpt":"Following an approach developed by Gieseker, Kn\\\"orrer and Trubowitz for discretized Schr\\\"odinger operators, we study the spectral theory of Harper operators in dimension two and one, as a discretized model of magnetic Laplacians, from the point of view of algebraic geometry. We describe the geometry of an associated family of Bloch varieties and compute their density of states. Finally, we also compute some spectral functions based on the density of states.\n  We discuss the difference between the cases with rational or irrational parameters: for the two dimensional Harper operator, the compa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.4136","kind":"arxiv","version":2},"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":"2026-05-18T02:37:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oUaPLX4Hu0giVm3YNRMTae2+y5yORr6QXrFUu5CY3Q3ULRhGXBnHwSNihl3xUhJwxWHe/V9G6h4wKb9Vsz2/Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T19:52:40.291891Z"},"content_sha256":"6bbc87e511a3e0674b574b528ed303970e20ff3b99f4b755b36dd4f44f99b40a","schema_version":"1.0","event_id":"sha256:6bbc87e511a3e0674b574b528ed303970e20ff3b99f4b755b36dd4f44f99b40a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/I7TSNRIL7ME24N4QFTM2EXAAJM/bundle.json","state_url":"https://pith.science/pith/I7TSNRIL7ME24N4QFTM2EXAAJM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/I7TSNRIL7ME24N4QFTM2EXAAJM/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":"2026-06-06T19:52:40Z","links":{"resolver":"https://pith.science/pith/I7TSNRIL7ME24N4QFTM2EXAAJM","bundle":"https://pith.science/pith/I7TSNRIL7ME24N4QFTM2EXAAJM/bundle.json","state":"https://pith.science/pith/I7TSNRIL7ME24N4QFTM2EXAAJM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/I7TSNRIL7ME24N4QFTM2EXAAJM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:I7TSNRIL7ME24N4QFTM2EXAAJM","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":"ee8b50515b5736deab3ae00f77eebe995698ca5c84d8c2207f35ddbe11c294e1","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-02-21T05:30:04Z","title_canon_sha256":"b0ec1a055a81e2f4abcead950a0ceb9d8b9c2dfd1f47a5cb564f90b66bc7c831"},"schema_version":"1.0","source":{"id":"1102.4136","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1102.4136","created_at":"2026-05-18T02:37:59Z"},{"alias_kind":"arxiv_version","alias_value":"1102.4136v2","created_at":"2026-05-18T02:37:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.4136","created_at":"2026-05-18T02:37:59Z"},{"alias_kind":"pith_short_12","alias_value":"I7TSNRIL7ME2","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_16","alias_value":"I7TSNRIL7ME24N4Q","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_8","alias_value":"I7TSNRIL","created_at":"2026-05-18T12:26:30Z"}],"graph_snapshots":[{"event_id":"sha256:6bbc87e511a3e0674b574b528ed303970e20ff3b99f4b755b36dd4f44f99b40a","target":"graph","created_at":"2026-05-18T02:37:59Z","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":"Following an approach developed by Gieseker, Kn\\\"orrer and Trubowitz for discretized Schr\\\"odinger operators, we study the spectral theory of Harper operators in dimension two and one, as a discretized model of magnetic Laplacians, from the point of view of algebraic geometry. We describe the geometry of an associated family of Bloch varieties and compute their density of states. Finally, we also compute some spectral functions based on the density of states.\n  We discuss the difference between the cases with rational or irrational parameters: for the two dimensional Harper operator, the compa","authors_text":"Dan Li","cross_cats":["math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-02-21T05:30:04Z","title":"The algebraic geometry of Harper operators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.4136","kind":"arxiv","version":2},"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:af66daee5f7e5e679f594d0b8e24d09879dd62b57fa100cb5cef1a396dd51d62","target":"record","created_at":"2026-05-18T02:37:59Z","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":"ee8b50515b5736deab3ae00f77eebe995698ca5c84d8c2207f35ddbe11c294e1","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-02-21T05:30:04Z","title_canon_sha256":"b0ec1a055a81e2f4abcead950a0ceb9d8b9c2dfd1f47a5cb564f90b66bc7c831"},"schema_version":"1.0","source":{"id":"1102.4136","kind":"arxiv","version":2}},"canonical_sha256":"47e726c50bfb09ae37902cd9a25c004b31f51eff7e080b83941f6bebc4abbad1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"47e726c50bfb09ae37902cd9a25c004b31f51eff7e080b83941f6bebc4abbad1","first_computed_at":"2026-05-18T02:37:59.108407Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:37:59.108407Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UndIs6FNf8pHcoLG90lcCUM+LzrUZcnh3NreyzJs6DcQjLVn8fKkIKfzsa4R07YW9ZOVpzxiUR5K93IYnSUQCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:37:59.108791Z","signed_message":"canonical_sha256_bytes"},"source_id":"1102.4136","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:af66daee5f7e5e679f594d0b8e24d09879dd62b57fa100cb5cef1a396dd51d62","sha256:6bbc87e511a3e0674b574b528ed303970e20ff3b99f4b755b36dd4f44f99b40a"],"state_sha256":"2eb9f1e6fdd407a861aa56bbd714447b0cb084488d1dbe274700c6f3b38dedfb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XbpseSLkRp+F7xEKSS8/RFAIWhVI82Q1TXwoX5lMxH+UEjo+MPvtoZAit4bVoBLSA7hOVXvoZaHHSb4S704mAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T19:52:40.295844Z","bundle_sha256":"c5c789b4f641647f147e4efdb24c8d5e6778be2e17caa9884900a9cfc0f85469"}}