{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:1999:6SQZAYG4BAKW2EUWJE6OOLTZMX","short_pith_number":"pith:6SQZAYG4","canonical_record":{"source":{"id":"math/9907165","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CA","submitted_at":"1999-07-25T05:15:50Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"2c7d750b23f38972f10cb6224c9a8dc65c7f8fd2912884e5ec9208bab3f6269d","abstract_canon_sha256":"adcff7fd8d0b7cc5dd7e3666bba271ed0c54c06af750295f84b8b6ddae459492"},"schema_version":"1.0"},"canonical_sha256":"f4a19060dc08156d1296493ce72e7965e031ae1990f7c05ac0bbbfb80db481eb","source":{"kind":"arxiv","id":"math/9907165","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9907165","created_at":"2026-07-04T14:41:15Z"},{"alias_kind":"arxiv_version","alias_value":"math/9907165v1","created_at":"2026-07-04T14:41:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9907165","created_at":"2026-07-04T14:41:15Z"},{"alias_kind":"pith_short_12","alias_value":"6SQZAYG4BAKW","created_at":"2026-07-04T14:41:15Z"},{"alias_kind":"pith_short_16","alias_value":"6SQZAYG4BAKW2EUW","created_at":"2026-07-04T14:41:15Z"},{"alias_kind":"pith_short_8","alias_value":"6SQZAYG4","created_at":"2026-07-04T14:41:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:1999:6SQZAYG4BAKW2EUWJE6OOLTZMX","target":"record","payload":{"canonical_record":{"source":{"id":"math/9907165","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CA","submitted_at":"1999-07-25T05:15:50Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"2c7d750b23f38972f10cb6224c9a8dc65c7f8fd2912884e5ec9208bab3f6269d","abstract_canon_sha256":"adcff7fd8d0b7cc5dd7e3666bba271ed0c54c06af750295f84b8b6ddae459492"},"schema_version":"1.0"},"canonical_sha256":"f4a19060dc08156d1296493ce72e7965e031ae1990f7c05ac0bbbfb80db481eb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:41:15.092117Z","signature_b64":"rYaUAowxuZZ/O5XjRUTvqx+BEcMqxyOrkhuQduxPTAAWxszGyZTmNJ1RaYbuCkzfRgNSy4F3V+2QBFr8zJbvBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f4a19060dc08156d1296493ce72e7965e031ae1990f7c05ac0bbbfb80db481eb","last_reissued_at":"2026-07-04T14:41:15.091745Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:41:15.091745Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/9907165","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":"2026-07-04T14:41:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qSWziChgMO+etHYMDR7XmGBPnCIgu2vo6B/U21Ba402OXyF+hVcQV0Gx67T5DEUkvWCRAvuUgwusyjfOx5MpAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T21:23:18.291741Z"},"content_sha256":"89d7c3102c90260afaafd15956fd4b7214c62511de1229382cf77fc1b97deaff","schema_version":"1.0","event_id":"sha256:89d7c3102c90260afaafd15956fd4b7214c62511de1229382cf77fc1b97deaff"},{"event_type":"graph_snapshot","subject_pith_number":"pith:1999:6SQZAYG4BAKW2EUWJE6OOLTZMX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Fredholm determinant formula for Toeplitz determinants","license":"","headline":"","cross_cats":["math.RT"],"primary_cat":"math.CA","authors_text":"Alexei Borodin, Andrei Okounkov","submitted_at":"1999-07-25T05:15:50Z","abstract_excerpt":"We prove a formula expressing a general n by n Toeplitz determinant as a Fredholm determinant of an operator 1-K acting on l_2({n,n+1,...}), where the kernel K admits an integral representation in terms of the symbol of the original Toeplitz matrix.\n  The proof is based on the results of one of the authors, see math.RT/9907127, and a formula due to Gessel which expands any Toeplitz determinant into a series of Schur functions.\n  We also consider 3 examples where the kernel involves the Gauss hypergeometric function and its degenerations."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9907165","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/9907165/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-04T14:41:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"g/ikAIe18fVytQZjN7PXdnhM0K2FEGrwXl1/pX60RwWmbDru06Qxb39HMWeiMNwBBIOfav4rLLFkP5QBbAtdBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T21:23:18.292298Z"},"content_sha256":"9849bf7bfd97126226d64c460dd0bfa92cc207dfe7710a87eed7acb68f623c4f","schema_version":"1.0","event_id":"sha256:9849bf7bfd97126226d64c460dd0bfa92cc207dfe7710a87eed7acb68f623c4f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6SQZAYG4BAKW2EUWJE6OOLTZMX/bundle.json","state_url":"https://pith.science/pith/6SQZAYG4BAKW2EUWJE6OOLTZMX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6SQZAYG4BAKW2EUWJE6OOLTZMX/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":"2026-08-14T21:23:18Z","links":{"resolver":"https://pith.science/pith/6SQZAYG4BAKW2EUWJE6OOLTZMX","bundle":"https://pith.science/pith/6SQZAYG4BAKW2EUWJE6OOLTZMX/bundle.json","state":"https://pith.science/pith/6SQZAYG4BAKW2EUWJE6OOLTZMX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6SQZAYG4BAKW2EUWJE6OOLTZMX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1999:6SQZAYG4BAKW2EUWJE6OOLTZMX","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":"adcff7fd8d0b7cc5dd7e3666bba271ed0c54c06af750295f84b8b6ddae459492","cross_cats_sorted":["math.RT"],"license":"","primary_cat":"math.CA","submitted_at":"1999-07-25T05:15:50Z","title_canon_sha256":"2c7d750b23f38972f10cb6224c9a8dc65c7f8fd2912884e5ec9208bab3f6269d"},"schema_version":"1.0","source":{"id":"math/9907165","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9907165","created_at":"2026-07-04T14:41:15Z"},{"alias_kind":"arxiv_version","alias_value":"math/9907165v1","created_at":"2026-07-04T14:41:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9907165","created_at":"2026-07-04T14:41:15Z"},{"alias_kind":"pith_short_12","alias_value":"6SQZAYG4BAKW","created_at":"2026-07-04T14:41:15Z"},{"alias_kind":"pith_short_16","alias_value":"6SQZAYG4BAKW2EUW","created_at":"2026-07-04T14:41:15Z"},{"alias_kind":"pith_short_8","alias_value":"6SQZAYG4","created_at":"2026-07-04T14:41:15Z"}],"graph_snapshots":[{"event_id":"sha256:9849bf7bfd97126226d64c460dd0bfa92cc207dfe7710a87eed7acb68f623c4f","target":"graph","created_at":"2026-07-04T14:41:15Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/math/9907165/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove a formula expressing a general n by n Toeplitz determinant as a Fredholm determinant of an operator 1-K acting on l_2({n,n+1,...}), where the kernel K admits an integral representation in terms of the symbol of the original Toeplitz matrix.\n  The proof is based on the results of one of the authors, see math.RT/9907127, and a formula due to Gessel which expands any Toeplitz determinant into a series of Schur functions.\n  We also consider 3 examples where the kernel involves the Gauss hypergeometric function and its degenerations.","authors_text":"Alexei Borodin, Andrei Okounkov","cross_cats":["math.RT"],"headline":"","license":"","primary_cat":"math.CA","submitted_at":"1999-07-25T05:15:50Z","title":"A Fredholm determinant formula for Toeplitz determinants"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9907165","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":"sha256:89d7c3102c90260afaafd15956fd4b7214c62511de1229382cf77fc1b97deaff","target":"record","created_at":"2026-07-04T14:41:15Z","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":"adcff7fd8d0b7cc5dd7e3666bba271ed0c54c06af750295f84b8b6ddae459492","cross_cats_sorted":["math.RT"],"license":"","primary_cat":"math.CA","submitted_at":"1999-07-25T05:15:50Z","title_canon_sha256":"2c7d750b23f38972f10cb6224c9a8dc65c7f8fd2912884e5ec9208bab3f6269d"},"schema_version":"1.0","source":{"id":"math/9907165","kind":"arxiv","version":1}},"canonical_sha256":"f4a19060dc08156d1296493ce72e7965e031ae1990f7c05ac0bbbfb80db481eb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f4a19060dc08156d1296493ce72e7965e031ae1990f7c05ac0bbbfb80db481eb","first_computed_at":"2026-07-04T14:41:15.091745Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T14:41:15.091745Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rYaUAowxuZZ/O5XjRUTvqx+BEcMqxyOrkhuQduxPTAAWxszGyZTmNJ1RaYbuCkzfRgNSy4F3V+2QBFr8zJbvBg==","signature_status":"signed_v1","signed_at":"2026-07-04T14:41:15.092117Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/9907165","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:89d7c3102c90260afaafd15956fd4b7214c62511de1229382cf77fc1b97deaff","sha256:9849bf7bfd97126226d64c460dd0bfa92cc207dfe7710a87eed7acb68f623c4f"],"state_sha256":"ab83f34322d93f7fff2cbf0e7c3fde0a84e23b1927276455249473e74829242b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6qJ4ZbGaTKzauLFl71jaXRgj3s3zZ+eKaMHEVS3LymsZ1XHChhY8rfGvkOlllPYi0WsXu5hTOb8JweY5eengCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-14T21:23:18.296232Z","bundle_sha256":"22720eec90d000ea1d023d00437582bc37d2e6564d8512ebbde306e10d533b78"}}