{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:1999:AY2LOZQCU3CLACX2RVA4BSPFCF","short_pith_number":"pith:AY2LOZQC","canonical_record":{"source":{"id":"math/9912236","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"1999-12-30T20:50:33Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"722cc597cfc4420cbee9029e049e32664ae5ed116db0f76527b2df99fc96dc1f","abstract_canon_sha256":"f9e0326f60c59297589061b20f8d0540a84f8b04f9c8c0d6f6d3f6513d51c6e1"},"schema_version":"1.0"},"canonical_sha256":"0634b76602a6c4b00afa8d41c0c9e5115009be4e5aeabc42ae4a23c600274373","source":{"kind":"arxiv","id":"math/9912236","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9912236","created_at":"2026-07-04T14:41:33Z"},{"alias_kind":"arxiv_version","alias_value":"math/9912236v1","created_at":"2026-07-04T14:41:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9912236","created_at":"2026-07-04T14:41:33Z"},{"alias_kind":"pith_short_12","alias_value":"AY2LOZQCU3CL","created_at":"2026-07-04T14:41:33Z"},{"alias_kind":"pith_short_16","alias_value":"AY2LOZQCU3CLACX2","created_at":"2026-07-04T14:41:33Z"},{"alias_kind":"pith_short_8","alias_value":"AY2LOZQC","created_at":"2026-07-04T14:41:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:1999:AY2LOZQCU3CLACX2RVA4BSPFCF","target":"record","payload":{"canonical_record":{"source":{"id":"math/9912236","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"1999-12-30T20:50:33Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"722cc597cfc4420cbee9029e049e32664ae5ed116db0f76527b2df99fc96dc1f","abstract_canon_sha256":"f9e0326f60c59297589061b20f8d0540a84f8b04f9c8c0d6f6d3f6513d51c6e1"},"schema_version":"1.0"},"canonical_sha256":"0634b76602a6c4b00afa8d41c0c9e5115009be4e5aeabc42ae4a23c600274373","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:41:33.333103Z","signature_b64":"m8jEGB23Mbrij/7u4wIkVdz/zPBJMCebARLM8VY9VNUaWaWKKuLtfdmN7A2wc9Gim7GXPXmREE1VQQgXRcN7BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0634b76602a6c4b00afa8d41c0c9e5115009be4e5aeabc42ae4a23c600274373","last_reissued_at":"2026-07-04T14:41:33.332701Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:41:33.332701Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/9912236","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":"2026-07-04T14:41:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8t7zAx1F15cs/r2YrhEM93IFWpEQN4lmhzuHOE4ah9J/Q0ODcj7/123+2axZlzu5pMQkbbx6KQpWB+fH9k4gCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-18T17:32:21.893366Z"},"content_sha256":"b45db0d36862546de3d379dbdb89acc1ebaf3b838276a4fc3cf6058ffb7147f6","schema_version":"1.0","event_id":"sha256:b45db0d36862546de3d379dbdb89acc1ebaf3b838276a4fc3cf6058ffb7147f6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:1999:AY2LOZQCU3CLACX2RVA4BSPFCF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Classification of positive definite lattices","license":"","headline":"","cross_cats":["math.GR"],"primary_cat":"math.NT","authors_text":"R. E. Borcherds","submitted_at":"1999-12-30T20:50:33Z","abstract_excerpt":"In this paper we describe an algorithm for classifying orbits of vectors in Lorentzian lattices. The main point of this is that isomorphism classes of positive definite lattices in some genus often correspond to orbits of vectors in some Lorentzian lattice, so we can classify some positive definite lattices. As an application we give the classification of the 665 25-dimensional unimodular positive definite lattices and the 121 even 25 dimensional positive definite lattices of determinant 2. We also use this algorithm to show that there is a unique 26 dimensional unimodular positive definite la"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9912236","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/9912236/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":"2026-07-04T14:41:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"R1Tegze3a/8If9MBEoSOeLMlP6rGF7g0cMHe2C2hKMsdBuc5jMKRs2NwKfj+nP+z+0bQhBbUJQuPoOd+D5K1Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-18T17:32:21.893739Z"},"content_sha256":"d9db8639ecea432eeb3165b94ce84e304b2109cd71b29f11641b75a035c1b871","schema_version":"1.0","event_id":"sha256:d9db8639ecea432eeb3165b94ce84e304b2109cd71b29f11641b75a035c1b871"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AY2LOZQCU3CLACX2RVA4BSPFCF/bundle.json","state_url":"https://pith.science/pith/AY2LOZQCU3CLACX2RVA4BSPFCF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AY2LOZQCU3CLACX2RVA4BSPFCF/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":"2026-08-18T17:32:21Z","links":{"resolver":"https://pith.science/pith/AY2LOZQCU3CLACX2RVA4BSPFCF","bundle":"https://pith.science/pith/AY2LOZQCU3CLACX2RVA4BSPFCF/bundle.json","state":"https://pith.science/pith/AY2LOZQCU3CLACX2RVA4BSPFCF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AY2LOZQCU3CLACX2RVA4BSPFCF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1999:AY2LOZQCU3CLACX2RVA4BSPFCF","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":"f9e0326f60c59297589061b20f8d0540a84f8b04f9c8c0d6f6d3f6513d51c6e1","cross_cats_sorted":["math.GR"],"license":"","primary_cat":"math.NT","submitted_at":"1999-12-30T20:50:33Z","title_canon_sha256":"722cc597cfc4420cbee9029e049e32664ae5ed116db0f76527b2df99fc96dc1f"},"schema_version":"1.0","source":{"id":"math/9912236","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9912236","created_at":"2026-07-04T14:41:33Z"},{"alias_kind":"arxiv_version","alias_value":"math/9912236v1","created_at":"2026-07-04T14:41:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9912236","created_at":"2026-07-04T14:41:33Z"},{"alias_kind":"pith_short_12","alias_value":"AY2LOZQCU3CL","created_at":"2026-07-04T14:41:33Z"},{"alias_kind":"pith_short_16","alias_value":"AY2LOZQCU3CLACX2","created_at":"2026-07-04T14:41:33Z"},{"alias_kind":"pith_short_8","alias_value":"AY2LOZQC","created_at":"2026-07-04T14:41:33Z"}],"graph_snapshots":[{"event_id":"sha256:d9db8639ecea432eeb3165b94ce84e304b2109cd71b29f11641b75a035c1b871","target":"graph","created_at":"2026-07-04T14:41:33Z","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/9912236/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper we describe an algorithm for classifying orbits of vectors in Lorentzian lattices. The main point of this is that isomorphism classes of positive definite lattices in some genus often correspond to orbits of vectors in some Lorentzian lattice, so we can classify some positive definite lattices. As an application we give the classification of the 665 25-dimensional unimodular positive definite lattices and the 121 even 25 dimensional positive definite lattices of determinant 2. We also use this algorithm to show that there is a unique 26 dimensional unimodular positive definite la","authors_text":"R. E. Borcherds","cross_cats":["math.GR"],"headline":"","license":"","primary_cat":"math.NT","submitted_at":"1999-12-30T20:50:33Z","title":"Classification of positive definite lattices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9912236","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:b45db0d36862546de3d379dbdb89acc1ebaf3b838276a4fc3cf6058ffb7147f6","target":"record","created_at":"2026-07-04T14:41:33Z","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":"f9e0326f60c59297589061b20f8d0540a84f8b04f9c8c0d6f6d3f6513d51c6e1","cross_cats_sorted":["math.GR"],"license":"","primary_cat":"math.NT","submitted_at":"1999-12-30T20:50:33Z","title_canon_sha256":"722cc597cfc4420cbee9029e049e32664ae5ed116db0f76527b2df99fc96dc1f"},"schema_version":"1.0","source":{"id":"math/9912236","kind":"arxiv","version":1}},"canonical_sha256":"0634b76602a6c4b00afa8d41c0c9e5115009be4e5aeabc42ae4a23c600274373","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0634b76602a6c4b00afa8d41c0c9e5115009be4e5aeabc42ae4a23c600274373","first_computed_at":"2026-07-04T14:41:33.332701Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T14:41:33.332701Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"m8jEGB23Mbrij/7u4wIkVdz/zPBJMCebARLM8VY9VNUaWaWKKuLtfdmN7A2wc9Gim7GXPXmREE1VQQgXRcN7BQ==","signature_status":"signed_v1","signed_at":"2026-07-04T14:41:33.333103Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/9912236","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b45db0d36862546de3d379dbdb89acc1ebaf3b838276a4fc3cf6058ffb7147f6","sha256:d9db8639ecea432eeb3165b94ce84e304b2109cd71b29f11641b75a035c1b871"],"state_sha256":"38e6fba28b5bc14b9eb684103996a0c9e863200da7cba15c94b6d705008d278c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qlYZ5jIDtOf0hkIhKwTo9kVU4j9cPnpUHGYwSx2rjwMQ+SAXnLDMlV0l/aUKiusZF50Rdt4dprvEOSao00PHAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-18T17:32:21.896110Z","bundle_sha256":"0ab2a36018847dcb82216d4382a2f516edf32b2058db650e272986ef2ac64d08"}}