{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:FQZ6GWZ6ORRZZ3H5YH2HSYOV7N","short_pith_number":"pith:FQZ6GWZ6","canonical_record":{"source":{"id":"1503.01328","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-03-04T14:57:44Z","cross_cats_sorted":[],"title_canon_sha256":"f52011175d384310aad0316513f2e3af74eb6c875efae7e413131b90b11f64a8","abstract_canon_sha256":"e7d7c46e5b6e99df0d6ca347fecd8109c77f6fb59bd61ddac1d674103930bab2"},"schema_version":"1.0"},"canonical_sha256":"2c33e35b3e74639cecfdc1f47961d5fb7d01b2d6908d2974453643b9eca72859","source":{"kind":"arxiv","id":"1503.01328","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.01328","created_at":"2026-05-18T02:25:34Z"},{"alias_kind":"arxiv_version","alias_value":"1503.01328v1","created_at":"2026-05-18T02:25:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.01328","created_at":"2026-05-18T02:25:34Z"},{"alias_kind":"pith_short_12","alias_value":"FQZ6GWZ6ORRZ","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_16","alias_value":"FQZ6GWZ6ORRZZ3H5","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_8","alias_value":"FQZ6GWZ6","created_at":"2026-05-18T12:29:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:FQZ6GWZ6ORRZZ3H5YH2HSYOV7N","target":"record","payload":{"canonical_record":{"source":{"id":"1503.01328","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-03-04T14:57:44Z","cross_cats_sorted":[],"title_canon_sha256":"f52011175d384310aad0316513f2e3af74eb6c875efae7e413131b90b11f64a8","abstract_canon_sha256":"e7d7c46e5b6e99df0d6ca347fecd8109c77f6fb59bd61ddac1d674103930bab2"},"schema_version":"1.0"},"canonical_sha256":"2c33e35b3e74639cecfdc1f47961d5fb7d01b2d6908d2974453643b9eca72859","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:25:34.790228Z","signature_b64":"Z/rIuoVRud4U6P8GuqBU4Gl31qF2oyXYqEOrYJT26wL0K8pCyg6gfH+yYInWMYmAUD/XnD6L76nI7XIpRNsxBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2c33e35b3e74639cecfdc1f47961d5fb7d01b2d6908d2974453643b9eca72859","last_reissued_at":"2026-05-18T02:25:34.789795Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:25:34.789795Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1503.01328","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":"2026-05-18T02:25:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GAQRv4Jg/MU3dQehHHD4YzpFo1Q1k+PE3he4eP49OyMzg/+fMU67AuKPuxqcOCgF8YP0o/jdwFbxfiiPOTCgBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T11:48:21.262856Z"},"content_sha256":"747d5dab480daacf7e846b5cd074812abe2cb1b83bed9b3f86818344755cb1e1","schema_version":"1.0","event_id":"sha256:747d5dab480daacf7e846b5cd074812abe2cb1b83bed9b3f86818344755cb1e1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:FQZ6GWZ6ORRZZ3H5YH2HSYOV7N","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Explicit Height Distribution and Expected Number of Local Maxima of Isotropic Gaussian Random Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Armin Schwartzman, Dan Cheng","submitted_at":"2015-03-04T14:57:44Z","abstract_excerpt":"The explicit formulae for the height distribution and expected number of local maxima have been obtained for isotropic Gaussian random fields on certain low-dimensional Euclidean space or low-dimensional spheres."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.01328","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":"2026-05-18T02:25:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Va98kXLXFtEe22WjSsRyEpbV8S0IGgldflAdQaPX9UZXocncSwMn5j7FwwcvRYjL5WMz1slKE+iJx/wMXQzpBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T11:48:21.263200Z"},"content_sha256":"84cb27db2044260348179ed232c5aafa2e6a8b5566e78d7fb22f200a3d2859ff","schema_version":"1.0","event_id":"sha256:84cb27db2044260348179ed232c5aafa2e6a8b5566e78d7fb22f200a3d2859ff"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FQZ6GWZ6ORRZZ3H5YH2HSYOV7N/bundle.json","state_url":"https://pith.science/pith/FQZ6GWZ6ORRZZ3H5YH2HSYOV7N/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FQZ6GWZ6ORRZZ3H5YH2HSYOV7N/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":"2026-06-01T11:48:21Z","links":{"resolver":"https://pith.science/pith/FQZ6GWZ6ORRZZ3H5YH2HSYOV7N","bundle":"https://pith.science/pith/FQZ6GWZ6ORRZZ3H5YH2HSYOV7N/bundle.json","state":"https://pith.science/pith/FQZ6GWZ6ORRZZ3H5YH2HSYOV7N/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FQZ6GWZ6ORRZZ3H5YH2HSYOV7N/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:FQZ6GWZ6ORRZZ3H5YH2HSYOV7N","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":"e7d7c46e5b6e99df0d6ca347fecd8109c77f6fb59bd61ddac1d674103930bab2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-03-04T14:57:44Z","title_canon_sha256":"f52011175d384310aad0316513f2e3af74eb6c875efae7e413131b90b11f64a8"},"schema_version":"1.0","source":{"id":"1503.01328","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.01328","created_at":"2026-05-18T02:25:34Z"},{"alias_kind":"arxiv_version","alias_value":"1503.01328v1","created_at":"2026-05-18T02:25:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.01328","created_at":"2026-05-18T02:25:34Z"},{"alias_kind":"pith_short_12","alias_value":"FQZ6GWZ6ORRZ","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_16","alias_value":"FQZ6GWZ6ORRZZ3H5","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_8","alias_value":"FQZ6GWZ6","created_at":"2026-05-18T12:29:22Z"}],"graph_snapshots":[{"event_id":"sha256:84cb27db2044260348179ed232c5aafa2e6a8b5566e78d7fb22f200a3d2859ff","target":"graph","created_at":"2026-05-18T02:25:34Z","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":"The explicit formulae for the height distribution and expected number of local maxima have been obtained for isotropic Gaussian random fields on certain low-dimensional Euclidean space or low-dimensional spheres.","authors_text":"Armin Schwartzman, Dan Cheng","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-03-04T14:57:44Z","title":"On the Explicit Height Distribution and Expected Number of Local Maxima of Isotropic Gaussian Random Fields"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.01328","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":"sha256:747d5dab480daacf7e846b5cd074812abe2cb1b83bed9b3f86818344755cb1e1","target":"record","created_at":"2026-05-18T02:25:34Z","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":"e7d7c46e5b6e99df0d6ca347fecd8109c77f6fb59bd61ddac1d674103930bab2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-03-04T14:57:44Z","title_canon_sha256":"f52011175d384310aad0316513f2e3af74eb6c875efae7e413131b90b11f64a8"},"schema_version":"1.0","source":{"id":"1503.01328","kind":"arxiv","version":1}},"canonical_sha256":"2c33e35b3e74639cecfdc1f47961d5fb7d01b2d6908d2974453643b9eca72859","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2c33e35b3e74639cecfdc1f47961d5fb7d01b2d6908d2974453643b9eca72859","first_computed_at":"2026-05-18T02:25:34.789795Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:25:34.789795Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Z/rIuoVRud4U6P8GuqBU4Gl31qF2oyXYqEOrYJT26wL0K8pCyg6gfH+yYInWMYmAUD/XnD6L76nI7XIpRNsxBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:25:34.790228Z","signed_message":"canonical_sha256_bytes"},"source_id":"1503.01328","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:747d5dab480daacf7e846b5cd074812abe2cb1b83bed9b3f86818344755cb1e1","sha256:84cb27db2044260348179ed232c5aafa2e6a8b5566e78d7fb22f200a3d2859ff"],"state_sha256":"c35dd43fb89dae715785c76222ab3b6ba4ecb5b931735d432febf46c64172634"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7dkGrMrLEgXFxy1/JcbpgC3r+2YafAvV6AFwsNc3lJQ8sFMyg9bIa7ZbULDvJiQ0wO383DWbZQx8i0CEtvxlBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T11:48:21.265157Z","bundle_sha256":"2126e89bbf43eb67b10b55a74a4802c6bd7603d10ad008317c84f6c7a058de3f"}}