{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2008:TKXAZI5EOAKOKJYX7JU6Z4JCXP","short_pith_number":"pith:TKXAZI5E","canonical_record":{"source":{"id":"0811.1416","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2008-11-10T09:20:45Z","cross_cats_sorted":[],"title_canon_sha256":"397e56986a24cd431a4d8f89b6acf326d374443195dbbddce50fcf1e7f2775e5","abstract_canon_sha256":"bab6aff9d07a06f1306769b2a062a8e354a1e538b3ab6aa687a3b1d6196d55ef"},"schema_version":"1.0"},"canonical_sha256":"9aae0ca3a47014e52717fa69ecf122bbc791e742eba99b9a96ba29d2d124418b","source":{"kind":"arxiv","id":"0811.1416","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0811.1416","created_at":"2026-05-18T04:41:25Z"},{"alias_kind":"arxiv_version","alias_value":"0811.1416v1","created_at":"2026-05-18T04:41:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0811.1416","created_at":"2026-05-18T04:41:25Z"},{"alias_kind":"pith_short_12","alias_value":"TKXAZI5EOAKO","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"TKXAZI5EOAKOKJYX","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"TKXAZI5E","created_at":"2026-05-18T12:25:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2008:TKXAZI5EOAKOKJYX7JU6Z4JCXP","target":"record","payload":{"canonical_record":{"source":{"id":"0811.1416","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2008-11-10T09:20:45Z","cross_cats_sorted":[],"title_canon_sha256":"397e56986a24cd431a4d8f89b6acf326d374443195dbbddce50fcf1e7f2775e5","abstract_canon_sha256":"bab6aff9d07a06f1306769b2a062a8e354a1e538b3ab6aa687a3b1d6196d55ef"},"schema_version":"1.0"},"canonical_sha256":"9aae0ca3a47014e52717fa69ecf122bbc791e742eba99b9a96ba29d2d124418b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:41:25.366003Z","signature_b64":"S1wtFx9YeznEvYYBNA2rhqyg7bV/bhpkiylJdTEkNKeEARWAu0iFO7w0vLDU63zUagRX1cnYXG22GugtDv5EAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9aae0ca3a47014e52717fa69ecf122bbc791e742eba99b9a96ba29d2d124418b","last_reissued_at":"2026-05-18T04:41:25.365250Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:41:25.365250Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0811.1416","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":"2026-05-18T04:41:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ey6YB5xAxXdnu+7/pppxGwMx01vsod0QErbYmlIMNpd28WoZ0RzRLbwPqkKmYsidXHz08JzohP1k3I/idMeSBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T23:32:43.172377Z"},"content_sha256":"53b6c8e62af7965c39d2c32c8c1aaa5daab31f72392f5e6208d60d0d993f2c27","schema_version":"1.0","event_id":"sha256:53b6c8e62af7965c39d2c32c8c1aaa5daab31f72392f5e6208d60d0d993f2c27"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2008:TKXAZI5EOAKOKJYX7JU6Z4JCXP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Spherical designs via Brouwer fixed point theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Andriy V. Bondarenko, Maryna S. Viazovska","submitted_at":"2008-11-10T09:20:45Z","abstract_excerpt":"For each N>=c_d*n^{2d*(d+1)/(d+2)} we prove the existence of a spherical n-design on S^d consisting of N points, where c_d is a constant depending only on $d$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0811.1416","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":"2026-05-18T04:41:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RnfKwVaVMQs2wx+PgPbQPpvVK5wWa7Y+G5FwdFhIL8yWIdnSCtpw0j3w0MyaAglPn7rFDaNzGNIpht1rlb83Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T23:32:43.172718Z"},"content_sha256":"a1678a5997419a13a46635edb06609d0a8278b47e9a4e04c97bb31dbdb63dafe","schema_version":"1.0","event_id":"sha256:a1678a5997419a13a46635edb06609d0a8278b47e9a4e04c97bb31dbdb63dafe"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TKXAZI5EOAKOKJYX7JU6Z4JCXP/bundle.json","state_url":"https://pith.science/pith/TKXAZI5EOAKOKJYX7JU6Z4JCXP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TKXAZI5EOAKOKJYX7JU6Z4JCXP/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":"2026-06-23T23:32:43Z","links":{"resolver":"https://pith.science/pith/TKXAZI5EOAKOKJYX7JU6Z4JCXP","bundle":"https://pith.science/pith/TKXAZI5EOAKOKJYX7JU6Z4JCXP/bundle.json","state":"https://pith.science/pith/TKXAZI5EOAKOKJYX7JU6Z4JCXP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TKXAZI5EOAKOKJYX7JU6Z4JCXP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:TKXAZI5EOAKOKJYX7JU6Z4JCXP","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":"bab6aff9d07a06f1306769b2a062a8e354a1e538b3ab6aa687a3b1d6196d55ef","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2008-11-10T09:20:45Z","title_canon_sha256":"397e56986a24cd431a4d8f89b6acf326d374443195dbbddce50fcf1e7f2775e5"},"schema_version":"1.0","source":{"id":"0811.1416","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0811.1416","created_at":"2026-05-18T04:41:25Z"},{"alias_kind":"arxiv_version","alias_value":"0811.1416v1","created_at":"2026-05-18T04:41:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0811.1416","created_at":"2026-05-18T04:41:25Z"},{"alias_kind":"pith_short_12","alias_value":"TKXAZI5EOAKO","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"TKXAZI5EOAKOKJYX","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"TKXAZI5E","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:a1678a5997419a13a46635edb06609d0a8278b47e9a4e04c97bb31dbdb63dafe","target":"graph","created_at":"2026-05-18T04:41:25Z","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 each N>=c_d*n^{2d*(d+1)/(d+2)} we prove the existence of a spherical n-design on S^d consisting of N points, where c_d is a constant depending only on $d$.","authors_text":"Andriy V. Bondarenko, Maryna S. Viazovska","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2008-11-10T09:20:45Z","title":"Spherical designs via Brouwer fixed point theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0811.1416","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":"sha256:53b6c8e62af7965c39d2c32c8c1aaa5daab31f72392f5e6208d60d0d993f2c27","target":"record","created_at":"2026-05-18T04:41:25Z","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":"bab6aff9d07a06f1306769b2a062a8e354a1e538b3ab6aa687a3b1d6196d55ef","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2008-11-10T09:20:45Z","title_canon_sha256":"397e56986a24cd431a4d8f89b6acf326d374443195dbbddce50fcf1e7f2775e5"},"schema_version":"1.0","source":{"id":"0811.1416","kind":"arxiv","version":1}},"canonical_sha256":"9aae0ca3a47014e52717fa69ecf122bbc791e742eba99b9a96ba29d2d124418b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9aae0ca3a47014e52717fa69ecf122bbc791e742eba99b9a96ba29d2d124418b","first_computed_at":"2026-05-18T04:41:25.365250Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:41:25.365250Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"S1wtFx9YeznEvYYBNA2rhqyg7bV/bhpkiylJdTEkNKeEARWAu0iFO7w0vLDU63zUagRX1cnYXG22GugtDv5EAA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:41:25.366003Z","signed_message":"canonical_sha256_bytes"},"source_id":"0811.1416","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:53b6c8e62af7965c39d2c32c8c1aaa5daab31f72392f5e6208d60d0d993f2c27","sha256:a1678a5997419a13a46635edb06609d0a8278b47e9a4e04c97bb31dbdb63dafe"],"state_sha256":"cead345e9633c34bbd07e0ef993e330e9f775b495f27e5b0a51a8d4afd6be752"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EYzXt24sy6CqTAdVIGq/Kop0f/plTHDdRhthxNdBHPwvYGaNWZ4vcl40b6Ci3US6s/C1murZg8WzR0OzS/TGDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T23:32:43.174671Z","bundle_sha256":"f41a20587e9922e21be83390dc73a66eb3ca31beee9641c30f5eedb03a33cf04"}}