{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:VCNGRKMR2WRVCXUBI66WL34USP","short_pith_number":"pith:VCNGRKMR","canonical_record":{"source":{"id":"2102.04599","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-02-09T01:45:25Z","cross_cats_sorted":["cs.NA","math.MG","math.NA","math.PR"],"title_canon_sha256":"2374cc65d2215395c23fac86967ac66b8ae588f63bda14500f24c17c9a6cf3c6","abstract_canon_sha256":"0a43965c8b946549b24a40424dbc3385aee9b0335f27f1c8630291cf094cd649"},"schema_version":"1.0"},"canonical_sha256":"a89a68a991d5a3515e8147bd65ef9493f9f2be91bf328b5bc2b70806bf43db2b","source":{"kind":"arxiv","id":"2102.04599","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2102.04599","created_at":"2026-07-05T04:06:14Z"},{"alias_kind":"arxiv_version","alias_value":"2102.04599v2","created_at":"2026-07-05T04:06:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2102.04599","created_at":"2026-07-05T04:06:14Z"},{"alias_kind":"pith_short_12","alias_value":"VCNGRKMR2WRV","created_at":"2026-07-05T04:06:14Z"},{"alias_kind":"pith_short_16","alias_value":"VCNGRKMR2WRVCXUB","created_at":"2026-07-05T04:06:14Z"},{"alias_kind":"pith_short_8","alias_value":"VCNGRKMR","created_at":"2026-07-05T04:06:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:VCNGRKMR2WRVCXUBI66WL34USP","target":"record","payload":{"canonical_record":{"source":{"id":"2102.04599","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-02-09T01:45:25Z","cross_cats_sorted":["cs.NA","math.MG","math.NA","math.PR"],"title_canon_sha256":"2374cc65d2215395c23fac86967ac66b8ae588f63bda14500f24c17c9a6cf3c6","abstract_canon_sha256":"0a43965c8b946549b24a40424dbc3385aee9b0335f27f1c8630291cf094cd649"},"schema_version":"1.0"},"canonical_sha256":"a89a68a991d5a3515e8147bd65ef9493f9f2be91bf328b5bc2b70806bf43db2b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:06:14.143664Z","signature_b64":"D8Zu+LrepkpbH+dsxCWjmH62pU1nD0t5eFwYTUy/GTO1VOAtkb3fXAPQK68+waw2xYK5NxWw2RpZl598mflHBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a89a68a991d5a3515e8147bd65ef9493f9f2be91bf328b5bc2b70806bf43db2b","last_reissued_at":"2026-07-05T04:06:14.143095Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:06:14.143095Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2102.04599","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":"2026-07-05T04:06:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GZ8VpCqzWMAW8z8t6vJzSmgy92WcN7kev1m6nBzPo+Gddj317jdx8wn3NNmwIv15Y3vyxrdmizpUrfVDCOqxBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T17:40:07.893839Z"},"content_sha256":"89f058ac97f8f77fdd1650fa69ea0ea6848a8ada53a1e5553262ce6ae42fc62b","schema_version":"1.0","event_id":"sha256:89f058ac97f8f77fdd1650fa69ea0ea6848a8ada53a1e5553262ce6ae42fc62b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:VCNGRKMR2WRVCXUBI66WL34USP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Minimax Spherical Designs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.NA","math.MG","math.NA","math.PR"],"primary_cat":"math.CO","authors_text":"Guanyang Wang, Jun Yan, Weibo Fu","submitted_at":"2021-02-09T01:45:25Z","abstract_excerpt":"Distributing points on a (possibly high-dimensional) sphere with minimal energy is a long-standing problem in and outside the field of mathematics. This paper considers a novel energy function that arises naturally from statistics and combinatorial optimization, and studies its theoretical properties. Our result solves both the exact optimal spherical point configurations in certain cases and the minimal energy asymptotics under general assumptions. Connections between our results and the L1-Principal Component analysis and Quasi-Monte Carlo methods are also discussed."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2102.04599","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2102.04599/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":"2026-07-05T04:06:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ntTgA0fApQsxgfjXNV6HBZTKMQFFWD2oZIfmMoeTNWHKZSXVoDdVVj/EeO28O+kPMflbvyVxi9flIO+56Y8mDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T17:40:07.894214Z"},"content_sha256":"7bd4b3f9ef315e32fa8d0a4e5e2298890f592713492903a950a6bb4d76d7023a","schema_version":"1.0","event_id":"sha256:7bd4b3f9ef315e32fa8d0a4e5e2298890f592713492903a950a6bb4d76d7023a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VCNGRKMR2WRVCXUBI66WL34USP/bundle.json","state_url":"https://pith.science/pith/VCNGRKMR2WRVCXUBI66WL34USP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VCNGRKMR2WRVCXUBI66WL34USP/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":"2026-07-06T17:40:07Z","links":{"resolver":"https://pith.science/pith/VCNGRKMR2WRVCXUBI66WL34USP","bundle":"https://pith.science/pith/VCNGRKMR2WRVCXUBI66WL34USP/bundle.json","state":"https://pith.science/pith/VCNGRKMR2WRVCXUBI66WL34USP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VCNGRKMR2WRVCXUBI66WL34USP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:VCNGRKMR2WRVCXUBI66WL34USP","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":"0a43965c8b946549b24a40424dbc3385aee9b0335f27f1c8630291cf094cd649","cross_cats_sorted":["cs.NA","math.MG","math.NA","math.PR"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-02-09T01:45:25Z","title_canon_sha256":"2374cc65d2215395c23fac86967ac66b8ae588f63bda14500f24c17c9a6cf3c6"},"schema_version":"1.0","source":{"id":"2102.04599","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2102.04599","created_at":"2026-07-05T04:06:14Z"},{"alias_kind":"arxiv_version","alias_value":"2102.04599v2","created_at":"2026-07-05T04:06:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2102.04599","created_at":"2026-07-05T04:06:14Z"},{"alias_kind":"pith_short_12","alias_value":"VCNGRKMR2WRV","created_at":"2026-07-05T04:06:14Z"},{"alias_kind":"pith_short_16","alias_value":"VCNGRKMR2WRVCXUB","created_at":"2026-07-05T04:06:14Z"},{"alias_kind":"pith_short_8","alias_value":"VCNGRKMR","created_at":"2026-07-05T04:06:14Z"}],"graph_snapshots":[{"event_id":"sha256:7bd4b3f9ef315e32fa8d0a4e5e2298890f592713492903a950a6bb4d76d7023a","target":"graph","created_at":"2026-07-05T04:06:14Z","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/2102.04599/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Distributing points on a (possibly high-dimensional) sphere with minimal energy is a long-standing problem in and outside the field of mathematics. This paper considers a novel energy function that arises naturally from statistics and combinatorial optimization, and studies its theoretical properties. Our result solves both the exact optimal spherical point configurations in certain cases and the minimal energy asymptotics under general assumptions. Connections between our results and the L1-Principal Component analysis and Quasi-Monte Carlo methods are also discussed.","authors_text":"Guanyang Wang, Jun Yan, Weibo Fu","cross_cats":["cs.NA","math.MG","math.NA","math.PR"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-02-09T01:45:25Z","title":"On the Minimax Spherical Designs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2102.04599","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":"sha256:89f058ac97f8f77fdd1650fa69ea0ea6848a8ada53a1e5553262ce6ae42fc62b","target":"record","created_at":"2026-07-05T04:06:14Z","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":"0a43965c8b946549b24a40424dbc3385aee9b0335f27f1c8630291cf094cd649","cross_cats_sorted":["cs.NA","math.MG","math.NA","math.PR"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-02-09T01:45:25Z","title_canon_sha256":"2374cc65d2215395c23fac86967ac66b8ae588f63bda14500f24c17c9a6cf3c6"},"schema_version":"1.0","source":{"id":"2102.04599","kind":"arxiv","version":2}},"canonical_sha256":"a89a68a991d5a3515e8147bd65ef9493f9f2be91bf328b5bc2b70806bf43db2b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a89a68a991d5a3515e8147bd65ef9493f9f2be91bf328b5bc2b70806bf43db2b","first_computed_at":"2026-07-05T04:06:14.143095Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T04:06:14.143095Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"D8Zu+LrepkpbH+dsxCWjmH62pU1nD0t5eFwYTUy/GTO1VOAtkb3fXAPQK68+waw2xYK5NxWw2RpZl598mflHBA==","signature_status":"signed_v1","signed_at":"2026-07-05T04:06:14.143664Z","signed_message":"canonical_sha256_bytes"},"source_id":"2102.04599","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:89f058ac97f8f77fdd1650fa69ea0ea6848a8ada53a1e5553262ce6ae42fc62b","sha256:7bd4b3f9ef315e32fa8d0a4e5e2298890f592713492903a950a6bb4d76d7023a"],"state_sha256":"808ba6730a6da625c24967a6e3f86ca152a9930826095cf80deff87866a32bfc"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qx7/o6fyRbvDcioCxXwmVwFTyiBGDLuZaOkvT6M2sH+jsdlj9LvXOj/2bBknSEDX71wBZjENejGAaanW75wbCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-06T17:40:07.896167Z","bundle_sha256":"978c1188912553800a45ed26fda71739b349a4ae0e42f93c0da225804b1eb41d"}}