{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:G7F2VW3SEIDFPJLOISXAF75J2G","short_pith_number":"pith:G7F2VW3S","canonical_record":{"source":{"id":"1105.0400","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-05-02T18:35:27Z","cross_cats_sorted":[],"title_canon_sha256":"953f3c0665e434f71fa2fa1882406ea5750ff8ff24d56f3f363cea855df683e3","abstract_canon_sha256":"cc3b5483112e7c47e6565763a2123146225319a8a503415033be59d007f50007"},"schema_version":"1.0"},"canonical_sha256":"37cbaadb72220657a56e44ae02ffa9d19e5191624408f02f723d556cf4dcdc8b","source":{"kind":"arxiv","id":"1105.0400","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1105.0400","created_at":"2026-05-18T04:23:11Z"},{"alias_kind":"arxiv_version","alias_value":"1105.0400v1","created_at":"2026-05-18T04:23:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.0400","created_at":"2026-05-18T04:23:11Z"},{"alias_kind":"pith_short_12","alias_value":"G7F2VW3SEIDF","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_16","alias_value":"G7F2VW3SEIDFPJLO","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_8","alias_value":"G7F2VW3S","created_at":"2026-05-18T12:26:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:G7F2VW3SEIDFPJLOISXAF75J2G","target":"record","payload":{"canonical_record":{"source":{"id":"1105.0400","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-05-02T18:35:27Z","cross_cats_sorted":[],"title_canon_sha256":"953f3c0665e434f71fa2fa1882406ea5750ff8ff24d56f3f363cea855df683e3","abstract_canon_sha256":"cc3b5483112e7c47e6565763a2123146225319a8a503415033be59d007f50007"},"schema_version":"1.0"},"canonical_sha256":"37cbaadb72220657a56e44ae02ffa9d19e5191624408f02f723d556cf4dcdc8b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:23:11.912843Z","signature_b64":"ez2mWbBnBJ0Mo1lRF7IvJtn7q1Z1QZ0PL+Hl8nr6dbJr8s+8ZqzB0Mo5jFWhYrj10jXcGIzD2nKZpotsj49fAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"37cbaadb72220657a56e44ae02ffa9d19e5191624408f02f723d556cf4dcdc8b","last_reissued_at":"2026-05-18T04:23:11.912272Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:23:11.912272Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1105.0400","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":"2026-05-18T04:23:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SY9VnvfmvoXKb31RCBpN00sWLRcs8SsqIa+jUSQmN0wVJRTrux7GKXTbasE72rAQU0FoaQQI5kzH4FFjn8GWDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T09:23:49.907384Z"},"content_sha256":"e248cca75ee3b70f56f3388b66e27d0630033dd2483f6d3657a09533573dba4f","schema_version":"1.0","event_id":"sha256:e248cca75ee3b70f56f3388b66e27d0630033dd2483f6d3657a09533573dba4f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:G7F2VW3SEIDFPJLOISXAF75J2G","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A countable set of directions is sufficient for Steiner symmetrization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Daniel A. Klain, Deane Yang, Erwin Lutwak, Gabriele Bianchi, Gaoyong Zhang","submitted_at":"2011-05-02T18:35:27Z","abstract_excerpt":"A countable dense set of directions is sufficient for Steiner symmetrization, but the order of directions matters."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.0400","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":"2026-05-18T04:23:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tTe9KJVyhspcUxDDslu2FxfY7nddCqUHUWMGjp/y+e4SLXzqys7uNO7/aI+haKGUgK0QcjWO9GmNLFYH1JU8AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T09:23:49.907745Z"},"content_sha256":"d21d2b74daa4dea5c9c4567f2cea4e101cd56b70ac2e4978ffbde6b62b1e391f","schema_version":"1.0","event_id":"sha256:d21d2b74daa4dea5c9c4567f2cea4e101cd56b70ac2e4978ffbde6b62b1e391f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/G7F2VW3SEIDFPJLOISXAF75J2G/bundle.json","state_url":"https://pith.science/pith/G7F2VW3SEIDFPJLOISXAF75J2G/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/G7F2VW3SEIDFPJLOISXAF75J2G/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":"2026-05-27T09:23:49Z","links":{"resolver":"https://pith.science/pith/G7F2VW3SEIDFPJLOISXAF75J2G","bundle":"https://pith.science/pith/G7F2VW3SEIDFPJLOISXAF75J2G/bundle.json","state":"https://pith.science/pith/G7F2VW3SEIDFPJLOISXAF75J2G/state.json","well_known_bundle":"https://pith.science/.well-known/pith/G7F2VW3SEIDFPJLOISXAF75J2G/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:G7F2VW3SEIDFPJLOISXAF75J2G","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":"cc3b5483112e7c47e6565763a2123146225319a8a503415033be59d007f50007","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-05-02T18:35:27Z","title_canon_sha256":"953f3c0665e434f71fa2fa1882406ea5750ff8ff24d56f3f363cea855df683e3"},"schema_version":"1.0","source":{"id":"1105.0400","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1105.0400","created_at":"2026-05-18T04:23:11Z"},{"alias_kind":"arxiv_version","alias_value":"1105.0400v1","created_at":"2026-05-18T04:23:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.0400","created_at":"2026-05-18T04:23:11Z"},{"alias_kind":"pith_short_12","alias_value":"G7F2VW3SEIDF","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_16","alias_value":"G7F2VW3SEIDFPJLO","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_8","alias_value":"G7F2VW3S","created_at":"2026-05-18T12:26:28Z"}],"graph_snapshots":[{"event_id":"sha256:d21d2b74daa4dea5c9c4567f2cea4e101cd56b70ac2e4978ffbde6b62b1e391f","target":"graph","created_at":"2026-05-18T04:23:11Z","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":"A countable dense set of directions is sufficient for Steiner symmetrization, but the order of directions matters.","authors_text":"Daniel A. Klain, Deane Yang, Erwin Lutwak, Gabriele Bianchi, Gaoyong Zhang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-05-02T18:35:27Z","title":"A countable set of directions is sufficient for Steiner symmetrization"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.0400","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:e248cca75ee3b70f56f3388b66e27d0630033dd2483f6d3657a09533573dba4f","target":"record","created_at":"2026-05-18T04:23:11Z","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":"cc3b5483112e7c47e6565763a2123146225319a8a503415033be59d007f50007","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-05-02T18:35:27Z","title_canon_sha256":"953f3c0665e434f71fa2fa1882406ea5750ff8ff24d56f3f363cea855df683e3"},"schema_version":"1.0","source":{"id":"1105.0400","kind":"arxiv","version":1}},"canonical_sha256":"37cbaadb72220657a56e44ae02ffa9d19e5191624408f02f723d556cf4dcdc8b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"37cbaadb72220657a56e44ae02ffa9d19e5191624408f02f723d556cf4dcdc8b","first_computed_at":"2026-05-18T04:23:11.912272Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:23:11.912272Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ez2mWbBnBJ0Mo1lRF7IvJtn7q1Z1QZ0PL+Hl8nr6dbJr8s+8ZqzB0Mo5jFWhYrj10jXcGIzD2nKZpotsj49fAw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:23:11.912843Z","signed_message":"canonical_sha256_bytes"},"source_id":"1105.0400","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e248cca75ee3b70f56f3388b66e27d0630033dd2483f6d3657a09533573dba4f","sha256:d21d2b74daa4dea5c9c4567f2cea4e101cd56b70ac2e4978ffbde6b62b1e391f"],"state_sha256":"fac3eebad9c8a8b3411fdda75b6835c3be45e028b6acba034c67c4bc7e12cebc"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gRq0sI9rihhcOYu05U4Szwz5ctGoKt/CELINLF7CZ4gJ+bi9kZ73E2WG+zjmAzwGhBAh0Gju/mGGhuX142TsCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T09:23:49.910279Z","bundle_sha256":"da52f4e9847f0c6ba82090cf4ae0912b3d9b2991e13b707f636fe3fb79055b38"}}