{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:HRXHQQSO3Y6HGRETHZY4SMGU24","short_pith_number":"pith:HRXHQQSO","canonical_record":{"source":{"id":"1412.6994","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2014-12-18T11:45:03Z","cross_cats_sorted":["math.HO"],"title_canon_sha256":"88e21c3909f14de7e359f00e03d19bb92e3c36f3dcb1cb2e88b5575300ca9a25","abstract_canon_sha256":"1bad50cd33fd9c13e13a3a62f37e7b3a9ebe08bfcffcc1bacf80780736e33457"},"schema_version":"1.0"},"canonical_sha256":"3c6e78424ede3c7344933e71c930d4d71245e112e3f82ade742fa5d5ed1fe85c","source":{"kind":"arxiv","id":"1412.6994","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.6994","created_at":"2026-05-18T02:30:45Z"},{"alias_kind":"arxiv_version","alias_value":"1412.6994v1","created_at":"2026-05-18T02:30:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.6994","created_at":"2026-05-18T02:30:45Z"},{"alias_kind":"pith_short_12","alias_value":"HRXHQQSO3Y6H","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_16","alias_value":"HRXHQQSO3Y6HGRET","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_8","alias_value":"HRXHQQSO","created_at":"2026-05-18T12:28:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:HRXHQQSO3Y6HGRETHZY4SMGU24","target":"record","payload":{"canonical_record":{"source":{"id":"1412.6994","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2014-12-18T11:45:03Z","cross_cats_sorted":["math.HO"],"title_canon_sha256":"88e21c3909f14de7e359f00e03d19bb92e3c36f3dcb1cb2e88b5575300ca9a25","abstract_canon_sha256":"1bad50cd33fd9c13e13a3a62f37e7b3a9ebe08bfcffcc1bacf80780736e33457"},"schema_version":"1.0"},"canonical_sha256":"3c6e78424ede3c7344933e71c930d4d71245e112e3f82ade742fa5d5ed1fe85c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:30:45.824165Z","signature_b64":"VpO8z6jYorVIulns24hpidP87z9afFycmEWXLLGoc4loLAue4uHExFekOQ4V1yEArcP+6TNphTQDsYQ/sHTEBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3c6e78424ede3c7344933e71c930d4d71245e112e3f82ade742fa5d5ed1fe85c","last_reissued_at":"2026-05-18T02:30:45.823750Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:30:45.823750Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1412.6994","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":"2026-05-18T02:30:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"73RLGa0y8fgAFzGE0Vzh+KUQTjG2cnfEjb65RMzowBPQRD7z7nAv3dwcCGBXQuOVCkjHzFZ7Cp0IndSdAr6dDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T12:28:03.721667Z"},"content_sha256":"37f4d4b7e32b63d4f3f4fd6c09b609b6773ad3cc479e65f695f80285b559eaf5","schema_version":"1.0","event_id":"sha256:37f4d4b7e32b63d4f3f4fd6c09b609b6773ad3cc479e65f695f80285b559eaf5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:HRXHQQSO3Y6HGRETHZY4SMGU24","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Topics on mathematical crystallography","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.HO"],"primary_cat":"math.MG","authors_text":"Toshikazu Sunada","submitted_at":"2014-12-18T11:45:03Z","abstract_excerpt":"In July 2012 the General Assembly of the United Nations resolved that 2014 should be the International Year of Crystallography, 100 years since the award of the Nobel Prize for the discovery of X-ray diffraction by crystals. On this special occasion, we address several topics in mathematical crystallography. Especially motivated by the recent development in systematic design of crystal structures by both mathematicians and crystallographers, we discuss interesting relationships among seemingly irrelevant subjects; say, standard crystal models, tight frames in the Euclidean space, rational poin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.6994","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":"2026-05-18T02:30:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SIb1f/h2HAjOTEfZkv5f6btO8Eb1aEaVrbbo9hhmEiAO0tuZwICsd6qIhl8bz/9ig8wbB8Y6OpXvbybLhRIIAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T12:28:03.722006Z"},"content_sha256":"b145c49a208ca16f54bb7b53c5fd6876486d15203afa2315d7c52360f794326a","schema_version":"1.0","event_id":"sha256:b145c49a208ca16f54bb7b53c5fd6876486d15203afa2315d7c52360f794326a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HRXHQQSO3Y6HGRETHZY4SMGU24/bundle.json","state_url":"https://pith.science/pith/HRXHQQSO3Y6HGRETHZY4SMGU24/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HRXHQQSO3Y6HGRETHZY4SMGU24/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":"2026-06-02T12:28:03Z","links":{"resolver":"https://pith.science/pith/HRXHQQSO3Y6HGRETHZY4SMGU24","bundle":"https://pith.science/pith/HRXHQQSO3Y6HGRETHZY4SMGU24/bundle.json","state":"https://pith.science/pith/HRXHQQSO3Y6HGRETHZY4SMGU24/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HRXHQQSO3Y6HGRETHZY4SMGU24/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:HRXHQQSO3Y6HGRETHZY4SMGU24","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":"1bad50cd33fd9c13e13a3a62f37e7b3a9ebe08bfcffcc1bacf80780736e33457","cross_cats_sorted":["math.HO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2014-12-18T11:45:03Z","title_canon_sha256":"88e21c3909f14de7e359f00e03d19bb92e3c36f3dcb1cb2e88b5575300ca9a25"},"schema_version":"1.0","source":{"id":"1412.6994","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.6994","created_at":"2026-05-18T02:30:45Z"},{"alias_kind":"arxiv_version","alias_value":"1412.6994v1","created_at":"2026-05-18T02:30:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.6994","created_at":"2026-05-18T02:30:45Z"},{"alias_kind":"pith_short_12","alias_value":"HRXHQQSO3Y6H","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_16","alias_value":"HRXHQQSO3Y6HGRET","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_8","alias_value":"HRXHQQSO","created_at":"2026-05-18T12:28:30Z"}],"graph_snapshots":[{"event_id":"sha256:b145c49a208ca16f54bb7b53c5fd6876486d15203afa2315d7c52360f794326a","target":"graph","created_at":"2026-05-18T02:30:45Z","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":"In July 2012 the General Assembly of the United Nations resolved that 2014 should be the International Year of Crystallography, 100 years since the award of the Nobel Prize for the discovery of X-ray diffraction by crystals. On this special occasion, we address several topics in mathematical crystallography. Especially motivated by the recent development in systematic design of crystal structures by both mathematicians and crystallographers, we discuss interesting relationships among seemingly irrelevant subjects; say, standard crystal models, tight frames in the Euclidean space, rational poin","authors_text":"Toshikazu Sunada","cross_cats":["math.HO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2014-12-18T11:45:03Z","title":"Topics on mathematical crystallography"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.6994","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":"sha256:37f4d4b7e32b63d4f3f4fd6c09b609b6773ad3cc479e65f695f80285b559eaf5","target":"record","created_at":"2026-05-18T02:30:45Z","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":"1bad50cd33fd9c13e13a3a62f37e7b3a9ebe08bfcffcc1bacf80780736e33457","cross_cats_sorted":["math.HO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2014-12-18T11:45:03Z","title_canon_sha256":"88e21c3909f14de7e359f00e03d19bb92e3c36f3dcb1cb2e88b5575300ca9a25"},"schema_version":"1.0","source":{"id":"1412.6994","kind":"arxiv","version":1}},"canonical_sha256":"3c6e78424ede3c7344933e71c930d4d71245e112e3f82ade742fa5d5ed1fe85c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3c6e78424ede3c7344933e71c930d4d71245e112e3f82ade742fa5d5ed1fe85c","first_computed_at":"2026-05-18T02:30:45.823750Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:30:45.823750Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VpO8z6jYorVIulns24hpidP87z9afFycmEWXLLGoc4loLAue4uHExFekOQ4V1yEArcP+6TNphTQDsYQ/sHTEBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:30:45.824165Z","signed_message":"canonical_sha256_bytes"},"source_id":"1412.6994","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:37f4d4b7e32b63d4f3f4fd6c09b609b6773ad3cc479e65f695f80285b559eaf5","sha256:b145c49a208ca16f54bb7b53c5fd6876486d15203afa2315d7c52360f794326a"],"state_sha256":"056a8862ef65f431877d1628654336d01913303af7423f90a01e05aec7f3e9c2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Cjav2T1oVu9WVM4pSoBheTuCh0H555y9DTeje+YogWY/2iP2jJ1BBHJ6nkdwQ0hw2TrV0xrB4KmJQIoH5IgGAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T12:28:03.723853Z","bundle_sha256":"99992620a6203346c0585266d1492e225a37b4a43c134b97054b60b58867d3e4"}}