{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:UFKCDUN6TDTPJK7G77GYRUB7KY","short_pith_number":"pith:UFKCDUN6","canonical_record":{"source":{"id":"1608.06842","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-08-24T14:36:15Z","cross_cats_sorted":[],"title_canon_sha256":"d9111441b115dcd76c60e11d5c2a23f4e37899720224267122e73ec65d0689c9","abstract_canon_sha256":"5e2f667a60aeaa5bc29f0333c070859bef4fa16fe540c415aa070b2f136677c6"},"schema_version":"1.0"},"canonical_sha256":"a15421d1be98e6f4abe6ffcd88d03f560115a5032da6876db2a159a0ede6cdf9","source":{"kind":"arxiv","id":"1608.06842","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.06842","created_at":"2026-05-18T01:07:57Z"},{"alias_kind":"arxiv_version","alias_value":"1608.06842v1","created_at":"2026-05-18T01:07:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.06842","created_at":"2026-05-18T01:07:57Z"},{"alias_kind":"pith_short_12","alias_value":"UFKCDUN6TDTP","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_16","alias_value":"UFKCDUN6TDTPJK7G","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_8","alias_value":"UFKCDUN6","created_at":"2026-05-18T12:30:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:UFKCDUN6TDTPJK7G77GYRUB7KY","target":"record","payload":{"canonical_record":{"source":{"id":"1608.06842","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-08-24T14:36:15Z","cross_cats_sorted":[],"title_canon_sha256":"d9111441b115dcd76c60e11d5c2a23f4e37899720224267122e73ec65d0689c9","abstract_canon_sha256":"5e2f667a60aeaa5bc29f0333c070859bef4fa16fe540c415aa070b2f136677c6"},"schema_version":"1.0"},"canonical_sha256":"a15421d1be98e6f4abe6ffcd88d03f560115a5032da6876db2a159a0ede6cdf9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:07:57.406017Z","signature_b64":"0mCiSAz5zIPIJaFRdvbXGtfFJMB90UdToimJdSrX1c/JMVK4BlGBgTEqJHyUi8wzmmEWrKRy+h+AZ1BJ1W0MDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a15421d1be98e6f4abe6ffcd88d03f560115a5032da6876db2a159a0ede6cdf9","last_reissued_at":"2026-05-18T01:07:57.405633Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:07:57.405633Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1608.06842","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":"2026-05-18T01:07:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sQztfBj6u4lVfwrlWjCdYISfH/Qgrj1Rabp4umlpNNnj6zsKDhz1kCnCz/q0jBVv+/VTpPBn06X9ZEo3XcFKCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T23:18:37.275730Z"},"content_sha256":"78989f2e5e788212b04065719ad9b6f9178a910def71ea408a2bc8fe70389528","schema_version":"1.0","event_id":"sha256:78989f2e5e788212b04065719ad9b6f9178a910def71ea408a2bc8fe70389528"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:UFKCDUN6TDTPJK7G77GYRUB7KY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Delone Sets: Local Identity and Global Symmetry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Nikolay Dolbilin","submitted_at":"2016-08-24T14:36:15Z","abstract_excerpt":"In the paper we present a proof of the local criterion for crystalline structures which generalizes the local criterion for regular systems. A Delone set is called a crystal if it is invariant with respect to a crystallgraphic group. So-called locally antipodal Delone sets, i.e. such sets in which all 2R-clusters are centrally symmetrical, are considered. It turns out that the local antipodal sets have crystalline structure. Moreover, if in a locally antipodal set all 2R-clusters are the same the set is a regular system, i.e. a Delone set whose symmetry group operates transitively on the set."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.06842","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":"2026-05-18T01:07:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3jcGhCgSNbFZmxK2FSTGEA5oVW6X+g+MCUGHs1Z33pI28nG8wEFP/vnViSLBOq6w0kaieQsfxDW+ZvLIrdj+Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T23:18:37.276461Z"},"content_sha256":"42050c2dea14e6067dcaf03a1101c089775d92708891dded4393a1034e442e6b","schema_version":"1.0","event_id":"sha256:42050c2dea14e6067dcaf03a1101c089775d92708891dded4393a1034e442e6b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UFKCDUN6TDTPJK7G77GYRUB7KY/bundle.json","state_url":"https://pith.science/pith/UFKCDUN6TDTPJK7G77GYRUB7KY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UFKCDUN6TDTPJK7G77GYRUB7KY/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":"2026-05-27T23:18:37Z","links":{"resolver":"https://pith.science/pith/UFKCDUN6TDTPJK7G77GYRUB7KY","bundle":"https://pith.science/pith/UFKCDUN6TDTPJK7G77GYRUB7KY/bundle.json","state":"https://pith.science/pith/UFKCDUN6TDTPJK7G77GYRUB7KY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UFKCDUN6TDTPJK7G77GYRUB7KY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:UFKCDUN6TDTPJK7G77GYRUB7KY","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":"5e2f667a60aeaa5bc29f0333c070859bef4fa16fe540c415aa070b2f136677c6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-08-24T14:36:15Z","title_canon_sha256":"d9111441b115dcd76c60e11d5c2a23f4e37899720224267122e73ec65d0689c9"},"schema_version":"1.0","source":{"id":"1608.06842","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.06842","created_at":"2026-05-18T01:07:57Z"},{"alias_kind":"arxiv_version","alias_value":"1608.06842v1","created_at":"2026-05-18T01:07:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.06842","created_at":"2026-05-18T01:07:57Z"},{"alias_kind":"pith_short_12","alias_value":"UFKCDUN6TDTP","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_16","alias_value":"UFKCDUN6TDTPJK7G","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_8","alias_value":"UFKCDUN6","created_at":"2026-05-18T12:30:46Z"}],"graph_snapshots":[{"event_id":"sha256:42050c2dea14e6067dcaf03a1101c089775d92708891dded4393a1034e442e6b","target":"graph","created_at":"2026-05-18T01:07:57Z","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 the paper we present a proof of the local criterion for crystalline structures which generalizes the local criterion for regular systems. A Delone set is called a crystal if it is invariant with respect to a crystallgraphic group. So-called locally antipodal Delone sets, i.e. such sets in which all 2R-clusters are centrally symmetrical, are considered. It turns out that the local antipodal sets have crystalline structure. Moreover, if in a locally antipodal set all 2R-clusters are the same the set is a regular system, i.e. a Delone set whose symmetry group operates transitively on the set.","authors_text":"Nikolay Dolbilin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-08-24T14:36:15Z","title":"Delone Sets: Local Identity and Global Symmetry"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.06842","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":"sha256:78989f2e5e788212b04065719ad9b6f9178a910def71ea408a2bc8fe70389528","target":"record","created_at":"2026-05-18T01:07:57Z","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":"5e2f667a60aeaa5bc29f0333c070859bef4fa16fe540c415aa070b2f136677c6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-08-24T14:36:15Z","title_canon_sha256":"d9111441b115dcd76c60e11d5c2a23f4e37899720224267122e73ec65d0689c9"},"schema_version":"1.0","source":{"id":"1608.06842","kind":"arxiv","version":1}},"canonical_sha256":"a15421d1be98e6f4abe6ffcd88d03f560115a5032da6876db2a159a0ede6cdf9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a15421d1be98e6f4abe6ffcd88d03f560115a5032da6876db2a159a0ede6cdf9","first_computed_at":"2026-05-18T01:07:57.405633Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:07:57.405633Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0mCiSAz5zIPIJaFRdvbXGtfFJMB90UdToimJdSrX1c/JMVK4BlGBgTEqJHyUi8wzmmEWrKRy+h+AZ1BJ1W0MDw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:07:57.406017Z","signed_message":"canonical_sha256_bytes"},"source_id":"1608.06842","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:78989f2e5e788212b04065719ad9b6f9178a910def71ea408a2bc8fe70389528","sha256:42050c2dea14e6067dcaf03a1101c089775d92708891dded4393a1034e442e6b"],"state_sha256":"ea3f37c01c6f95ad04a4c574af0037950c29e474f66e93399775b82f038d1471"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"t0aVMY248xUlyd7F4WYRIWEtg1P/p+EqDxpZj7cL3Ii39F08Vp0Ea4t/Sov/YgECCIbznRjnq9IdPIszV2FPBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T23:18:37.279869Z","bundle_sha256":"68d1b10488ebbf021fe3633b3f28321112f134ff9d64b31d53fd223df44e7d21"}}