{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:MTDOAUNLSP5VCP45SV5EYQIIL2","short_pith_number":"pith:MTDOAUNL","canonical_record":{"source":{"id":"1608.08215","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-08-29T20:00:00Z","cross_cats_sorted":["cond-mat.mtrl-sci","hep-th","math.MG","math.MP"],"title_canon_sha256":"6ae1cdca8a9dab1937880e34189b358ff74bde9b1d5feaf0c3264401a7d38bbb","abstract_canon_sha256":"681d0c8e22e36d8b6e6a16f4a68cb7f853f2fa0b490bb7296c91fa8b7a50ae4d"},"schema_version":"1.0"},"canonical_sha256":"64c6e051ab93fb513f9d957a4c41085ea71b121b43d2d6003113d96b7a031123","source":{"kind":"arxiv","id":"1608.08215","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.08215","created_at":"2026-07-05T05:12:01Z"},{"alias_kind":"arxiv_version","alias_value":"1608.08215v4","created_at":"2026-07-05T05:12:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.08215","created_at":"2026-07-05T05:12:01Z"},{"alias_kind":"pith_short_12","alias_value":"MTDOAUNLSP5V","created_at":"2026-07-05T05:12:01Z"},{"alias_kind":"pith_short_16","alias_value":"MTDOAUNLSP5VCP45","created_at":"2026-07-05T05:12:01Z"},{"alias_kind":"pith_short_8","alias_value":"MTDOAUNL","created_at":"2026-07-05T05:12:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:MTDOAUNLSP5VCP45SV5EYQIIL2","target":"record","payload":{"canonical_record":{"source":{"id":"1608.08215","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-08-29T20:00:00Z","cross_cats_sorted":["cond-mat.mtrl-sci","hep-th","math.MG","math.MP"],"title_canon_sha256":"6ae1cdca8a9dab1937880e34189b358ff74bde9b1d5feaf0c3264401a7d38bbb","abstract_canon_sha256":"681d0c8e22e36d8b6e6a16f4a68cb7f853f2fa0b490bb7296c91fa8b7a50ae4d"},"schema_version":"1.0"},"canonical_sha256":"64c6e051ab93fb513f9d957a4c41085ea71b121b43d2d6003113d96b7a031123","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:12:01.514167Z","signature_b64":"FyXZx+T0/aTv5NUB/qk4Vuvqh7jFEnYQhzEsrnsONjB8D+7ThKZt1TXIUH4C1sUhIj5UD5n/p6EfbaBAHHsJAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"64c6e051ab93fb513f9d957a4c41085ea71b121b43d2d6003113d96b7a031123","last_reissued_at":"2026-07-05T05:12:01.513684Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:12:01.513684Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1608.08215","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T05:12:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"u6sGYV3tfgzoY0Ic1SncB1fXi6N8hQv3lQHX7ukegIrQZO7avmJnG7o8QO8cakCLuQWiuPupPdQS6zrkoJTsDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T07:34:11.198305Z"},"content_sha256":"8b7d27ccf7b70bd45d5ae5e990b6b589bdd2b10ab594c0b17cf70822f26f4034","schema_version":"1.0","event_id":"sha256:8b7d27ccf7b70bd45d5ae5e990b6b589bdd2b10ab594c0b17cf70822f26f4034"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:MTDOAUNLSP5VCP45SV5EYQIIL2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Coxeter Pairs, Ammann Patterns and Penrose-like Tilings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.mtrl-sci","hep-th","math.MG","math.MP"],"primary_cat":"math-ph","authors_text":"Latham Boyle, Paul J. Steinhardt","submitted_at":"2016-08-29T20:00:00Z","abstract_excerpt":"We identify a precise geometric relationship between: (i) certain natural pairs of irreducible reflection groups (``Coxeter pairs\"); (ii) self-similar quasicrystalline patterns formed by superposing sets of 1D quasi-periodically-spaced lines, planes or hyper-planes (``Ammann patterns\"); and (iii) the tilings dual to these patterns (``Penrose-like tilings\"). We use this relationship to obtain all irreducible Ammann patterns and their dual Penrose-like tilings, along with their key properties in a simple, systematic and unified way, expanding the number of known examples from four to infinity. F"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.08215","kind":"arxiv","version":4},"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/1608.08215/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":"2026-07-05T05:12:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"h0FF4oUroHjmIq6Saz8Fkj5kdyzfWKjwNqs2ky50P/SXTrBC9vAw4Mi8HYn/AaAs90GtWujn+hBOxdYhl6IFAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T07:34:11.199010Z"},"content_sha256":"b51c395c99e0f2375a7d356bd4d44900a5b861a5589e19f72503a9abfd3bc05d","schema_version":"1.0","event_id":"sha256:b51c395c99e0f2375a7d356bd4d44900a5b861a5589e19f72503a9abfd3bc05d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MTDOAUNLSP5VCP45SV5EYQIIL2/bundle.json","state_url":"https://pith.science/pith/MTDOAUNLSP5VCP45SV5EYQIIL2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MTDOAUNLSP5VCP45SV5EYQIIL2/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":"2026-08-11T07:34:11Z","links":{"resolver":"https://pith.science/pith/MTDOAUNLSP5VCP45SV5EYQIIL2","bundle":"https://pith.science/pith/MTDOAUNLSP5VCP45SV5EYQIIL2/bundle.json","state":"https://pith.science/pith/MTDOAUNLSP5VCP45SV5EYQIIL2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MTDOAUNLSP5VCP45SV5EYQIIL2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:MTDOAUNLSP5VCP45SV5EYQIIL2","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":"681d0c8e22e36d8b6e6a16f4a68cb7f853f2fa0b490bb7296c91fa8b7a50ae4d","cross_cats_sorted":["cond-mat.mtrl-sci","hep-th","math.MG","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-08-29T20:00:00Z","title_canon_sha256":"6ae1cdca8a9dab1937880e34189b358ff74bde9b1d5feaf0c3264401a7d38bbb"},"schema_version":"1.0","source":{"id":"1608.08215","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.08215","created_at":"2026-07-05T05:12:01Z"},{"alias_kind":"arxiv_version","alias_value":"1608.08215v4","created_at":"2026-07-05T05:12:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.08215","created_at":"2026-07-05T05:12:01Z"},{"alias_kind":"pith_short_12","alias_value":"MTDOAUNLSP5V","created_at":"2026-07-05T05:12:01Z"},{"alias_kind":"pith_short_16","alias_value":"MTDOAUNLSP5VCP45","created_at":"2026-07-05T05:12:01Z"},{"alias_kind":"pith_short_8","alias_value":"MTDOAUNL","created_at":"2026-07-05T05:12:01Z"}],"graph_snapshots":[{"event_id":"sha256:b51c395c99e0f2375a7d356bd4d44900a5b861a5589e19f72503a9abfd3bc05d","target":"graph","created_at":"2026-07-05T05:12:01Z","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/1608.08215/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We identify a precise geometric relationship between: (i) certain natural pairs of irreducible reflection groups (``Coxeter pairs\"); (ii) self-similar quasicrystalline patterns formed by superposing sets of 1D quasi-periodically-spaced lines, planes or hyper-planes (``Ammann patterns\"); and (iii) the tilings dual to these patterns (``Penrose-like tilings\"). We use this relationship to obtain all irreducible Ammann patterns and their dual Penrose-like tilings, along with their key properties in a simple, systematic and unified way, expanding the number of known examples from four to infinity. F","authors_text":"Latham Boyle, Paul J. Steinhardt","cross_cats":["cond-mat.mtrl-sci","hep-th","math.MG","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-08-29T20:00:00Z","title":"Coxeter Pairs, Ammann Patterns and Penrose-like Tilings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.08215","kind":"arxiv","version":4},"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":"sha256:8b7d27ccf7b70bd45d5ae5e990b6b589bdd2b10ab594c0b17cf70822f26f4034","target":"record","created_at":"2026-07-05T05:12:01Z","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":"681d0c8e22e36d8b6e6a16f4a68cb7f853f2fa0b490bb7296c91fa8b7a50ae4d","cross_cats_sorted":["cond-mat.mtrl-sci","hep-th","math.MG","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-08-29T20:00:00Z","title_canon_sha256":"6ae1cdca8a9dab1937880e34189b358ff74bde9b1d5feaf0c3264401a7d38bbb"},"schema_version":"1.0","source":{"id":"1608.08215","kind":"arxiv","version":4}},"canonical_sha256":"64c6e051ab93fb513f9d957a4c41085ea71b121b43d2d6003113d96b7a031123","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"64c6e051ab93fb513f9d957a4c41085ea71b121b43d2d6003113d96b7a031123","first_computed_at":"2026-07-05T05:12:01.513684Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:12:01.513684Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FyXZx+T0/aTv5NUB/qk4Vuvqh7jFEnYQhzEsrnsONjB8D+7ThKZt1TXIUH4C1sUhIj5UD5n/p6EfbaBAHHsJAA==","signature_status":"signed_v1","signed_at":"2026-07-05T05:12:01.514167Z","signed_message":"canonical_sha256_bytes"},"source_id":"1608.08215","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8b7d27ccf7b70bd45d5ae5e990b6b589bdd2b10ab594c0b17cf70822f26f4034","sha256:b51c395c99e0f2375a7d356bd4d44900a5b861a5589e19f72503a9abfd3bc05d"],"state_sha256":"e5f7b164cb103c4edf5e18b4b76bb4e7920fd0ce811f0b984444ad2af830725d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Y3G70oltBI1l8RcKop+LkPy5oC/VfBsHBXNvJH8mqPhA4fujEeRuius5hiZdM6+dsFhviGFYBss276hqzcwVAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-11T07:34:11.203584Z","bundle_sha256":"cba55888df7c6e116e0d0f3377fe4634d2713f34309690080387698266e48b0d"}}