{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:L5LYRB43GMA2MRN23Z4RIXOJSD","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":"82754aafe5bd5d7c8a4a0b324391bb5e6c04af482ee537233d74493ffe84b38e","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-10-15T10:14:03Z","title_canon_sha256":"4dc5e920d5d215b6b471fadf5956b719063a648aa4b64f666841ee9fb3c220cc"},"schema_version":"1.0","source":{"id":"1210.3967","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.3967","created_at":"2026-05-18T03:42:43Z"},{"alias_kind":"arxiv_version","alias_value":"1210.3967v1","created_at":"2026-05-18T03:42:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.3967","created_at":"2026-05-18T03:42:43Z"},{"alias_kind":"pith_short_12","alias_value":"L5LYRB43GMA2","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_16","alias_value":"L5LYRB43GMA2MRN2","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_8","alias_value":"L5LYRB43","created_at":"2026-05-18T12:27:14Z"}],"graph_snapshots":[{"event_id":"sha256:22c66f1e7e5057179a3a06c986d93199ed29e730c06e5fb4cb02ca1bd24df0f9","target":"graph","created_at":"2026-05-18T03:42:43Z","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":"Aperiodic tilings with a small number of prototiles are of particular interest, both theoretically and for applications in crystallography. In this direction, many people have tried to construct aperiodic tilings that are built from a single prototile with nearest neighbour matching rules, which is then called a monotile. One strand of the search for a planar monotile has focussed on hexagonal analogues of Wang tiles. This led to two inflation tilings with interesting structural details. Both possess aperiodic local rules that define hulls with a model set structure. We review them in comparis","authors_text":"Franz G\\\"ahler (Bielefeld), Michael Baake (Bielefeld), Uwe Grimm (Milton Keynes)","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-10-15T10:14:03Z","title":"Hexagonal inflation tilings and planar monotiles"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.3967","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":"sha256:4a4e90427f9336eb3c4445548cbbe45c8be5b34327926fae5749cde7ed83b366","target":"record","created_at":"2026-05-18T03:42:43Z","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":"82754aafe5bd5d7c8a4a0b324391bb5e6c04af482ee537233d74493ffe84b38e","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-10-15T10:14:03Z","title_canon_sha256":"4dc5e920d5d215b6b471fadf5956b719063a648aa4b64f666841ee9fb3c220cc"},"schema_version":"1.0","source":{"id":"1210.3967","kind":"arxiv","version":1}},"canonical_sha256":"5f5788879b3301a645bade79145dc990d6bd2977bdfde638599c3f53dbef4ad2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5f5788879b3301a645bade79145dc990d6bd2977bdfde638599c3f53dbef4ad2","first_computed_at":"2026-05-18T03:42:43.817173Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:42:43.817173Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HpKoaoybbCkk7mCVBHbGlxcUDAba4lUBrd8oyGpzhPSSweekuWdr+MFRMsb6kUWVV2f7xB5X+UJ9OgQXkY5QAw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:42:43.817771Z","signed_message":"canonical_sha256_bytes"},"source_id":"1210.3967","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4a4e90427f9336eb3c4445548cbbe45c8be5b34327926fae5749cde7ed83b366","sha256:22c66f1e7e5057179a3a06c986d93199ed29e730c06e5fb4cb02ca1bd24df0f9"],"state_sha256":"924065afea1de8cb233e178eb5aeff96c7d3e28618de0f334668c52172c8d8b1"}