{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:ZLJPHGTC232CPX43RGGFYEI37O","short_pith_number":"pith:ZLJPHGTC","canonical_record":{"source":{"id":"2109.09384","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.MG","submitted_at":"2021-09-20T09:06:17Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"136bb242bf95ee4970d82c8e2268898cb820b63d2d6f059c07b3fef22d9f4504","abstract_canon_sha256":"d9925378df812820d62934fc72422f7ff8abe865b90dbe5cd785e292d75ef67d"},"schema_version":"1.0"},"canonical_sha256":"cad2f39a62d6f427df9b898c5c111bfba6f9326cd0c6fb15d899e0c2d6de6873","source":{"kind":"arxiv","id":"2109.09384","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2109.09384","created_at":"2026-07-05T11:25:36Z"},{"alias_kind":"arxiv_version","alias_value":"2109.09384v2","created_at":"2026-07-05T11:25:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2109.09384","created_at":"2026-07-05T11:25:36Z"},{"alias_kind":"pith_short_12","alias_value":"ZLJPHGTC232C","created_at":"2026-07-05T11:25:36Z"},{"alias_kind":"pith_short_16","alias_value":"ZLJPHGTC232CPX43","created_at":"2026-07-05T11:25:36Z"},{"alias_kind":"pith_short_8","alias_value":"ZLJPHGTC","created_at":"2026-07-05T11:25:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:ZLJPHGTC232CPX43RGGFYEI37O","target":"record","payload":{"canonical_record":{"source":{"id":"2109.09384","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.MG","submitted_at":"2021-09-20T09:06:17Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"136bb242bf95ee4970d82c8e2268898cb820b63d2d6f059c07b3fef22d9f4504","abstract_canon_sha256":"d9925378df812820d62934fc72422f7ff8abe865b90dbe5cd785e292d75ef67d"},"schema_version":"1.0"},"canonical_sha256":"cad2f39a62d6f427df9b898c5c111bfba6f9326cd0c6fb15d899e0c2d6de6873","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:25:36.032835Z","signature_b64":"K7qzBpX6d54K5W0atbYk7FX53Gu8D+qbvuJ+APaToSOicQWhFvGI7x+KQk/or31Sz8BHE0c4/d8c/BJkg8AUAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cad2f39a62d6f427df9b898c5c111bfba6f9326cd0c6fb15d899e0c2d6de6873","last_reissued_at":"2026-07-05T11:25:36.032275Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:25:36.032275Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2109.09384","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T11:25:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/KHhDNokgojRtqzwBnyMhO5nYXLjdqD6gdlRp3VYwdisXu7M5Jz5K2v9zw/rQKjRNpOYJ7ikxzPZsFUlumCMBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-18T05:02:37.539782Z"},"content_sha256":"a250e11cc108ca5f24c92f28ac967070ecfb0967d446af6423a4bf6aa9d59b27","schema_version":"1.0","event_id":"sha256:a250e11cc108ca5f24c92f28ac967070ecfb0967d446af6423a4bf6aa9d59b27"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:ZLJPHGTC232CPX43RGGFYEI37O","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Forcing nonperiodic tilings with one tile using a seed","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.MG","authors_text":"Bernhard Klaassen","submitted_at":"2021-09-20T09:06:17Z","abstract_excerpt":"The so-called \"einstein problem\" (a pun playing with the famous scientist's name and the German term \"ein Stein\" for \"one stone\") asks for a simply connected prototile only allowing nonperiodic tilings without need of any matching rule. So far, researchers come only close to this demand by defining decorated prototiles forcing nonperiodicity of any generated tiling using matching rules. In this paper a class of spiral tilings (and one non-spiral example) is linked to a weaker form of the einstein problem where one or several seed tiles are used. Furthermore, the classical types of matching rul"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.09384","kind":"arxiv","version":2},"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/2109.09384/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":"2026-07-05T11:25:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KjOFceHZxg3BiDi6MSAxBzs/OZErYI/xu8VpUOw7lX1k7hAE4MxIGg/q7isjDZK8qXthJyfQnRJZnguduA0qDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-18T05:02:37.540334Z"},"content_sha256":"a33ab7d72f58b8a26cb00ed9aa5cb74874bba4ea93f33985490d208aef18a3ab","schema_version":"1.0","event_id":"sha256:a33ab7d72f58b8a26cb00ed9aa5cb74874bba4ea93f33985490d208aef18a3ab"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZLJPHGTC232CPX43RGGFYEI37O/bundle.json","state_url":"https://pith.science/pith/ZLJPHGTC232CPX43RGGFYEI37O/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZLJPHGTC232CPX43RGGFYEI37O/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":"2026-08-18T05:02:37Z","links":{"resolver":"https://pith.science/pith/ZLJPHGTC232CPX43RGGFYEI37O","bundle":"https://pith.science/pith/ZLJPHGTC232CPX43RGGFYEI37O/bundle.json","state":"https://pith.science/pith/ZLJPHGTC232CPX43RGGFYEI37O/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZLJPHGTC232CPX43RGGFYEI37O/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:ZLJPHGTC232CPX43RGGFYEI37O","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":"d9925378df812820d62934fc72422f7ff8abe865b90dbe5cd785e292d75ef67d","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.MG","submitted_at":"2021-09-20T09:06:17Z","title_canon_sha256":"136bb242bf95ee4970d82c8e2268898cb820b63d2d6f059c07b3fef22d9f4504"},"schema_version":"1.0","source":{"id":"2109.09384","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2109.09384","created_at":"2026-07-05T11:25:36Z"},{"alias_kind":"arxiv_version","alias_value":"2109.09384v2","created_at":"2026-07-05T11:25:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2109.09384","created_at":"2026-07-05T11:25:36Z"},{"alias_kind":"pith_short_12","alias_value":"ZLJPHGTC232C","created_at":"2026-07-05T11:25:36Z"},{"alias_kind":"pith_short_16","alias_value":"ZLJPHGTC232CPX43","created_at":"2026-07-05T11:25:36Z"},{"alias_kind":"pith_short_8","alias_value":"ZLJPHGTC","created_at":"2026-07-05T11:25:36Z"}],"graph_snapshots":[{"event_id":"sha256:a33ab7d72f58b8a26cb00ed9aa5cb74874bba4ea93f33985490d208aef18a3ab","target":"graph","created_at":"2026-07-05T11:25:36Z","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/2109.09384/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The so-called \"einstein problem\" (a pun playing with the famous scientist's name and the German term \"ein Stein\" for \"one stone\") asks for a simply connected prototile only allowing nonperiodic tilings without need of any matching rule. So far, researchers come only close to this demand by defining decorated prototiles forcing nonperiodicity of any generated tiling using matching rules. In this paper a class of spiral tilings (and one non-spiral example) is linked to a weaker form of the einstein problem where one or several seed tiles are used. Furthermore, the classical types of matching rul","authors_text":"Bernhard Klaassen","cross_cats":["math.CO"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.MG","submitted_at":"2021-09-20T09:06:17Z","title":"Forcing nonperiodic tilings with one tile using a seed"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.09384","kind":"arxiv","version":2},"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:a250e11cc108ca5f24c92f28ac967070ecfb0967d446af6423a4bf6aa9d59b27","target":"record","created_at":"2026-07-05T11:25:36Z","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":"d9925378df812820d62934fc72422f7ff8abe865b90dbe5cd785e292d75ef67d","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.MG","submitted_at":"2021-09-20T09:06:17Z","title_canon_sha256":"136bb242bf95ee4970d82c8e2268898cb820b63d2d6f059c07b3fef22d9f4504"},"schema_version":"1.0","source":{"id":"2109.09384","kind":"arxiv","version":2}},"canonical_sha256":"cad2f39a62d6f427df9b898c5c111bfba6f9326cd0c6fb15d899e0c2d6de6873","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cad2f39a62d6f427df9b898c5c111bfba6f9326cd0c6fb15d899e0c2d6de6873","first_computed_at":"2026-07-05T11:25:36.032275Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:25:36.032275Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"K7qzBpX6d54K5W0atbYk7FX53Gu8D+qbvuJ+APaToSOicQWhFvGI7x+KQk/or31Sz8BHE0c4/d8c/BJkg8AUAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T11:25:36.032835Z","signed_message":"canonical_sha256_bytes"},"source_id":"2109.09384","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a250e11cc108ca5f24c92f28ac967070ecfb0967d446af6423a4bf6aa9d59b27","sha256:a33ab7d72f58b8a26cb00ed9aa5cb74874bba4ea93f33985490d208aef18a3ab"],"state_sha256":"a7a88f8fcb491c7b825e91414dc8a1757a93a0e89f63ab2df4fec1d0ebbac30b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"C0jJdEauDrIoxk94o8GbUq5RcHXiurR+NSPXO4yS7m4+MyQ15HonGAvvpvOXS394lhQU6+I5Cmrg18imjDUTCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-18T05:02:37.545904Z","bundle_sha256":"b6b9e0856b249ae2bb2e5b3b3d1a1e05c80e2fd55bff515fefd3c09d2fe6be41"}}