{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:7Y4LJMRKEE7VUYXN7YNS3CVVD7","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":"eb6ba95dd76b8da60632a5e697938b2b7359d358b95094d0de10d092a48f5dea","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-04-08T09:31:20Z","title_canon_sha256":"af516aa65f1b3839cd01cc0b71904ace43583cbb1f4f95673094cfdeca8c28ba"},"schema_version":"1.0","source":{"id":"1904.03907","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.03907","created_at":"2026-05-17T23:42:02Z"},{"alias_kind":"arxiv_version","alias_value":"1904.03907v2","created_at":"2026-05-17T23:42:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.03907","created_at":"2026-05-17T23:42:02Z"},{"alias_kind":"pith_short_12","alias_value":"7Y4LJMRKEE7V","created_at":"2026-05-18T12:33:12Z"},{"alias_kind":"pith_short_16","alias_value":"7Y4LJMRKEE7VUYXN","created_at":"2026-05-18T12:33:12Z"},{"alias_kind":"pith_short_8","alias_value":"7Y4LJMRK","created_at":"2026-05-18T12:33:12Z"}],"graph_snapshots":[{"event_id":"sha256:a1da8aadea63e2dd0eff08564743a6db88160d6eede1d527c65809732a91b925","target":"graph","created_at":"2026-05-17T23:42:02Z","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":"We consider a set of necessary conditions which are efficient heuristics for deciding when a set of Wang tiles cannot tile a group.\n  Piantadosi gave a necessary and sufficient condition for the existence of a valid tiling of any free group. This condition is actually necessary for the existence of a valid tiling for an arbitrary finitely generated group.\n  We then consider two other conditions: the first, also given by Piantadosi, is a necessary and sufficient condition to decide if a set of Wang tiles gives a strongly periodic tiling of the free group; the second, given by Chazottes et. al.,","authors_text":"Benjamin Hellouin de Menibus, Hugo Maturana Cornejo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-04-08T09:31:20Z","title":"Necessary conditions for tiling finitely generated amenable groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.03907","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:f1181941adeee8fa81330b64ff65527bfb8d52ac59adcf9888afd0768a90cf72","target":"record","created_at":"2026-05-17T23:42:02Z","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":"eb6ba95dd76b8da60632a5e697938b2b7359d358b95094d0de10d092a48f5dea","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-04-08T09:31:20Z","title_canon_sha256":"af516aa65f1b3839cd01cc0b71904ace43583cbb1f4f95673094cfdeca8c28ba"},"schema_version":"1.0","source":{"id":"1904.03907","kind":"arxiv","version":2}},"canonical_sha256":"fe38b4b22a213f5a62edfe1b2d8ab51fddcb556e993a54839ed2bffe8e23825e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fe38b4b22a213f5a62edfe1b2d8ab51fddcb556e993a54839ed2bffe8e23825e","first_computed_at":"2026-05-17T23:42:02.303182Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:42:02.303182Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"DBCk9xB4rDd/8bNfUIqHZQd5JGmfIvCGPxgZiYLXujhMIaOwfwi24ykj6N7dWO7WBBwVUCxkebVfqh8RxrwKBg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:42:02.303704Z","signed_message":"canonical_sha256_bytes"},"source_id":"1904.03907","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f1181941adeee8fa81330b64ff65527bfb8d52ac59adcf9888afd0768a90cf72","sha256:a1da8aadea63e2dd0eff08564743a6db88160d6eede1d527c65809732a91b925"],"state_sha256":"2450e43c300c42fccb0c87384980d14bf43fe957e593845c69487232da6c6985"}