{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:44J57QQR2QFYSWWILZFXXML6SV","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":"8ff58f29ab7e09eb0c89471cb23d3cfa1ff6ce6f2444b24e43925bb98e72cf1d","cross_cats_sorted":["math.DS","math.PR"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2023-03-14T14:58:41Z","title_canon_sha256":"ce51ef22fc3ef4151c0ee330190aaf1bcb67b7aaa4f8007e442ee2acbe1499cf"},"schema_version":"1.0","source":{"id":"2303.07956","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2303.07956","created_at":"2026-07-05T05:51:13Z"},{"alias_kind":"arxiv_version","alias_value":"2303.07956v1","created_at":"2026-07-05T05:51:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2303.07956","created_at":"2026-07-05T05:51:13Z"},{"alias_kind":"pith_short_12","alias_value":"44J57QQR2QFY","created_at":"2026-07-05T05:51:13Z"},{"alias_kind":"pith_short_16","alias_value":"44J57QQR2QFYSWWI","created_at":"2026-07-05T05:51:13Z"},{"alias_kind":"pith_short_8","alias_value":"44J57QQR","created_at":"2026-07-05T05:51:13Z"}],"graph_snapshots":[{"event_id":"sha256:6c4bbfe7128943692c5db8d31806cd849e2e40e9adb57b809e8f12f320ad4929","target":"graph","created_at":"2026-07-05T05:51:13Z","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/2303.07956/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"It is proved that the number of subsets of $[n]^d$ that tile $\\mathbb{Z}^d$ is $\\left(3^{\\frac{1}{3}}\\right)^{n^d \\pm o(n^d)}$.","authors_text":"Elad Tzalik, Gady Kozma, Itai Benjamini","cross_cats":["math.DS","math.PR"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2023-03-14T14:58:41Z","title":"The number of tiles of $\\mathbb{Z}^d$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2303.07956","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:bf27a54147f86203ba04767cf25f5e128a886a57cd15e806bc54b8c4998896a8","target":"record","created_at":"2026-07-05T05:51:13Z","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":"8ff58f29ab7e09eb0c89471cb23d3cfa1ff6ce6f2444b24e43925bb98e72cf1d","cross_cats_sorted":["math.DS","math.PR"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2023-03-14T14:58:41Z","title_canon_sha256":"ce51ef22fc3ef4151c0ee330190aaf1bcb67b7aaa4f8007e442ee2acbe1499cf"},"schema_version":"1.0","source":{"id":"2303.07956","kind":"arxiv","version":1}},"canonical_sha256":"e713dfc211d40b895ac85e4b7bb17e9542cbd9e80a8a53da9a821edb1a4d5fb4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e713dfc211d40b895ac85e4b7bb17e9542cbd9e80a8a53da9a821edb1a4d5fb4","first_computed_at":"2026-07-05T05:51:13.422634Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:51:13.422634Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tdrJlTejuVaf9ClVanpmVLmMxdXw1IRqlV2HpGJ0h5BtxyozoqGLGYiCp1YB1/GYyYGLjRjzFonOypB9FMO1Cw==","signature_status":"signed_v1","signed_at":"2026-07-05T05:51:13.423096Z","signed_message":"canonical_sha256_bytes"},"source_id":"2303.07956","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bf27a54147f86203ba04767cf25f5e128a886a57cd15e806bc54b8c4998896a8","sha256:6c4bbfe7128943692c5db8d31806cd849e2e40e9adb57b809e8f12f320ad4929"],"state_sha256":"6da9560020c10d64cc587639afe4a70f9bb448164fcfb9cf513e10cc5aed057d"}