{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:C7KIRTOV5ICME2N3S2XHB4RN5T","short_pith_number":"pith:C7KIRTOV","canonical_record":{"source":{"id":"1602.05784","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-18T13:02:10Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"c5a9864d723235be7d7c0a95875ee0bec6dfd964ffcbd06ef27e048a24355830","abstract_canon_sha256":"b2f0f4df2d97a24e67bcb73b4bc99ad4901ee6cbb9541395fa253a8eb1d026e9"},"schema_version":"1.0"},"canonical_sha256":"17d488cdd5ea04c269bb96ae70f22decc7afc1b4274135f274dca86e3341f437","source":{"kind":"arxiv","id":"1602.05784","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.05784","created_at":"2026-05-18T01:12:19Z"},{"alias_kind":"arxiv_version","alias_value":"1602.05784v3","created_at":"2026-05-18T01:12:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.05784","created_at":"2026-05-18T01:12:19Z"},{"alias_kind":"pith_short_12","alias_value":"C7KIRTOV5ICM","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_16","alias_value":"C7KIRTOV5ICME2N3","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_8","alias_value":"C7KIRTOV","created_at":"2026-05-18T12:30:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:C7KIRTOV5ICME2N3S2XHB4RN5T","target":"record","payload":{"canonical_record":{"source":{"id":"1602.05784","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-18T13:02:10Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"c5a9864d723235be7d7c0a95875ee0bec6dfd964ffcbd06ef27e048a24355830","abstract_canon_sha256":"b2f0f4df2d97a24e67bcb73b4bc99ad4901ee6cbb9541395fa253a8eb1d026e9"},"schema_version":"1.0"},"canonical_sha256":"17d488cdd5ea04c269bb96ae70f22decc7afc1b4274135f274dca86e3341f437","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:12:19.803925Z","signature_b64":"WNiC2VGWKFQT1z+VLoeAkHejQIy6aUsYRKyBoa0KgMRCcPKyl0hEF4sOCXNl2tLWV3MwrSs7So03095Vo9brDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"17d488cdd5ea04c269bb96ae70f22decc7afc1b4274135f274dca86e3341f437","last_reissued_at":"2026-05-18T01:12:19.803583Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:12:19.803583Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1602.05784","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:12:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+AMUjIa/MzqkxejLbbwuO3pAzIvxsZoai5l5sQpNJ4G+y1fpkjxed8OdxosKXfCbEOQ8U7u51PVa3+CiMJIvBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T09:52:00.169942Z"},"content_sha256":"54453e9c5df885576ab5c49e1e14300e705e6ae021f6f1f41f33e3e06ebdc963","schema_version":"1.0","event_id":"sha256:54453e9c5df885576ab5c49e1e14300e705e6ae021f6f1f41f33e3e06ebdc963"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:C7KIRTOV5ICME2N3S2XHB4RN5T","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Subtilings of Polyomino Tilings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.CO","authors_text":"Jacob Turner","submitted_at":"2016-02-18T13:02:10Z","abstract_excerpt":"We consider a problem concerning tilings of rectangular regions by a finite library of polyominoes. We specifically look at rectangular regions of dimension $n\\times m$ and ask whether or not a tiling of this region can be rearranged so that tiling of the $n\\times m$ rectangle can be realized as a tiling of an $n\\times m'$ rectangle and an $n\\times m\"$ rectangle, $m=m'+m\"$. We call this a subtiling. We show that the associated decision problem is $\\mathsf{NP}$-complete when restricted to rectangular polyominoes. We also show that for certain finite libraries of polyominoes, if $m$ is sufficien"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.05784","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"2026-05-18T01:12:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qdElwCi5st0G6+qzzdkDB9RsvkkvIFtBtuFk8E64fhm6f46xRLOid25cH+Ep4GvwPG5dfZxH/g+BzFW9aiHPAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T09:52:00.170285Z"},"content_sha256":"2611cecb73f07fe4f5403b63c0a92d32610196dd8257d7a1a12d3fc691b4e453","schema_version":"1.0","event_id":"sha256:2611cecb73f07fe4f5403b63c0a92d32610196dd8257d7a1a12d3fc691b4e453"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/C7KIRTOV5ICME2N3S2XHB4RN5T/bundle.json","state_url":"https://pith.science/pith/C7KIRTOV5ICME2N3S2XHB4RN5T/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/C7KIRTOV5ICME2N3S2XHB4RN5T/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":"2026-06-30T09:52:00Z","links":{"resolver":"https://pith.science/pith/C7KIRTOV5ICME2N3S2XHB4RN5T","bundle":"https://pith.science/pith/C7KIRTOV5ICME2N3S2XHB4RN5T/bundle.json","state":"https://pith.science/pith/C7KIRTOV5ICME2N3S2XHB4RN5T/state.json","well_known_bundle":"https://pith.science/.well-known/pith/C7KIRTOV5ICME2N3S2XHB4RN5T/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:C7KIRTOV5ICME2N3S2XHB4RN5T","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":"b2f0f4df2d97a24e67bcb73b4bc99ad4901ee6cbb9541395fa253a8eb1d026e9","cross_cats_sorted":["math.AC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-18T13:02:10Z","title_canon_sha256":"c5a9864d723235be7d7c0a95875ee0bec6dfd964ffcbd06ef27e048a24355830"},"schema_version":"1.0","source":{"id":"1602.05784","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.05784","created_at":"2026-05-18T01:12:19Z"},{"alias_kind":"arxiv_version","alias_value":"1602.05784v3","created_at":"2026-05-18T01:12:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.05784","created_at":"2026-05-18T01:12:19Z"},{"alias_kind":"pith_short_12","alias_value":"C7KIRTOV5ICM","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_16","alias_value":"C7KIRTOV5ICME2N3","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_8","alias_value":"C7KIRTOV","created_at":"2026-05-18T12:30:09Z"}],"graph_snapshots":[{"event_id":"sha256:2611cecb73f07fe4f5403b63c0a92d32610196dd8257d7a1a12d3fc691b4e453","target":"graph","created_at":"2026-05-18T01:12:19Z","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 problem concerning tilings of rectangular regions by a finite library of polyominoes. We specifically look at rectangular regions of dimension $n\\times m$ and ask whether or not a tiling of this region can be rearranged so that tiling of the $n\\times m$ rectangle can be realized as a tiling of an $n\\times m'$ rectangle and an $n\\times m\"$ rectangle, $m=m'+m\"$. We call this a subtiling. We show that the associated decision problem is $\\mathsf{NP}$-complete when restricted to rectangular polyominoes. We also show that for certain finite libraries of polyominoes, if $m$ is sufficien","authors_text":"Jacob Turner","cross_cats":["math.AC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-18T13:02:10Z","title":"On Subtilings of Polyomino Tilings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.05784","kind":"arxiv","version":3},"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":"sha256:54453e9c5df885576ab5c49e1e14300e705e6ae021f6f1f41f33e3e06ebdc963","target":"record","created_at":"2026-05-18T01:12:19Z","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":"b2f0f4df2d97a24e67bcb73b4bc99ad4901ee6cbb9541395fa253a8eb1d026e9","cross_cats_sorted":["math.AC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-18T13:02:10Z","title_canon_sha256":"c5a9864d723235be7d7c0a95875ee0bec6dfd964ffcbd06ef27e048a24355830"},"schema_version":"1.0","source":{"id":"1602.05784","kind":"arxiv","version":3}},"canonical_sha256":"17d488cdd5ea04c269bb96ae70f22decc7afc1b4274135f274dca86e3341f437","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"17d488cdd5ea04c269bb96ae70f22decc7afc1b4274135f274dca86e3341f437","first_computed_at":"2026-05-18T01:12:19.803583Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:12:19.803583Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WNiC2VGWKFQT1z+VLoeAkHejQIy6aUsYRKyBoa0KgMRCcPKyl0hEF4sOCXNl2tLWV3MwrSs7So03095Vo9brDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:12:19.803925Z","signed_message":"canonical_sha256_bytes"},"source_id":"1602.05784","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:54453e9c5df885576ab5c49e1e14300e705e6ae021f6f1f41f33e3e06ebdc963","sha256:2611cecb73f07fe4f5403b63c0a92d32610196dd8257d7a1a12d3fc691b4e453"],"state_sha256":"c58e20636059abf30366cd3476a7fe0f4dc8a3ff769b5b6064b286436a763748"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xd3DG0wonj3dvfu38Xab+WPtNSfE84XaGxDDf3Z++PcWCEiqSpQRuy+z75cJuksKeYT6d5RuZ48hUnIs7arMDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-30T09:52:00.172135Z","bundle_sha256":"2b74f16849b5974af67140e608a6693c3fd08a5dfe4da9977e8e9efb27ee331c"}}