{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:USMWBTMQZ43NBU4EPKS47PUUXA","short_pith_number":"pith:USMWBTMQ","canonical_record":{"source":{"id":"2506.19295","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-06-24T04:00:41Z","cross_cats_sorted":["cs.CG","math.MG"],"title_canon_sha256":"a503ab857d6a578ea60cea504b95a168d8f423f9b0f7b6562023c59b05166e98","abstract_canon_sha256":"c4cc16405c3a2d1954d15a4355d48ea2eefc2e4f8bdc0358a091479f0ad2ba99"},"schema_version":"1.0"},"canonical_sha256":"a49960cd90cf36d0d3847aa5cfbe94b833ea771427049e30963a30fda840a637","source":{"kind":"arxiv","id":"2506.19295","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.19295","created_at":"2026-07-05T11:26:12Z"},{"alias_kind":"arxiv_version","alias_value":"2506.19295v1","created_at":"2026-07-05T11:26:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.19295","created_at":"2026-07-05T11:26:12Z"},{"alias_kind":"pith_short_12","alias_value":"USMWBTMQZ43N","created_at":"2026-07-05T11:26:12Z"},{"alias_kind":"pith_short_16","alias_value":"USMWBTMQZ43NBU4E","created_at":"2026-07-05T11:26:12Z"},{"alias_kind":"pith_short_8","alias_value":"USMWBTMQ","created_at":"2026-07-05T11:26:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:USMWBTMQZ43NBU4EPKS47PUUXA","target":"record","payload":{"canonical_record":{"source":{"id":"2506.19295","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-06-24T04:00:41Z","cross_cats_sorted":["cs.CG","math.MG"],"title_canon_sha256":"a503ab857d6a578ea60cea504b95a168d8f423f9b0f7b6562023c59b05166e98","abstract_canon_sha256":"c4cc16405c3a2d1954d15a4355d48ea2eefc2e4f8bdc0358a091479f0ad2ba99"},"schema_version":"1.0"},"canonical_sha256":"a49960cd90cf36d0d3847aa5cfbe94b833ea771427049e30963a30fda840a637","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:26:12.854424Z","signature_b64":"19QqwXeWMFSfdOQS65+eJc3LMZhpIVl26jaPRiogBzC7pjj5dzMphy7KN2ybVfqi8nTkJQ8iN5/ubqGXpCuaCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a49960cd90cf36d0d3847aa5cfbe94b833ea771427049e30963a30fda840a637","last_reissued_at":"2026-07-05T11:26:12.853988Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:26:12.853988Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2506.19295","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T11:26:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SuRXdZksW2bntrEoqsYxQ2xPX2DtCvdDfURpBf7M5i+QM+PIpPOe9sFHBZVNhYKHZdNwIup/tSety0nV9MCEBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-22T05:04:59.489258Z"},"content_sha256":"b4d0ccfb2ec1fba71f194e23c70f918ee18c175652e0b65d5bd3ee0d7f88b7e9","schema_version":"1.0","event_id":"sha256:b4d0ccfb2ec1fba71f194e23c70f918ee18c175652e0b65d5bd3ee0d7f88b7e9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:USMWBTMQZ43NBU4EPKS47PUUXA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Undecidability of Translational Tiling of the Plane with Four Tiles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG","math.MG"],"primary_cat":"math.CO","authors_text":"Chao Yang, Zhujun Zhang","submitted_at":"2025-06-24T04:00:41Z","abstract_excerpt":"The translational tiling problem, dated back to Wang's domino problem in the 1960s, is one of the most representative undecidable problems in the field of discrete geometry and combinatorics. Ollinger initiated the study of the undecidability of translational tiling with a fixed number of tiles in 2009, and proved that translational tiling of the plane with a set of $11$ polyominoes is undecidable. The number of polyominoes needed to obtain undecidability was reduced from $11$ to $7$ by Yang and Zhang, and then to $5$ by Kim. We show that translational tiling of the plane with a set of $4$ (di"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.19295","kind":"arxiv","version":1},"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/2506.19295/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":"2026-07-05T11:26:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ojTlFq9NrD+Z8MIz5JoGRRHh0ogQO5+wTYbLtB9VOMvh87+YzteXLZsYmo2MrtXS3D9RZyM+h7/w52Up4/gjBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-22T05:04:59.489657Z"},"content_sha256":"84d8d37072911c2df8fc4d8d02eae7377961a540d990486da49d6db73fa8944c","schema_version":"1.0","event_id":"sha256:84d8d37072911c2df8fc4d8d02eae7377961a540d990486da49d6db73fa8944c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/USMWBTMQZ43NBU4EPKS47PUUXA/bundle.json","state_url":"https://pith.science/pith/USMWBTMQZ43NBU4EPKS47PUUXA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/USMWBTMQZ43NBU4EPKS47PUUXA/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":"2026-08-22T05:04:59Z","links":{"resolver":"https://pith.science/pith/USMWBTMQZ43NBU4EPKS47PUUXA","bundle":"https://pith.science/pith/USMWBTMQZ43NBU4EPKS47PUUXA/bundle.json","state":"https://pith.science/pith/USMWBTMQZ43NBU4EPKS47PUUXA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/USMWBTMQZ43NBU4EPKS47PUUXA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:USMWBTMQZ43NBU4EPKS47PUUXA","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":"c4cc16405c3a2d1954d15a4355d48ea2eefc2e4f8bdc0358a091479f0ad2ba99","cross_cats_sorted":["cs.CG","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-06-24T04:00:41Z","title_canon_sha256":"a503ab857d6a578ea60cea504b95a168d8f423f9b0f7b6562023c59b05166e98"},"schema_version":"1.0","source":{"id":"2506.19295","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.19295","created_at":"2026-07-05T11:26:12Z"},{"alias_kind":"arxiv_version","alias_value":"2506.19295v1","created_at":"2026-07-05T11:26:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.19295","created_at":"2026-07-05T11:26:12Z"},{"alias_kind":"pith_short_12","alias_value":"USMWBTMQZ43N","created_at":"2026-07-05T11:26:12Z"},{"alias_kind":"pith_short_16","alias_value":"USMWBTMQZ43NBU4E","created_at":"2026-07-05T11:26:12Z"},{"alias_kind":"pith_short_8","alias_value":"USMWBTMQ","created_at":"2026-07-05T11:26:12Z"}],"graph_snapshots":[{"event_id":"sha256:84d8d37072911c2df8fc4d8d02eae7377961a540d990486da49d6db73fa8944c","target":"graph","created_at":"2026-07-05T11:26:12Z","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/2506.19295/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The translational tiling problem, dated back to Wang's domino problem in the 1960s, is one of the most representative undecidable problems in the field of discrete geometry and combinatorics. Ollinger initiated the study of the undecidability of translational tiling with a fixed number of tiles in 2009, and proved that translational tiling of the plane with a set of $11$ polyominoes is undecidable. The number of polyominoes needed to obtain undecidability was reduced from $11$ to $7$ by Yang and Zhang, and then to $5$ by Kim. We show that translational tiling of the plane with a set of $4$ (di","authors_text":"Chao Yang, Zhujun Zhang","cross_cats":["cs.CG","math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-06-24T04:00:41Z","title":"Undecidability of Translational Tiling of the Plane with Four Tiles"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.19295","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:b4d0ccfb2ec1fba71f194e23c70f918ee18c175652e0b65d5bd3ee0d7f88b7e9","target":"record","created_at":"2026-07-05T11:26:12Z","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":"c4cc16405c3a2d1954d15a4355d48ea2eefc2e4f8bdc0358a091479f0ad2ba99","cross_cats_sorted":["cs.CG","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-06-24T04:00:41Z","title_canon_sha256":"a503ab857d6a578ea60cea504b95a168d8f423f9b0f7b6562023c59b05166e98"},"schema_version":"1.0","source":{"id":"2506.19295","kind":"arxiv","version":1}},"canonical_sha256":"a49960cd90cf36d0d3847aa5cfbe94b833ea771427049e30963a30fda840a637","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a49960cd90cf36d0d3847aa5cfbe94b833ea771427049e30963a30fda840a637","first_computed_at":"2026-07-05T11:26:12.853988Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:26:12.853988Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"19QqwXeWMFSfdOQS65+eJc3LMZhpIVl26jaPRiogBzC7pjj5dzMphy7KN2ybVfqi8nTkJQ8iN5/ubqGXpCuaCg==","signature_status":"signed_v1","signed_at":"2026-07-05T11:26:12.854424Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.19295","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b4d0ccfb2ec1fba71f194e23c70f918ee18c175652e0b65d5bd3ee0d7f88b7e9","sha256:84d8d37072911c2df8fc4d8d02eae7377961a540d990486da49d6db73fa8944c"],"state_sha256":"87775f20ee9497da8d9bb28149fc6c5bbf1f934b13cb8bb41f91482363cad06b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qZMVVHG73NrQy8kzmHk1/99Y9rI0rLcncdb7qDWNY+68JA2+A+gMtyvNbe1yUl4n3hNIJJFmq4FxTsdsKPxGCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-22T05:04:59.493354Z","bundle_sha256":"dfb1bffa444b9293ffcb80641b8964543d02a5071cb8ab2ef739073a8a1b5c0c"}}