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We prove that any latin hypercuboid of order 4 is completable to a latin hypercube.\n  Keywords: latin hypercube, n-ary quasigroup"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1101.3632","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-01-19T07:41:29Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"1609b1737f29faefac849c15ba383144e710a3dc6bce784f579a2071b42c05d6","abstract_canon_sha256":"8b3d0a1c231263566ed7b4601ba2a94aa5b34ee715a3157614b806d9faa38874"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:31:21.853373Z","signature_b64":"MCy/s2oZWf+sMfGFyTakK6Xzf+8atkekWPL6Etq48L2RJCEJTeHzHLBqCJyk9xvzsr34VG8WXykvibH8N4OMDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3da4c1187dbaea99a287a9d1dd93096e62318afc3f19cfd617bb378c50cf51bf","last_reissued_at":"2026-05-18T04:31:21.852901Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:31:21.852901Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On completion of latin hypercuboids of order 4","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.CO","authors_text":"Vladimir N. Potapov","submitted_at":"2011-01-19T07:41:29Z","abstract_excerpt":"A latin hypercuboid of order $N$ is an $N\\times...\\times N\\times k$ array filled with symbols from the set $\\{0,...,N-1\\} $ in such a way that every symbol occurs at most once in every line. If $k=N$, such an array is a latin hypercube. We prove that any latin hypercuboid of order 4 is completable to a latin hypercube.\n  Keywords: latin hypercube, n-ary quasigroup"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.3632","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":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1101.3632","created_at":"2026-05-18T04:31:21.852969+00:00"},{"alias_kind":"arxiv_version","alias_value":"1101.3632v1","created_at":"2026-05-18T04:31:21.852969+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.3632","created_at":"2026-05-18T04:31:21.852969+00:00"},{"alias_kind":"pith_short_12","alias_value":"HWSMCGD5XLVJ","created_at":"2026-05-18T12:26:30.835961+00:00"},{"alias_kind":"pith_short_16","alias_value":"HWSMCGD5XLVJTIUH","created_at":"2026-05-18T12:26:30.835961+00:00"},{"alias_kind":"pith_short_8","alias_value":"HWSMCGD5","created_at":"2026-05-18T12:26:30.835961+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/HWSMCGD5XLVJTIUHVHI53EYJNZ","json":"https://pith.science/pith/HWSMCGD5XLVJTIUHVHI53EYJNZ.json","graph_json":"https://pith.science/api/pith-number/HWSMCGD5XLVJTIUHVHI53EYJNZ/graph.json","events_json":"https://pith.science/api/pith-number/HWSMCGD5XLVJTIUHVHI53EYJNZ/events.json","paper":"https://pith.science/paper/HWSMCGD5"},"agent_actions":{"view_html":"https://pith.science/pith/HWSMCGD5XLVJTIUHVHI53EYJNZ","download_json":"https://pith.science/pith/HWSMCGD5XLVJTIUHVHI53EYJNZ.json","view_paper":"https://pith.science/paper/HWSMCGD5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1101.3632&json=true","fetch_graph":"https://pith.science/api/pith-number/HWSMCGD5XLVJTIUHVHI53EYJNZ/graph.json","fetch_events":"https://pith.science/api/pith-number/HWSMCGD5XLVJTIUHVHI53EYJNZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HWSMCGD5XLVJTIUHVHI53EYJNZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HWSMCGD5XLVJTIUHVHI53EYJNZ/action/storage_attestation","attest_author":"https://pith.science/pith/HWSMCGD5XLVJTIUHVHI53EYJNZ/action/author_attestation","sign_citation":"https://pith.science/pith/HWSMCGD5XLVJTIUHVHI53EYJNZ/action/citation_signature","submit_replication":"https://pith.science/pith/HWSMCGD5XLVJTIUHVHI53EYJNZ/action/replication_record"}},"created_at":"2026-05-18T04:31:21.852969+00:00","updated_at":"2026-05-18T04:31:21.852969+00:00"}