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Two cubes $[0,1)^d+t$, $[0,1)^d+s$ are called a twin pair if their closures have a complete facet in common, that is if $|t_j-s_j|=1$ for some $j\\in [d]=\\{1,..., d\\}$ and $t_i=s_i$ for every $i\\in [d]\\setminus \\{j\\}$. In 1930, Keller conjectured that in every cube tiling of R^d there is a twin pair. Keller's conjecture is true for dimensions $d\\leq 6$ and false for all dimensions $d\\geq 8$. For $d=7$ the conjecture is still open. Let $x\\in R^d$, $i\\in [d]$, "},"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":"1304.1639","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2013-04-05T08:35:50Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"2247dbb0a0c54f8b044f0d44d76aac5effce83a328d788e4d7dcede536cf5b30","abstract_canon_sha256":"3e95738e6c3e3113c303d380ab573b90bd26dc0d3400ea43a3597be39e42156f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:30:32.653998Z","signature_b64":"S+nxmATMzGyXVlU30PwVhuJsOGjCNXFZXZO5EhptB6hDKGtZNA0LZI2k4fe1WdxlwV+U6HcUAg55QxiodMjcAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"06467ef195afb33b9036beb14bac6d74fc8e48ba798eff992b21f004e7e51719","last_reissued_at":"2026-05-18T02:30:32.653590Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:30:32.653590Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rigid polyboxes and Keller's conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.MG","authors_text":"Andrzej P. Kisielewicz","submitted_at":"2013-04-05T08:35:50Z","abstract_excerpt":"A cube tiling of R^d is a family of pairwise disjoint cubes $[0,1)^d+T=\\{[0,1)^d+t:t\\in T\\}$ such that $\\bigcup_{t\\in T}([0,1)^d+t)=R^d$. Two cubes $[0,1)^d+t$, $[0,1)^d+s$ are called a twin pair if their closures have a complete facet in common, that is if $|t_j-s_j|=1$ for some $j\\in [d]=\\{1,..., d\\}$ and $t_i=s_i$ for every $i\\in [d]\\setminus \\{j\\}$. In 1930, Keller conjectured that in every cube tiling of R^d there is a twin pair. Keller's conjecture is true for dimensions $d\\leq 6$ and false for all dimensions $d\\geq 8$. For $d=7$ the conjecture is still open. 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