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Two cubes $[0,1)^d+t$, $[0,1)^d+s$ are called a twin pair if $|t_j-s_j|=1$ for some $j\\in [d]=\\{1,\\ldots, d\\}$ and $t_i=s_i$ for every $i\\in [d]\\setminus \\{j\\}$. In $1930$, Keller conjectured that in every cube tiling of $\\mathbb{R}^d$ 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 $x\\in \\mathbb{R}^d$, $i\\in [d]$, and let "},"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":"1701.07155","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-01-25T04:36:32Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"e3cec9fc0ace1be971025bff0313d2cde7230738723f3a7eeb0e9fdb97118a05","abstract_canon_sha256":"51c3b1fa388c9f697879fefad564658054ca1cb241f6671d6b6063490d11f899"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:52:07.324639Z","signature_b64":"FKLCEdNqN0rE3UhB9CRGi+77auUg+pRTTU/mhtoQ2G/3T9kQaoTo2k5oUuaI3ibQaEuoU/NXAY0TkGaYVe9mBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4e5189e3029aab1c8d51e5432b6dba54915f567cc1fa916de5db9cd508216d37","last_reissued_at":"2026-05-18T00:52:07.324099Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:52:07.324099Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Towards Resolving Keller's Cube Tiling Conjecture in Dimension Seven","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CO","authors_text":"Andrzej P. Kisielewicz","submitted_at":"2017-01-25T04:36:32Z","abstract_excerpt":"A cube tiling of $\\mathbb{R}^d$ is a family of pairwise disjoint cubes $[0,1)^d+T=\\{[0,1)^d+t\\colon t\\in T\\}$ such that $\\bigcup_{t\\in T}([0,1)^d+t)=\\mathbb{R}^d$. Two cubes $[0,1)^d+t$, $[0,1)^d+s$ are called a twin pair if $|t_j-s_j|=1$ for some $j\\in [d]=\\{1,\\ldots, d\\}$ and $t_i=s_i$ for every $i\\in [d]\\setminus \\{j\\}$. In $1930$, Keller conjectured that in every cube tiling of $\\mathbb{R}^d$ 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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