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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 $L(T,x"},"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":"1401.4689","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2014-01-19T16:14:34Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"ea3e30ead3d4d09c525bc8db85d1dffca73cfd9773076a5a533cd3882ccfba39","abstract_canon_sha256":"675dae314dbc97653ab9250db85af3c9633effd146d8db871a9fe70c02e19a17"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:51:14.895656Z","signature_b64":"pW9SQdYkcOi/lIww8dUlOiZ0WQ1uv/JSeBWcrnrxggoPDiMDEKfYwq9g9qsy6Ghqhh8QQS66E2zjhpvlDKKiDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f0a4a0dc345c5e4a59e2358039435d5f986c419bd4dc6ec7efe3ea54f93f9317","last_reissued_at":"2026-05-18T02:51:14.895091Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:51:14.895091Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Keller's conjecture in dimension seven","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.MG","authors_text":"Andrzej P. Kisielewicz, Magdalena {\\L}ysakowska","submitted_at":"2014-01-19T16:14:34Z","abstract_excerpt":"A cube tiling of $\\mathbb{R}^d$ is a family of pairwise disjoint cubes $[0,1)^d+T=\\{[0,1)^d+t:t\\in T\\}$ such that $\\bigcup_{t\\in T}([0,1)^d+t)=\\mathbb{R}^d$. 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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