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A miniature $M$ of $P$ is said to be average-sized (resp. normal-sized) if the volume of $M$ is equal to the limit of the sequence whose $n$-th term is the average of the volumes of all miniarures (resp. all horizontal miniatures) whose vertices belong to $(n^{-1}\\mathbb Z)^d.$ We prove that, for any lattice"},"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":"2501.00459","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-12-31T14:17:31Z","cross_cats_sorted":[],"title_canon_sha256":"e92551e3ec0521aef26a2a84a6893f3ddd91dbcbde36fe3fc76a726903701a6b","abstract_canon_sha256":"c293bea73645b96d0a99a07f936e54e8cdcb8426fadfd24035d5b645bfbde3fc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:05:26.662289Z","signature_b64":"9v2Hk3uvxvyrvu1uNXjDGg1Np1RZ7gofWzXjeHZD+ZTJZ8NwKChv1nIcCEsNuVCrs9lkIkCwsse4FpZYTR8jCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ce649316ca8931f858ac7190767bafb335797e9d6488488c584c61eb6462db40","last_reissued_at":"2026-05-20T00:05:26.661524Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:05:26.661524Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Average-sized miniatures and normal-sized miniatures of lattice polytopes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Takashi Hirotsu","submitted_at":"2024-12-31T14:17:31Z","abstract_excerpt":"Let $d \\geq 0$ be an integer and $P \\subset \\mathbb R^d$ be a $d$-dimensional lattice polytope. 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