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As an arithmetic consequence, any two-level design whose $J$-characteristics vanish in orders one, two, three, five, and seven but which has a nonzero odd-order $J$-characteristic must have at least $256$ runs. This settles, uniformly in the number of factors, a conjecture of Eendebak, Schoen, Vazquez, and Goos (2023) on the none"},"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":"2607.09478","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-10T14:53:04Z","cross_cats_sorted":["math.ST","stat.TH"],"title_canon_sha256":"ccf87bfbaadd6e8912cd44410a98edcba277b3908254f5729db0ae63e20386fd","abstract_canon_sha256":"6c262cf2ddc670af10101001b4537d8224b827db26c53d807fc4cba49ed962cb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-13T01:20:12.853588Z","signature_b64":"iSiSS2N8MpywWWKOSoV9+Hrg8jqRQUEXnlYmcE9sGbWGJvZRb8y47piZGD21xvYSi8LPZIwAeOBCexCkuQyKBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c7a45950119284a1e49231c69e87cea75832549499e6aad5493254bf6258016f","last_reissued_at":"2026-07-13T01:20:12.852297Z","signature_status":"signed_v1","first_computed_at":"2026-07-13T01:20:12.852297Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A divisibility theorem for odd $J$-characteristics of two-level designs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.CO","authors_text":"Pieter Thijs Eendebak","submitted_at":"2026-07-10T14:53:04Z","abstract_excerpt":"We prove a divisibility theorem for the signed $J$-characteristics of two-level designs: if the number of factors $n$ is odd and every $J$-characteristic of a proper odd-cardinality subset of factors vanishes, then the top $J$-characteristic is divisible by $2^{n-1}$. 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