{"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 $2^{n-1}$. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.09478","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.09478/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}