{"paper":{"title":"The Endpoint Cardinality of Discrete Cube Skeleta","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.IT","math.IT","math.MG"],"primary_cat":"math.CO","authors_text":"Dean Menezes","submitted_at":"2026-07-16T23:12:32Z","abstract_excerpt":"We determine the minimum order of a finite lattice set that contains a filled axis-parallel cube skeleton about every point of some $N$-point set of centers. For fixed integers $0\\leq k<n$, the answer for $N$ centers is $N^{1-(n-k)/(2n^2)}$, up to constants depending on $n$ and $k$. Thornton proved every smaller exponent and gave a construction of this order; the endpoint lower bound was left open when $k\\geq1$. Our proof combines a midpoint estimate, a labelled form of Shearer's projection inequality, and a strong induction that balances large and small radii without a dyadic pigeonhole loss."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.15502","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/2607.15502/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"}