{"paper":{"title":"An explicit construction of two completely independent spanning trees in the four-dimensional dual-cube","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Jitendra Prajapati","submitted_at":"2026-08-02T00:00:20Z","abstract_excerpt":"Lalou, Mbarek, Skender and Togni (arXiv:2607.25917) proved that the $n$-dimensional dual-cube $F_n$ admits two completely independent spanning trees for every $n\\ge 5$, observed that none exist for $n\\le 3$, and identified $F_4$ as the first unresolved case, reporting more than 700 hours of inconclusive computation. We settle this case affirmatively by an explicit construction, completing the classification: $F_n$ admits two completely independent spanning trees if and only if $n\\ge 4$. The internal-vertex sets of the two trees are the level sets of a single ten-term cubic polynomial over $\\ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.00900","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/2608.00900/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"}