{"paper":{"title":"A Beckmann boundary form of Talagrand's conjecture on the discrete cube","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.FA","math.PR"],"primary_cat":"math.CA","authors_text":"Haonan Zhang, Paata Ivanisvili, Xinyuan Xie","submitted_at":"2026-06-30T17:02:21Z","abstract_excerpt":"We introduce the Beckmann boundary of a Boolean function \\[\n  \\mathsf{B}(f)=\\inf_{\\operatorname{div} V=Lf}\\mathbb E\\|V(x)\\|_2. \\] Here \\[\n  L=\\sum_iD_i,\\qquad D_i f(x)=\\frac{f(x)-f(x^{\\oplus i})}{2}, \\] and $\\operatorname{div} V(x)=\\sum_i (V_{i}(x)-V_{i}(x^{\\oplus i}))$. This nonlocal quantity is no larger than the usual two-sided, one-sided, colored, optimized colored, or optimized fractional colored boundaries. Nevertheless, every nonconstant Boolean $f$ satisfies \\[\n  \\mathsf{B}(f)\\gtrsim \\operatorname{Var}(f)\n  \\sqrt{\\log\\!\\left(1+\\frac{1}{\\sum_i\\operatorname{Inf}_i(f)^2}\\right)}. \\] We al"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.31961","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/2606.31961/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"}