{"paper":{"title":"Optimal Thresholds for Monotone Non-Boolean Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Allen Lin, Saba Lepsveridze","submitted_at":"2025-09-08T21:43:42Z","abstract_excerpt":"Let $[q] = \\{0,1,\\ldots,q-1\\}$, let $\\Delta[q]$ denote the simplex of probability measures on $[q]$, and let $\\gamma$ denote the Lebesgue measure normalized on $\\Delta[q]$. We prove that for any symmetric monotone function $f \\colon[q]^n \\to [q]$ and any $a \\in [q]$ we have \\begin{equation*}\n  \\gamma(\\{\\mu \\in \\Delta[q]\\;\\vert\\;\\mathbb{P}_{x\\sim\\mu^{\\otimes n}}[f(x)=a] \\in (\\varepsilon,1-\\varepsilon)\\}) = O(1/\\log n)\\text{.} \\end{equation*} We also show that this bound is tight. This improves Kalai and Mossel's previous bound of $O(\\log \\log n/\\log n)$ and answers their question completely."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.07246","kind":"arxiv","version":2},"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/2509.07246/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"}