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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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2509.07246","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2025-09-08T21:43:42Z","cross_cats_sorted":[],"title_canon_sha256":"9dc92cdf1ec255e73f8569fc9a94172da94006ccdb3b85012f4a2efe67baca69","abstract_canon_sha256":"5038a5be3a7b01054b416b4b90d9c7845d2e4918a7a67decacef3a3c1c90d323"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T01:04:57.143751Z","signature_b64":"wjFHmcOIh4ac3okbiB3Fi5KfszTiw7AqUQfdHyAvyhXlUYx0uTBSAjReluQSdztBUoHe767Womiy4XQ0jVsDAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ab7fd04b61bbd82512b355de653dd9de590790ceb60fd2b09ae9ee0f5e69cfaa","last_reissued_at":"2026-05-20T01:04:57.142790Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T01:04:57.142790Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2509.07246","created_at":"2026-05-20T01:04:57.142937+00:00"},{"alias_kind":"arxiv_version","alias_value":"2509.07246v2","created_at":"2026-05-20T01:04:57.142937+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.07246","created_at":"2026-05-20T01:04:57.142937+00:00"},{"alias_kind":"pith_short_12","alias_value":"VN75AS3BXPMC","created_at":"2026-05-20T01:04:57.142937+00:00"},{"alias_kind":"pith_short_16","alias_value":"VN75AS3BXPMCKEVT","created_at":"2026-05-20T01:04:57.142937+00:00"},{"alias_kind":"pith_short_8","alias_value":"VN75AS3B","created_at":"2026-05-20T01:04:57.142937+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/VN75AS3BXPMCKEVTKXPGKPOZ3Z","json":"https://pith.science/pith/VN75AS3BXPMCKEVTKXPGKPOZ3Z.json","graph_json":"https://pith.science/api/pith-number/VN75AS3BXPMCKEVTKXPGKPOZ3Z/graph.json","events_json":"https://pith.science/api/pith-number/VN75AS3BXPMCKEVTKXPGKPOZ3Z/events.json","paper":"https://pith.science/paper/VN75AS3B"},"agent_actions":{"view_html":"https://pith.science/pith/VN75AS3BXPMCKEVTKXPGKPOZ3Z","download_json":"https://pith.science/pith/VN75AS3BXPMCKEVTKXPGKPOZ3Z.json","view_paper":"https://pith.science/paper/VN75AS3B","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2509.07246&json=true","fetch_graph":"https://pith.science/api/pith-number/VN75AS3BXPMCKEVTKXPGKPOZ3Z/graph.json","fetch_events":"https://pith.science/api/pith-number/VN75AS3BXPMCKEVTKXPGKPOZ3Z/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VN75AS3BXPMCKEVTKXPGKPOZ3Z/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VN75AS3BXPMCKEVTKXPGKPOZ3Z/action/storage_attestation","attest_author":"https://pith.science/pith/VN75AS3BXPMCKEVTKXPGKPOZ3Z/action/author_attestation","sign_citation":"https://pith.science/pith/VN75AS3BXPMCKEVTKXPGKPOZ3Z/action/citation_signature","submit_replication":"https://pith.science/pith/VN75AS3BXPMCKEVTKXPGKPOZ3Z/action/replication_record"}},"created_at":"2026-05-20T01:04:57.142937+00:00","updated_at":"2026-05-20T01:04:57.142937+00:00"}