{"paper":{"title":"Symmetric Distributions from Shallow Circuits","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Anthony Ostuni, Daniel M. Kane, Kewen Wu","submitted_at":"2025-11-18T04:23:42Z","abstract_excerpt":"We characterize the symmetric distributions that can be (approximately) generated by shallow Boolean circuits. More precisely, let $f\\colon \\{0,1\\}^m \\to \\{0,1\\}^n$ be a Boolean function where each output bit depends on at most $d$ input bits. Suppose the output distribution of $f$ evaluated on uniformly random input bits is close in total variation distance to a symmetric distribution $\\mathcal{D}$ over $\\{0,1\\}^n$. Then $\\mathcal{D}$ must be close to a mixture of the uniform distribution over $n$-bit strings of even Hamming weight, the uniform distribution over $n$-bit strings of odd Hamming"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2511.14127","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/2511.14127/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"}