{"paper":{"title":"A Quadratic Lower Bound for Noncommutative Circuits","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Every fan-in 2 noncommutative arithmetic circuit for the palindrome polynomial requires size Omega of n squared.","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Pratik Shastri","submitted_at":"2026-04-22T13:55:01Z","abstract_excerpt":"We prove that every fan-in $2$ noncommutative arithmetic circuit computing the palindrome polynomial has size $\\Omega(nd)$. In particular, when $d=n$ we obtain an $\\Omega(n^2)$ lower bound. The proof builds on and refines a previous work of the author. Key ideas in the proof were generated by Gemini 3.1 Pro."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"every fan-in 2 noncommutative arithmetic circuit computing the palindrome polynomial has size Ω(n^2)","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The proof relies on refining techniques from the author's prior work; without the full text the precise load-bearing assumptions in the refinement step cannot be identified.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Every fan-in 2 noncommutative arithmetic circuit computing the palindrome polynomial has size Ω(n²).","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Every fan-in 2 noncommutative arithmetic circuit for the palindrome polynomial requires size Omega of n squared.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"4690fd502ab8bbdf0bd914e6caafead2e3c524b2ba61c8a964a4d127fb2b313b"},"source":{"id":"2604.20575","kind":"arxiv","version":3},"verdict":{"id":"040f0f01-ae9d-424f-a72f-a14a0736f34c","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-09T22:42:13.925226Z","strongest_claim":"every fan-in 2 noncommutative arithmetic circuit computing the palindrome polynomial has size Ω(n^2)","one_line_summary":"Every fan-in 2 noncommutative arithmetic circuit computing the palindrome polynomial has size Ω(n²).","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The proof relies on refining techniques from the author's prior work; without the full text the precise load-bearing assumptions in the refinement step cannot be identified.","pith_extraction_headline":"Every fan-in 2 noncommutative arithmetic circuit for the palindrome polynomial requires size Omega of n squared."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.20575/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"}