{"paper":{"title":"On the Failure of the Upper Bound in the Refined BMV Conjecture and a Pinching Correction","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":["math.FA","math.OA"],"primary_cat":"math.CO","authors_text":"Trung Hoa Dinh","submitted_at":"2026-05-18T03:02:04Z","abstract_excerpt":"We analyze why the refined Bessis--Moussa--Villani conjecture fails. The refined conjecture proposed that the normalized trace average over all words with prescribed numbers of letters \\(A\\) and \\(B\\) should be bounded above by the clustered word \\(\\Tr(A^nB^m)\\). Recent counterexamples of Cha show that this upper bound is false already for \\(3\\times3\\) positive semidefinite matrices when \\(n=m=5\\). We explain the failure from the viewpoint of commutative common parts. The term \\(\\Tr(A^nB^m)\\) is not the canonical common part of the pair \\((A,B)\\); it is only one clustered word. After pinching "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.17782","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/2605.17782/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"}