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Each of Van den Bergh's double brackets, Fairon--McCulloch's right double brackets, and also Ginzburg--Schedler's wheeled Poisson brackets induces a $\\operatorname{GL}_N$-invariant Poisson structure on the representation scheme $\\operatorname{Rep}_N(A)$ parametrizing $N$-dimensional representations of $A$, thereby satisfying the Kontsevich--Ros"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We establish a bijection between pairs of coupled double Poisson brackets and wheeled Poisson brackets of Ginzburg and Schedler. On free polynomial algebras, we establish a one-to-one correspondence between linear coupled double Poisson brackets and Poisson-left-pre-Lie algebras, and describe quadratic ones via solutions of the associative and classical Yang-Baxter equations satisfying a compatibility condition.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The cross-Jacobi identity is imposed as part of the definition of a coupled pair; if this identity fails to hold for natural choices of the two component brackets, the bijection with wheeled Poisson brackets would not apply to those choices.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Introduces coupled double Poisson brackets, proves bijection to wheeled Poisson brackets, and gives correspondences to Poisson-left-pre-Lie algebras and Yang-Baxter solutions on free polynomial algebras.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Pairs of coupled double Poisson brackets stand in bijection with wheeled Poisson brackets.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"02d79858fae00ad6b08d6183790344f41ce7afa9eaf8ee95f62de6f0edd5c8ef"},"source":{"id":"2605.17696","kind":"arxiv","version":1},"verdict":{"id":"e37582fb-46a6-4e0a-96e6-942403cb9f05","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T21:52:44.995520Z","strongest_claim":"We establish a bijection between pairs of coupled double Poisson brackets and wheeled Poisson brackets of Ginzburg and Schedler. On free polynomial algebras, we establish a one-to-one correspondence between linear coupled double Poisson brackets and Poisson-left-pre-Lie algebras, and describe quadratic ones via solutions of the associative and classical Yang-Baxter equations satisfying a compatibility condition.","one_line_summary":"Introduces coupled double Poisson brackets, proves bijection to wheeled Poisson brackets, and gives correspondences to Poisson-left-pre-Lie algebras and Yang-Baxter solutions on free polynomial algebras.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The cross-Jacobi identity is imposed as part of the definition of a coupled pair; if this identity fails to hold for natural choices of the two component brackets, the bijection with wheeled Poisson brackets would not apply to those choices.","pith_extraction_headline":"Pairs of coupled double Poisson brackets stand in bijection with wheeled Poisson brackets."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.17696/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T22:01:19.476348Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T22:01:02.441737Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"cited_work_retraction","ran_at":"2026-05-19T21:51:57.115196Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"shingle_duplication","ran_at":"2026-05-19T21:49:43.945878Z","status":"skipped","version":"0.1.0","findings_count":0},{"name":"citation_quote_validity","ran_at":"2026-05-19T21:49:43.744588Z","status":"skipped","version":"0.1.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T21:33:23.518430Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T21:21:57.427583Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"0e7625ce81b130abcb44c1aab0db74228173ec61c2bb507b91d8a8a6db1df482"},"references":{"count":115,"sample":[{"doi":"","year":2013,"title":"Classification of constant solutions for associative Yang-Baxter equation on $gl(3)$","work_id":"5fc49273-f8d5-4f88-acdd-bd2b3a0898a5","ref_index":1,"cited_arxiv_id":"1212.6421","is_internal_anchor":true},{"doi":"","year":2003,"title":"Letters in Mathematical Physics , volume=","work_id":"9e5cfe6a-0127-4dd7-9499-f061c26bbe37","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2008,"title":"Transactions of the American Mathematical Society , volume=","work_id":"7dc4fa9a-f468-4ed3-b39c-1bc4b0d4269d","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2007,"title":"Noncommutative geometry and Cayley-smooth orders , author=. 2007 , publisher=","work_id":"4d7e23ad-0816-4913-9e52-a41de4f41dac","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"Universal localizations, Atiyah conjectures and graphs of groups , author=. 2024 , eprint=","work_id":"50951db0-959c-41dd-95de-2251c81dc7ca","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":115,"snapshot_sha256":"f5ff46da54329cee6cf135b8e41105bd1324101a1e39a3e8d1e285991ad58ca5","internal_anchors":25},"formal_canon":{"evidence_count":2,"snapshot_sha256":"0136bff3e89a19be9ece52f87bafdf4844d8da527191d6e3725dd78d0e60c6ae"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}