{"paper":{"title":"At Most Two Infinite Blue Clusters in the CMR Representation of the Edwards-Anderson Spin Glass","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The blue subgraph in the two-replica CMR representation of the Edwards-Anderson spin glass has at most two infinite connected components.","cross_cats":["math-ph","math.MP"],"primary_cat":"cond-mat.dis-nn","authors_text":"Yan Ru Pei","submitted_at":"2026-05-17T09:15:39Z","abstract_excerpt":"The two-replica Chayes-Machta-Redner (CMR) representation is one of the main proposed geometric signatures of spin-glass order in the short-range Edwards-Anderson model. Mean-field arguments and recent numerics suggest that the low-temperature phase should exhibit two macroscopic blue clusters carrying opposite overlap signs. We prove a rigorous structural constraint in this direction. For any translation-invariant joint Gibbs measure on disorder, two spin replicas, and CMR bond variables on Z^d, the blue subgraph contains at most two infinite connected components; if two exist, then they lie "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For any translation-invariant joint Gibbs measure on disorder, two spin replicas, and CMR bond variables on Z^d, the blue subgraph contains at most two infinite connected components; if two exist, then they lie in a common infinite grey cluster and belong to opposite overlap-parity classes.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The blue-bond process lacks insertion tolerance and positive association, so the proof must rely on the full joint measure together with a finite-box merge operation and the mass-transport bound on ends of translation-invariant subgraphs rather than standard Burton-Keane or random-cluster arguments (abstract, paragraph on main obstacle).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Proves that the blue subgraph in the CMR representation of the short-range Edwards-Anderson model has at most two infinite clusters, which must lie in one grey cluster with opposite overlap parity if both are infinite.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The blue subgraph in the two-replica CMR representation of the Edwards-Anderson spin glass has at most two infinite connected components.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"c5ce70c01d8f28e12cb4a648d0e3c6f724f372357cc10682a376fbad69bbd16b"},"source":{"id":"2605.17338","kind":"arxiv","version":1},"verdict":{"id":"90c8c680-c404-4fa5-87ad-0291815bb397","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T22:46:24.075405Z","strongest_claim":"For any translation-invariant joint Gibbs measure on disorder, two spin replicas, and CMR bond variables on Z^d, the blue subgraph contains at most two infinite connected components; if two exist, then they lie in a common infinite grey cluster and belong to opposite overlap-parity classes.","one_line_summary":"Proves that the blue subgraph in the CMR representation of the short-range Edwards-Anderson model has at most two infinite clusters, which must lie in one grey cluster with opposite overlap parity if both are infinite.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The blue-bond process lacks insertion tolerance and positive association, so the proof must rely on the full joint measure together with a finite-box merge operation and the mass-transport bound on ends of translation-invariant subgraphs rather than standard Burton-Keane or random-cluster arguments (abstract, paragraph on main obstacle).","pith_extraction_headline":"The blue subgraph in the two-replica CMR representation of the Edwards-Anderson spin glass has at most two infinite connected components."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.17338/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T23:01:19.674469Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T22:52:24.793560Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T21:41:57.804042Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T21:33:23.739037Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"3790d07e71ca9778d4f41e36f8ba4c1ac38c64636a39ee22f3247d5845bc0597"},"references":{"count":12,"sample":[{"doi":"","year":1998,"title":"Oliver Redner, Jon Machta, and Lincoln F. Chayes. Graphical representations and cluster algorithms for critical points with fields.Physical Review E, 58(3):2749–2752, 1998","work_id":"0b691a20-96e1-4163-99ce-72a87ef348c9","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2007,"title":"J. Machta, C. M. Newman, and D. L. Stein. The percolation signature of the spin glass transition.Journal of Statistical Physics, 130:113–128, 2007","work_id":"089eeaa9-10a4-4018-a658-3e44a29a24de","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2023,"title":"Cluster percolation in the two-dimensional Ising spin glass.Physical Review E, 107(5):054103, 2023","work_id":"90673ce2-e728-4d3c-83a4-83bc2ed6603c","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2026,"title":"Cluster percolation in the three-dimensional±Jrandom- bond Ising model.Physical Review E, 113(2):024139, 2026","work_id":"b73f6a36-0ba1-4b87-9b97-13d3746e1b58","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1998,"title":"Graphical representations for Ising systems in external fields.Journal of Statistical Physics, 93:17–32, 1998","work_id":"953f748d-0011-46e7-98d5-e4cc3e6278b5","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":12,"snapshot_sha256":"3df44d84390f2186d993cb90ee5274979021142e15adacfeb635c0211ea62884","internal_anchors":0},"formal_canon":{"evidence_count":1,"snapshot_sha256":"0d1c724862889404e4b4114d6ac373c37b99aa2b9236c01ac312a73d5dae40b6"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}