{"paper":{"title":"Color--Phase Separation for Mixed Random Operators in Two-Speed Stochastic Klein--Gordon Systems","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"In two-speed stochastic Klein-Gordon systems, color labels organize contractions while phase labels separate Duhamel channels via speed-induced frequency gaps.","cross_cats":[],"primary_cat":"math.PR","authors_text":"Guangqian Zhao","submitted_at":"2026-04-23T17:28:45Z","abstract_excerpt":"We study a two-component stochastic Klein-Gordon system on \\(\\mathbb T^3\\) with fixed distinct speeds, pure cross interaction \\(u_1u_2\\), and diagonal independent space-time white noises. The mixed paracontrolled random operators exhibit a color-phase separation mechanism: color labels determine Wick contractions and covariance blocks, while phase labels record the Duhamel-source phase difference between the outer propagator and the stochastic high-frequency leg.\n  In the present two-speed model this phase difference is produced by the low-high bound \\(|\\omega_i(\\ell+q)-\\omega_j(\\ell)|\\gtrsim "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The mixed paracontrolled random operators exhibit a color-phase separation mechanism: color labels determine Wick contractions and covariance blocks, while phase labels record the Duhamel-source phase difference between the outer propagator and the stochastic high-frequency leg produced by the low-high bound |ω_i(ℓ+q)−ω_j(ℓ)| ≳ N^α for i≠j.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The model is restricted to diagonal independent colors, fixed distinct speeds, and pure cross interaction u1u2; the phase difference bound |ω_i(ℓ+q)−ω_j(ℓ)| ≳ N^α for i≠j is assumed to hold and is used to separate same-color and cross-color terms (abstract, paragraph on color-phase separation).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Proves local paracontrolled solutions and Galerkin convergence for a two-speed stochastic Klein-Gordon system with cross interaction via color-phase separation of mixed operators, for 12/13 < α ≤ 1.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"In two-speed stochastic Klein-Gordon systems, color labels organize contractions while phase labels separate Duhamel channels via speed-induced frequency gaps.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"a4dcd5817f0b758548a2d76a741256377db0d04804829f4c0ec6112406661b71"},"source":{"id":"2604.21884","kind":"arxiv","version":4},"verdict":{"id":"284430bf-eb22-419e-ba83-33bcbe2bb245","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T16:52:10.721250Z","strongest_claim":"The mixed paracontrolled random operators exhibit a color-phase separation mechanism: color labels determine Wick contractions and covariance blocks, while phase labels record the Duhamel-source phase difference between the outer propagator and the stochastic high-frequency leg produced by the low-high bound |ω_i(ℓ+q)−ω_j(ℓ)| ≳ N^α for i≠j.","one_line_summary":"Proves local paracontrolled solutions and Galerkin convergence for a two-speed stochastic Klein-Gordon system with cross interaction via color-phase separation of mixed operators, for 12/13 < α ≤ 1.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The model is restricted to diagonal independent colors, fixed distinct speeds, and pure cross interaction u1u2; the phase difference bound |ω_i(ℓ+q)−ω_j(ℓ)| ≳ N^α for i≠j is assumed to hold and is used to separate same-color and cross-color terms (abstract, paragraph on color-phase separation).","pith_extraction_headline":"In two-speed stochastic Klein-Gordon systems, color labels organize contractions while phase labels separate Duhamel channels via speed-induced frequency gaps."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.21884/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":17,"sample":[{"doi":"","year":2003,"title":"Da Prato and A","work_id":"ab6a199f-0d1c-422e-94ad-ec5d90d94bcb","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2014,"title":"Hairer,A theory of regularity structures, Invent","work_id":"3fc41afb-ce53-412e-afcb-abec94a62834","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2015,"title":"M. Gubinelli, P. Imkeller, and N. Perkowski,Paracontrolled distributions and singular PDEs, Forum Math. Pi3(2015), e6","work_id":"818caab8-2144-46f2-8db7-52bcbe39d583","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"M. Gubinelli, H. Koch, and T. Oh,Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity, J. Eur. Math. Soc.26(2024), 817–874","work_id":"55d71c52-b18d-4335-868f-ff05a36a1c1d","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2022,"title":"Y. Deng, A. R. Nahmod, and H. Yue,Random tensors, propagation of randomness, and nonlinear dispersive equations, Invent. Math.228(2022), 539–686","work_id":"6396ff91-7f19-4d44-a3d6-5f49661832bf","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":17,"snapshot_sha256":"dd84d49ad74bcc8e89b6954ef3b9e7dade107c961146924d16c22aca8e030f68","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"}