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We formulate a sufficient condition -- based on the signs of the constant and leading coefficients of the characteristic polynomial of the linearized Jacobian scaled by the diffusion coefficients -- that guarantees a Turing-like instability to spatially inhomogeneous solutions on appropriately chosen domains $\\Omega$. We also present a specific condition on the domain size $|\\Omega|$ required to trigger this instability. As a consequence of employing a"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We formulate a sufficient condition -- based on the signs of the constant and leading coefficients of the characteristic polynomial of the linearized Jacobian scaled by the diffusion coefficients -- that guarantees a Turing-like instability to spatially inhomogeneous solutions on appropriately chosen domains Ω.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The spatially homogeneous reaction network admits a monomial steady-state parameterization (explicitly invoked in the abstract and used to obtain algebraic inequalities); if this parameterization does not exist or is not monomial, the reduction to polynomial inequalities in rate constants and diffusion coefficients fails.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Sufficient algebraic conditions on the signs of the constant and leading coefficients of the diffusion-scaled characteristic polynomial are derived to guarantee Turing instability on suitable domains, then applied to a two-site phosphorylation network to obtain a condition involving only four rate-1","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For reaction networks admitting a monomial steady-state parameterization, the signs of the constant and leading coefficients in the diffusion-scaled characteristic polynomial supply a sufficient condition for Turing-like spatial instability","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"c0ff8f653c99cd46dd42c66c9dd39cb73206512a820758ab7d2a20632d73442f"},"source":{"id":"2605.16049","kind":"arxiv","version":1},"verdict":{"id":"229e072c-f533-4af3-971f-0c8f57ab2741","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T18:45:43.187729Z","strongest_claim":"We formulate a sufficient condition -- based on the signs of the constant and leading coefficients of the characteristic polynomial of the linearized Jacobian scaled by the diffusion coefficients -- that guarantees a Turing-like instability to spatially inhomogeneous solutions on appropriately chosen domains Ω.","one_line_summary":"Sufficient algebraic conditions on the signs of the constant and leading coefficients of the diffusion-scaled characteristic polynomial are derived to guarantee Turing instability on suitable domains, then applied to a two-site phosphorylation network to obtain a condition involving only four rate-1","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The spatially homogeneous reaction network admits a monomial steady-state parameterization (explicitly invoked in the abstract and used to obtain algebraic inequalities); if this parameterization does not exist or is not monomial, the reduction to polynomial inequalities in rate constants and diffusion coefficients fails.","pith_extraction_headline":"For reaction networks admitting a monomial steady-state parameterization, the signs of the constant and leading coefficients in the diffusion-scaled characteristic polynomial supply a sufficient condition for Turing-like spatial instability"},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.16049/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T19:01:18.986635Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T18:51:54.907895Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T17:33:41.556223Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T16:41:55.530040Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"d55c1be9fbd86e95c044ade5567f76c0128d298871a0ab50d9e7ab18a1156828"},"references":{"count":38,"sample":[{"doi":"","year":2022,"title":"Turing patterns, 70 years later , Nature Computational Science 2 (2022), no. 8, 463–464","work_id":"62e6fa2a-a13c-4047-8afa-2eb069a41ad1","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1993,"title":"Ashbaugh and Rafael D","work_id":"97593aa0-a677-4410-9a9b-f1b172e53085","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2026,"title":"C. 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