Social media fact-checking has long been challenged by evidence-level and aggregation-level conflicts, where erroneous evidence mimics authoritative news sources. To capture this challenge and support conflict verification tasks, we present ContraNote, a large-scale real-world dataset curated from X's Community Notes system. It includes 33,686 posts for evaluating evidence-level conflict resolution, and 54,474 instances for evaluating aggregation-level prioritization. Additionally, we propose CoVer, a factual adjudication framework with three-stage pipelines: evidence schema normalization, factual consensus and support verification. This prioritizes evidence over noise to prevent it from compromising the final verdict. Technical evaluations show that CoVer achieves strong performance compared with state-of-the-art baselines across ContraNote (86.0% Acc., 68.0% mac. F1, 64.5 bal. Acc. on Conflict; and 88.5% Acc., 88.5 mac. F1 and 89.2 bal. Acc. on Prioritization), CONFACT-HumC (88.4% Acc.) and CONFACT-ModC (89.4% Acc.).

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