Evidence-grounded fact-checking requires predicting claim veracity while returning faithful evidence at fine granularity, including exact sentences, table cells, and complete multi-document chains. Although large language models enable decomposition, planning, and multi-agent verification, they can still produce convincing rationales with weak provenance, especially under heterogeneous evidence and multi-hop requirements. We propose GAVEL, a multi-agent debate framework that enforces evidence grounding throughout inference. GAVEL introduces an Evidence Contract that requires debaters to state atomic subclaims and bind each to explicit evidence units, and a Mechanized Chain of Scrutiny in which a neutral Scrutinizer audits outputs and performs deterministic validation of cited identifiers and quoted spans. A Judge then selects a sufficient evidence set and produces the final decision. Experiments on FEVEROUS and HOVER in an open-book setting show that GAVEL improves provenance-aware metrics that jointly require correct labels and correct, complete evidence over strong recent baselines. Ablations confirm that both evidence binding and mechanized citation validation are key to the gains.