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Brief Hospital Course (BHC) narratives must be clinically useful yet faithful to fragmented EHR evidence. LLM-based clinical summarizers still introduce unsupported statements, and alignment can encourage omissions ("say-less" degeneration). We introduce VERI-DPO, which uses claim verification to mine preferences and distill them into the summarizer with Direct Preference Optimization (DPO). On MIMIC-III-Ext-VeriFact-BHC (100 ICU patients; patient-level splits), we train a retrieval-augmented verifier to label claim-evidence pairs as Supported, Not Supported, or Not Addressed via a single-token format. The verifier scores sentence-level claims from sampled BHC candidates and aggregates margins into a coverage-aware utility to mine length-controlled, contradiction-anchored preference pairs. On held-out patients, verifier-mined preferences separate candidates by contradiction density, and VERI-DPO reduces Not Supported claim rates from 10.7% to 1.9% (local verifier judge) and from 11.6% to 6.4% (GPT-4o judge), while improving validity from 76.7% to 82.5% and maintaining informative length.