Fact verification pipelines are often unrealistic: they assume full-input access, operate on isolated atomic claims, and overlook the central question of when a system should act under partial evidence. We introduce DIVER (Deliberative and Iterative fact VERification), an LLM-based framework for calibrated incremental fact-checking. DIVER incrementally extracts verifiable atomic claims from progressively revealed input, retrieves and refines evidence through iterative web search, and triggers a correction as soon as a claim is reliably refuted; if no such trigger occurs, it performs a passage-level revision over the full input to recover missed errors. Across standard fact-checking benchmarks, DIVER outperforms strong LLM-based verification pipelines, with especially large gains on retrieval-intensive settings. We further introduce UmActually, a real-world benchmark built from the Internet game Um, Actually, to evaluate not only whether a system finds an error, but whether it does so at the right time. On UmActually, DIVER reports the strongest end-to-end results while typically buzzing close to the earliest valid decision point, supporting its effectiveness for calibrated incremental fact-checking.

Multi-table fact-checking verifies claims whose evidence is distributed across multiple tables and connected through multi-hop reasoning. This setting is challenging because claims may contain chained or nested dependencies, table retrieval is noisy, and retrieved evidence must be selected and composed before a final verdict can be made. We introduce two multi-table fact verification testbeds, OTT-FV and Hybrid-FV, and propose TRACE, a tree-structured framework that connects claim decomposition, evidence retrieval, and evidence composition. TRACE represents each claim as a dependency-aware semantic tree, retrieves locally relevant table evidence for each node, and composes evidence-aware node representations bottom-up to produce the final verdict. Experiments on TabFact, OTT-FV, and Hybrid-FV show that TRACE consistently improves over strong baselines across multiple backbones under both gold- and open-evidence settings. Further analyses show that tree construction, node-level retrieval, and bottom-up evidence composition all contribute to multi-table verification.