Agentic Web search can retrieve relevant information without establishing that the retrieved content actually supports the claims used in an answer. Existing agents typically keep search and evidence recording in a linear interaction trace and optimize primarily for final-answer correctness, providing limited supervision for intermediate grounding. We present EviGraph, a deep-search framework that separates search execution from evidence recording while using a shared policy for the trainable roles. An executor plans concise queries, a frozen evidence verifier inspects source pages and returns verbatim evidence items with an explicit polarity, and the policy maps those items to add/support graph requests that are checked by a deterministic structural validator. The resulting graph serves both as persistent working memory and as a source of dense process rewards, enabling reinforcement learning to directly supervise evidence construction rather than only the final answer. On BrowseComp-Plus, a Qwen3-8B EviGraph agent achieves 35.9% accuracy under a matched interaction budget, compared with 26.9% for the same dual-role architecture without reinforcement learning and 2.7% for a monolithic agent, while generating fewer tokens per rollout. Consistent gains on BrowseComp, GAIA, and XBench indicate that explicitly structuring and rewarding evidence recording improves agentic search

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