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Data-free self-evolving search agents offer a scalable route to systems that generate their own questions, answer them, and improve from their own feedback without human annotations. Yet, without verifiable evidence, this loop can reward fluent but unsupported examples, turning the self-generated curriculum into an opaque and potentially unreliable training signal. We argue that evidence verifiability is a prerequisite for trustworthy self-evolution in search agents: each generated instance should include not only an answer but also a source-grounded span whose contribution to that answer can be measured. We introduce EVE-Agent, an Evidence-Verifiable Self-Evolving Agent that operationalizes this principle through a modification to the proposer--solver framework. The proposer generates a question, an answer, and a verbatim evidence span. 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We introduce EVE-Agent, an Evidence-Verifiable Self-Evolving Agent that operationalizes this principle through a modification to the proposer--solver framework. The proposer generates a question, an answer, and a verbatim evidence span. 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