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LLM agents increasingly operate in long-horizon, verifier-rich environments, but current self-improvement mechanisms often write episode-derived lessons directly into memory, making local success an unreliable signal for reusable and safe knowledge. This paper addresses the core question of when an experience-derived behavioral update should be admitted into persistent agent state. We propose Falsification-Gated Self-Evolution (FGSE), a verifier-grounded framework that represents each candidate lesson as a structured hypothesis with an explicit precondition, behavior change, expected effect, and verifier. FGSE applies the hypothesis only in a temporary state, tests it on target-transfer, falsification, and archived-regression probes, and then commits, refines, or rejects it according to measured gain and risk. Across web, app, tool-use, and long-term memory benchmarks, FGSE achieves strong task performance, including 39.8 WebArena success rate, 78.8 AppWorld average completion, and 0.724/0.487 Pass$^1$ on $τ$-Retail/$τ$-Airline, while reducing falsification failures to 5.0%, regression damage to 1.3%, and harmful committed updates to 3.6%. These results suggest that testing memory updates as hypotheses supports more reliable long-term adaptation.