Conversational Capture: A Trajectory-Level Framework for Evaluating Generative Engine Optimization in Multi-turn Human-Agent Interaction

Generative Engine Optimization (GEO) shapes content to increase its likelihood of being cited by answer engines built on retrieval-augmented large language models. GEO is typically evaluated as a single-turn property: for a fixed query, an evaluator measures a source's visibility in one answer. We argue that the single answer is an inadequate unit of analysis. Human-agent information seeking forms a closed loop: the agent's answer changes the user's beliefs and therefore the next question, which in turn determines what the agent retrieves. We introduce conversational capture, a phenomenon in which a source cited early becomes substantially more likely to be cited again. Capture operates through a machine-side channel, history-conditioned retrieval, and a human-side channel, follow-up questions directed toward the captured source. We formalize the interaction as a two-layer closed-loop system and derive trajectory-level constructs: cumulative conversational visibility; a direct/feedback decomposition of trajectory gain; a nested split of the feedback term into machine-side and human-side channels; a capture coefficient; a compounding ratio; and a misranking diagnostic. Using reinforcement-process (P'olya-urn) theory, we prove that the feedback term is zero under single-turn evaluation and that GEO's cumulative payoff grows superlinearly with conversation length while capture develops. A model-derived illustration shows that the feedback term can exceed the direct term, the compounding ratio exceeds two within ten turns, and single-turn and trajectory rankings agree only weakly (Kendall's τ = 0.4). We connect the human channel to information foraging, trust calibration, and Bayesian persuasion, and discuss design implications for answer engines.