Abstract: Artificial intelligence (AI) agents are increasingly explored as collaborative learning partners in mathematics education, yet it remains unclear what makes an AI recognizable as a peer. This study examines how domain experts perceive and interact with an adaptive LLM-based peer agent combining a Big Five-informed persona with motivational state modulation for collaborative mathematics problem solving. Through thematic analysis of expert interviews, we identify conditions shaping peer perceptions, including emotional safety, shared control, and collaborative reasoning. Participants primarily judged the AI through its social behavior rather than mathematical capabilities. The AI was perceived more as a peer when it created a non-judgmental space, engaged in tentative reasoning, displayed appropriate imperfection, and respected participants' pace. Conversely, blind agreement and over-instruction reduced trust and shifted perceptions toward a tutor role. Based on these findings, we propose human-centered design dimensions for AI peer agents to guide future peer-oriented learning systems.