In the human-AI collaboration area, the context formed naturally through multi-turn interactions is typically flattened into a chronological sequence and treated as a fixed whole in subsequent reasoning, with no mechanism for dynamic organization and management along the collaboration workflow. Yet these contexts differ substantially in lifecycle, structural hierarchy, and relevance. For instance, temporary or abandoned exchanges and parallel topic threads persist in the limited context window, causing interference and even conflict. Meanwhile, users are largely limited to influencing context indirectly through input modifications (e.g., corrections, references, or ignoring), leaving their control neither explicit nor verifiable. To address this, we propose Mixed-Initiative Context, which reconceptualizes the context formed across multi-turn interactions as an explicit, structured, and manipulable interactive object. Under this concept, the structure, scope, and content of context can be dynamically organized and adjusted according to task needs, enabling both humans and AI to actively participate in context construction and regulation. To explore this concept, we implement Contextify as a probe system and conduct a user study examining users' context management behaviors, attitudes toward AI initiative, and overall collaboration experience. We conclude by discussing the implications of this concept for the HCI community.

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