AI assistants are increasingly embedded in computational notebooks, scientific workflows, software engineering tools, and learning systems, yet human-AI interactions are often treated as transient prompt-response exchanges rather than durable data products, hindering reproducibility, explainability, replay, behavioral analysis, and adaptive AI support. We argue that these interactions form a new data class requiring formal representation and introduce the Human-AI Interaction Description Language (HAIDL), a formal interaction model with a Backus-Naur Form-inspired concrete syntax. HAIDL encodes each interaction as a structured object capturing computational context, user request, prompt payload, AI response, feedback, problem context, provenance, and model metadata. We integrate this vision into HAIDL-X, a middleware for JupyterLab that instantiates HAIDL by capturing user-bot state, building contextualized prompts, interfacing with AI providers, collecting feedback, and storing interactions in a repository. We add a learning layer over HAIDL that maps interaction objects and notebook execution traces into reproducible measures of user behavior, including engagement, performance, help-seeking, error-oriented use, and learning personas. HAIDL-X provides a foundation for treating human-AI collaboration as structured, queryable, replayable, analyzable data.

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