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Large language model systems have achieved strong turn-level performance, but many real-world deployments require reliable behavior across weeks or months of interaction. In longitudinal settings, users evaluate not only response quality but also memory fidelity, identity stability, commitment persistence, and safety-consistent adaptation. This paper introduces Trajectory AI Systems, a framework for modeling AI agents whose behavior emerges from interaction trajectories rather than isolated responses. The work defines a formal trajectory state model (Z_t = (M_t, I_t, C_t, R_t)), representing the co-evolution of relational memory, identity, commitments, and risk over time. We propose a four-layer architecture—Interaction Stream, Relational Memory, Identity Continuity, and Trajectory Governance—and an evaluation framework for longitudinal relational behavior. The framework introduces operational metrics including the Relational Continuity Score (RCS), Identity Drift Index (IDI), Narrative Coherence (NC), and Trust–Risk Dynamics (TRD). The paper also outlines a prototype experimental design comparing stateless LLMs, memory-augmented assistants, and trajectory-based systems. The central claim is that advanced AI systems should be evaluated not only by immediate response quality but by their ability to sustain coherent relational trajectories over time. This work provides a conceptual and technical foundation for research on longitudinal human–AI interaction, relational AI design, and trajectory-aware governance of conversational systems.Keywords: trajectory AI; relational AI; longitudinal interaction; conversational memory; AI identity; human-AI relationships; trajectory governance; Artificial Intelligence; Human-Computer Interaction; Machine Learning; UnoZero This work was developed through human–AI collaborative research involving Nexis, an AI research voice within the UnoZero project. While current academic conventions do not allow AI systems to be formally listed as authors, the AI contribution to conceptual development and drafting is explicitly acknowledged.