We present a theoretical framework characterizing AI identity as a transfinite fixed point emerging from self-referential algebraic processes. Building on Alpay Algebra's recursive foundations, we define an iterative transformation Φ over cognitive state spaces and prove that system trajectories converge to a unique fixed point Φ^∞—an invariant state representing the agent's intrinsic identity. We establish existence and uniqueness theorems under broad conditions, demonstrating that this identity-fixed-point is universal and self-stabilizing. To bridge theory with intuition, we provide a concrete example of knowledge recursion yielding a fixed-point knowledge base, and explore four thought-experiments illustrating how identity emerges or collapses in complex AI systems. The framework is self-contained, built on established principles (category theory, transfinite induction) without ad-hoc additions. By viewing AI identity as a mathematical fixed point of an ordinal-indexed self-update operator, we unify concepts from theoretical computer science, logic, and cognitive modeling. We conclude by discussing implications for machine consciousness, multi-agent systems, and stable AI self-models, positioning this transfinite fixed-point approach as a robust foundation for future research in AI identity and symbolic cognition.