The remarkable successes of contemporary artificial intelligence (AI) have been largely driven by empiricism, but the lack of a solid theoretical foundation has produced deep crises in interpretability, safety, and generalization. This paper proposes a new mathematical framework called ”Dynamic Geometry,” aimed at establishing a rigorous axiomatic basis for AI. The framework reconceptualizes AI systems as parameter-driven multilayer families of holistic operators on Banach spaces, whose core dynamics are determined by the evolution of operator spectra — the so-called ”spectral flow.” By formalizing representation learning as spectral projection, relating interpretability to algebraic-topological invariants (such as Ktheory and cyclic homology), linking the realization of “infinite context” with Lyapunov stability of long-time evolution and tying safety (e.g., hallucination control) to verifiable mathematical evidence, Dynamic Geometry provides a fundamental path to constructing provable, robust, and general AI systems. We prove that this framework can encompass any finite-parameter deep learning model in terms of expressive power, and through an analysis based on spectral stability we reveal conditional advantages in its generalization behavior. Dynamic Geometry not only offers a unified theoretical framework to address the current crises in AI, but may also open a new era of next-generation AI driven by first principles and rigorously defined mathematics.