Mathematical Foundation of Artificial Intelligence : Geometric, Physical, Causal, and Autonomous AI

This is the second volume in a two-part series. The first volume developed the language needed to think about differential manifolds, metrics, curvature and structures of intelligence. This volume asks what follows when that language is used to reconstruct AI from the ground up around four themes: geometry, physics, causality and autonomy.

Mathematical Foundations of Artificial Intelligence: Geometric, Physical, Causal, and Autonomous AI is the second volume in a two-part series. The first volume, Mathematical Foundations of Artificial Intelligence: Basics of Manifold Theory (CRC Press, 2026), developed the geometric language needed to think rigorously about differential manifolds, metrics, curvature, and related structures of intelligence. That volume established the vocabulary. This volume asks what follows when that language is used to reconstruct artificial intelligence from the ground up around four foundational themes: geometry, physics, causality, and autonomy. Artificial intelligence is moving beyond pretraining alone toward the mechanisms of intelligence. Many branches of AI that appear separate can be understood as different aspects of one broader mathematical program. In this program, intelligence is not treated merely as disembodied symbol manipulation or statistical pattern matching. It is modeled as motion and autonomous evolution on structured spaces, constrained by geometry, physics, causality, information, and decision-making. Representation becomes manifold learning; reasoning becomes structured flow on latent spaces; control becomes the steering of these flows under uncertainty; memory becomes graph-structured persistence across time; and autonomy becomes the verified execution of decisions in the world. Key Features• Develops the physical transformer, motivated by the gap between digital AI and physical AI. • Interprets transformer computation through spin systems, neural differential manifolds, Hamiltonian mechanics, Hamilton-Jacobi-Bellman optimal control, episodic semantic workspaces, and graph-theoretic planning. • Extends the geometric viewpoint into biology, especially single-cell dynamics and RNA velocity on manifolds. • Introduces tools from Lie theory, differential geometry, control-affine systems, and controllability theory to study how cell states may be steered across biological manifolds. • Develops mechanics-omics coupled models that connect rod mechanics, energy flow, and reaction-diffusion gene dynamics using differential geometry, Lie groups, Cartan moving frames, and Hamiltonian structure. • Develops a theory of spatial intelligence that fuses perception, reasoning, and geometric control rather than treating any one of them in isolation.