Excerpt from the foreword The study of straightening — the systematic transformation of geometric bodies or point configurations toward an ideal, simplified, or canonical shape — stands at the intersection of geometry, control theory, and computational mathematics. Though seemingly straightforward in concept, the mathematical challenges inherent in designing algorithms that can effect such transformations reveal deep and elegant structures, inviting exploration far beyond practical application. This book presents an abstract, rigorous examination of straightening algorithms, emphasizing their mathematical foundations rather than their physical instantiation. Our goal is to develop a theoretical framework that demonstrates how artificial intelligence (AI) can engage directly with programming languages or existing codebases through mathematically driven instructions. In other words, AI need not be confined to opaque “black-box” learning; instead, it can actively learn, interpret, and adjust programs via explicit mathematical principles encoded in direct language commands.