The book Observability and Mathematics Quantum Yang-Mills Theory and Modelling, written by [Author name], delves into the intricacies of Quantum Yang-Mills theory and its mathematical foundation, providing a comprehensive understanding of the subject. The book begins by exploring the current state of elementary particle theory and the predictions made by physicists through experiments. However, despite these discoveries, the theoretical foundations of Quantum Yang-Mills theory remain unclear, specifically the "mass gap" property that has been observed experimentally but not yet understood from a theoretical perspective. To address this challenge, the author presents their unique approach to solving this problem based on Mathematics with Observers, which involves removing the arithmetic infinity idea and considering Observers dependent ascending chain of embedded sets of finite decimal fractions with arithmetic operations. This innovative approach enables the development of new calculus, differential geometry, and other interpretations of vector fields and differential forms, ultimately leading to the proof of the existence of the Yang-Mills theory and the mass gap.

''