The emerging, multi-disciplinary field of Complex Adaptive Systems (CAS) is an alternative to linear, reductionist thinking. It is based on the observations that real-world systems, regardless of scale, are emergent, complex, adaptive, and evolutionary. In this research the scale of CAS examined range from distances of Planck’s constant to Gigaparsecs. CAS has also heavily leveraged the interpretations of several recent Nobel Laureates and assumes too that the world is random, indeterministic, and chaotic. But randomness, chaos, and indeterminism can hardly create the progressive, increasingly harmonious world that we are a part of. At the heart of this issue lies confusion around what innovation in CAS really is. The essential approach to arriving at a mathematical basis of innovation for CAS here has been to view systems from the outside-in as opposed to from the inside-out and the bottom-up. In this approach innovation is conceptualized as existing in every single space-time point-instant in a system. There is a process of precipitation by which this innovation may express itself through a series of quaternary-based architectural forces that are the prime sources of innovation. These series or arrays of forces may further precipitate by informing organizational signatures. Organizations can be thought of as formations with a unique signature at their center, and can vary in complexity and scale. The unique signature for each organization is usually hidden though by common surface dynamics, and “to innovate” is to work through and change the habitual and common patterns in order to allow the deeper founts of innovation to become active at the surface level. When this happens, it is then that innovation occurs. Once that is more clearly seen then the erected probabilistic and uncertainty functions assumed to be true of the fundamental layers of nature, will be relegated to their place as interim devices in model building. The nature of innovation can be progressively elaborated through inductive reasoning to arrive at a mathematical framework for innovation in CAS. Rather than assume a chaotic, random, indeterministic world as a starting point, this framework can be built assuming a purposeful, ordered world characterized by qualified determinism. Equations to provide insight into the inherent innovation bias of our system, the nature of each point in the system, the broad architectural forces behind the development of organizations, the inherent uniqueness of each organization, the way to think about varying cultures or organizations, and the inherent dynamism of our system, form the edifice of this framework. The resulting model can then be used deductively to reinforce observations, and predictively to suggest directions and / or steps to emerging trends. This research hence, through deriving mathematical equations, and by further applying these to various domains ranging from the quantum, to the atomic, to the cellular, to the astrophysical, has been able to provide mathematical contributions to the theory of CAS and to various CAS application areas. With respect to the theory of CAS, mathematical contributions have been made to understanding the underlying directional bias of CAS activity, understanding the nature of each point in any CAS, and creating mathematical sets for architectural forces that are posited to be behind the development of any CAS. Further, mathematical contributions have been made to understanding the inherent dynamics in any CAS, the dynamics of stagnation and growth in CAS, and the balance of randomness and determinism of any CAS. Mathematical contributions also extend to framing complexity in CAS, understanding what can drive sustainability of CAS, and arriving at a general set of mathematical operators true of any CAS. In terms of application areas in the organizational space, mathematical contributions have been made to understanding uniqueness of organizations, the emergence of uniqueness in organizations, and what constitutes varying culture of organizations. Further, existing work done by Nobel Laureate Ilya Prigogine and Alan Turing have been leveraged to further frame organizational transitions, and to frame and model shifts in innovations, respectively. Further mathematical contributions have been made in a range of CAS areas at different scale and level of complexity. Hence, a series of equations have been derived for the electromagnetic spectrum. Quantum, atomic, and cellular wave equations have been derived building off Schrodinger’s existing Wave Equation. Further qualifications have been derived for Heisenberg’s Uncertainty Principle and an equation has been derived for the integration of different layers of CAS also using Heisenberg’s Uncertainty Principle. Equations for space and time alteration as per Einstein’s Theory of Relativity have also been derived. Additionally, equations for the architectures of quantum particles, periodic table elements, and molecular plans at the cellular level have also been derived. Finally, equations for dark matter and dark energy, non-probabilistic quantum states in quantum computing, and the emergence of CAS in the universe have been derived. In all over 225 equations in 25 different areas have been derived in this dissertation. In fact, as suggested by the CAS equation derived for a unified field, everything, from unseen energy fields, to quantum particles, to atoms, to molecules, to cells, and therefore to all animate and even inanimate and even unseen objects, and therefore even any CAS system regardless of scale would have a high-degree of quaternary intelligence embedded in it and exist simultaneously. Quoting Schrodinger: “What we observe as material bodies and forces are nothing but shapes and variations in the structure of space. Particles are just schaumkommen (appearances). The world is given to me only once, not one existing and one perceived. Subject and object are only one. The barrier between them cannot be said to have broken down as a result of recent experience in the physical sciences, for this barrier does not exist.” This implicit quaternary-based intelligence likely sheds new light on properties such as distributed control, uncertainty, paradox, co-evolution, emergence, amongst others, seen as fundamental to CAS. Thinking about CAS as purposeful, and animated by a mathematically-framed engine of innovation, allows existence to potentially be considered as a unified field. Further, it allows insight and additional solutions to a host of complex problems regardless of scale – at the quantum, cellular, human, organizational, sociotechnical, market, economical, political, and social levels - to be conceptualized, designed, elaborated, and managed differently.