The past 30 years have witnessed a number of attempts to develop theories based on quaternion and octonion hypernumber coordinates. In the past three years this author has developed a Cosmos Theory that combines hypernumbers with Quantum Field Theory and General Relativity (GR) in intricate detail that uncovered many hitherto unknown features of spaces. The spaces were based on 10 Cayley numbers that include quaternions, octonions, sedenions, and so on The Cosmos Theory that resulted goes far beyond qualitative "ideas." The essence of the approach lies in the creation and annihilation operators appearing in Fourier expansions of new PseudoFermion wave functions that are made into Cayley number operators.

These operators map to arrays of dimensions that become the space-time dimensions, and the Internal Symmetries representation dimensions, of the ten spaces. This book develops a new PseudoFermion wave function formalism, which uses a two coordinate systems Fourier field expansions to generate the dimension array of each space Geometrically - including the dimensions of Internal Symmetries.

This formalism is the basis of an SU(8) Geometric ProtoCosmos, broken to SU(4) subgroups, from which all things arise. PseudoQuantum Theory is generalized to a four field formalism. The SU(8) formalism leads to a U(1) HyperUnification space Geometric ProtoCosmos.

Each space, with space-time dimensions ranging from zero to 18, thus becomes a geometric construct. The book explores the PseudoFermion construction of the 10 spaces. Its detailed study reveals unforeseen consequences. The derived dimension arrays furnish coordinates for space-times and internal symmetries. They also map on a one-to-one basis with the known, and anticipated, fermion particle spectrum - a "Mendeleyev-like periodic table."

Cosmos Theory has a number of related consequences: a parton substructure, Regge-like trajectories of spaces, a Veneziano formulation, and now a new Fibonacci-like structuring exemplifying a form of space growth like plants, a new Ramsey Theory approximation yielding projections for high Ramsey Numbers beyond current computation. (Ramsey Theory is important for fundamental Physics and Mathematics.)

One previously unforeseen consequence of Cosmos Theory is the appearance of a HyperUnification space for each of the ten spaces. These spaces enable the full unification of space-time and internal symmetries in each of the ten spaces through solely GR transformations. These transformations also govern fermion wave functions. They were explored in the author's 1970s peer-reviewed, published papers.

Another consequence is a further unification of all ten spaces and their HyperUnification spaces in a 42 space-time dimension space: Full HyperUnification Space. This space opens the door to interspace communication (and possibly ultimately to travel) given suitable interactions.

This book focuses on the Geometric origin of spaces, and their arrays of dimensions, from the creation and annihilation operators of a set of four PseudoFermion wave functions in a PseudoQuantum formulation. Cosmos Theory leads to the Unified SuperStandard Theory (UST), which includes the Standard Model of Elementary Particles

Cosmos Theory's Fibonacci-like spaces structure and its Ramsey number approximations are presented with noteworthy consequences. A major leap in Cosmos Theory!