This paper presents a fully discrete mathematical framework in which the universe is constructed from integer quaternions. Using relative primality relations among quaternion components, a Particle Type Function is defined, encoding particle identities and empty space. From this function, a hierarchy of higher dimensional representations named T4, T5, T5e, T6, and T6e is introduced. These representations successively encode translation invariance, directional structure, locality, and probabilistic field perception.A wave function concept emerges naturally from preserved arithmetic distributions rather than from imposed dynamical equations. Before collapse, probability is distributed globally across all possible origins and directions. Collapse is interpreted as a redistribution of probability rather than the selection of a single origin or reference frame. All physical fields are unified and interacting because they are all defined from the same underlying Particle Type Function. This framework proposes a discrete arithmetic substrate from which observed physical field behavior emerge.