This third book in the series on Unity Root Matrix Theory (URMT) advances the subject into mainstream physics by detailing how it relates to such topics as the classical harmonic oscillator, The Special Theory of Relativity and some related cosmology. The book starts by extending URMT's mathematical methods to handle arbitrary real and complex vectors, and then proceeds to show how oscillators and Special Relativity can be formulated in the language of URMT. Among the results is the embodiment of Einstein's relativistic energy-momentum equation in a 5D formulation, with mass emergent from a scalar potential. There are also some cosmological implications stemming from a relativistic Doppler solution, notably the Hubble expansion law - all quite an achievement given URMT's origins in number theory and Diophantine equations. Additionally, using URMT's unique variational methods, a 4D formulation naturally produces a quadratic, harmonic potential, with a consequent solution for the harmonic oscillator. Other topics include Lorentz transformations and some mechanics. The book finishes by showing how these real and complex formulations can be recast in integers, i.e. a return to URMT's integer foundations.
This book marks a significant advance in the practical applications of URMT, and is subtitled Volume I in the knowledge that more URMT physics lies ahead.