The mathematical foundations of Unity Root Matrix Theory (URMT) are presented in the book by way of six, progressive research papers:

1. Unity Root Matrix Theory Foundations
2. Pythagorean Triples as Eigenvectors and Related Invariants
3. Geometric and Physical Aspects of Pythagorean Triples as Eigenvectors
4. Solving Unity Root Matrix Theory
5. Unifying Concepts in Pythagorean Triples as Eigenvectors
6. A General, Non-unity Eigenvalue in Unity Root Matrix Theory

Unity root matrix theory is the study of a special type of integer, unity root matrix arising from the application of a physical, invariance principle to an abstract dynamical conservation equation. The theory has many similarities to concepts in modern, theoretical physics, such as transformation invariance, symmetry, duality and geometric structure. It is also formulated entirely in integers and therefore has a natural quantisation built in from the beginning.

Simplifications reduce the theory to quadratic, Pythagorean and hyperbolic conservation equations with numerous integer invariants. Not least, Pythagorean triples appear in abundance and show the subject of Pythagoras is far from exhausted, continuing to provide new insights.

Developed on surprisingly simple but fundamental concepts, it provides a rich mathematical and physical structure, justifying it as a subject to be studied in its own right by physicists and mathematicians alike.

Ultimately, it is thought that unity root matrix theory may provide an alternative reformulation of some fundamental concepts in physics and an integer-based escape from the current, unification impasse.