The Matrix — Parameters and Dynamics: Foundation of CPQ Theory

This book describes the matrix — the elastic substrate from which space, time, and all physical phenomena emerge in the CPQ framework. Unlike most modern physical theories, which describe how things behave without asking what they are made of, CPQ provides an ontology: a concrete medium composed of spacetime quanta with two orthogonal deformation modes (torsion T⃗T T and volume VV V), shared saturation (Ω2≤1² 1 Ω2≤1), and topological protection of localized structures. The book derives the complete nonlinear Lagrangian of the matrix from four axioms and three free parameters, obtains the Euler–Lagrange equations of motion, and presents numerical verification of six analytical limits: linear wave propagation at exactly cc c, spontaneous genesis of ΔV V ΔV mass wells, stationary vortex profiles with topological protection, 3D Hopfion stability, impedance reflection (96% agreement with prediction), and de Broglie trail of a moving defect. This is the foundation stone of the CPQ framework. Six follow-up books — CPQ Theory (Books I–III) and CPQ Applications (Books 01–03) — derive from the matrix described here: Maxwell's and Einstein's equations, special and general relativity, quantum mechanics, particle structure, confinement, galactic dynamics without dark matter, cosmic expansion without dark energy, and the CMB spectrum. All from one medium, one Lagrangian, three free parameters. Keywords: spacetime quantum, elastic substrate, nonlinear field theory, saturation, topological defects, Hopfion, impedance, confinement, emergent spacetime, ontological foundation Related works: Porschová, A. (2026). CPQ Theory — A Unified Framework from Spacetime Quanta to Cosmology (Books I–III). Zenodo. https: //doi. org/10. 5281/zenodo. 18686137 Porschová, A. (2026). CPQ Field Theory — Applications (Books 01–03). Zenodo. https: //doi. org/10. 5281/zenodo. 19100819