AbstractThis thesis establishes the definitive unified mathematical framework for the Hyper-Matrix field.We move beyond the “Crisis of Distinction” by modeling the number field as a π-centered periodicmanifold [0, 2π] anchored at the invariant Mean Center (Mn = 0.5). We derive the Invariant SystemRadius (rbase) from the H¨older Mean convergence of ϕ and √e and define the n-th Order RadicalEngine (Rn). Through Relativistic Wave Dynamics, we prove that prime numbers are stasis nodeswhere the manifold’s velocity vanishes (vef f → 0) relative to the Mean Center. Finally, we resolve theRiemann Hypothesis as the mandatory stability condition for Asymptotic Phase Freezing required toprevent structural collapse at the manifold’s infinity axes. [1-5]