Abstract This thesis establishes the Hyper-Matrix as a self-correcting harmonic engine where the distribution of any complex number system is governed by a dynamic barycenter at 0.5. We define “Stretching” (space acceleration) as the linear growth flux of a generalized Radical Engine and “Pulling” (gravity) as the logarithmic dampening required for structural closure. By analyzing the Stability Functional H(r), we identify specific rhythmic checkpoints at π, 3π/2, and −π/2 that maintain the manifold’s integrity across a full-wave bipolar reset. We conclude that all integers are functionally prime relative to their local mean center, and that gravity and acceleration are the two halves of a singular numeric respiration.