Z-Geometric Dynamics II: Geometric Entropy and High-Redshift Predictions

Building upon the Z-Geometric Dynamics (ZGD) theory, this paper introduces a geometric entropy framework (ZGE) to unify cosmic and quantum scales. Starting from three physical intuitions—the electron as a self-interfering mode on a 2-dimensional world-surface, the speed of light as an emergent quantity rather than a fundamental constant, and the "brick" analogy distinguishing gravity from electromagnetism—we postulate the geometric entropy axiom: defining the cosmic quantum number N≡Rp/lP, and asserting N=e^ (1/α), where Rp=4. 4×10²6 m is the present particle horizon radius, lP=√ (ℏG/c³) the Planck length, and α a dimensionless geometric entropy constant. This axiom relates the largest cosmic scale to the smallest quantum scale via an exponential relation and reveals the geometric origin of the speed of light: c is not an independent constant but emerges from the interplay of the cosmic scale, the Planck scale, and the geometric entropy α. Combining the ZGD rotation velocity formula v= (GMc²/Rp) ^ (1/4) with the redshift evolution of the particle horizon (in the matter-dominated approximation, Rp (z) ∝ (1+z) ^ (-3/2) ), we predict that the rotation velocity of galaxies grows with redshift as v (z) ∝ (1+z) ^ (3/8) ; at z=10 the velocity is about 2. 4 times larger than today. This prediction is testable with JWST and ALMA within the next few years. Remarkably, the geometric entropy constant computed from the axiom, α≈0. 00707, is remarkably close to the fine-structure constant αEM≈0. 007297, hinting at a profound geometric kinship between gravity and electromagnetism. ZGE and ZGD are naturally linked through the shared particle horizon Rp, offering a unified geometric perspective on dark matter, dark energy, and the origin of the speed of light.