Geometric Origin of Dark Matter: The Cosmological Quantization Condition and Resolution of the Hubble Tension

In this paper we consider an effective cosmological model in which the FriedmannRobertson-Walker (FRW) metric of general relativity (GR) is kept intact, while onlythe time evolution of the scale factor a(t) is given by a Newton-Mach (NM) type ordinary differential equation. As starting assumptions we impose three requirements:(i) Newtonian mechanics, (ii) a Mach-type relation c^2 = GM(r)/r interpreted as a holographic saturation condition, and (iii) the local equivalence principle in an FRWbackground. Combining these with the Planck force F_P = c^4/G and the dynamics of the physical radius r(t) = a(t)χ_0 of a representative comoving sphere χ = χ_0,we derive the NM equation ˙r^2 = c^2[1 − 2 ln(r/r0)]. A crucial finding of this paper is the introduction of a phenomenological ”Cosmological Quantization Condition”linking the macroscopic cosmic expansion to the microscopic electromagnetic finestructure constant α_EM . By calibrating our model with local measurements by theSH0ES collaboration (H0 ≃ 73.04 km s^−1 Mpc^−1), we empirically determine the gravitational fine-structure parameter to be α ≈ 16.67. We demonstrate that thisparameter suggests the existence of N ≈ 727 effective holographic degrees of freedom. Furthermore, the model explicitly identifies the origin of Dark Matter as a relic of pre-Planckian geometric energy, naturally unifying it with dark energy without requiring a cosmological constant Λ. Finally, supported by recent advances in quantum holography and Absolutely Maximally Entangled (AME) states, we suggest that this framework resolves the horizon problem without inflation and provides a geometric solution to the Hawking singularity and information loss paradoxes.