Planck Meets Hubble within Cosmological Natural Units and Frequency-Dependent Yukawa Potentials Bridging Gravity, QCD, and Nuclear Scales

We extend the classical $cG\hbar$ natural units by incorporating the present-day Hubble constant $H_0$, yielding Planck-Hubble (PH) units that connect quantum-gravitational and cosmological scales. These hybrid units exhibit a striking numerical alignment with quantum chromodynamics (QCD) characteristic values ($\sim$ fm lengths, $\sim$100--300 MeV energies). We derive explicit formulas for base and derived quantities and propose an empirical relation tying the gravitational constant $G$ to QCD mass scales modulated by cosmic expansion.

Building on a related phenomenological framework \cite{papou2026b}, we discuss how frequency-dependent Yukawa potentials mediated by a scalar field may provide a dynamical mechanism for unifying long-range gravity with short-range residual nuclear interactions, potentially explaining the PH--QCD coincidence and offering insights into emergent gravity or scale-dependent couplings. Numerical comparisons use CODATA 2018 and Planck 2018 data, with implications for the Hubble tension explored.