Toward a MOND-Like Solution to the Galaxy Dark Matter Problem from Minimum Unruh Temperature and Geometric-Mean CMB Temperature

A recurring geometric-mean structure appears in several horizon-thermodynamic approaches to cosmology. In particular, recent work has proposed that the cosmic microwave background temperature may be expressible as a geometric mean between the minimum and maximum Hawking temperatures available in the Hubble sphere, $T_{\rm cmb}=\sqrt{T_{\min}T_{\max}}$, or, in a closely related formulation, as the geometric mean of minimum and maximum Unruh temperatures, $T_{\rm cmb}=\sqrt{T_{U,\min}T_{U,\max}}$. Since the Unruh temperature is linear in acceleration, $T_U=\frac{\hbar a}{2\pi k_B c}$, a geometric mean in Unruh-temperature space is automatically a geometric mean in acceleration space. This paper explores whether the deep-galaxy acceleration relation $g_{\rm obs}\simeq \sqrt{g_N g_{\min}}$ can be interpreted as the gravitational analogue of the same geometric-mean horizon principle. Here \(g_N=GM/r^2\) is the Newtonian acceleration generated by baryonic matter, and \(g_{\min}\) is a universal minimum or horizon acceleration scale. The resulting low-acceleration law immediately yields $v^4=GMg_{\min}$, which has the baryonic Tully--Fisher form. This approach does not begin with a modified velocity law; instead, the geometric-mean acceleration is taken as the fundamental low-acceleration postulate.