An apparent relation between the Newtonian gravitational constant and the Planck constant in a geometry-based energetic expression derived from a hypothetical system

The hypothetical system considered here comprises only the so-called internal energy, without free energy. Thus, in the canonical form, the partition function Z of the system has a unity value. As a first further specification, the system, in terms of energy distribution, exists in two states with the amounts of energy E₁ and E₂ = 2E₁. The mean energy of the system may thus be expressed as a weighted average of E₁ and E₂, i.e., p₁E₁ + p₂E₂.  Given E₂ = 2E₁ and Z = 1, the state probabilities p₁ = 1/phi and p₂ = 1/phi^2 where the geometric-ratio-constant phi = [1+5^(1/2)]/2. Because this condition corresponds to an asymmetric partition of a probability space into two portions with a ratio of phi, a thermodynamic solution of the expression for the mean energy is given by (-1/beta)[(1/phi)ln(1/phi)+(1/phi^2)ln(1/phi^2)] = (1/beta)(1+1/phi^2)ln(phi). A hypothetical example of such portioned systems is described in the main text with more details. As a second further specification, the amount of mean energy in the system is defined as the average of comparable amounts of gravitational energy and photon energy. The relative portions of these two types of energy have the constant near-unity ratio that is the same as the ratio between the significant digits of the physical constants G and h. Under the condition that the amounts of energy specified in these two ways are equal at the presently testable precision, the following relation is reached: G = (1 + 1/phi^2)ln(phi^2)*10^-10 kg^-1 m^3 s^-2 - h*10^23 kg^-2 m s^-1, in which G is related to h by the mathematical constants 1, 2, e and phi, along with the necessary physical units and their associated exponential scalars. A calculation of G using this expression to the sixth significant digit from h of an exact value yields 6.67430*10^-11 kg^-1 m^3 s^-2, which matches the accepted G that was recommended by CODATA in 2018. This calculation is readily verifiable. It will be interesting to learn the limit of precision at which the above apparent equality remains valid, or starts to break down, when an independently reproducible experimental estimate of G with more significant digits becomes available in the future.