Dear Branko,
I think there is both truth and mystery in your essay.
You start with the notion of a part and the Whole which can be expressed as a ratio. This idea is fundamental to Mach's principle, as you mentioned in another article. You choose a logarithmic expression for the ratio. This is needed to see the Planck scale as a centre for logarithmic symmetry through the geometric mean.
Next you take three well-known physical constants - the mass of the proton, and the product of two related dimensionless quantities: the fine structure constant and the proton to electron mass ratio. From this pair of highly accurate physical data, some reasoning and some basic physical and mathematical formulas, you produce highly accurate estimates of unrelated physical constants, perhaps even better than observation - the gravitation constant being one possible example.
Dimensional analysis can sometimes make significant predictions, but there would not seem much to start with here. Interesting.
I was wondering about the accuracy of any physical constant. Depending on the type of measurement, I wonder if the result of a measurement performed on Earth might be influenced by Earth's gravitational field. A first order effect would be about 10^(-9), which could possibly limit the accuracy of some physical constants to nine significant digits. This should not be a problem with your dimensionless constants.
I was really impressed by the correspondence between our quantum harmonic oscillators.
If this is a trick, it is a good one.
Best regards,
Colin