Steve Dufourny
I must develop the idea and explain how I found this number 1.583 with the logarithms and the mass and radius of real physical quantum and cosmological objects. The number β ≈ 1.583 is a dimensionless constant that emerges from the relationship between mass and radius of all spherical objects, from quantum scales to cosmological scales. It appears as the product of logarithms of radius and mass ratios of spherical objects, depending on how we define the logarithmic combination.
It is a Mass–Radius Power Law, so we consider, for quantum, gravitational, and cosmological objects, several parameters for the slope exponent, and it depends on physical forces across the scales. That is why it must be calculated as a function of the scale of analysis for the spherical objects, using their radii and mass. It is there that the logarithmic compression is important, with some adaptations, but here is the formula:
log R = k log M + C.
Maybe I must improve it, I don’t know, but we have many orders of magnitude. The ratio of logs gives a dimensionless slope, and the references can be chosen. Real data confirmation appears in the correct calculations and references when the ratios are computed.
β for all these spherical objects seems to be a number consistently appearing around 1.583. I believe that it supports my theory of spherisation about the necessity of having stable spherical objects; they are obliged to obey constraints of stability. That is why even the forces must be balanced. So it must act at all scales, for all spherical objects.
Logarithms compress multiplicative differences in mass and radius into a scale-independent number. This creates an emergent universal slope β, independent of scale in fact.
It is dimensionless, and this is important, and it reflects the hierarchical spherical structure of matter from nucleons to the universe. I don’t know how to consider the ranking of all spherical systems when we consider these radii and masses and their evolution in log space, and with the densities and the evolution, and even if we add the information aspects.
It is not a constant like G, c, or ħ, so it is not yet a physical constant, but it is interesting to correlate with my theory of spherisation and with possible derivations from first principles.
For any spherical object:
β = log(R_object / R_ref) / log(M_object / M_ref) ≈ 1.583
or equivalently, for predictive purposes:
R_object = R_ref (M_object / M_ref)β
It seems it can work for protons, electrons, nuclei, planets, stars, black holes, galaxies, clusters, halos, and the Hubble sphere.
The spherical stability and this universal scaling law with this number seem relevant and empirically robust across scales, linking quantum to cosmology. It reflects a hidden hierarchy of the universe, compressible via logarithms.
I must work on spherical topological geometric algebras and try to make derivations from these first principles and correlate them with the theory of spherisation and with the evolution of the universe. Quantum gravitation and the primary information are also relevant to analyze because information seems essential, but where is this primary information? I have a mass–energy–information equivalence, but do we have, as I explained, DE as informational,a fifth antigravitational force encoding photons and DM to create ordinary matter? How should we consider the negentropic stabilities in this case and the evolution? Or do we have information in GR, or in strings in 1D at the Planck scale connected with a 1D cosmic field of GR, or the same logic with geometrodynamics? I don’t know, and I don’t assert. But there is a logic.
But returning to the dimensionless ratio and the mass and radii across all scales in a spherical logic, this can reveal hidden structures in my humble opinion. It is the orders of magnitude here that are interesting, and the spherical logic. This invariant slope is intriguing; it is like a dimensionless constant in fact. There is also a relevant link with the evolution and observations and measurements and changes in densities across scales. The spherical stability conditions are relevant. We have paths to analyze a deeper kind of universal, not yet explained pattern.
My theory tells that nature prefers spherical stable systems, so they are geometrically natural and permit the physics and properties, and others like quantum confinement, gravitational equilibrium, and scaling relations. So even mathematically they are important; that is why the Poincaré conjecture proved by Perelman is also important, I think. In all cases, the empirical observations, the measured values of mass and radii, the fact that the ratio is dimensionless, that it is reproducible, revealing a new pattern,all this has the potential for deeper theoretical implications and even unification of spherical objects, supporting my theory of spherisation.
