—————————— read pre-print on research gate (new window) ——————————

                    TERMS         INPUTS         SYMBOLS         CONDORDANCE ANALYSIS        

The fundamental expression can be resolved with respect to the MQ expressions which describe the fundamental measures, the latter describing more precise versions of Planck's unit expressions.  The fundamental expression describes the simplest relation between the three dimensions. With respect to the Internal Frame of the universe, the expression is also known as the Planck momentum.

In principle, the fundamental expression is a simplification of the three fundamental measure expressions. Importantly, the fundamental measures differ from those of Planck's unit expressions. The fundamental measures are more precise, accounting for a length contraction effect associated with discrete measure associated with the Internal Frame of the universe. Structurally, both the fundamental measures and the fundamental expression do not contain the reduced Planck constant. 

Notably a solution to the fundamental expression cannot be solved when using existing classical expressions. The reason for this, from a general point of view, is that expressions in classical mechanics are drafted with respect to the discrete Internal Frame of the universe. And in practice, this is why Planck was unable to resolve a physically instantiated derivation of the Planck unit expressions as a function of more fundamental terms.

MQ succeeds with the introduction of the non-discrete System Frame of the universe. Using the Pythagorean theorem to describe the relation between the discrete Reference and Internal Frames and non-discrete System Frame, we offer a new description of gravitational curvature. This is used to resolve an expression for G in terms of the fundamental measures. And along with the expression for the speed of light and that of Heisenberg's uncertainty principle, we have all the component features needed to derive the fundamental expression and each of the fundamental measures without self-referencing source measures.

We call expressions written in terms of the fundamental measures example of the Measurement Quantization (MQ) approach. Notably, it is the associated counts of these fundamental measures that is most significant. Considering the Planck scale bound, a reduction of the uncertainty principle demonstrates that all measure terms cancel leaving only the counts. That is, the bound identifies a geometry of the expanding frame of the universe. While this does not necesarrily establish that all measure is discrete, it sets forth a physically significant means to derive the fundamental measures. And with that we can then derive the physical constants. By example, the inverse fine structure constant is shown to be the difference between the discrete and non-discrete product of 42θ_si. And where the physical constants are properly derived as the difference between these two frames we establish a framework of understanding whereby a discrete Internal Frame and a non-discrete System Frame is a physically significant construct of our universe.

For this reason, we identify comparisons of dimensions as a function of their associated counts, that being, their:

• length frequency c=l_f/t_f
• mass frequency m_f/t_f
• frequency 1/t_f
• length-to-mass frequency l_f/m_f.

Yes, we will sometimes describe frequency as a function of the associated measure terms, but the use of the term 'frequency' is specifically a description of the associated counts for each reference dimension.

Most physical constants can be shown to be a composite of the fundamental measures. The fundamental measures include the fundamental constantθ_si. Example physical constants written using only the fundamental measures include:

• G=(l_f/t_f) (l_f/t_f) (l_f/t_f) (t_f/m _f)
• ħ=2θ_sil_f
• H_U=2θ_si.

The fundamental expression cannot be reduced further as it consists entirely of references. That said, there are circumstances whereby all terms reduce to a logical unity (i.e., 1=1). This is often a result of a System Frame definition, as in such cases one is working only with counts.