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• θ_si can be measured as the polarization angle of quantum entangled X-rays at the degenerate frequency of a maximal Bell state. As an angle θ_si=3.26239 rad ± 2 μrad; as a momentum θ_si=3.26239030392(48) kg m s^-1 and with respect to the Target Frame, θ_si has no units. The relation of angle and mass is mathematically demonstrated, as well, by No-Ping Chen, et. al.

• n_L, n_M and n_T are physically significant discrete counts of l_f, m_f and t_f respectively.
• l_f, m_f and t_f are the fundamental measures, more precise expressions for Planck’s units – length, mass, and time – that consider the effects of length contraction associated with discrete measure.

The most direct confirmation of count bounds would be a description of the effective mass of a galaxy. In that the mass of most galaxies exceeds that upper count bound of mass, we find gravity constrained, proportionate to the bound mass, a function of the mass density profile along the orbital path of a star.

We can use Newton's expression to replace the modeled velocity v_m with a classical set of terms, G, M and R, but as values are difficult to present in classical form, we use the modeled velocity v_m as a proxy. The solution occurs only where v_m=v_e such that v_m describes the path of least energy. For the Milky Way, this produces an effective velocity curve that matches the modeled velocity with a 1.394 km/s standard deviation.

Importantly, the expression is valid only so long as v_m describes the least energy orbital path. Had we used the corresponding terms - G, M and R - the expression would be valid only where the path of least energy is resolved at each increment in elapsed time t_f.

Firstly, we clarify what is meant when we use the term frequency to describe the notions of length, mass, and time. Specifically, we recognize not the dimension, but the count of the corresponding reference measure with respect to the count of another reference measure. This is to say that Measurement Quantization (MQ) does not approach the description of phenomena as a function of measure (i.e. SI Units), but as a description of counts.

Thus, length frequency - the count of the reference measure for length with respect to a count of the reference measure for time n_L/n_T=1 is such a well-known count bound that it is perhaps the most overlooked and significant frequency in modern theory. The relation is typically measured in the context of the distance traveled by light in vacuum with respect to elapsed time. Another relation of importance is the count of a reference mass n_M when compared to a count of a reference time n_T. We call the upper count bound of n_M/n_T the mass frequency. The count n_T of t_f inversed is the frequency, also known as the Planck frequency. And finally the upper count bound of l_f with respect to a count of m_f we call the length-to-mass frequency.

While length frequency is a well understood phenomenon, little research has been devoted to mass frequency. Its significance can be elusive, but in the study of Informativity - that field of science in which MQ is applied to the description of physical phenomena - we refine its description as a physically significant upper bound to the count of discrete units of mass with respect to time. The effect is commonly referred to as the dark matter phenomenon. It isn't that the effects of a mass of M=n_Mm_f experienced by a star do not vary in count respective of the total, but rather the ability for an observer (or a star) to physically distinguish a count of fundamental units of mass in excess of the upper count bound to the mass frequency is not possible. It is no more possible than the ability of that same observer to observe a count of length measures in excess of the upper count bound to the length frequency (i.e. a phenomenon moving faster than the speed of light). And, as such, when expressed as a function of universal expansion and accounting for variations in mass distribution, then the velocity of stars is described.

In contrast, we mention briefly that count frequencies are not what divide each of the mass/energy distributions we observe, for instance, with respect to a CMB power spectrum. Using the MQ approach to describe spacetime, we find these distributions better described as that which can never be observed due to the metric expansion of space (dark mass), that which will or has been observed (observable mass) and what is presently visible (visible mass). The dark matter distribution is the observable minus the visible. Different from ΛCDM, MQ descriptions require the presence of only one measured value, θ_si. For greater precision, this value can be resolved as a measure of the fine structure constant.

The last quality of measurement frequency is unique to the MQ approach. Specifically, we bring to your attention that the upper counts of length and mass with respect to time are equal. This can be very useful in some expressions as substitution from one count to another is possible where the associated value is known to be equal. By example, we can have an expression such as n_Lt_f which we are unable to reduce or consolidate. Because the magnitude of n_L=n_T, we can make a substitution and then reduce to the dimension of time, t=n_Tt_f. That is, the substitution of a value equivalent count for the purposes of comparison with respect to another count can be carried out without issue, enabling us to draw comparisons with a dimension not specifically described at the outset. We might call this the 'principle of quantized equivalence'.