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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 the upper count bound to mass per increment of elapsed time t_f, 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 the physical constants, G, M and R, but as these values are difficult to measure today, we use the modeled velocity v_m as a proxy for demonstration of the relation between bound mass and orbital velocity. The solution occurs only where the modeled and effective velocities are equal v_m=v_e. For the Milky Way, this produces an effective velocity curve that matches the modeled velocity having a 1.394 km/s standard deviation.

Firstly, we clarify what is meant when we use the term frequency when describing comparisons of measure for length, mass, and time. Specifically, we recognize not the distance, 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 of fundamental units of measure, where those fundamental measures are resolved as an emergent property of the expanding sphere of information identifying the universe.

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 significant frequency in modern theory. The relation is typically measured in the context of the distance traveled by light in vacuum relative to elapsed time. Another relation of importance is the count of a reference mass n_M when compared to a count of elapsed time n_T. We call the upper count bound of n_M/n_T the mass frequency. The inverse count n_T of t_f is known as 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 identify as the length-to-mass frequency.

While the length frequency bound is a well understood phenomenon, little research has been devoted to the mass frequency bound. 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 a discrete unit of fundamental 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 that which 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 are the upper counts of length and mass with respect to time when 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 in magnitude. 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'.

Remember, relativity describes the measure of length, mass, and time as a function of the observer. MQ recognizes this effect and it is applicable when describing an observation between two frames within the Internal Frame of the universe. Conversely, the fundamental measures are an emergent property of the expanding Internal Frame of the universe. Such that it is shown that the radial rate of expansion is constant, the fundamental measures are also constant, references which define fixed values in the local frame of all observers. It is important to avoid confusing the fundamental measures - which are emergent properties of the universe - with relative measure between frames internal to the universe (i.e., between you and me).