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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.

• 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.
• Q_L is the fractional portion of a count of l_f when engaging in a more precise calculation.
• Q_Ln_Lr, also known as the Informativity differential describes the length contraction associated with discrete measure.

Over the past 100 years frames of reference have become central to precisely resolving the measure of phenomena. Two frames are recognized, that associated with the observer and that with the phenomenon. A third frame is implicit. Relations are anchored in relation to the speed of light.

Measurement Quantization (MQ) explicitly introduces the third frame using a shared nomenclature for all three. Without, formulating physically correlated expressions can be difficult and often not possible.

Before we begin, though, we briefly review what MQ is. MQ is a nomenclature applied to existing expressions in modern theory. Terms that make up existing classical expressions are broken down into their fundamental measures - l_f, m_f and t_f - and multiplied by counts of those measures - n_L, n_M and n_T. For example, when applying an MQ nomenclature to an example expression for Heisenberg's uncertainty principle we find that all the measure terms cancel out leaving only the counts. With this observation along with the expressions for the speed of light and escape velicity and the solution for discrete gravity, we are able to resolve the values of the count terms associated with the fundamental measures. We are also able to resolve the properties of measure: discreteness, countability and in reference to three frames of reference.

With respect to the inertial frame of the observer, the traditional two frame approach is all that is needed to describe phenomena. But, when resolving expressions that describe universal expansion, the age of the universe, its diameter and more complex phenomena, for instance properties of the Cosmic Microwave Background, we find that the frame of the universe as a system of reference is important. We will consider Hubble's constant as just one example, a measure of expansion in units of km s^-1 Mpc^-1.

Notably, the reference distance, the megaparsec, is somewhat arbitrary. Moreover, as an arbitrary unit of measure, it is unconstrained with respect to the system we are describing (the universe). Lacking a fixed physical relation to the universe mitigates our ability to recognize important relationships.

To correct this practice, we define the rate of expansion with respect to the universe such that H_U=2θ_si.

Several properties of our universe are then more apparent. For one, we discover that the rate of expansion is constant. We also find that the rate of expansion is correlated to the three measures: l_fm_f=2θ_sit_f. And we find that θ_si is the only constant necessary to resolve each of the mass/energy distributions presently described by ΛCDM. We also find that θ_si describes gravitational curvature with quantum precision throughout the entire measurement domain. That is to say, physical support for θ_si as a physically significant feature of our universe covers nearly every discipline from gravity, optics, quantum mechanics to cosmology.

To better describe phenomena, we have enhanced the existing nomenclature with additional terminology. For one, the upper count relation demonstrated by the phenomenon of light, we call length frequency, the upper count bound of fundamental units of length with respect to a count of fundamental units of time l_f/t_f. Equally important is mass frequency, the upper count of fundamental units of mass per fundamental unit of time m_f/t_f. And finally there is the length-to-mass frequency, l_f/m_f. Each are physically significant, for example mass frequency is essential to resolving an expression for the orbital velocity of stars in a galaxy.

For our present purposes, we note that these expressions are defined with respect to the universe. In MQ form, the universe is considered a system and allows for expressions that describe properties of the universe with respect to an observer outside or inside the universe. For instance, those expressions that describe the universe are the same as those that describe a fundamental unit of mass m_f. Expressions describing fundamental mass are from the perspective of an external frame and expressions describing the universe are from the perspective of an internal frame.

Finally, we will address the expansion parameter 2θ_si briefly and its role with respect to the third frame. We can, for instance reformulate the fundamental expression into a form known as a unity expression^{(2,Eq. 102)}. In this form, we can visualize the expansion of the universe as a length count change assoicated with universal expansion n_L² relative to the upper count bound associated with the speed of light n_Lc². You may also recognize this as the speed parameter v²/c² common in many relativity expressions.

You will notice that the expansion parameter θ_si appears nondimensionalized in the above expression (i.e. carries no units). Up to this point θ_si has carried the units of momentum. That is the applicable dimension with respect to most expressions. We call such expressions self-referencing in that such expressions describe phenomena relative to the Measurement Frame of the observer.

But, for phenomena that are defined with respect to the Target Frame of the universe (self-defining) as is the case in the above unity expression, θ_si has no units. That is to say, it is dimensionless. Recognition of this trait is important in MQ and central to the concept of references. Specifically, the self-referencing expressions we use to describe nearly all interactions between an observer and a phenomenon are dimensional. In contrast, the universe has no external reference and as such phenomena defined relative to the Target Frame are non-discrete.

Finally, we bring to your attention that the constants and laws of nature are best described as the difference between the Target and Measurement frames. Gravity offers one example of the mathematical approach to resolving this difference. The fundamental constant θ_si (which is radian with respect to the measure of entangled photons), the gravitaitonal constant, Planck's reduced constant and H_U are all examples of values resolved as a function of this difference.

In that θ_si is central to frames of reference we lastly note that the value can be measured with respect to the polarization of X-rays in specific Bell states and at their degenerate frequency. Those experiments were carried out and published by Shwartz and Harris in their 2011 paper, 'Polarization Entangled Photons at X-Ray Energies', the results matching the predictions of MQ to the same precision.

Notably, we have used the Informativity differential distance adjusted value for the reduced Planck constant in the last of these expressions, which is important when describing a macroscopic phenomenon with a quantum measured value such as ħ. Alternately, as described in the table above, we can also resolve θ_si as a function of the macroscopically measured value G without needing to apply the Informativity differential. Further reading on the importance of the Informativity differential and the physical support for this effect are provided in the same named discussion as well as that regarding the fundamental measures, among several other notable experiments where accounting for this effect resolves calculation differences with up to thirteen digits of correspondence.