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There are no inputs needed to resolve these expressions.

• n_L, n_M and n_T are physically significant discrete counts of l_f, m_f and t_f respectively.
• n_Lr describes the count of l_f representative of the position of an observable with respect to the frame of a center of mass.
• n_Lo is the count of l_f observed with respect to the observer’s frame of reference.
• n_Ll is the count of l_f measured with respect to the observer’s frame of reference.

Before we can discus spacetime curvature, we review the physical significance of the fundamental measures. Without this understanding, it is not possible to discuss what curved space means. Notably, the notion of fundamental measures already exists, the most notable of these are the Planck Units. But Planck Units are not precise predictors of measured phenomena nor are they physically instantiated and derived. Rather, they are an organization of the physical constants c, G and hbar so as to expose each of the three dimensions singularly. For these reasons, the Planck Units have never enjoyed classification as physically significant descriptions of observed phenomena.

Conversely, the fundamental measures are derived from physically correlated measures, that is, from first principles. Using the Measurement Quantization (MQ) expression for gravitational curvature along with expressions for Heisenberg's uncertainty principle (specifically set to the Planck scale bound), escape velocity and the speed of light, the properties of measure with respect to the Internal Frame - discreteness and countability - are resolved. They are, in this way, more precise versions of the Planck Units, improved to account for a length contraction effect associated with the discrete Internal Frame of the universe. And with that we have a physically instantiated first principles approach to resolve new properties of measure.

Those properties are, that measure is discrete and countable with respect to the Internal Frame of the universe, that place where observers reside. Using the expressions for discrete gravity we can demonstrate that the notion of measure is a function of references. And as such, a better description of measure is as counts n_L, n_M, and/or n_T of the fundamental references l_f, m_f, and/or t_f. Expressions for these reference values can be resolved known only the radial rate of expansion of the sphere of information which defines our universe.

Importantly, discrete measure carries with it length contraction, an MQ feature of length only. We can physically assess length contraction with respect to existing CODATA measurements, therein providing one means of physically assessing the discreteness of measure. And therein we also demonstrate that measure with respect to the Internal Frame is discrete. Importantly, as the System Frame of the universe has no external reference, we recognize that measure with respect to the System Frame is non-discrete. And finally, considering the difference between these frames allows us to resolve expressions and values for the physical constants.

We should clarify, these descriptions present that measure within the Internal Frame is discrete and count bound. By example, a description of observable mass shows that the observation of mass per increment of elapsed time is limited to a count of the Planck frequency per increment of elapsed time. This doesn't that the observation of mass has an upper bound over elapsed time. But it does mean that the effects of gravity are bounded and cannot present an effective mass in excess of the mass frequency bound. When combined with an MQ description of expansion, these two effects resolve an expression for the motion of stars about a galactic core. By example, the standard deviation from Stacey McGaugh's MOND data for the first 85,000 lightyears of the Milky Way show a 1.394 km/s standard deviation.

With this and many other avenues of physical support, if the notion of measure is better described as a count of fundamental references, then we ask how could space be curved? The most significant feature of a reference is that it has no additional features. By example, if a length reference carried with it a physically significant property of curvature, then by definition of curvature, the reference would not be the smallest length feature of the reference phenomenon and therein there would exist a smaller length reference. But, as demonstrated in the introductory paper entitled Measurement Quantization, the fundamental measure does describe the smallest length reference having physical significance. It follows that space cannot be curved and more explicitly, the notion of curvature cannot be of physical significance to an observer.

One might propose, can't we consider the notion of curvature with respect to the non-discrete System Frame of the universe? In that phenomena measured relative to the System Frame have significance non-discretely, one could argue that space could carry with it the property of curvature.

Granted, the System Frame of the universe is non-discrete and as such it could be argued that there is a physically significant feature of the universe that demonstrates curvature. Such an argument might follow such that: because the physical constants derive from the difference between the discrete and non-discrete frames, if the non-discrete frame had curavture, then that feature would be reflected in the frame of the observer as an emergent property of the universe.

But we remind the reader that the notion of curvature was envisioned to better describe the motion of a phenomenon near a gravitational mass. As that motion is described by the difference between the discrete and non-discrete frames, we no longer benefit from envisioning a property of curvature with respect to the construct of space.