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• 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.
• 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.
• n_Lc describes the count of l_f representative of a change in position of light measured with respect to the observer’s frame of reference.
• 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.
• H is the value of H0 m s-1Mpc-1 when resolved with respect to universal expansion (i.e., the Target Frame).
• H_U is the rate of universal expansion with respect to the diameter of the universe (per D_U). This differs from stellar expansion (i.e., Hubble’s description).
• M_acr is the rate of mass accretion in the universe.
• A_ee is the dilated age of the universe as measured from the current expansionary epoch. Its value is a function of expansion with respect to the total elapsed time representing the quantum epoch.
• A_qe is the non-dilated age of the universe corresponding to the quantum epoch.

We present expressions that provide support for an upper bound to the dimension of length. Therein, we offer the fundamental expression, which provides a description of the three measures in their most reduced form. That relation can be modified to isolate the notion of length relative to mass and time, thus revealing that space has at most three physically significant dimensions.

Before presenting this argument, several prerequisite claims must be established. This, we achieve with Measurement Quantization (MQ). MQ is a nomenclature, a paring of each of the three dimensions as two component terms, a fundamental measure and some count of that measure. The nomenclature was developed as a result of analysis of Heisenberg's uncertainty principle, which when applying the MQ approach allows a reduction of the expression such that all measure terms cancel out.

This analysis is crucial to unraveling the values of the count terms and then the values of the fundamental measures. We are, with this able to establish the physical significance of measure. Significant challenges in modern theory are resolved as a result of the model, for instance a better understanding of optical experiments regarding quantum entanglement, a description of discrete gravity and resolution of differences between theory and measure of the gravitational constant and the reduced Planck constant.

For our first example, we consider an expression we identify as the unity expression. The expression has been organized such that it is now in a more familiar form, a relativistic relaiton with the beta term in the numerator.

The expression H_U=t_f / l_fm_f=2θ_si describes universal expansion such that H_U is the rate of expansion when defined with respect to the universe (i.e. as opposed to per megaparsec). Taking the cube root (i.e. the measure of length in each of the three dimensions) provides us the Pythagorean correlation to side a, which when summed with the square of side b must equal 1. Notably, side b which is made up of a change count n_L representing the change in length count corresponding to t₀ of the expansionary epoch divided by the change in length count traveled by light describes the corresponding bound ratio for the expansion.

With this we can now consider what a universe with other dimensions might look like by changing the exponent respectively. That is, a four dimensional universe - as regards spatial dimensions - would require a fourth root, a five dimensional universe a fifth root and so forth. The relative values of the fundamental measures would still have correspondence to the bound ratio on side b. But, the number of those solutions would be limited, constrained to the cubic relation afforded by the Pythagorean theorem. It is on this ground that we lean against multidimensional models in favor of a three dimensional spatial framework.

For completeness, we also consider the expression for mass accretion.

Notice that the rate of accretion is a function of θ_si where H_U=2θ_si cubed (i.e. once for each spatial dimension). In this example it becomes evident that any factorization of dimensions is just that. While not mathematically incorrect, factorizations of the spatial frame of reference do not afford a greater understanding of the laws of nature.